ACM 2 - Multiphysics Modelling and Simulation for ...

Applied Condition Monitoring

Mohamed Haddar · Mohamed Slim Abbes Jean-Yves Choley · Taoufik Boukharouba Tamer Elnady · Andrei Kanaev Mounir Ben Amar · Fakher Chaari Editors

Multiphysics Modelling and Simulation for Systems Design and Monitoring Proceedings of the Multiphysics Modelling and Simulation for Systems Design Conference, MMSSD 2014, 17–19 December, Sousse, Tunisia

Applied Condition Monitoring Volume 2

Series editors Mohamed Haddar, National School of Engineers of Sfax, Tunisia Walter Bartelmus, Wroclaw University of Technology, Poland Fakher Chaari, National School of Engineers of Sfax, Tunisia e-mail: [email protected] Radoslaw Zimroz, Wroclaw University of Technology, Poland

About this Series The book series Applied Condition Monitoring publishes the latest research and developments in the field of condition monitoring, with a special focus on industrial applications. It covers both theoretical and experimental approaches, as well as a range of monitoring conditioning techniques and new trends and challenges in the field. Topics of interest include, but are not limited to: vibration measurement and analysis; infrared thermography; oil analysis and tribology; acoustic emissions and ultrasonics; and motor current analysis. Books published in the series deal with root cause analysis, failure and degradation scenarios, proactive and predictive techniques, and many other aspects related to condition monitoring. Applications concern different industrial sectors: automotive engineering, power engineering, civil engineering, geoengineering, bioengineering, etc. The series publishes monographs, edited books, and selected conference proceedings, as well as textbooks for advanced students. More information about this series at http://www.springer.com/series/13418

Mohamed Haddar · Mohamed Slim Abbes Jean-Yves Choley · Taoufik Boukharouba Tamer Elnady · Andrei Kanaev Mounir Ben Amar · Fakher Chaari Editors

Multiphysics Modelling and Simulation for Systems Design and Monitoring Proceedings of the Multiphysics Modelling and Simulation for Systems Design Conference, MMSSD 2014, 17–19 December, Sousse, Tunisia

ABC

Editors Mohamed Haddar National School of Engineers of Sfax Sfax Tunisia

Tamer Elnady Faculty of Engineering ASU Sound and Vibration Lab Ain Shams University Cairo Egypt

Mohamed Slim Abbes National School of Engineers of Sfax Sfax Tunisia

Andrei Kanaev Laboratory of Materials Sci. & Process University of Paris 13 Villetaneuse France

Jean-Yves Choley Superior Engineering Institute of Paris Saint-Ouen France

Mounir Ben Amar University of Paris 13 Laboratory of Materials Sci & Process Villetaneuse France

Taoufik Boukharouba Laboratory for Advanced Mechanics University of Sciences and Technology Houari Boumediene (USTHB) Alger Algeria

ISSN 2363-698X Applied Condition Monitoring ISBN 978-3-319-14531-0 DOI 10.1007/978-3-319-14532-7

Fakher Chaari National School of Engineers of Sfax Sfax Tunisia

ISSN 2363-6998

(electronic)

ISBN 978-3-319-14532-7

(eBook)

Library of Congress Control Number: 2014958074 Springer Cham Heidelberg New York Dordrecht London c Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com)

Preface

Multiphysics problems are usually encountered when the behaviour of a system is affected by several physical phenomena. The interaction between these phenomena leads to a completely different behaviour than the one resulting if each physical phenomenon would be treated separately from the others. In recent years, studying multiphysics systems has become a hot research field and has attracted researchers from different areas. This book is the proceedings of MMSSD 2014 (Multiphysics Modelling and Simulation for Systems Design and Monitoring) which was held on December 17–19, 2014, in Sousse, Tunisia. The first edition of the conference attracted about 140 participants who had the opportunity to discuss a number of important topics in the field of multiphysics modelling and simulation, and establish important collaborations. A few chapters in this book are devoted to modeling and simulation of multiphysics systems. They describes several kind of interactions, such as thermomechanical and electromechanical interactions, fluid-structure interactions, acoustic-structure interactions, and thermofluidic interactions. Specific contributions deal with mechatronic design and their applications, for example, in the field of energy harvesting or in robotics. Other contributions describe advances in systems engineering, mainly in the area of complex systems design. They reports on the implementation of advanced numerical and analytical simulation methods, and include case studies. Another topic is concerning with material characterization of multiphysics systems. The corresponding chapters discuss several structural applications, giving a special emphasis to the analysis of material behavior. A few other chapters cover monitoring methods of multiphysics systems. Failure characterization and faults identification represent the mostly-discussed subjects in this section.

VI

But, what is the future of multiphysics systems? After three days of lively discussion all the participants concurred on the necessity of increasing the level of collaboration between researchers and professionals with different expertise. This synergy is fundamental to deal with the increasing complexity of future systems. December 2014

Mohamed Haddar (Sfax-Tunisia) Mohamed Slim Abbes (Sfax-Tunisia) Jean-Yves Choley (Paris - France) Taoufik Boukharouba (Algiers, Algeria) Tamer Elnady (Cairo, Egypt) Andrei Kanaev (Paris - France) Mounir Ben Amar (Paris - France) Fakher Chaari (Sfax-Tunisia)

Contents

Influence of the Plies Orientation on the Rigidity of the Laminated Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abdelghani Belkadi, Ali Ahmed Benyahia, Nourdine Ouali

1

Simplified Calculation Methods on Smoke and Temperature Stratification in Ventilated Compartments . . . . . . . . . . . . . . . . . . . . . . . . . . Souhila Agred, Abdallah Benarous, Djamel Karmed, Larbi Loukarfi

9

Analysis of the Notched Specimens on the Fracture Behavior by the Volumetric Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mustafa Moussaoui, Salim Meziani

19

Temperature Effect on Shear Flow and Time Dependant Modelling of Cutting Oil Emulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Larbi Hammadi, Nasr-Eddine Boudjenane, Mansour Belhadri

27

A Comparison between Two Reliability Based Optimization Strategies of Tuned Mass Damper Parameters under Stochastic Loading . . . . . . . . . Elyes Mrabet, Mohamed Guedri, Samir Ghanmi, Mohamed Soula, Mohamed Ichchou Modeling and Simulation for Vertical Rail Vehicle Dynamic Vibration with Comfort Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mortadha Graa, Mohamed Nejlaoui, Ajmi Houidi, Zouhaier Affi, Lotfi Romdhane Localization of Impact on a Beam by Time Reversal Method . . . . . . . . . . Omar Chaterbache, Abdelhamid Miloudi

37

47

59

VIII

Contents

Early Detection of Gear Failure by Vibration Analysis . . . . . . . . . . . . . . . Mustapha Merzoug, Khalid Ait-Sghir, Abdelhamid Miloudi, Jean Paul Dron, Fabrice Bolaers

69

Effect of Water Hammer on Relatively Long Inclined Pipelines . . . . . . . . Manel Dalleli, Ezzeddine Hadj-Taïeb, Christian Schmitt, Zitouni Azari

81

Experimental Study of the 60◦ PBT6 Pitching Blade Effect with a PIV Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bilel Ben Amira, Zied Driss, Mohamed Salah Abid

91

Incidence Angle Effect on the Aerodynamic Structure of an Incurved Savonius Wind Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Zied Driss, Olfa Mlayeh, Dorra Driss, Makram Maaloul, Mohamed Salah Abid Experimental Characterization of a NACA2415 Airfoil Wind Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Zied Driss, Tarek Chelbi, Ahmed Kaffel, Mohamed Salah Abid DFMM Approach: Modelling, Formalization and Choice of Material and Manufacturing Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Mohamed Saidi, Achraf Skander, Slim Kaddeche Study of the Aerodynamic Structure around an Obstacle with Inclined Roof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Slah Driss, Zied Driss, Imen Kallel Kammoun Contribution to Inverse Kinematic Modeling of a Planar Continuum Robot Using a Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . 141 Ammar Amouri, Chawki Mahfoudi, Abdelouahab Zaatri Experimental and Numerical Study of Beams under Low-Velocity Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Mahmoud Neder, Abdelhamid Miloudi Modeling of Cutting Forces and Roughness by the Response Surface Method (RSM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Fayçal Bentaleb, Idriss Amara Analysis of Inelastic Behavior of Amorphous Polymers . . . . . . . . . . . . . . . 169 Nourdine Ouali, Ali Ahmed Benyahia

Contents

IX

First Principal Calculations of Optical Properties of InGaN2 Using in Solar Cells Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Hamza Bennacer, S. Berrah, A. Boukortt, Mohamed Issam Ziane Agent-Based Approach for the Optimal Design of Mechatronic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Amir Guizani, Moncef Hammadi, Jean-Yves Choley, Thierry Soriano, Mohamed Slim Abbes, Mohamed Haddar A Digital Pattern Approach to the Design of an Automotive Power Window by means of Object-Oriented Modelling . . . . . . . . . . . . . . . . . . . . 199 Stanislao Patalano, Ferdinando Vitolo, Antonio Lanzotti Numerical Study of the External Excitation Amplitude Effect on Liquid Sloshing Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Bouabidi Abdallah, Zied Driss, Abid Mohamed Salah The Study of the Hardening Precipitates and the Kinetic Precipitation. Its Influence on the Mechanical Behavior of 2024 and 7075 Aluminum Alloys Used in Aeronautics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Ahmed Ben Mohamed, Amna Znaidi, Moez Baganna, Rachid Nasri Multidisciplinary Design Optimization in Small Satellite . . . . . . . . . . . . . . 229 Abdelmadjid Boudjemai, Abdelmoumen Bacetti, Mohammed Amine Zafrane, Rachida Hocine Mechanical Vibration Cancellation Using Impact Absorber . . . . . . . . . . . 239 Kaouther Chehaibi, Charfeddine Mrad, Rachid Nasri RFLP Approach in the Designing of Power-Trains for Road Electric Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Clemente Capasso, Moncef Hammadi, Stanislao Patalano, Ruixian Renaud, Ottorino Veneri Mechanical Vibration Elimination Using Friction Absorber . . . . . . . . . . . 259 Aymen Nasr, Charfeddine Mrad, Rachid Nasri AC Flashover: An Analysis with Influence of the Pollution, Potential and Electric Field Distribution on High Voltage Insulator . . . . . . . . . . . . 269 Hani Benguesmia, Nassima M’Ziou, Ahmed Boubakeur Evaluation by Wide-Angle X-ray Scattering of the Particle Evolution in Polypropylene Processed by Equal Channel Angular Extrusion . . . . . . 281 Ramdane Boulahia, Taoufik Boukharouba, Jean-Michel Gloaguen A Multi-site Supply Chain Planning Using Multi-stage Stochastic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Houssem Felfel, Omar Ayadi, Faouzi Masmoudi

X

Contents

A System Engineering Conception of Multi-objective Optimization for Multi-physics System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Mian Chen, Omar Hammami Falling Film in a Heated Micro-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Sabrine Mejri, Olivier Fudym, Jalila Sghaier, Ahmed Bellagi Interfacial Stresses in FRP-Plated RC Beams: Effect of Adherend Shear Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Abderezak Rabahi, Belkacem Adim, Selma Chargui, Tahar Hassaine Daouadji A Sensitivity Analysis of Multi-objective Cooperative Planning Optimization Using NSGA-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Wafa Ben Yahia, Omar Ayadi, Faouzi Masmoudi Transient Flow Analysis through Centrifugal Pumps . . . . . . . . . . . . . . . . . 339 Issa Chalghoum, Sami Elaoud, Mohsen Akrout, Ezzeddine Hadj-Taïeb Vibro-Acoustic Analysis of Laminated Double-Wall: Finite Element Formulation and Reduced-Order Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 Walid Larbi, Jean-François Deü, Roger Ohayon Time Step Size Effect on the Hydrodynamic Structure around a Water Darrieus Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Ibrahim Mabrouki, Ahmed Samet, Zied Driss, Mohamed Salah Abid Simulation of the Dynamic Behavior of Multi-stage Geared Systems with Tooth Shape Deviations and External Excitations . . . . . . . . . . . . . . . 369 Hassen Fakhfakh, Jérome Bruyère, Philippe Velex, Samuel Becquerelle Fuzzy Modeling and Control of Centrifugal Compressor Used in Gas Pipelines Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Ahmed Hafaifa, Guemana Mouloud, Belhadef Rachid Burst Test and J-Integral Crack Growth Criterion in High Density Poly-Ethylene Pipe Subjected to Internal Pressure . . . . . . . . . . . . . . . . . . . 391 Mohamed Amine Guidara, Mohamed Ali Bouaziz, Christian Schmitt, Julien Capelle, Ezzeddine Hadj-Taïeb, Zitouni Azari, Said Hariri Solving the Three-Dimensional Time-Harmonic Maxwell Equations by Discontinuous Galerkin Methods Coupled to an Integral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Nabil Gmati, Stéphane Lanteri, Anis Mohamed

Contents

XI

Iterative Methods for Steady State Looped Network Analysis . . . . . . . . . . 409 Zahreddine Hafsi, Sami Elaoud, Mohsen Akrout, Ezzeddine Hadj-Taïeb Investigation and Modeling of Surface Roughness of Hard Turned Aisi 52100 Steel: Tool Vibration Consideration . . . . . . . . . . . . . . . . . . . . . . 419 Ikhlas Meddour, Mohamed Athmane Yallese, Hamdi Aouici Finite Element Simulation of Fatigue Damage Accumulation for Repaired Component by Cold Expansion Method . . . . . . . . . . . . . . . . . . . 433 Abdelkrim Aid, Mostefa Bendouba, Mohamed Benguediab, Abdewahab Amrouche Geometrically Non-linear Free Vibration of Fully Clamped Symmetrically Laminated Composite Skew Plates . . . . . . . . . . . . . . . . . . . 443 Hanane Moulay Abdelali, Bilal Harras, Rhali Benamar Scratch Adhesion Characteristics of PVD Cr/CrN Deposited on High Speed Steel and Silicon Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 Kaouthar Khlifi, Hafedh Dhifelaoui, Ahmed Ben Cheikh Larbi Load Sharing Behavior in Planetary Gear Set . . . . . . . . . . . . . . . . . . . . . . . 459 Ahmed Hammami, Miguel Iglesias Santamaria, Alfonso Fernandez Del Rincon, Fakher Chaari, Fernando Viadero Rueda, Mohamed Haddar Low Velocity Impact Behavior of Glass Fibre-Reinforced Polyamide . . . . 469 Jamel Mars, Mondher Wali, Remi Delille, Fakhreddine Dammak Burst Pressure Estimation of Corroded Pipeline Using Damage Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 Djebbara Benzerga Numerical Simulation of Incremental Sheet Metal Forming Process . . . . 489 Lotfi Ben Said, Mondher Wali, Fakhreddine Dammak A Higher Order Shear Strain Enhanced Solid-Shell Element for Laminated Composites Structures Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 497 Abdessalem Hajlaoui, Abdessalem Jarraya, Mondher Wali, Fakhreddine Dammak New Approch of High Cycle Fatigue Behaviour of Defective Material under Multiaxial Loading in 1045 Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 Hassine Wannes, Anouar Nasr, Chokri Bouraoui Determination of Stress Concentration for Orthotropic and Isotropic Materials Using Digital Image Correlation (DIC) . . . . . . . . . . . . . . . . . . . . 517 Mhalla Mohamed Makki, Bouraoui Chokri

XII

Contents

The Extended Finite Element Method for Cracked Incompressible Hyperelastic Structures Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 Mehrez Zaafouri, Mondher Wali, Said Abid, Mohammad Jamal, Fakhreddine Dammak Displacement Influence on Frequencies and Modal Deformations of a Sandwich Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 Idris Chenini, Youssef Abdelli, Rachid Nasri Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

Influence of the Plies Orientation on the Rigidity of the Laminated Composites Abdelghani Belkadi, Ali Ahmed Benyahia, and Nourdine Ouali Laboratoire de Mécanique Avancée (LMA) – USTHB – Bab Ezzouar / Alger {belkhadi_pg,abenyahiaa,nourouali}@yahoo.fr

Abstract. The overall properties of a laminated composite structure depend on design thereof, i.e., they depend on the thickness, on the orientation and the material of each ply in the stacking sequence, that is why optimization tools are used to assist decision making and designing structures with optimized properties. In this study, the design variables are only the plies orientation because the plies number, the material and the thickness are already selected. This problem will be treated as an optimization problem under constraints, whose objective function is the modulus of transverse stiffness of laminated composite to maximize, and the optimization variables are the plies orientation. The Monte Carlo method is applied as a constrained stochastic optimization, based on stratification parameters for the determination of optimal fiber orientations laminate structures to find the maximum longitudinal modulus of rigidity (Ex). Validation results of this approach have been compared with the results of Girard [4] who has used a genetic algorithm to optimize laminates by using equations of the classical theory of laminates and, by referring to the results of Luersen [3]. Keywords: Laminated composites, optimization, stiffness, plies orientation.

1

Introduction

The mechanical properties of a laminated structure are related to the design parameters, including the number of layers, thickness, orientation, and the material of each fold and the stacking sequence of these folds. This is very advantageous compared to conventional materials because it gives the possibility of designing structures according to the properties that meet the needs and requirements. Typical problems of optimal design concern particularly a laminate, the rigidity, strength, critical load, the natural frequencies and the minimization of thermal expansion coefficients. However, mechanical properties of a laminate cannot generally be predicted trivially, even less optimized. Pedersen (1987) showed that in term of fibers orientation, the composite optimization problems tend to have multiple local optima which make them difficult to approach by using local optimition methods based on gradients. Luersen et al. (2004) used an optimization method © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_1

1

2

A. Belkadi, A.A. Benyahia, and N. Ouali

called Globalized and Bounded Nelder-Mead. As for Girard (2006), he has used a genetic algorithm for maximizing the longitudinal rigidity module with constraints on the shear modulus and Poisson's ratio. The design variables are the ply orientations which vary in a continuous or discrete manner. The purpose of this study is to determine the optimal fibers orientations, for each ply of a laminated structure, which give the modulus of longitudinal stiffness with maximum constraints on the shear modulus and Poisson's ratio. The study will use the Monte Carlo method as a method of stochastic optimization under constraints

2

Modelling

For a laminate consisting of n layers numbered from the bottom to the top, average surface is chosen as the reference plane. The constitutive equation of a laminated plate expressing the resultants and moments based on the membrane deformations and bends is given by:

(1)

Where and k correspond to vectors of deformations and bends of the average plan. This constitutive equation can also be written in the following contracted form: ….

….

…..

….

(2)

Where A is the membrane stiffness matrix, D flexural stiffness matrix and B matrix membrane-bending-torsion coupling and their terms are given by the following expressions: •

∑

•

1/2 ∑

•

1/3 ∑

(3)

The relationships of the off-axis rigidity constants and the main axes for orthotropic, symmetrical and balanced material can be written in the following forms:

Influence of the Plies Orientation on the Rigidity of the Laminated Composites

1 1 0 0

0 0 0 0 1 0 0 1

0 0

0 0 1 1 0 0

0 0

0

0

0

0

(4)

0 0 0 0 1 0 0 1

0 0

3

0

0

0

0

(5)

With the parameters of lamination: /

1 , , ,

cos 2

, cos 4

, sin 2

, sin 4

cos 2

, cos 4

, sin 2

, sin 4

/ / /

, , ,

6

Furthermore, for a design of a balanced and decoupled laminate of which the material is chosen beforehand, stratification parameters define fully this design by searching the optimal orientation of plies in the interval -90° to 90°. Thus, the optimal solutions can be determined from the stratification parameters, as follows: 1

3

1 and 2ξ

1

ξ

1

Method of Resolution

The optimization method used to solve this problem is that of Monte-Carlo, but with reduction of interval research. The method involves introducing a reduction factor of the interval research (the best solution found neighbourhood) in which are carried out random tests. At the beginning of the process, the size of the neighbourhood is made wide enough but when the number of iterations increases, there is a narrowing of this neighborhood. This allows well explore the field of research in the initial phase of the calculations. The reduction comes to allow refinements of the solution. On the other hand, one introduces a number of successive chess (E) which will be counted since the last improvement. E will remain small as long as the process of optimization improves the solution, but when E begins to take

4


large values, this means that the process brings no improvement, then, it will be more profitable to reduce the research interval to allow refinements by introducing a pawhich is a threshold of decision-making. The effectiveness of the rerameter _ duction technique of research interval depends on essentially the wise choice of the reduction factor of research interval and of the failures threshold parameter. Consider the example treated by Girard (2006). It is a symmetrical and balanced laminate with 16 glass/epoxy unidirectional plies (UD). The laminate must be designed to have a modulus of elasticity longitudinal (maximum) while ensuring that its module of shear ( ) higher or equal to 12 GPa, and that its poisson’s ratio ( ) is smaller than 0,5. The elastic properties of the glass-epoxy UD plies are: E1 = 45 GPa, E2 = 10 GPa, G12 = 4.5 GPa and = 0.31. Since the laminate is symmetrical and balanced, so there are 4 possible orientations of the UD plies limited between 0° and 90°. Thus, the optimization problem can be posed as follows: Maximize the modulus of rigidity plies under the constraints:

-

4

by changing the orientation of the 16

symmetrical and balanced Stacking. With : 12 and ν

0.5

Results and Discussions

The idea of solving this problem is to work in the search space , as shown in figure 1 and then back to the directions of the plies. If one takes ξ and ξ as design variables, the field of feasibility is a parable and each point in this area (point lamination) corresponds to a tensor A. Indeed, the shear modulus and the poisson’s ratio of the optimal solution were respectively 12 GPa and 0.5, either the lower and upper limits of these constraints. In this case, a solution violating one of the constraints will probably have a longitudinal Young modulus greater than the optimal solution. The mode of evolution of the Young module based on the change in the orientation of the plies is illustrated by figure 2. That is why it must penalise enough this solution, this violates one of the constraints, in order to make it less attractive compared to solutions that do not violate any constraints. Otherwise, the algorithm will focus on a no-workable solution and, therefore, will generate a solution violated one or more constraints. In the case of this study, a programming our algorithm was made under MATLAB, where we have considered 0.85 as value of reduction factor, with a maximum number of chess factor _ 250. The program was rolled out 100 times to ensure the optimum values. Optimal solutions obtained by the proposed algorithm are given in table 1.


5

Fig. 1 Influence de l’orientation des plis sur le module de rigidité longitudinal

This table includes, in addition to results concerning the laminate studied by Girard (2006), results of optimization for two other laminates of the same type (symmetric and balanced) but with 4 plies and 8 plies. Table 1 Optimization results by algorithm proposed plies number

04

08

16

Strafication sequence optimized

Ex max.

[ ± 48.0047 ]

12.8259

[ ± 48.0048 ]

12.8258

[ ± 48.0056 ]

12.8256

[ ± 53.1224 ± 39.3220 ]

14.5268

[ ± 53.1239 ± 39.3366 ]

14.5216

[ ± 39.3249 ± 53.1152 ]

14.5262

[ ±53.9459 ±35.1962 ±45.9894 ±49.6245]

14.5225

[±35.4489 ±45.3720 ±54.2212 ±49.6980]

14.5276

[±58.3783 ±40.9708 ±42.2751 ± 43.5982]

14.5239

A comparison between the results, obtained for a 16 plies, by our algorithm and those obtained by Girard are grouped in table 2

6


Table 2 Comparison of the results obtained with those reported by Girard [4]

To check the performance of our algorithm, we will apply it to another laminate which has been studied by Vanucci (2007). It's a unidirectional carbon/epoxy laminate of 8 folds 128Gpa, E = 13 GPa, = 6.4 GPa, ν = 0.3) where we want to have a Young's modulus maximal by changing the orientation of the plies while complying with the following conditions: 25 and. 1 . For optimization, Vanucci has used the method of Miki (1982), which is a graphoanalytical method based on the use of lamination parameters. The results of our optimization are compared to those found by Vanucci (2007) in table 3. Table 3 Optimization of the sequence of an 8 ply carbon/epoxy laminate

5

Conclusion

In this work a stratified structure optimization technique based on the parameters of stratification (lamination parameters) is presented, this technique is based on the decomposition of the parameters that influence the behavior of an elastic laminate invariant parts and dependent parts of the stacking (the parameters of stratification). Thus, everything that depends on the material is separated of what is essentially geometric (sequence of stacking and orientations). For this purpose, the base material is preselected, then the parameters that influence the change in elastic behavior are precisely the lamination parameters, which are the only parameters which describe the design of the structure, because the stacking sequence, the thickness and the number of layers are already selected. To this end, the Monte Carlo method is applied as a constrained stochastic optimization, based on stratification parameters for the determination of optimal fiber orientations laminate


7

structures to find the maximum longitudinal modulus of rigidity (Ex ). Validation results of this approach have been compared with the results of Girard [4] who has used a genetic algorithm to optimize laminates by using equations of the classical theory of laminates and, by referring to the results of Luersen [3]. The obtained results are also compared with the results of Vanucci [8], who has used the method of Miki (1982) which is a grapheme method - analytical based on the use of lamination parameters. This method also shows that for a desired modulus of elasticity, there may be several sequences stacking, which allows handling of the manufacturing processes for the composite structures

References Pedersen, P.: On sensitivity analysis and optimal design of specially orthotropic laminates. Engineering Optimization 11, 305–316 (1987) Vanucci, P.: Méthodes Avancées en Optimisation des Structures, Conception optimale des composites. Université de Versaille (2007) Luersen, M.A., Le Riche, R., Guyon, F.: A constrained, globalized, and bounded NelderMead method for engineering optimization. Thèse de doctorat, Ecole Doctorale SPMI, pp. 52–59 (Décembre 2004) Girard, F.: Optimisation de stratifiés en utilisant un algorithme génétique. Thèse de maîtrise, Université LAVAL, pp. 9–20, 49–69 (2006) Voyiadjis, G.Z., Kattan, P.I.: Mechanics of Composite Materials with MATLAB. Louisiana State University, pp. 9–190 (2005) Irisari, F.-X.: Stratégies de calcul pour l’optimisation multiobjectif des structures composites. Thèse de doctorat, l’Université Toulouse III (2009) Allaire, G.: Conception optimale de structures. Springer (2007) Vannucci, P.: Matériaux composites structuraux: Méthode d’analyse et de conception. Université de Versailles et Saint-Quentin-en-Yvelines (2007)

Simplified Calculation Methods on Smoke and Temperature Stratification in Ventilated Compartments Souhila Agred1, Abdallah Benarous2,*, Djamel Karmed3, and Larbi Loukarfi4 1

LCEMSM Laboratory, Hassiba Benbouali University, Po Box 151, UHB Chlef, Algeria [email protected] 2 Mechanical Department, Faculty of Technology, UHB Chlef, Algeria [email protected] 3 ISAE-ENSMA, Po Box 40109, Futuroscope Chasseneuil, France [email protected] 4 LCEMSM Laboratory, UHB Chlef, Algeria [email protected]

Abstract. Using different calculation methods with respect to smoke and heat evacuation systems (SHEVS), a comparative study is presented with regards to enclosure fire in large single storey compartments. A focus is made on European and American manual methods for which a set of graphical Matlab routines are developed. The results obtained with several modeling approaches are discussed in this paper. Keywords: smoke and heat evacuation, calculation methods, natural/mechanical ventilation, fire engineering software.

1

Introduction

A fire can be defined as undesirable burning of materials with release of heat and toxic gases, causing hazards to people and structures. In this context, it has been shown that human causalities are much more due to smoke exposure than burns by flames (Stec et al., 2010). Consequently, some efforts are needed to explore fire safety engineering issues, more particularly those related with heat and smoke dynamics. In this direction, there is a specific need for fire simulation softwares, as well as predictive tools for the design of smoke and heat evacuating systems (SHEVS). These devices are supposed to allow for thermal comfort under safe (no fire) conditions while providing optimal circumstances for survival and egress of occupants. Design procedures for SHEVS use manual-based methods, which obey to regionally standardizations and safety regulations. *

Corresponding author.

© Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_2

9

10

S. Agred et al.

Although the calculation methods deal with analytical /empirical formulations, they have the ability to meet quickly specifications for fire safety projects and help providing start-up data for zone and field modelling (Paranthoen et al., 2010). In the present work, one proposes to explore some formulations involving SHEVS design for enclosure fires. Two standard calculation methods (European, American) are applied in the case of single storey compartments subjected to natural and mechanical ventilation (CR 12010-5, 2005), (Klote et al., 2012). For comparison purposes, additional calculation using a zone modelling software (CFAST) is performed (Peacock et al., 2011). Towards a technological reliability regarding fire safety engineering, calculation methods are automated in a form of computer routines for which a set of graphical user interfaces (GUI) are developed.

2

Calculation Methodology

2.1

Input Variables and Calculation Assumptions

For the calculation methods, temperature and smoke behaviour is supposed to rely on a steady state assumption. Since the convective and radiative transfer to walls are not considered, the formulation is not suitable for studies on fire resistance of structures. In fact, thermal losses from the flame to the plume are only quantified by a relative fraction α R of the heat release rate Q f . Prior to any SHEVS design, a conventional fire should be defined in terms of heat release ( HRR = Q f ) and pool dimensions ( A f , D f , Pf ). The compartment is characterized by the floor area, ceiling height ( H ceiling ) as well as opening area ( Ai ) for gates /windows. Depending on the user choice, stratification is characterised by the smoke (hot layer) thickness ( δ s ) or the temperature rise ΔT of the cold layer. These informations are indicated by means of the main graphical user interface (Fig.1).

Fig. 1 The main graphical user interface (GUI)

Simplified Calculation Methods on Smoke and Temperature Stratification

2.2

11

The European Method

A threshold value of 20% is specified for radiative losses. The convective thermal power is given by:

Q c = (1 − α R )Q f = 0.8Q f

(1)

If this value induces an absolute thermal rise ΔT , the mass flow rate of the smoke can be expressed by: M s =

Q c

(2)

cΔ T

where c is the specific heat capacity for the air-smoke mixture. Owing to buoyancy forces, a stratification of the hot layer is created and the smoke lies at a vertical distance Ys from the floor:

(

Ys = M s / C e Pf

)

2/3

This value is retained if the regularity condition Ys ≤

(3) A f is fulfilled, so that

the smoke layer thickness is given by:

δ s = H ceiling − Ys

(4)

For large compartments, the empirical constant C e is set to 0.19. If the numerical value of Ys exceeds the ceiling height, one considers that the smoke flow rate is too high and the user may impose an “appropriate” value for the thermal rise (Fig.2).

Fig. 2 Graphical user interface for smoke calculation

12

S. Agred et al.

As the static pressure is supposed to be constant throughout the hot and cold layers, the smoke volumetric flow rate can be written as: Vs = ( M s .Ts ) /( ρ ∞ T∞ )

(5)

where Ts = ΔT + T∞ stands for the static temperature within the smoke layer. t

In the case of natural ventilation, a set of N NV vents whose total area ANV are expected in the compartment, so that: M s Ts

t

ANV C v = 2

2 ρ ∞ gδ s T∞ ΔT −

2 M s Ts .T∞

(6)

( Ai C i )2

where C v , C i denote loss coefficients of ventilation holes and air openings (gates, windows) respectively. The cross sectional area of each vent is easily obtained from the following relation : n

t

ANV = ANV / N NV

(7)

In the case of a mechanical ventilation, a set of N MV ventilators /jet fans are to be expected for the compartment. The corresponding mass flow rate for each decrit vice should not exceed a critical value M to avoid the plug-holing phenomeMV

non (Klote et al., 2012). Depending on whether the component is considered as ‘far’ or ‘adjacent’ to the wall, two expressions for the threshold flow rate are used:

⎧1.3 δ s2

crit M MV = ⎨

⎩ Ts

crit M MV

g .δ s T∞ ΔT if X VW < Dv

⎧ 2.05 δ s2 ρ ∞ =⎨ g .T∞ ΔT D v if X VW ≥ Dv ⎩ Ts

(8)

Dv , X VW are respectively the hydraulic diameter and the distance to the wall of each device. The number of elements (Fig.3) that should be set to extract the smoke is given by: crit N MV ≥ M s / M MV

(9)


13

Fig. 3 SHEVS calculation GUI (mechanical ventilation case)

2.3

The American Method

The American method specifies two rates of radiative losses depending on the presence (20%) or absence (50%) of sprinklers, so that the convective thermal power is expressed as: Q c = (1 − α R )Q f

; α R = 0.2 or 0.5

(10)

The same procedure is used for the calculation of the smoke layer thickness, according to a slight different value for the constant C e = 0.188 . The total area required for a natural ventilation case is obtained by solving the nonlinear equation:

t

ANV C v =

M s

⎛ At C ⎜ NV v Ts2 + ⎜ ⎝ Ai C i

ρ∞

2

⎞ ⎟ T T ⎟ ∞ s ⎠

2 g δ s T s ΔT

(11)

For a mechanical ventilation case, a critical volume flow rate is imposed to avoid plug-holing. For each device (ventilator, jet-fan) the volume flow rate is given by: 2 crit VMV = Ts

5

gδ s T∞ ΔT

(12)

The expected number of components (Fig.4) is obtained from: crit N MV ≥ Vs / VMV

(13)

14

S. Agred et al.

Fig. 4 Graphical user interface for SHEVS calculation

3


The first test case concerns a polyvalent hall which dimensions are 95 m length, 66m width and 11m height. The hall features ten gates whose total opening area is 109 m2. The fire source is taken to be a square 9m×9m pool (category 2) with a total heat release rate of HRR = 2.02 Megawatts. A natural ventilation configuration is considered here. Basing on 20% radiative loss, the temperature rise is estimated at ΔT = 150 °C for both methods, with a corresponding smoke flow rate of M = 108 kg/s. In this case, the average smoke temperature is almost 170 °C s

which shows an overestimation of 42% from that of the hot layer, as predicted by the CFAST solver (Fig.5). The important discrepancy is certainly, due to the fact that the “manual” formulations do not account for thermal losses to the ceiling wall. Furthermore, the CFAST solver makes use of a well-known entrainment model that accurately predicts the air-smoke mixing (Mc Caffrey, 1983). A model that justifies the moderate temperature, in the vicinity of the ceiling.


15

180

160 European + American method CFAST calculation

Hot layer temperature (C°)

140

120

100

80

60

40

20 0

50

100

150

200

250

300

350

400

450

500

550

Time (s)

Fig. 5 Hot layer temperature (natural ventilation)

Consequently, the steady state temperature (120°C) of the hot layer seems to be most revealing of the thermal behaviour. Regarding the smoke stratification, the European as well as the American methods recover approximately the same value Ys ≈ 6.3 m against Ys = 8.0 m for the CFAST solver (Fig.6).

Fig. 6 Hot layer height for various calculation methods (natural ventilation)

16

S. Agred et al.

Figure 6 shows a relative deviation of 27% on the Ys values obtained by the CFAST software. The discrepancy was expected since a one-dimensional (1D) development is assumed for the smoke dynamics in the zone modelling solver. This assumption is strongly questionable for such geometrical (large compartment) and physical (natural convection) configurations. t

The two “manual” methods succeed in predicting the value ANV = 53 m2 for the total ventilation area. This quantity is mandatory to build a geometrical model for any field-based (CFD) calculation. The second test case is a simplified supermarket model of length 70m, width 35m and 4m height. Gates and windows occupy an area of 84 m2. The design fire of 4.5 Megawatts represents a square pool of 3m×3m, yielding a thermal power of 500 Kw/m2. Owing to the important heat release flux, a mechanical ventilation design will be considered. On a basis of a prescribed temperature rise 150 °C, a mass flow rate M s = 24 kg/s is obtained for both methods. However, equation (3) yields a non realistic value Ys = 4.8 m of the smoke layer height, since the ceiling height is only 4m. This result indicates that a stable thermal steady state is unlikely. According to the regulatory condition (see eqn. 3), the value Ys = 3 m will be retained in the following calculations. crit The European method gives a global critical mass flow rate M MV = 3.65 kg/s.

If one assumes that the ventilation components are sufficiently remote from the walls, a minimum of N MV = 7 ventilators having 1m diameter, are required within the compartment. The American method considers the existence of a threshold for the volume crit flow rate. The calculated value VMV = 3 m3/s corresponds to a minimum of N MV = 11 ventilators. This latter value seems more realistic since ventilator characteristics are based on a constant volume flow rate rather than a mass flow rate, which is a fluid-dependent quantity. Both methods significantly overestimate the smoke layer temperature by 55% comparing with the CFAST solver calculation (Fig.7). Figure 7 depicts the time evolution of the average hot layer temperature. It is noticed that an established steady state cannot be reached throughout the curve. A slight asymptotic behaviour is observed after 10 minutes of burning. The limiting hot layer temperature (110°C) seems to be fairly moderate thanks to ventilation devices that contribute to improve the mixing of air and smoke. In this mechanical ventilation case, the CFAST solver exhibits a slight overestimation of only 09% on the smoke layer height, comparing with manual methods (Fig.8).


17

180

160 European + American method CFAST calculation

Hot layer temperature (C°)

140

120

100

80

60

40

20 0

50 100 150 200 250 300 350 400 450 500 550 600 650 700 750

Time (s)

Fig. 7 Hot layer temperature (mechanical ventilation)

Fig. 8 Hot layer height for various calculation methods

Although the compartment has a large surface where flow of air and smoke may develop in a three dimensional pattern, the mechanical ventilation contributes to favour a spreading (blowing) direction. Consequently, the ceiling flow (at least at the blowing level) can be considered as quasi one-dimensional. The present configuration certainly enhances the applicability of the zone modelling calculation while allowing for a reduction of fire simulations cost.

18

4

S. Agred et al.

Conclusion

Different calculation methods have been applied for two test cases of large single storey compartments. Both European and American method recover almost the same values for SHEVS characteristics. Indeed, these “manual” procedures contain quasi-identical formulations while basing upon a similar design philosophy. It has been shown that the American method seems more conservative with respect to critical volume flow rate, which is a more appropriate yardstick of a ventilator performance. Mixing of air and smoke in the plume region were accurately quantified by means of an entrainment sub-model that exists in the zone modelling solver. Smoke and temperature stratification were predicted fairly good using the CFAST solver. This was not the case of the manual methods for which a common drawback is the steady state assumption. Several fire issues can be investigated by means of zone/field modelling formulations where much pertinent informations can be provided. Nevertheless, field/zone calculations are much time and resource consuming and require more skills from the user to deal with realistic applications.

References CR 12010-5: Guidelines on functional recommendations and calculation methods for smoke and heat evacuating systems. CEN Technical Report No.191, Brussels, Belgium (2005) Klote, J.H., Milke, J.A., Turnbull, P.G., Kashef, A., Ferreira, M.J.: The handbook of smoke control engineering, 1st edn. American Society of Heating, Refrigerating and Air Conditioning Engineers (ASHRAE), Washington (2012) Mc Caffrey, B.J.: Momentum implications for buoyant diffusion flames. Combustion and Flame (1983), doi: 0010-2180(83) 90129-3 Paranthoen, P., Gonzalez, M.: Mixed convection in a ventilated enclosure. International Journal of Heat and Fluid Flow (2010), doi:10.1016/2010.01.006 Peacock, R.D., Forney, G.P., Reneke, P.A.: Consolidated model for fire growth and smoke transport. NIST Technical Report No.1026 R1, USA (2011) Stec, A.A., Hull, T.R.: Fire toxicity, Part 2, 1st edn. Woodhead Publishing, Cambridge (2010)

Analysis of the Notched Specimens on the Fracture Behavior by the Volumetric Method Mustafa Moussaoui and Salimpop Meziani Laboratory of Mechanic, University Constantine 1, Campus Chaab Erssas, Constantine 25000 – ALGERIA {moussaoui_must,meziani_salim}@yahoo.fr [email protected]

Abstract. A single edge notched specimen is commonly used in materials of steel construction for the determination the notch stress intensity factor. CT- specimen is considered as a finite element model containing an elliptical notch under a uniform uni-axial tensile loading. The volumetric method is applied in perfect elasticplastic behavior. Changing made to notch parameters influences the evaluation of the effective stress accordingly the results of the notch stress intensity factor. Keywords: Weight function, Notch stress intensity factor, Notch effect, Effective stress.

1

Introduction

Voluntarily created or induced inadvertently, the geometric discontinuities always exist in mechanical components and engineering structures. They can take different forms such as the notches, the voids, the micro-cracks, etc… They induce strong local stresses and strains in their roots. Accordingly, the cracks initiate at their roots and they become responsible for failures of the notched components. It is well established that the concentration of localized stresses associated to a geometric discontinuity, reduces the static and cyclic strength of a structure (Shin. C 1994). Accordingly, the evaluation of the stress intensity factor associated to the various types of geometrical discontinuities or notch becomes paramount for the engineer during the design in order to guard against the risk of sudden fracture. The stress distribution near the tip of such a cutting is characterized by a stress singularity. The dimensional quantities of elliptical notch are major axis and minor axis, affect the stress intensity factor values. Local approaches are introduced to describe the magnitude of stress field, and evaluate the factors that influence the behaviour of structures under various loading conditions. According to this study, a new concept, based on the volumetric method focus on the notches effect. Analysis of stress distribution at a notch’s root shows a pseudo-singularity stress distribution governed by notch stress intensity factor (NSIF), KΙρ (Pluvinage et al


19

20

M. Moussaoui and S. Meziani

1999). The result of this works and others indicate that this approach gives a good description in relation with the notch effect. Under a plane stress mode, two methods are used especially volumetric approach and ones of Irwin for analysis of the stresses field using FEM methods. This present study focus on the effect of the variation of dimensional quantities of the notch on the fracture behaviour of notched specimens by the calculation of notch stress intensity factor and analysis the stresses field that reigns at the notch root.

2

Finite Element Modeling of Elliptical Notch

A semi-elliptical notch is applied to a plain CT specimen (Fig. 1) having the dimensions 20x20 [mm], subjected to a uniform uniaxial tension loading σ at its two ends of value 125 [MPa]. Four notch configurations are used, with different values of the minor axis (b=0.5, 4 and 5[mm]) and the major axis ‘a’ takes all the following values 0.5,1,2,3,5,6,8,10 [mm]. To get a better accuracy in the results, a much more refined mesh was created around the edge of the notch.

Fig. 1 Finite element model

Considering that the material has perfect elastic plastic behavior and having the following mechanical properties in Table 1. Table 1 Mechanical characteristics of the material Young's modulus E [MPa] 230E03

yield strength σE [MPa] 670

Poisson’s coefficient ν 0.293

Analysis of the Notched Specimens on the Fracture Behavior

3

21

Behavior Analysis of Elliptical Notch Root by Analytical Methods

Irwin’s approach has set a goal: the determination of the stress intensity factor in mode I, taking into account the presence of a plastic zone that is assumed to be small compared to the crack length. The resulting value of this factor, KI , will establish a critical “intrinsic” value corresponding to the elastic–plastic fracture (Irwin 1964). Irwin therefore assumes an extension of the fracture of ‘re 2 . The curve corresponding to the optimized values are below the curve corresponding to the failure probability evaluated with the optimum parameters when the DPO approach is applied. The relative error between Pf*, D and Pf , D is given in Fig.4 (b). The inspection of this figure shows that a minimum of error equal to 3% is reached for ψ ≈ 1.2 and the maximum error is less than 6% over the entire range of ψ . The comparison of the obtained results in Fig.3 (b) and Fig.4 (b) demonstrate clearly that the RBO using the DPO approach is more robust than the RBO using the displacement approach. The robustness should be understood in the sense that the RBO using the DPO can provide the results obtained with the RBO using the displacement approach with a maximum error, over the considered range of ψ , less than 6% which is less that the obtained error of 9.5% in the reciprocal case.

Failure probability

8

x 10

-3

RBO (displacement approach) Pf,D (Diss.power optimum)

6

4

2

0 0.5

( a) 1

1.5

2

ψ

2.5

3

3.5

Relative error (%)

6

5

(b)

4

3 0.5

1

1.5

2

ψ

2.5

3

3.5

Fig. 4 (a) Pf*, D obtained using the displacement approach compared with those evaluated using the optimum values obtained with the DPO approach; (b) the relative error.

5

Concluding Remarks

In this paper two RBO problem of TMD parameters under stochastic loading are presented and compared. The selection of one of these strategies totally depends on the designer aim. The first strategy is based on a displacement approach so that the optimal TMD parameters are obtained by minimizing the failure probability

46

E. Mrabet et al.

explicitly related to the covariance responses of the primary structure. The second RBO strategy is a new approach in RBO problems. The approach uses the DPO in the primary structure. The introduction of this strategy is motivated by the strictly relationship between the DPO in the primary structure and the damage level of the vibrating structure. The expression of the failure probability, for the DPO approach, is presented and a comparison between the presented RBO strategies is made. The obtained results show that both strategies are strongly earthquake dependent on values of earthquake frequencies close to main system frequency and they are less sensitive outside of this frequency. The correlation analysis made between the presented strategies showed that the RBO with DPO approach is more robust in the sense that it can provide the optimized TMD parameters if the RBO displacement is applied with a maximum relative error less than 6%. Besides, both RBO strategies are equivalent, from robustness point of view, for frequencies close to the main system one because the relative error is roughly the same.

References Yu, H., Gillot, F., Ichchou, M.N.: Reliability based robust design optimization for tuned mass damper in passive vibration control of deterministic/uncertain structures. J. Sound Vib. 332, 2222–2238 (2013) Marano, G.C., Greco, R., Sgobba, S.: A comparison between different robust optimum design approaches: Application to tuned mass dampers. Probab. Eng. Mech. 25, 108–118 (2010) Chakraborty, S., Roy, B.K.: Reliability based optimum design of tuned mass damper in seismic vibration control of structures with bounded uncertain parameters. Probab. Eng. Mech. 26, 215–221 (2011) Gupta, S., Manohar, C.S.: Probability Distribution of Extremes of Von Mises Stress in randomly vibrating structures. J. Vib. Acoust. 127, 547–555 (2005) Crandall, S.H.: First crossing probabilities of the linear oscillator. J. Sound Vib. 12(3), 285–299 (1970) Li, J., Chen, J.: Stochastic dynamics of structures. John Wiley, Asia (2009) Greco, R., Marano, G.C.: Optimum design of Tuned Mass Dampers by displacement and energy perspectives. Soil Dyn. Earthq. Eng. 49, 243–253 (2013)

Modeling and Simulation for Vertical Rail Vehicle Dynamic Vibration with Comfort Evaluation Mortadha Graa1, Mohamed Nejlaoui1, Ajmi Houidi2, Zouhaier Affi1, and Lotfi Romdhane2,3 1

Laboratoire de Génie Mécanique(LGM), Ecole Nationale d’Ingénieurs de Monastir, Université de Monastir, Tunisie 2 Laboratoire de Mécanique de Sousse (LMS), Ecole Nationale d’Ingénieurs de Sousse, Université de Sousse, Tunisie 3 On leave at the College of Engineering, The American University of Sharjah, UAE

Abstract. Investigation of vibration is an important topic for the purposes of ride comfort in railway engineering. The vibration of rail vehicles becomes very complex because it is affected by the condition of vehicles, including suspensions and wheel profile, condition of track sections, including rail profile, rail irregularities, cant and curvature. The present study deals with the effects of railway track imperfections on dynamic behavior, and investigates the effect of vehicle speed and the rail irregularity on ride comfort through numerical simulation. The numerical simulation of the vertical dynamic behavior of a typical railroad vehicle will be performed using Largrangian dynamics. The model consists of 17 degrees of freedom with 4 wheelsets, 2 bogies and a car body. For the assessment of the ride comfort, the Sperling ride index (ISO2631) is calculated using filtered RMS accelerations. The ride characteristics of the vehicle provide an assessment of the dynamic behavior of the vehicle through the analysis of the accelerations at the vehicle body, whereas the ride comfort assesses the influence of the vehicle dynamic behavior on the human body. A parametric study was carried out to suggest design modifications in order to improve the level Sperling index. Keywords: Sperling index, ride quality, rail vehicle, dynamic behavior, ride comfort, vertical dynamic vibration.

1

Introduction

Rail transport is one of the major modes of transportation, so it must offer a high comfort level for passengers and crew. However, the comfort that passengers experience is usually perceived differently from one individual to another. In several research works, noise and vibration have been identified as the most important factors for high comfort. The main sources of vibration in a train are track defects © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_6

47

48

M. Graa et al.

in welding or rolling defects, rail joints, etc. The nature of vibration itself is random and covers a wide frequency range [1]. The improvement of the passenger comfort while travelling has been the subject of intense interest for many train manufacturers, researchers and companies all over the world. Although new techniques in manufacturing and design ensure better ride quality in railway carriages, it is sometimes impossible to completely eliminate track defects or various ground irregularities. The dynamic behavior of a train also depends on the load and and its mechanical components, such as springs, dampers, etc., which interact with the wheels, the train body and bogies. The dynamic performance of a rail road vehicle as related to safety and comfort is evaluated in terms of specific performance indices. The quantitative measurement of the ride quality is one of such performance indices. Ride quality is interpreted as the capability of the rail road suspension to maintain the motion within the range of human comfort. Sperling’s ride index is a measure of the ride quality and ride comfort used by ISO 2631[2]. Due to the complex dynamics that exists between the rail and wheel, rail vehicle dynamics are often difficult to model accurately. This velocity-dependent dynamics justifies the importance of the track input to railcar modeling. In the physical system, the input comes from the actual track. In a model, a user-defined input is used to predict the actual track characteristics. The user-defined input can be created analytically or can be based on actual measurements. Measured track data are obtained by running a specialized railcar down the track. Analytic track data are created using mathematical shapes, such as cusps, bends, and harmonic functions, to represent the track geometry [3]. There have been several studies, which dealt with the dynamic analysis of rail vehicles in order to enhance the ride comfort while travelling. Nejlaoui et al [4] optimized the structural design of passive suspensions in order to ensure simultaneously passenger safety and comfort. Abood et al [5] investigated the Railway carriage simulation model to study the influence of vertical secondary suspension stiffness on ride comfort of railway car body. Kumar and Sujata [2] presented the numerical simulation of the vertical dynamic behavior of a railway vehicle and calculated Sperling ride index for comfort evaluation. Nielsen and Igeland [6] investigated the vertical dynamic behavior for a railway bogie moving on a rail which is discretely supported by sleepers resting on an elastic foundation. Effects of imperfections on the running surfaces of wheel and rail were studied by assigning irregularity functions to these surfaces.

2

Modeling of Rail Road Vehicle

To analyze the dynamic behavior of railway vehicles, usually the vehicle (and if necessary the environment) is represented as a multi body system. A multi body system consists of rigid bodies, interconnected via massless force elements and joints. Due to the relative motion of the system’s bodies, the force elements generate applied forces and torques. Typical examples of such force elements are springs, dampers, and actuators combined in primary and secondary suspensions of railway vehicles.

Modeling and Simulation for Vertical Rail Vehicle Dynamic Vibration

2.1

49

Assumptions

The assumptions made in formulating the model are as follows: • • • • • • •

2.2

Bogie and car body component masses are rigid. The springs and dampers of the suspension system elements have linear characteristics. Friction does not exist between the axle and the bearing. The vehicle is moving with constant velocity on a rigid and constant gauge. All wheel profiles are identical from left to right on a given axle and from axle to axle and all wheel remain in contact with the rails. Straight track. An irregularity in the vertical direction with the same shape for left and right rails.

Rail Road Vehicle Model

Figure.1 illustrates the train vehicle model adopted in this study. It consists of a vehicle body, two bogies frames and four wheelsets. Each bogie consists of the bogie frame, and two wheel sets. The car body is modeled as a rigid body having a mass Mc; and having moment of inertia Jbx and Jcy about the transverse and longitudinal axes, respectively. Similarly, each bogie frame is considered as a rigid body with a mass Mb (Mb1 and Mb2) with moment of inertia Jbx and Jby about the transverse and longitudinal axes, respectively. Each axle along with the wheel set has a mass Mw (for four axles Mw1; Mw2; Mw3 and Mw4). The spring and the shock absorber in the primary suspension for each axle are characterized by a spring stiffness Kp and a damping coefficient Cp, respectively. Likewise, the secondary suspension is characterized by a spring stiffness Ks and a damping coefficient Cs, respectively. As the vehicle car body is assumed to be rigid, its motion may be described by the vertical displacement (bounce or Zc) and rotations about the transverse horizontal axis (pitch or Φc) and about the longitudinal horizontal axis (roll or θc). Similarly, the movements of the three bogie units are described by three degrees of freedom Zb; θb and Φb, each about their centers. Each axle set is described by two degrees of freedom Zw; and Φw. about their centers. Totally, 17 degrees of freedom have been considered in this study for the vehicle model shown in Figure.1. The detailed parameters regarding the moment of inertia and mass of different component are given in Table 1. Table 1 Detailed parameter of rigid bodies Name of Rigid Bodies Car Body Bogie-I and II Wheel-set-I,II,III and IV

Mass ( Kg ) 6,7 ×105 105 4000

Moment of Inertia( Kg.m² ) IXX IZZ 105 106 105 105 4000 4000

50

M. Graa et al.

Some parameters regarding the rigid bodies are already given in Table 1; however, the other parameters, which are essential for the simulation of the vehicle, are presented in Table 2. 2Lb

2b2 Y

2Ld

h3 Ks

Cs

h2 h1 R

Z

Kp

K Hertizian

2b1

y(lateral)

Cp

Scondary Suspension Primary Suspension

2a

(pitching)

Z Speed V

x (logitudinal) (rolling) (yawing) z(vertical)

Fig. 1 physical model of railway vehicle

A typical rail road vehicle system is composed of various components such as car body, springs, dampers, Bogies, Wheel-set, and so forth. When such dynamic systems are put together from these components, one must interconnect rotating and translating inertial elements with axial and rotational springs and dampers, and also appropriately account for the kinematics of the system structure.

3

Track Inputs to Rail Road Vehicle

The dynamic wheel loads generated by a moving train are mainly due to various wheel/track imperfections. These imperfections are considered as the primary source of dynamic track input to the railroad vehicles. Normally, the imperfections that exist in the rail-track structure are associated with the vertical track profile, cross level, rail joint, wheel flatness, wheel/rail surface corrugations and sometimes uneven support of the sleepers. In actual practice different types of periodic, a-periodic or random track irregularities may exist on the track, but in the present study bump type of irregularity is considered as shown in Figure.2 [3]. The shape of the irregularity is assumed to be similar on the left and the right rails.


51

Table 2 Vehicle parameters Parameter Primary spring stiffness Secondary spring stiffness Primary damping coefficient Secondary damping coefficient Vertical hertz spring stiffness Longitudinal distance between bogies I and II and car body mass center Longitudinal distance between wheel-set and corresponding bogie origin Lateral distance between a longitudinal primary suspension and corresponding wheel-set Lateral distance between longitudinal secondary suspension and corresponding bogie origin Lateral distance between contact point of wheel–rail and corresponding wheel-set origin Lateral distance between vertical primary suspension and corresponding wheel-set origin Lateral distance between vertical secondary suspension and car body mass center Nominal wheel radius Vertical distance between wheel-set and bogie mass centers Vertical distance between bogie mass center and lateral secondary suspension Vertical distance between lateral secondary suspension and car body mass center

Nomenclature Kp Ks Cp Cs Khz Lb Ld dp ds a b1 b2 R1 h1 h2 h3

Values 106 N/m 1,7×10 6N/m 6×104 Ns/m 105 Ns/m 35×109 N/m 6m 1,4 m 1m 1m 0,7163 m 1m 1m 0,61 m 0,3 m 0,2 m 1,3 m

Z X H

L Fig. 2 Model of track irregularity

The bump excitations of the left wheels (Figure. 3) of leading bogies are as follows:

⎧H ⎪ Z ri = ⎨ 2 ⎪0 ⎩

⎡ ⎛ V ⎞⎤ ⎢1 − cos ⎜⎝ 2π L ( t − tdi ) ⎟⎠⎥ ⎣ ⎦

for

tdi ≤ t ≤

L V

+ tdi

(i = 1..4) (1)

otherwise

⎡ 2 Ld 2 Lb 2 Lb + 2 Ld ⎤ ⎢⎣ V , V , ⎥⎦ V

Where [t d 1 , t d 2 , td 3 , td 4 ] = 0,

(2)

In the present study, H is taken as 0.03m and L is taken as 1m.

4

Simulation Study

The models were built in the MATLAB/ Simulink® environment. The fixed step solver ODE-45 (Dormand-Prince) was utilized, with the sampling time Ts=0.0001. Ts is smaller than the fastest half-car Active Vehicle Suspension Systems (AVSS) model dynamics, enabling observation of all model dynamics [7, 8].

52

M. Graa et al.

Dynamic analysis was carried out for the vehicle at different speeds: 15m/s, 30m/s, 45m/s and 60m/s. 0.03

2.5

Zr2

0.025

x 10

-3

15 30 45 60

2

Zr3

0.02

Car body displacement (m)

Bum p e xcita tion (m )

Zr1

Zr4 Zr5

0.015

Zr6

0.01

0.005 0 0

m/s² m/s² m/s² m/s²

1.5 1 0.5 0 -0.5

0.2

0.4

0.6 Time(s)

0.8

1

-1 0

0.5

1

1.5

2

2.5

Time(s)

Fig. 3 The bump excitations of the left wheels vehicle speed of 15m/s

4.1

Fig. 4 The vertical Car body displacement for differents vehicle speeds

Dynamics Analysis

The following output parameters are evaluated: •

Vertical displacement and acceleration at the floor of the car-body center of mass.

Vertical acceleration at the front and the rear bogie center pivot. The displacement and acceleration responses of the carbody at speeds of 15m/s, 30m/s, 45m/s, 60m/s are shown, respectively, in figure.4 and figure.5. Plots show that initially, the value of acceleration is nearly equal to 0 m/s2, which is mainly the acceleration without gravity. Finally it goes to zero, when the vibration of the car body ceases and stabilizes. The acceleration is generally within an acceptable range and does not show any instability. The acceleration response of front and the rear bogie with time is presented at different velocities of vehicle in figure.6.a) and figure.6.b), respectively. It is clear from the plots that initially the wheels of the front bogies come in contact with the track irregularity and the vibration starts in the front bogie and later these vibrations are shifted to the rear bogies. The amplitude of the vehicle vibration also increased with the vehicle speed.

4.2

Sperling Ride Index

Sperling’s ride index is defined as [9, 10, 11]: 1

⎛ n f 10 ⎞10 WZ = ⎜⎜ ∑ WZi ⎟⎟ ⎝ i =1 ⎠

(3)


53

Where nf is the total number of discrete frequencies of the acceleration response of the railway vehicle identified by the FFT and WZi is the comfort index corresponding to the itch discrete frequency, given by: 1

WZ = ⎡⎣ ai2 B( fi ) 2 ⎤⎦ 6.67

(4)

Where ai denotes the amplitude of the peak acceleration response (in cm/s²) measured on the floor of the itch frequency identified by the FFT and B(fi) a weighting factor, given by:

1.911 f 2 + ( 0.25 f 2 )

B( f ) = k

2

(5)

(1 − 0.277 f 2 )2 + (1.563 f − 0.0368 f 3 )2

Where k = 0.737 for horizontal vibration and 0.588 for vertical vibration.

4.3

Comfort Evaluation

Acceleration frequency response plots were generated for car body at vehicle speeds of 15m/s to 60 m/s and are shown in figure.5.b) to calculate the Sperling ride comfort index. The FFT plot is generated for a frequency range between 0 to 25 Hz, as the human beings are most sensitive in the frequency range of 4 to 12.5 Hz. Ride comfort analysis has been performed for speeds ranging from 15m/s to 60m/s. The analysis has been performed on the system model to calculate the vertical acceleration of the system. FFT output is taken to get the peak acceleration frequency component. Comfort index has been calculated through Eqs[3-5], which are presented in Table.3.

1000

Car body Acceleration (m/s²)


Car body acceleration (cm/s²)

15 30 45 60

1

0.5

0

15 30 45 60


500

0

-0.5 -500

0

0.2

0.4

0.6

0.8 Time(s)

(a)

1

1.2

1.4

1.6

10

0

1

10 Frequency (Hz)

10

2

10

3

(b)

Fig. 5 The vertical Car body acceleration in time domain (a) and frequency domain (b) for vehicle speed of 15m/s, 30m/s, 45m/s and 60m/s

54

M. Graa et al.

Table 3 Sperling’s ride comfort index an evaluation for different vehicle velocities Vehicle Speed (m/s) 15 30 45 60

Sperling Index (Wz) 2.53 2.03 1.72 1.62

Ride comfort evaluation More pronounced but not unpleasant Clearly noticeable Just noticeable Just noticeable 200

200

Front bogie acceleration (m/s²)

100


50 0 -50 -100

100


50 0 -50 -100 -150

-150 -200 0

15 30 45 60

150 Rear bogie acceleration (m/s²)

15 30 45 60

150

0.05

0.1

0.15

0.2 Time(s)

0.25

0.3

0.35

0.4

-200 0

0.5

1

1.5

Time(s)

(a)

(b)

Fig. 6 The vertical Front (a) and Rear (b) bogie acceleration for vehicle speed of 15m/s, 30m/s, 45m/s and 60m/s

The maximum and minimum ISO Sperling Index values are respectively 2.53 and 1.62 for the rail vehicle speed respectively 15 m/s and 60 m/s. These values respectively indicate “The more pronounced but not unpleasant” and “just noticeable” zones. Figure 7 show that the comfort index decreases as the vehicle speed increases while maintaining an acceptable level of comfort. This means that the passengers are not much affected by the vibration as they are exposed to low level of vibrations. 2.6 More pronounced but not unpleasant

Ride index Wz

2.4

2.2

2 Clearly noticeable

1.8

1.6 10

20

30 40 Speed(m/s)

50

Fig. 7 The Sperling index comfort variation for different speeds

60


4.4

55

Parametric Study

A parametric study was undertaken to find the influence of different suspension parameters on the Sperling index (Wz). The parameters considered for the analysis were primary stiffness, secondary stiffness, primary damping and secondary damping. The results of the parametric study have been plotted in terms of performance index such as the Wz index versus vehicle parameter. Figures 8 and 9 show the variation of Wz index as a function of stiffness and damping respectively (primary and secondary). The increase in the primary suspension stiffness reduces the Wz index at the carbody marginally up to a speed of 60 m/s. The influence of secondary stiffness has been found to be just the opposite of the primary stiffness. A reduction of the secondary stiffness value from the present value reduces the Wz index at the carbody at all the speeds. The variation of the primary damping is seen to have little influence on the Wz index at the carbody. At speeds above 45 m/s, increase in the primary damping is shown to produce marginal reduction in the Wz index at the carbody. The secondary damping has great influence at speeds higher than 30 m/s.

2.8

C arb o d y Sp erlin g In d ex W z

Carbody Sperling Index Wz

2.8 2.6 2.4

15 30 45 60

2.2

m/s m/s m/s m/s

2 1.8 1.6 1.4 0

500

1000

1500

Primary stiffness Kp (N/mm)

2000

2500

2.6 2.4 2.2 2

15 30 45 60

1.8 1.6 1.4 1.65

1.7

1.75

1.8

1.85

Secondary stiffness Ks (N/mm)

a)

m/s m/s m/s m/s 1.9 4 x 10

b)

Fig. 8 The influence of Primary a) and secondary b) stiffness

An increase in the secondary damping reduces the Wz index at speeds greater than 30 m/s, whereas up to 30 m/s speed, the secondary damping has little influence. It has also been observed that all the primary damping have very limited influence at low speeds.

56

M. Graa et al. 2.6

3 Carbody Sperling index Wz

Carbody Sperling index Wz

2.4 2.2 2 15 30 45 60

1.8 1.6 1.4

m/s m/s m/s m/s

2.5 15 30 45 60

2

m/s m/s m/s m/s

1.5

1

1.2 1 500

600 700 800 900 Primary damping Cp (Ns/mm)

1000

0.5 6

8

10 12 14 16 18 Secondary damping (Ns/mm)

a)

20

22

b)

Fig. 9 The influence of Primary a) and secondary b) damping

5

Conclusion

Vertical dynamic analysis has been carried out for a Railway Vehicle. A 17 degree of freedom model is used for the analysis. Velocity input at all the wheelset is given by considering similar bump irregularities at both right and left rail. A vertical acceleration response at the car body has been calculated in the frequency domain. Sperling Ride index has been calculated for the above vehicle. The Sperling Ride Index values at different speeds are presented. The calculated values of the Sperling index are found well in the satisfactory limits defined by the ISO 2631 standard which means that the passengers are not much affected by the vibration as they are exposed to low level of vibrations. A parametric study has been carried out with emphasis on better ride index. The parametric study has brought out possible design changes required in different parameters to deliver better Sperling comfort index. It should be noticed that the parametric study was carried out to suggest design modifications to improve Wz index, but others dynamic and control behaviors like stability, control of secondary suspension were to be considered when implementing the design modifications.

References [1] Skarlatos, D., et al.: Railway fault diagnosis using a fuzzy logic method. Applied Acoustic 65(10), 951–966 (2004) [2] Kumar, H., Sujata: Vertical dynamic analysis of a typical indian rail road vehicle. In: Proceedings of Computational Mechanics and Simulation, IIT, India, pp. 8–10 (2006) [3] Dukkipati, V., Amyot, J.: Computer Aided Simulation in Railway Dynamics. Marcel Dekker, New York (1988) [4] Nejlaoui, M., et al.: Multiobjective robust design optimization of rail vehicle moving in short radius curved tracks based on the safety and comfort criteria. Simulation Modelling Practice and Theory 30, 21–34 (2013)


57

[5] Abood, K.H.A., Khan, R.A.: The Railway carriage simulation model to study the influence of vertical secondary suspension stiffness on ride comfort of railway carboy. Journal of Mechanical Engineering Science 225, 1349–1359 (2011) [6] Nielsen, J., Igeland, A.: Vertical dynamic interaction between train and trackinfluence of wheel and track imperfections. Journal of Sound and Vibration 187, 25–26 (1995) [7] Chen, H., et al.: Application of Constrained H∞ Control to Active Suspension Systems on Half-Car Models. Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME 127(3), 345–354 (2005) [8] Pedro, J., Dahunsi, O.: Neural Network-based Feedback Linearization Control of a Servo-Hydraulic Vehicle Suspension System. International Journal of Applied Mathematics and Computer Science 21(1), 137–147 (2011) [9] Forstberg, J.: Ride comfort and motion sickness in tilting trains (2000) - Human responses to motion environments in train and simulator experiments. Master’s Thesis, KTH/ FKT/D-00/28-SE, Division of Railway Technology, KTH (2000) [10] Carg, V.K.: Dynamics of Railway Vehicle Systems. Academic Press, NY (1984) [11] B153/RP21 Application of ISO Standard to Railway Vehicles: Comfort Index Nmv Comparison with the ISO/SNCF Comfort Note and with the Wz (European Rail Research Institute) (1993)

Localization of Impact on a Beam by Time Reversal Method Omar Chaterbache1 and Abdelhamid Miloudi2 1

Laboratory for Advanced Mechanics (LMA), USTHB, BP32 El Alia, Bab Ezzouar, Algiers, Algeria [email protected] 2 Laboratory for Advanced Mechanics (LMA), USTHB, BP32 El Alia, Bab Ezzouar, Algiers, Algeria [email protected]

Abstract. This paper deals with a numerical simulation of time reversal (TR) process applied to a cantilever beam. To conduct this process we assume a finite difference scheme based on Euler-Bernoulli theory of transversal vibration of the beam. The TR process is then used to localize the position of an impact on the beam. To complete the TR process we need one or more measurements of the field on different positions. We had study the effect of the measurements number and their positions on the localization with the TR process. The objective of this study is to reduce the number of measurements and hopefully use it in an experiment with few measuring instruments. Keywords: Transverse vibrations, finite difference, impact source, time reversal, localization.

1

Introduction

The time reversal idea is based on the time invariance of theory formulation of a physical phenomenon (Sachs, 1987). Robert G. Sachs (Sachs, 1987) has explained in his book ”The Physics of Time Reversal” that the time invariance concept is purely theoretical and only applicable to a physical governing ”equations”. Also the time invariance can’t have a physical sense only if it’s tested experimentally (Sachs, 1987). Mathias Fink was interested on ultrasonic field time invariance (Fink et al., 1989) and he used it to ultrasound field focusing on an object through an inhomogeneous (Fink, 1992). First he supposes the existence of substance acting like an acoustic source when it receives an ultrasonic field. So, the reflected pressure from that is measured at different positions and then remitted in the inverted sense (i.e. time reversed) at the same measurement positions simultaneously. This last step should focus all the pressure on the body (Fink, 1992). Our work was inspired from the study by G. Ribay (Ribay, 2006) who compares the time © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_7

59

60

O. Chaterbache and A. Miloudi

reversal to a simple correlation. When a database exists on the behavior of an impact plate at several positions. A step similar to the D.O.R.T method proposed by C. Prada (Prada et al., 1996). By determining the correlation coefficients between the response to an impact in any position and the signals of the database G. Ribay establishes a map of the correlation coefficients as a function of position and notes that this ratio is close to 1 near the position of the original source. Recently D. Francoeur and A. Berry tried to carry out the time-reversal mirror experimentally in the case of an infinite beam (Francoeur and Berry, 2008). Using a network of exciters piezoceramic (PZT) and piezopolymer sensors (PVDF) to simulate the time reversal process. Then the DORT method is used to locate heterogeneities (point masses). An acoustic source at the audible frequency was used. R. Ernst and J. Dual uses the time reversal method to localize an acoustic source applied to a simply supported beam (Ernst and Dual, 2014). Using a single measurement, he located discontinuities in the section of the beam.

2

Numerical Model

Let’s consider a finite beam of length L, thickness h, a cross section A, density ρ , a moment of inertia I and Young modulus E. The response of the cantilever beam subjected to a transverse impact force applied at position x = x0 is the displacement w(x, t). It’s a solution to the Euler-Bernoulli equation (Graff 1975) with the corresponding boundary conditions:

⎧ ∂ 2 w( x, t ) ⎪⎪M x = − E I ∂x 2 ⎨ 2 2 ⎪ ρ A ∂ w( x, t ) = ∂ Mx( x, t ) + δ ( x − x ) δ (t ) 0 ⎪⎩ ∂x 2 ∂t 2

(1)

The system (1) contains only second order derivatives. It will be discretized with fourth order finite difference in space and second order in time. The discretezation detail is reported to the appendix. The impact force is of the form shown in figure 1. As needed in our work we construct the function (2) of the impact force to be as close as possible to a real impact (caused by a hammer impactor for example (Fig. 2)) : 2

⎛ −8t ⎞ ⎟ τ ⎠

⎛ 4 t ⎞ ⎜⎝ ⎟ e ⎝τ ⎠

δ (t ) = e 2 ⎜ τ is the impact duration.

(2)

Localization of Impact on a Beam by Time Reversal Method

Fig. 1 Impact form

τ = 4 ms

61

Fig. 2 Hammer impact form

The figure 3 shows the response to an impact at the middle of the beam in terms of transverse displacement. The impact duration was τ = 0.2 ms

Fig. 3 Transversal displacement at L / 2 (steal beam, L = 1 m, h = 6 mm, A = 180 mm²)

In the following we noted that working in terms of accelerations (Fig. 4) gave better results. Acceleration is more sensitive to the vibrations than displacement.

62


Fig. 4 Transversal acceleration at L / 2 (steal beam, L = 1 m, h = 6 mm, A = 180 mm²)

3

Time Reversal Principle

The time reversal process consists of three steps : (1) perform the impulse responses hij(t) between each impact positions xi and measurement positions xj to the same impact force and time interval T ; (2) time reverse hij(t) to obtain hij(T-t) (noted hij(-t) in the following) ; (3) and the last step is to send back hij(-t) from the same positions (initially measuring positions). The response after the third step at a position xl is given by:

wl ( xl , t ) = ∑ hik (−t ) ∗ hkl (t ) k

(3)

* represent the convolution product. Equation (3) shows that the time reversal gives a maximum at the initial impact position.

wmax ( xl ) = max( wl ( xl , t ) )

4

(4)

Impact Localization

The trace of the responses’ maxima after retransmission versus the position allows us to determine the initial impact position. First we define a set of measurement positions as the “measuring network” with m positions on the beam and an “excitation network” with n positions. For both defined networks the positions are


63

equally spaced. A localization of an impact at position x0 = 0.2 m is performed with same impact and measuring positions for both networks (Fig. 5). We note a maximum at the impact position and two maxima around at a distance, d, from the maximum representing the half-wavelength tion τ . The distance d is given by:

d=

λ 2

=4

EI ρA

λ

2

corresponding to the impact dura-

πτ

(3)

Fig. 5 Impact localization at x0 = 0.2 m

The measuring network (Fig. 5) was formed by a measuring position every 0.01 m (i.e. m = 100) which can be considered as an ideal case for localization. But for further experiments consideration it was in our interest to reduce the number of measuring points. (See sections 5 and 6)

5

Influence of the Measuring Positions Number

To study the effect of measuring position number on localization we propose a comparison of three cases (Fig. 6). 1st case: the positions are same for both networks which means m = n. In this case a square transfer matrix is built containing all the impulse responses hij(t); 2nd case: a transfer matrix is built with only two points of measurement and n impact positions; 3rd case: Only one measuring point is chosen here. So we get a (1 x n) transfer matrix.

64


Fig. 6 Impact localization at x0 = 0.2 m with three different measuring networks

Figure 6 shows the emergence of other maxima; parasites. Given the nature of the problem, the impact spreads in both directions giving rise to two types of waves; progressive and evanescent. When the measurement is made on either sides of the impact position (second case with two measurement points, for example), the parasites are less apparent. In the case of a single measurement point the information on the impact position is not lost but the parasites are more important.

6

Influence of the Measuring Position

Here we are interested on the third case with only one measuring point. We study the influence of the measuring position from the initial impact position; left or right side, and the effect of the single measuring position itself.

Fig. 7 Impact localization at x0 = 0.2 m with a single point measurement


65

Figure 7 shows two cases for a single measuring point; left (x = 0.1 m) and right (x = 0.8 m) of the impact position (x0 = 0.2 m). In both cases we distinguish the position of impact. The maxima parasites probably are related to the interaction between the wave caused by the impact and boundary conditions. The reflections from the boundaries act as virtual sources.

Fig. 8 Impact localization with measuring at x = 0:5m

Using a single measuring point doesn’t affect the time reversal localization but we need to select carefully its position. In the case of transverse waves in a beam and because of the reciprocity principle (hij = hji) the impact localization fails if the measurement is made at x = 0.5 m (the middle of the beam). Figure 8 shows that for this particular point localization is completely disrupted; two maxima of the same level have emerged.

7

Conclusion

This study allowed us to analyze the time reversal process for the impact localization on a cantilever beam. We demonstrate the effect of the number of measurement points as well as the choice of their positions and allow us to capitalize on the efficiency of time reversal. The time reversal method shows its robustness for impact localization even with partial information about the behavior of the beam (one or two measuring points). The ideal case is to measure the response of the beam along its entire length which needs advanced experimental technique of measuring. So by reducing the number of measurement points to one we can consider an experiment with a single accelerometer.

66


References Ernst, R., Dual, J.: Acoustic emission localization in beams based on time reversed dispersion. Ultrasonics 54(6), 1522–1533 (2014) Fink, M.: Time reversal of ultrasonic fields. i. basic principles. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control (1992) Fink, M., Prada, C., Wu, F., Cassereau, D.: Self focusing in inhomogeneous media with time reversal acoustic mirrors. In: Proceedings of the IEEE Ultrasonics Symposium (1989) Francoeur, D., Berry, A.: Time reversal of flexural waves in a beam at audible frequency. The Journal of the Acoustical Society of America 124(2), 1006–1017 (2008) Graff, K.: Wave Motion in Elastic Solids. Dover Publications (1975) Prada, C., Manneville, S., Spoliansky, D., Fink, M.: Decomposition of the time reversal operator: Detection and selective focusing on two scatterers. The Journal of the Acoustical Society of America 99(4), 2067–2076 (1996) Ribay, G.: Localisation de source en milieu réverbérant par Retournement Temporel. PhD thesis (2006) Sachs, R.: The Physics of Time Reversal. University of Chicago Press (1987)

Appendix Numeric Scheme Let’s differentiate the system of equations (1) using a centered scheme with a constant time step Δt and space step Δx . We write then in second order and using

the notation wi = w(k Δt , i Δx ) : k

∂ 2 w( x, t ) wik +1 − 2wik + wik −1 = + O Δt 2 ∂t 2 Δt 2

( )

(6)

And in fourth order: ∂ 2 w( x, t ) − wik− 2 + 16wik−1 − 30wik + 16wik+1 − wik+ 2 = + O Δx 4 12 Δx 2 ∂x 2

( )

(7)

The system (1) leads to the following iterative process :

⎧ k − wik−2 + 16wik−1 − 30wik + 16wik+1 − wik+2 ⎪Mxi = − E I 12 Δx 2 ⎪ ⎪ wk +1 − 2wik + wik −1 = ⎪ρ A i Δt 2 ⎨ ⎪ − Mx k + 16Mx k − 30Mx k + 16Mx k − Mx k i −2 i −1 i i +1 i+2 ⎪ 12 Δx 2 ⎪ ⎪ k ⎩+ f i

(8)


67

This iterative process is to assess the displacement at time k + 1 knowing the displacement and bending moment at time k (and k-1 for displacement) and the force applied at a point i.

Stability Criteria The stability condition for this process is determined using the Fourier method by k

replacing wi by

ξ k e I kx i Δx

. The condition

ξ i ', kt ) represent the control decision while the inventory ( S ikt ) and backorder size ( BDkt ) as well as the amounts of finished products shipped to the customer (TRi −>CUS ,kt ) are the consequences or state variables. In this problem, we assume that there is enough information available of customer demand at the beginning of each stage that allows the decision maker to adjust the decision variables ( Pikt ) and (TRi −>i ',kt ) for the overall demand scenarios. On the other hand, the consequences

decisions variables ( Sikt ) , ( BDkt ) and (TRi −>CUS ,kt ) should be fixed for the different demand scenarios at each stage of the scenario tree.

4

Multi-stage Stochastic Mathematical Formulation

Indices

Li ST j

Set of direct successor plant of plant i.

i, i’

Production plant index (i,i’ = 1, 2 , ...,I) where i belongs to stage n and i’ belongs to stage n+1.

Set of stages (j= 1,2, ..., N).

A Multi-site Supply Chain Planning Using Multi-stage Stochastic Programming

k t S

Product index (k = 1,2, ..., K). Period index (t = 1,2, ..., T). Scenario index (s = 1,2,. . .,S).

tn

Set of time periods related to node n.

n, m

Node of scenario tree (n = 1,2,. . .,Nn) where

293

⎛ Pre(n), t − 1∉ tn m=⎜ t − 1 ∈ tn ⎝ n, Pre()

Predecessor of the node n in the scenario tree.

Decision variables

Pikt (n)

Production amounts of product k at plant i in period t at node n.

Amounts of end of period inventory of product k for scenario s at plant i in period t at node n. Amounts of end of period inventory of semi-finished product k s JSikt (n) for scenario s at plant i in period t at node n. Backorder amounts of finished product k for scenario s in period t s BDkt (n) at node n. TRi −>i ',kt (n) Amounts of product k transported from plant i to i’ in period t at node n. Amounts of product k transported from the last plant i to customer s TRi −>CUS ,kt (n) for scenario s in period t at node n. Amounts of product k received by plant i for scenario s in period t Qi ,k ( n) at node n.

S ikts ( n )

Parameters s kt

D

Demand of finished product k for scenario s in period t. The occurrence probability of scenario s for the random demand

π

S

s

where

∑π

s

=1

s =1

πn

Probability of the node n in the scenario tree

The multi-stage stochastic model Nn

S

n =1

s =1

T

K

I

Max E [ NPV ] = ∑π n ∑π s ∑∑∑ prk TRis−>CUS ,kt (n) − cbk BDks,t (n) t =1 k =1 i =1

−csik (S (n) + JS (n)) − cti −>CUS ,kTRis−>CUS ,kt (n) − cti −>i ',k TRi −>i ',kt (n) s ikt

s ikt

Nn

T

K

I

n =1

t =1 k =1 i =1

−∑π n ∑∑∑ cpik Pikt (n)

(1)

294

H. Felfel, O. Ayadi, and F. Masmoudi

Siks ,t (n) = Siks ,t −1 (m) + Pikt (n) − ∑ TRi −>i ',kt (n), ∀i ∈ STj < N , ∀k , t, s, n

(2)

i '∈Li

I

I

i =1

i =1

∑ Siks ,t (n) = ∑ Siks ,t −1 (m) + Pikt (n) − TRis−>CUS ,kt (n),

(3)

∀i ∈ ST j= N , k , t , s, n JSiks ,t ( n) = JSiks ,t −1 ( m) + Qikt ( n) − Pikt ( n),

∀i, k , t , s, n

BDkts ( n) = BDks,t −1 ( m) + Dkts ( n) − TRis−>CUS ,kt ( n), ∀k , t , s, n

Qi ' k ,t + DL (n) = ∑ TRi −>i ',kt (n),

∀i, k , t , s, n

(4) (5) (6)

i '∈Li

K

∑b P

k ikt

(n) ≤ cappit ,

∀i, t , n

(7)

k =1

K

∑S

s ikt

(n) + JSikts (n) ≤ capsit ,

k =1 K

∑ TR k =1 s ikt

i −> i ', kt

(n) ≤ captrit ,

∀i, t , s, n ∀i, t , s, n

Pikt ( n), S ( n), JSikts ( n), TRi −>i ',kt ( n), TRis−>CUS ,kt ( n), Qi ,k ( n), BDkts ( n) ≥ 0,

∀i , k , t , s , n

(8) (9)

(10)

The objective function (1) aims to maximize the expected profit for different scenarios of product demand by subtracting the total expected cost from the expected revenue. The equation (2) provides the balance for the inventory of products in every production stage except for the last stage. Constraint (3) represents the balance for the inventory for the last production stage, considering customer demands. Constraint (4) is the inventory balance equation for the semi-finished products. Constraint (5) provides the balance equation for shortage in end product demand. Equation (6) represents the balance equations for the transportation between the different plants. The set of constraints (7), (8) and (9) makes sure that the production capacity, the storage capacity and the transportation capacity are respected. Constraint (10) is the non-negativity restriction on the decision variables.


5

295

Illustrative Example

In this section, an illustrative example involving a multi-site supply chain network is presented in order to show the effectiveness of the proposed model. The considered supply chain network consists of 5 production stages with 8 serial and parallel plants and 2 products. The planning horizon includes 10 periods and the length of a period is one week. Four scenarios are considered for the random demand of two finished products P1 and P2 under period T6, T8 and T10 as reported in Table 1. The total number of demand scenarios is equal to 64 scenarios ( 43 = 64 ). The proposed model is then solved and the quality of the obtained solutions is compared as detailed in section 4.1. The experiments are conducted using LINGO 14.0 package program and MS-Excel 2010 with an INTEL(R) Core (TM) and 2 GB RAM. Table 1 Random finished product demand with corresponding probability Scenario

T6

T8

T10

P1

P2

PROB

P1

P2

PROB

P1

P2

PROB

S1

3350

3230

0.25

3300

3310

0.25

3430

3210

0.2

S2

2650

2520

0.35

2730

2770

0.4

2720

2530

0.3

S3

1860

1550

0.2

1930

2040

0.15

1950

1860

0.25

S4

1240

1020

0.2

1180

1060

0.2

1010

930

0.25

5.1

Results of Stochastic and Deterministic Programming Models

In this section, we compare the solution of the four-stage stochastic programming model (4SM) to the solution of the three-stage (3SM) and two-stage stochastic programming model (2SM) as well as the solution of deterministic model (DTM).The solution of deterministic model is calculated based on the mean value of the uncertain demand. For the four-stage programming model, the planning horizon is clustered into 3 levels and the multi-stage decision process is approximated by a four-stage one. The first stage is the time period zero, the second stage consists of periods 1-6, the third stage includes periods 7-8 and the fourth period includes periods 9-10. For the three-stage programming model, the first stage consists of the time period zero, the second stage consists of periods 1-6, the third stage includes periods 7-10. However, for the two-stage programming approach all the time periods are considered as a one stage. In the two-stage stochastic model, the production amounts in each plant and the product amounts to be transported between upstream and downstream plants are taken “here and now” before the realization of the uncertainty. Other decision variables such as inventory and backorder size, flow of finished products to be shipped to the customer are made in a “wait and see” mode until the realization of random demand.

296


According to Fig. 3 and Table 2, the expected net profit of the supply chain production plan of (4SM), (3SM) and (2SM) models are significantly higher than those of the deterministic model. Moreover, the expected inventory and backorder costs of the deterministic model are considerably greater than those of the (4SM), (3SM) and (2SM) models. It is worth remarking that when the number of stages increases in the demand scenario tree, the expected net profit increases too. Indeed, the expected profit of (4SM) model is higher than those of (3SM) and (2SM) models. Furthermore, Fig. 3 shows that the expected profit of (4SM) and (3SM) models are close to each other. So, there is no need to consider more stages in the scenario tree.

Fig. 3 Expected values of different supply chain planning models

In order to evaluate the influence of uncertainty on the planning decisions two stochastic measures was used. The first metric is the expected value of perfect information (EVPI). This parameter helps to determine the expected profit loss under uncertainty (Birge and Louveaus 1997). It can be calculated as:

EVPI= WS –2SP

(11)

Where WS represents the objective value of the “wait and see” model and 2SP is the objective value of two-stage stochastic programming model. The second measure used in order to evaluate the performance of multi-stage programming model is the relative value of the multi-stage stochastic solution (RVMSS) (Nickel et al. 2012). This measure shows how much saving can be realized using the stochastic programming model in comparison with the deterministic model. It is formulated as follows:


RVMSS = 100

297

SP − EEV EEV

(12)

Where EEV represents the optimal solution of deterministic model. The obtained results are reported in Table 2 and Table 3. As shown in Table 2, the EVPI/WS ratio shows a big influence of demand uncertainty on the obtained solution (19.26%). So, it is worth to have better forecast about the demand scenarios. We can see also from Table 3 that the relative value of the multi-stage stochastic solution RVMSS of the 4SM, 3SM and 2SM models are 20.33%, 19.19% and 16.53%, respectively. It is clear that the multi-stage programming solution of 4SM and 3SM outperforms the two-stage programming solution for the supply chain planning problem. Table 2 Expected net profit of different models

WS

EEV

2SP

3SP

4SP

141720

98193.06

114419.9

117035.02

118153.26

Table 3 Stochastic programming metrics

6

EPVI

EPVI/WS

RVMSS (2SP)

RVMSS (3SP)

RVMSS (4SP)

27300.09

19.26%

16.53%

19.19%

20.33%

Conclusion

In this paper, we have developed a multi-site supply chain production planning model under demand uncertainty. A multi-stage stochastic programming approach is applied to incorporate the effects of uncertainty in the production planning problem. The computational results show that the quality of the four-stage and three stage stochastic model solutions is higher than those of the two-stage stochastic and deterministic models. Managing risk by incorporating a risk metric into the stochastic programming model is one of the directions for future research. Acknowledgements. We would like to acknowledge the financial support provided by the Mobility of researchers and research for the creation of value (MOBIDOC) as well as LINDO Systems, Inc for giving us a free educational research license of the extended version of LINGO 14.0 software package.

References Ahmed, S., King, A., Parija, G.: A multi-stage stochastic integer programming approach for capacity expansion under uncertainty. Journal of Global Optimization 26, 3–24 (2003) Birge, J.R., Louveaux, F.: Introduction to stochastic programming. Springer, New York (1997)

298


Felfel, H., Ayadi, O., Masmoudi, F.: Multi-objective Optimization of a Multi-site Manufacturing Network. In: Mechatronic Systems: Theory and Applications. Springer (2014) Gupta, A., Maranas, C.D.: Managing demand uncertainty in supply chain planning. Computers and Chemical Engineering 27(8-9), 1219–1227 (2003) Kall, P., Wallace, S.W.: Stochastic Programming. John Wiley and Sons, New York (1994) Kazemi Zanjani, M., Nourelfath, M., Ait-Kadi, D.: A multi-stage stochastic programming approach for production planning with uncertainty in the quality of raw materials and demand. International Journal of Production Research 48(16), 4701–4723 (2010) Leung, S.C.H., Tsang, O.S., Ng, W.L., Wu, Y.: A robust optimization model for multi-site production planning problem in an uncertain environment. European Journal of Operational Research 181, 224–238 (2007) Leung, S.C.H., Wu, Y., Lai, K.K.: A stochastic programming approach for multi-site aggregate production planning. Journal of the Operational Research Society 57, 123–132 (2006) Lin, J.T., Chen, Y.Y.: A multi-site supply network planning problem considering variable time buckets– A TFT-LCD industry case. The International Journal of Advanced Manufacturing Technology 33(9-10), 1031–1044 (2006) Lin, J.T., Chen, T.L., Chu, H.C.: A stochastic dynamic programming approach for multisite capacity planning in TFT-LCD manufacturing under demand uncertainty. International Journal of Production Economics 148, 21–36 (2014) Mirzapour Al-e-hashem, S.M.J., Baboli, A., Sazvar, Z.: A stochastic aggregate production planning model in a green supply chain: considering flexible lead times, nonlinear purchase and shortage cost functions. European Journal of Operational Research (ISI) 230, 26–41 (2013) Nagar, L., Jain, K.: Supply chain planning using multi-stage stochastic programming. Supply Chain Management: An International Journal 13(3), 251–256 (2008) Nickel, S., Saldanha-da-Gamac, F., Zieglera, H.P.: A multi-stage stochastic supply network design problem with financial decisions and risk management. Omega 40(5), 511–524 (2012) Shah, N.K., Ierapetritou, M.G.: Integrated production planning and scheduling optimization of multisite, multiproduct process industry. Computers and Chemical Engineering 37, 214–226 (2012) Wang, R.C., Fang, H.H.: Aggregate production planning in a fuzzy environment. International Journal of Industrial Engineering-Theory, Applications, and Practice 7(1), 5–14 (2000)

A System Engineering Conception of Multiobjective Optimization for Multi-physics System Mian Chen1 and Omar Hammami2 1

IRT SystemX, France [email protected] 2 ENSTA ParisTech, France [email protected]

Abstract. At present, a system involves more and more domains and becomes more and more complex. In this paper, a system engineering conception of multiobjective optimization for a multi-physics system is formed. An example of multiphysics system design illustrates the presented approach. Keywords: System Engineering, Multi-Physics Simulation, Multi-Objective Optimization.

1

Introduction

A system recently deals with more and more domains like mechanical, electronic and thermal etc. In addition, the interaction and information exchanges among them increase rapidly. Therefore, the complexity of system is increasing gradually. As the complexity of a system augments, the work of system optimization becomes not only heavier but also more difficult. Besides, the cost and time for the whole system engineering design augment [9]. System engineering flow plays a crucial role because it should have a global view of the whole system and optimize the whole system in consideration of multiple objectives in different domains. Also, it ought to insure that the system satisfies some requirements [8]. For the moment, in order to develop a multi-physics system, the experts in different domains have to hold a lot of meetings to reach agreement. However, a method of automatic system engineering working flow with multi-objective optimization simultaneously is expected.

2

Multi-objective Optimization System Engineering Flow for a Multi-physics System

In Fig.1, a multi-objective system engineering flow for multi-physics system is represented. At the top level, by SysML, a system engineering model which includes all relative system information is defined. At the bottom level, there are a lot of different physical system models and each model could be simulated separately. At central level, a model of multi-objective optimization is built and it connects the top level and the bottom level.


299

300

M. Chen and O. Hammami

Fig. 1 Multi-objective optimization system engineering flow

In order to realize the expected system engineering flow and take into account the feasibility, several software are chosen as Fig.2. Rhapsody is chosen as a tool to deal with system engineering model which presents the system’s requirements and describes all the structural, parametric information by language SysML [6]. At the bottom level, several different software could be applied. Catia is chosen to simulate the geometrical model, Dymola is used to simulate the mechatronic model and Comsol is chosen to do the mechanical and thermal simulation. Certainly, other software of physical simulation would be added in this multiobjective system engineering working flow to realize different simulations. Besides, the software ModelCenter is taken use to do the multi-objective optimization, which would harmonize multiple simulations in different software and on the other side, it could collaborate with Rhapsody [7].

Fig. 2 Multi-objective optimization system engineering flow with different software

3

Multi-objective Optimization System Engineering Flow for a Study Case – Power Closure System

In this paper, a study case of Power Closure System [10] is chosen to illustrate the proposed multi-objective optimization system engineering working flow. The goal of this system is that by electronic button control, two little motors could drive

A System Engineering Conception of Multi-objective Optimization

301

respectively two jacks to open or close the whole liftgate automatically which could replace traditionally manual operation.

3.1

Rhapsody Model

In Fig.3, among all of the system engineering model, only the system structure definition is shown. Overall, the power closure system has four parts: control part, mechanical part, motor and liftgate. And the liftgate includes a piece of glass and a frame made of metal [11] [12].

Fig. 3 Rhapsody model

3.2

Catia Model

For the geometrical model, according to those existing liftgate in market, a very simple 3D CAD model of liftgate is designed and its form is shown in Fig.4. This model has two different parts, one part is the frame which is made of metal and the other part is the window which is made of transparent glass.

Fig. 4 Catia model

302

3.3


Comsol Model

To create the mechanical model in Comsol, the geometrical model of Catia could be imported directly in Comsol. In Comsol, a moment when the liftgate has already opened to the maximal angle is simulated. Supporting that at the moment, the force is still perpendicular to the principal liftgate and the value is 200N. Then, the stress and displacement distribution of the whole liftgate could be seen in Fig.5 and Fig.6.

Fig. 5 Comsol model - Stress distribution

3.4

Fig. 6 Comsol model - Displacement distribution

Dymola Model

The motor PLE-29 is chosen as model which is furnished by Johnson Electric and applied in Power Lift Gate System. According to the parameters sheet and several important indices, a model of Motor is obtained and shown in Fig.7.

Fig. 7 Dymola model - Motor


303

In order to do the liftgate model, multi-body library is used. A reduction gear box and a ball screw is chosen as one part of the jack. The whole liftgate with two arms is shown in Fig.8.

Fig. 8 Dymola model - Liftgate

Fig. 9 Dymola model - 3D Animation open- Fig. 10 Dymola model - 3D Animation ing closing

After modeling of motor and liftgate, those two parts could be combined. In the Dymola model, the motor control part is simplified, a constant tension is supposed to supply those two motors which drive the whole liftgate. Finally, a 3D animation simulation has been done. The whole process of opening or closing is between 7 seconds and 8 seconds (Fig.9 and Fig.10).

304

4


Multi-objective Optimization

Before running the whole system multi-objective optimization, one important conception should be emphasized: an optimization problem with more than one objective, there is no single optimum solution. It exists a number of solutions which are all optimal. Without any further information, no solution from the set of optimal solutions can be said to be better than any other.

4.1

NSGA-II

Compared with other multi-objective algorithms like Multi-Objective Simulated Annealing (MOSA) [5], Multi-Objective Optimization using Evolutionary Algorithms (MOEA) [4] and Multi-Objective Genetic Algorithm II (MOGA-II) [3], an algorithm called Non-dominated Sorting Genetic Algorithm II (NSGA-II) [1] [2] is selected to achieve the multi-objective optimization in this paper. The algorithm NSGA-II involves with three main conceptions: non-dominated sort, crowding distance and genetic operators. Its main loop of optimization process is presented in Fig.11.

Fig. 11 NSGA-II algorithm

4.2

ModelCenter Model

After imposing the Rhapsody model into ModelCenter, the model lists all the input variables and output variables. In the tool ModelCenter, for running the optimization process, at first, according to the different goals, the objective function should be defined. Secondly, the constraints are indicated as Fig.12. Thirdly, the decision variables should be defined. Finally, the NSGA-II has been selected as algorithm to do the optimization. After defining all the parameters of multi-objective optimization as above, the optimization process could be launched, and the results would be presented in Fig.13. The red points represent infeasible solutions (which don’t meet constraints), the green ones represent feasible Pareto solutions and the blue ones represent the optimal Pareto (which represents optimal solutions and no one could be better than the others).


305

Fig. 12 NSGA-II objectives and constrains

Fig. 13 Results

5

Conclusion

This work proposes a multi-objective optimization system engineering working flow for multi-physics system which could be regarded as an automatic new methodology to reduce design cost and cycle time. Although there are still some details which have to be completed and improved, the main conception has been realized.

306


Even though the main conception is realized in this paper, from the industrial viewpoint, this study case is too simple. Therefore, in the future work, some more connections problems between those software and models synchronization should be solved. Acknowledgment. For the members in the laboratory U2IS and SUPMECA, here we express all our esteem and sincere acknowledgement to them for their encouragement, friendship moral support and their scientific assistants for all the time.

References [1] Deb, K.: Multi-Objective Optimization Using Evolutionary Algorithms, Chichester, U.K. (2001) [2] Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II. IEEE Trans. Evol., Comput. 6(2), 182–197 (2002) [3] Poles, S.: MOGA-II An improved Multi-Objective Genetic Algorithm. Esteco, Italy (2003) [4] Seshadri, A.: Multi-Objective Optimization using Evolutionary Algorithms (MOEA) [5] Rigoni, E.: MOSA Multi-Objective Simulated Annealing, Esteco, Italy (2003) [6] Systems Engineering Tutorial for Rational Rhapsody, IBM (2009) [7] MBSE PAK for Rhapsody USER GUIDE, Phoenix Integration, Blacksburg, USA (2014) [8] Hammami, O.: SYNSYS-ME: Seamless System Engineering to mechanical flow through multiobjective optimization and requirements analysis. In: 8th Annual IEEE Systems Conference (SysCon), Ottawa, Canada (2014) [9] Hammadi, M., Choley, J., Penas, O., Riviere, A., Louati, J., Haddar, M.: A new multi-criteria indicator for mechatronic system performance evaluation in preliminary design level. In: 9th France-Japan Mechatronics, Paris, France (2012) [10] Raka, S.: Methodes et outils ensemblistes pour le pre-dimensionnement de systemes mecatroniques. Ensea, Cergy-Pontoise, France (2011) [11] Peak, R.S., Burkhart, R., Friedenthal, S., Wilson, M., Bajaj, M., Kim, I.: SimulationBased Design Using SysML Part 1: A Parametrics Primer. In: INCOSE International Symposium, San Diego, U.S.A. (2007) [12] Peak, R., Burkhart, R.M., Friedenthal, S.A., Wilson, M.W., Bajaj, M., Kim, I.: Simulation-Based Design Using SysML Part 2: Celebrating Diversity by Example. In: INCOSE Intl. Symposium, San Diego, U.S.A. (2007) [13] Hammadi, M.: Contribution a l’integration de la modelisation et la simulation multiphysique pour la conception des systemes mecatroniques. In: LISMMA and SUPMECA, Paris, France (2012)

Falling Film in a Heated Micro-channel Sabrine Mejri1, Olivier Fudym2, Jalila Sghaier1, and Ahmed Bellagi1 1

Research unit of heat and thermodynamics of industrial processes National Engineering School of Monastir Rue Ibn Al Jazzar 5000 Monastir, Tunisia [email protected] 2 Research Center of Albi in Process Engineering, Divided Solids, Energy and Environment Ecole des Mines d'Albi Campus Jarlard - Route de Teillet 81013 Albi CT Cedex 09, France [email protected]

Abstract. The main objective of this work is to study experimentally and numerically a falling film in a micro-channel. The experimental section involves in creating a temperature gradient within the liquid, while monitoring the temperature using an infrared camera. A numerical model is established and solved by a semianalytical method called the thermal quadrupole method. Finally, we conclude with a comparison between the experiments and the numerical study. Keywords: falling film, micro-channel, thermal quadrupole.

1

Introduction

Miniaturization is currently a promising field of both theoretical and experimental research (Aiello et al. 2009; Foerster et al. 2013). It is to reduce the dimensions of some industrial equipment in order to reduce the technological constraints in favor of environmental chemistry and processes safety (Brendner et al. 2013). It is targeting, in general, refrigeration equipments, air conditioning and heat pumps which have their effects on the environment, the layer ozone and global warming (Luke et al. 2010). Thus, comes the idea of increasing processes. This means: developing and creating methods and devices offering improved production quality, reducing the size / capacity, lowering effluent quantities and / or catalysts, and reducing costs and energy losses with a greater respect for environmental constraints (Mhiri.N 2006). Micro-reactors operate on the principle of a continuous process and are highly different from those several traditional characteristics such that the increase of the exchange surface through the ratio S / V (Perry et al. 2010), the improvement of heat and mass transfer and a high temperature gradient and pressure (Baldassari et al. 2013; Conti et al. 2012). They help maintain better control of reaction conditions, improved security conditions, the gain in energy and reducing congestion (Hung et al. 2013). The falling films are generally used as an intermediary for the transfer of heat © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_32

307

308

S. Mejri et al.

and mass transfer (Zhang 2008; Bu et al 2011) in industrial equipment such as condensers and the evaporation towers (Marenzana et al. 2004; Buonomo et al. 2012). An extension of thermal quadrupole formalism is proposed for the modeling of heat conduction in heterogeneous environments, where the change in thermal properties is one-dimensional (Fudym 2002). The main problem was to find an intrinsic relationship between generalized temperature and heat flux at the boundary of a heterogeneous medium.

2

Experimental Set Up and Procedure

Our experiment consists in: creating a temperature gradient, in the back of the plate, and then managing the liquid flow (water and ethanol). An infrared camera is used in order to visualize the effect of the temperature gradient on the flow.

2.1

Preparation

First of all, the plate was cleaned well and then sprayed with a black spray to create a black thin layer in order to maximize radiation, since an infrared camera is used for the visualization of the heat transfer on the surface of the falling film. Then, an electrical resistance is implanted in the back of the plate to generate heat.

Fig. 1 Experimental Set up

Falling Film in a Heated Micro-channel

309

In order to create a temperature gradient, a heat exchanger is placed in contact with the back face of the plate to maintain a low temperature. The infrared camera is a FLIR A320 (COMPACT range 19,600 pixels), which records the various infrared radiation emitted by the body and which vary with their temperature. This thermal imaging camera does not capture what is behind a wall or an obstacle. Its spectral range is from 12 to 8 microns, and since water is almost opaque in this field then it is certain that the radiations captured by this camera are those of the liquid surface. To achieve the flow in the micro-channel, we used a syringe and a very thin tube. In this experiment we used two liquids: water and ethanol. The flow of ethanol was smoother because its surface tension is lower than that of water.

2.2

Results of the Experiments

After processing the images, the results obtained are as follows: In figures (Fig. 2 and Fig. 3), we visualized the ethanol falling film flow in a micro-channel, wherein it is noted that initially the temperature of the ethanol at room temperature (20 ° C), increases, once the flow reaches the heating zone to 32 ° C and then begins to decrease through the cooling zone to 23 ° C.

Fig. 2 IR image: ethanol falling film on a channel of the micro-structured plate

S. Mejri et al.

Ethanol surface Temperature in a micro-channel 40 30 20 10 0 2.1875 3.0625 3.9375 4.8125 5.6875 6.5625 7.4375 8.3125 9.1875 10.0625 10.9375 11.8125 12.6875 13.5625 14.4375

Ethanol surface temperature (°C)

310

Ethanol surface Temperature in a micro-channel

Lenght (cm)

Fig. 3 Curve of the surface temperature of the ethanol falling film in a single channel

Fig. 4 IR image: water falling film on a channel of the micro-structured plate

For the water (Fig. 4 and Fig. 5), the temperature reached 33°C, and then went down to 18°C. In both experiments, we calculated the temperature gradient and found it 2.8°C per centimeter which is close to 3°C/cm, the value we wanted to reach.

311

Water surface Temperature in a micro-channel 40 30 20 10 0 0 0.924 1.848 2.772 3.696 4.62 5.544 6.468 7.392 8.316 9.24 10.164 11.088 12.012 12.936

Water surface temperature (°C)


Water surface Temperature in a micro-channel

Lenght (cm)

Fig. 5 Curve of the surface temperature of the falling film of ethanol for one channel

3

Mathematical Model

3.1

Governing Equations

The plate we are using is a stainless micro-plate in which the micro-structured channels have the following dimensions: Width: 1000 microns Depth: 150 microns Length: 15 cm The fluid flow is in a free fall flow governed by gravity and in permanent regime. The flow assumptions are as follows: • • • • •

2D permanent regime. Ideal fluid (no shear stresses). Constant physical properties. No effect of surface tension. Very thin liquid film.

Applying the Navier-Stocks equations on our system and taking the boundary conditions, we get these equations:

y u = um (1 − (1 − )²)

(1)

∂ ²T ∂ ²T ∂T ) + = α( ∂x ² ∂y ² ∂x

(2)

δ

u

312

S. Mejri et al.

∂ ²T ∂ ²T =0 + ∂x² ∂y ²

(3)

Equation (2) is the result of the energy equation in the falling film and Equation (3) is the result of the energy equation in the micro-channel.

3.2

Solving with the Thermal Quadrupoles Method

We will adopt the method of thermal quadrupoles to solve this problem, which is mainly finding the relationship between flow and temperature at input and output by applying the Fourier Transform (Fig. 6).

Fig. 6 Input and output of the system

Transforming all the equations previously obtained along with the boundary conditions gave us these results: In the plate:

⎛ cos pl ⎛θ 0 ⎞ ⎜ ⎜⎜ ⎟⎟ = ⎜ ⎝ φ 0 ⎠ ⎜ − λ p sin p l ⎝

⎞ sin pl ⎟⎛ θ l ⎞ λ p ⎟⎜⎜ ⎟⎟ φ cos pl ⎟⎠⎝ l ⎠ 1

(4)

In order to use the thermal quadrupoles method on the liquid, we proceed by dividing the film into N layers having each one the same “Eq. 1”. So the matrix in each layer is:


313

(5) And finally we have the matrix of our system:

⎛ ⎛θw ⎞ ⎜ cos pl ⎜⎜ ⎟⎟ =⎜ ⎝φw ⎠ ⎜−λ psin pl ⎝

1 δ δ⎞ ⎞ ⎛ sin pl⎟ N ⎜ coshKi N λK sinhKi N⎟⎛θδ ⎞ i ⎟⎜ ⎟ pl ⎟*∏⎜ δ δ ⎟⎜⎝φδ ⎟⎠ i=1 ⎜ ⎟ coshKi cos pl ⎠ ⎜λKi sinhKi ⎟ N N ⎠ ⎝

1

(6)

With the inverse Fourier transform, we find the expression of the surface temperature of the falling liquid film which is simulated then to make the profile:

T∂ = Re(θ ∂ 0 + ∑ θ ∂* exp( pxm (ix ))) / L

(7)

Fig. 7 Simulated water surface temperature evolution

The Figure (Fig. 7) shows the curve of the temperature profile at the surface of the flow. We can see the increase in the temperature in the heating zone.

314

S. Mejri et al.

Fig. 8 Simulated ethanol surface temperature evolution

Note that the temperature rises from 22.8 ° C, the input channel temperature and the beginning of the heating zone, at x = 0.018 m, reached its peak at x = 0.052 m with a temperature of 27.4 ° C, and begins to fall back to its initial temperature. This drop is due to the periodicity of the proper function (temperature), since the resolution was made by the integral Fourier transform. Same thing for the ethanol graph (Fig. 8), rising from a temperature close to 22°C, the liquid exceeded 28°C in the heating zone and then come down to the cooling temperature. We studied also the changes in temperature within the liquid along the y axis. We note that there is not a great change throughout the y axis which means we can avoid the division of our multilayer medium, that is to say we can keep a single mesh. There is, also, a resemblance between the numerically determined temperature profile and that determined experimentally. From the figure that the temperature profile in the flow, we can conclude that the independence with respect to y means that the temperature can be considered as a single unit with one average speed. We managed to achieve a temperature gradient close to 3 ° C / cm; this number was calculated by the laboratory of Nancy LRGP for diabatic distillation.


4

315

Conclusion

In this work, we conducted experiments of flowing liquid on our micro-structured plate, then on a single channel, and using an infrared camera, we filmed this flow to compare the experimental curves with digital ones. In the second part, we have developed a computer code using Matlab software, which allowed us to describe the temperature profile on the surface of the flow based on the semi-analytical method of thermal quadrupoles. The code developed will allow us to address several problems, such as the control problem, which is essentially to determine the thermal conditions necessary to achieve a desired temperature profile. That is to say, the flux or seek to impose the temperature from the temperature profile at the surface. Acknowledgements. This work is partially supported by l’Ecole des mines d’AlbiCarmaux and the Research Center of Albi in Process Engineering, Divided Solids, Energy and Environment. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.

References Zhang, F., Wub, Y.T., Gengb, J., Zhangb, Z.B.: An investigation of falling liquid films on a vertical heated/cooled plate. International Journal of Multiphase Flow 34, 13–28 (2008), doi:10.1016/j.ijmultiphaseflow.2007.08.003 Mhiri, N.: Etude d’un micro-absorbeur Gaz/Liquide à film tombant. Récents progrès en génie des procédés, Dissertation, Institut national polytechnique de Lorraine (2006) Fudym, O., Ladevie, B., Batsale, J.C.: A seminumerical approach for heat diffusion in heterogeneous media: one extension of analytical quadrupole method. Numerical Heat Transfer, Part B 42, 325–348 (2002), doi:10.1080/10407790190053978 Bu, X., Ma, W., Huang, Y.: Numerical study of heat and mass transfer of ammonia-water in falling film evaporator. Heat Mass Transfer 48, 725–734 (2011), doi:10.1007/s00231011-0923-4 Perry, I., Jannot, Y., Maillet, D., Fiers, B.: Effect of velocity distribution on external wall temperature field for a flat microchannel. Experimental Heat Transfer 23, 27–43 (2010), doi:10.1080/08916150903402724 Baldassari, C., Marengo, M.: Flow boiling in micro-channels and micro-gravity. Progress in Energy and Combustion Science 39, 1–36 (2013), doi:10.1016/j.pecs.2012.10.001 Hung, T.C., Huang, Y.X., Yan, W.M.: Thermal performance analysis of porousmicrochannel heat sink with different configuration designs. International Journal of Heat and Mass Transfer 66, 235–243 (2013), doi:10.1016/j.ijheatmasstransfer.2013.07.019 Wiesegger, L.E., Knauss, R.P., Guntsching, G.E., Brandner, J.J.: Vapor-liquid phase separation in micro-/ministructured devices. Chemical Engineering Science 93, 32–46 (2013), doi:10.1016/j.ces.2013.01.030 Conti, A., Lorenzini, G., Jeluria, Y.: Transient conjugate heat transfer in straight microchannels. International Journal of Heat and Mass Transfer 55, 7532–7543 (2012), doi:10.1016/j.ijheatmasstransfer.2012.07.046

316

S. Mejri et al.

Verma, N., Mewes, D., Luke, A.: Lattice Boltzman study of velocity, temperature, and concentration in micro-reactors. International Journal of Heat and Mass Transfer 53, 3175–3185 (2010), doi:10.1016/j.ijheatmasstransfer.2010.03.009 Foerster, M., Lam, K.F., Sorensen, E., Gavriilidis, A.: In situ monitoring of microfluidic distillation. Chemical Engineering Journal 227, 13–21 (2013), doi:10.1016/j.cej.2012.11.125 Marenzana, G., Perry, I., Maillet, D.: Mini and micro-channels: influence of axial conduction in the walls. International Journal of Heat and Mass Transfer 47, 3993–4004 (2004), doi:10.1016/j.ijheatmasstransfer.2004.04.016 Buonomo, B., Manca, O.: Transient natural convection in a vertical micro-channel heated at uniform heat flux. International Journal of Thermal Sciences 56, 35–47 (2012), doi:10.1016/j.ijthermalsci.2012.01.013 Aiello, G., Ciafolo, M.: Natural convection cooling of a hot vertical wall wet by a falling liquid film. International Journal of Heat and Mass Transfer 52, 5954–5961 (2009), doi:10.1016/j.ijheatmasstransfer.2009.08.007

Interfacial Stresses in FRP-Plated RC Beams: Effect of Adherend Shear Deformations Abderezak Rabahi1, Belkacem Adim1, Selma Chargui2, and Tahar Hassaine Daouadji1 1

Département de Génie Civil, Université Ibn Khaldoun, BP 78 Zaaroura, 14000 Tiaret, Algérie [email protected], [email protected], [email protected] 2 Département de Génie Civil, Université Hassiba Benbouali, Chlef, Algérie [email protected]

Abstract. A recently popular method for retrofitting reinforced concrete (RC) beams is to bond fibre reinforced polymer (FRP) plates to their tensile faces. An important failure mode of such plated beams is the debonding of the FRP plates from the concrete due to high level of stress concentration in the adhesive at the ends of the FRP plate. This paper presents an improved solution for interfacial stresses in a concrete beam bonded with the FRP plate by including the effect of the adherend shear deformations. The analysis is based on the deformation compatibility approach where both the shear and normal stresses are assumed to be invariant across the adhesive layer thickness. In the present theoretical analysis, the adherend shear deformations are taken into account by assuming a parabolic shear stress through the thickness of both the concrete beam and the bonded plate. Numerical results from the present analysis are presented both to demonstrate the advantages of the present solution over existing ones and to illustrate the main characteristics of interfacial stress distributions. Keywords: FRP composites, Interfacial stresses, Concrete Strengthening.

1

Introduction

The structural plating of reinforced concrete (RC) beams with fibre reinforced polymer (FRP) laminates represents a new technology in the civil engineering field. In fact, until today, the most common method in structural strengthening has been the hermitian technique using the application of steel plates. The new method has many advantages: • • •

High strength to weight ratio; Lightness; Corrosion resistance.


317

318

A. Rabahi et al.

The transferring of stresses from concrete to the FRP reinforcement is central to the reinforcement effect of FRP-strengthened concrete structures. This is because the stresses are susceptible to cause the undesirable premature and brittle failure. In strengthening reinforced concrete beams with FRP strips, different failure modes have been observed. The importance of including the shear-lag effect of the adherends was shown by several authors such as Adams and Wake, Jones and Callinan and Tsai et al. in adhesive lap joints. Tounsi has extended this theory to study concrete beams strengthened by FRP plates. The basic assumption in these two studies is a linear distribution of shear stress across the thickness of the adherends. However, it is well known that, in beam theory, this distribution is parabolic through the depth of beam. The objective of the present investigation is to improve the method developed by Tounsi by assuming a parabolic shear stress across the depth of both FRP plate and RC beams. In view of this, it is desirable that a solution methodology be developed where the effect of adherend shear deformations can be included in a better manner so that the accuracy of Tounsi’s solution can be properly assessed. With this in mind, the objectives of this paper are first to present an improvement to Tounsi’s solution to obtain a new closed-form solution which accounts for the parabolic adherend shear deformation effect in both the beam and bonded plate and second to compare quantitatively its solution against the new one developed in this paper by numerical illustrations. Numerical examples and a parametric study are presented to illustrate the governing parameters that control the stress concentrations at the edge of the FRP strip. Finally, the adopted improved model describes better the actual response of the FRP- RC hybrid beams and permits the evaluation of the adhesive stresses, the knowledge of which is very important in the design of such structures. It is believed that the present results will be of interest to civil and structural engineers and researchers.

2

Method of Solution

2.1

Basic Assumptions

The following assumptions were made in the analytical study: 1. All materials considered are linear elastic. 2. The beam is simply supported and shallow, i.e., plane sections remain plane in bending. 3. No slip is allowed at the interface of the bond (i.e., there is a perfect bond at the adhesive steel or FRP plate interface). 4. Bending deformations of the adhesive are neglected. 5. Stresses in the adhesive layer do not change with the thickness. 6. The shear stress analysis assumes that the curvatures in the beam and plate are equal (since this allows the shear stress and peel stress equations to be uncoupled). However, this assumption is not made in the peel stress solution. When

Interfacial Stresses in FRP-Plaated RC Beams: Effect of Adherend Shear Deformations

3119

the beam is loaded, vertical separation occurs between RC beam and FRP platte. This separation creates an n interfacial normal stress in the adhesive layer. We noote that this assumption is ussed in several works, e.g., Tounsi et al., Smith and Tenng and Stratford and Cadei. 7. Aparabolic shear sttress distribution through the depth of both the concreete beam and the bonded platte is assumed. 8. The section propertiies of the RC beam were based on the uncracked section, excluding the conventionaal steel reinforcement. It is known that for uncracked secction, the concrete can susstain tension. However, for cracked section, the concreete cannot sustain tension and d this is why the effect of steel reinforcement in concreete is not neglected.

Fig. 1 Simply supported beaam strengthened with bonded FRP plate

Fig. 2 Forces in infinitesimal element of a soffit-plated beam

2.2

Shear Stress Diistribution along the FRP–Concrete Interfacee

A differential section, dx x, can be cutout from the FRP- strengthened RC beam m (Fig. 2), as shown in Fig. 3. The strains in the RC beam near the adhesive interfacce and the external FRP reinforcement can be expressed, respectively as

ε1 (x) =

N t dτ a du1 (x) y1 d M1 (x) + 1 + 1 = dx E1I1 E1 A1 4G1 dx

(1)

320

A. Rabahi et al.

ε 2 (x) =

du2 (x) − y2 N 5t dτ a = M2 (x) + 2 − 2 dx E2 I 2 E2 A2 12G2 dx

(2)

Where N(x) is the axial force in each adherend and A is the cross- sectional area. The shear stress in the adhesive can be expressed as follows:

τ a = τ ( x ) = K s [u 2 ( x ) − u1 ( x )]

(3)

Moment equilibrium of the differential segment of the plated beam in Fig. 3 gives

MT (x) = M1(x) +M2(x) + N(x)(y1 + y2 +ta )

(4)

Where MT(x) is the total applied moment. The bending moment in each adherend, expressed as a function of the total applied moment and the interfacial shear stress, is given as x ⎤ R ⎡ (5) M1 (x) = ⎢MT (x) − b2 ∫τ (x)( y1 + y2 + ta )dx⎥ R +1 ⎣ 0 ⎦ x ⎤ 1 ⎡ (6) M ( x) = M ( x ) − b τ ( x )( y + y + t )dx 2

⎢ R +1 ⎣

T

2

∫

1

2

0

a

⎥ ⎦

The first derivative of the bending moment in each adherend gives:

dM1(x) R [VT (x) − b2τ (x)(y1 + y2 + ta )] = R +1 dx

(7)

dM1 (x) R [VT (x) − b2τ (x)(y1 + y2 + ta )] = dx R +1

(8)

Substituting eqs. (1) and (2) into Eq. (3) and differentiating the resulting equation yields ⎛ − y dM2 (x) ⎛ t 5t ⎞ d 2τ (x) d 2τ (x) 1 dN2 (x) y1 dM1(x) 1 dN1 (x) ⎞ ⎟⎟ − Ks ⎜⎜ 2 + 1 ⎟⎟ = Ks ⎜⎜ 2 + − − 2 E2 A2 dx E1I1 dx E1 A1 dx ⎠ dx ⎝ E2 I2 dx ⎝ 4G2 12G1 ⎠ dx²

⎛ y +y ⎞ ⎛ ( y + y )( y + y + t ) 1 1 ⎞ d 2τ (x) ⎟⎟τ ( x) + K1⎜⎜ 1 2 ⎟⎟VT ( x) = 0 + − K1b2 ⎜⎜ 1 2 1 2 a + + E I E I E A E A dx2 1 1 2 2 1 1 2 2⎠ ⎝ E1I1 + E1I1 ⎠ ⎝

(9)

(10)

Where K1 =

1

(11)

⎛ ta t 5t ⎞ ⎜⎜ + 1 + 2 ⎟⎟ G 4 G 12 G2 ⎠ 1 ⎝ a

General solution to Eq. (10) is given by

τ ( x ) = B1 cosh( λ x ) + B 2 sinh( λ x ) + m1VT ( x )

(12)

Interfacial Stresses in FRP-Plated RC Beams: Effect of Adherend Shear Deformations

⎛ ( y1 + y2 )(y1 + y2 + ta ) 1 1 ⎞ ⎟ + + E1I1 + E2 I 2 E1 A1 E2 A2 ⎟⎠ ⎝

λ2 = K1b2 ⎜⎜

m1 =

321

(13)

K1 ⎛ y1 + y2 ⎞ ⎜ ⎟ λ2 ⎜⎝ E1 I1 + E2 I 2 ⎟⎠

(14)

The expression for the interfacial shear stress at any point − λx

⎤ qe ⎡ m2 a ( L − a) − m1 ⎥ ⎦ λ ⎣ 2

τ ( x) = ⎢

⎛L ⎞ + m1q⎜ − a − x ⎟ ⎝2 ⎠

0 ≤ x ≤ LP

(15)

Where q is the uniformly distributed load and x; a; L and Lp are defined in Fig. 1.

m2 =

2.3

K1 y1 E1I1

(16)

Interfacial Normal Stress Distribution along the FRP– Concrete Interface

The interfacial normal stress in the adhesive can be expressed as follows:

σn (x) = KnΔw(x) = Kn [w2 (x) − w1(x)]

(17)

Where Kn is the normal stiffness of the adhesive per unit length and can be written as Kn =

σ n ( x) Δ w( x )

=

σ n ( x) ⎛ 1 ⎞

E ⎜ ⎟= a Δw( x) / t a ⎜⎝ t a ⎟⎠ t a

(18)

W1(x) and W2(x) are the normal displacements of adherends 1 and 2, respectively. Differentiating Eq. (17) twice results in ⎡ d 2 w 2 ( x ) d 2 w1 ( x ) ⎤ d 2σ n ( x ) = − K n⎢ 2 dx 2 dx 2 ⎥⎦ ⎣ dx

(19)

Adherend 1:

d 4 w1 ( x) 1 y dτ ( x) q = b2σ ( x) + 1 b2 + 4 dx E1I1 E1 I1 dx E1 I1

(20)

Adherend 2:

dτ ( x) y d 4 w2 ( x) 1 =− b2σ ( x) + 2 b2 4 dx E2 I2 dx E2 I 2

(21)

322

A. Rabahi et al.

Substitution of Eqs. (20) and (21) into the fourth derivation of the interfacial normal stress obtainable from Eq. (19) gives the following governing differential equation for the interfacial normal stress:

d 4σ n ( x) Eab2 ⎛ 1 Eb ⎛ y y ⎞ dτ ( x) qEa 1 ⎞ ⎜⎜ ⎟⎟σ n ( x) + a 2 ⎜⎜ 1 − 2 ⎟⎟ + + + =0 4 dx ta ⎝ E1I1 E2 I 2 ⎠ ta ⎝ E1I1 E2 I 2 ⎠ dx ta E1I1

(22)

General solution to Eq. (22) is given by

σ n ( x) = e−βx [C1 cos(βx) + C2 sin(βx)] + eβx [C3 cos(βx) + C4 sin(βx)] − n1

dτ ( x) − n2q (23) dx

For large values of x it is assumed that the normal stress approaches zero and, as a result, C3 = C4 = 0. The general solution therefore becomes

σ n ( x) = e−βx [C1 cos(βx) + C2 sin(βx)] − n1

dτ ( x) − n2 q dx

(24)

Where:

E a b2 ⎛ 1 1 ⎞ ⎟ ⎜⎜ + 4t a ⎝ E1 I1 E 2 I 2 ⎟⎠

β =4

⎛ y E I − y2 E1I1 ⎞ ⎟⎟ n1 = ⎜⎜ 1 2 2 ⎝ E1I1 + E2 I 2 ⎠

n2 =

(25)

(26)

E2 I 2 b2 ( E1 I1 + E 2 I 2 )

(27)

As is described by Tounsi 2006, the constants C1 and C2 in Eq.(24) are determined using the appropriate boundary conditions and they are written as follows:

C1 =

Ea [VT (0) + βMT (0)] − n3 3 τ (0) + 2β ta E1I1 2β 3

(28)

n1 ⎛ d 4τ (0) d 3τ (0) ⎞ ⎟ ⎜ +β 3 ⎜ 4 2β ⎝ dx dx3 ⎟⎠ C2 =

n1 d 3τ (0) Ea ( 0 ) M − T 2β 2 t a E1 I1 2 β 2 dx 3

(29)

Where:

n3 =

Eat2 ta

⎛ y1 y2 ⎜⎜ − ⎝ E1 I 1 E 2 I 2

⎞ ⎟⎟ ⎠

(30)


3

323

Results and Discussion

The effect of the adherend shear deformations using the as sumption of parabolic shear stress distribution across the thickness of the adherends on the maximum shear and normal stress is examined by comparing the results obtained with the present theory and those determined by two other theories [Smith S.T 2001 a,b,c]. This difference is due to the assumption used in the present theory which is in agreement with the beam theory (parabolic distribution of shear stresses through the depth of the beam). Hence, it is apparent that the adherend shear deformation reduces the interfacial stresses concentration and thus renders the adhesive shear distribution more uniform. The interfacial normal stress is seen to change sign at a short distance away from the plate end.

Fig. 3 Comparison of interfacial shear/normal stress for CFRP-plated RC beam.

4

Parametric Study

4.1

Effect of Plate Thickness

The thickness of the FRP plate is an important design variable in practice. Fig. 4 shows the effect of the thickness of the FRP plate on the interfacial stresses. Here, the values of the thickness, 2, 4, 6, 8, 10 and 12mm, are considered. It is shown that the level and concentration of interfacial stress are influenced considerably by the thickness of the FRP plate. The interfacial stresses increase as the thickness of FRP plate increases.

324

A. Rabahi et al.

Fig. 4 Effect of plate thickness on interfacial stresses in CFRP-strengthened beam

4.2

Effect of Plate Material on Interfacial Stresses in Strengthened Beam

Fig. 5 gives interfacial normal and shear stresses for the RC beam bonded with a CFRP plate and GFRP plate, respectively, which demonstrates the effect of plate material properties on interfacial stresses. The length of the plate is Lp = 2400mm, and the thickness of the plate and the adhesive layer are both 4mm. The results show that, as the plate material becomes softer (from CFRP to GFRP), the interfacial stresses become smaller, as expected. This is because, under the same load, the tensile force developed in the plate is smaller, which leads to reduced interfacial stresses. The position of the peak interfacial shear stress moves closer to the free edge as the plate becomes less stiff.

Fig. 5 Effect of plate material on interfacial stresses in strengthened beam


4.3

325

Effect of Adherend Thickness

Fig. 6 shows the effect of the thickness of adherend on the interfacial stresses. Here, the values of the thickness, 2, 4, 6, and 8 mm, are considered. It is shown that the level and concentration of interfacial stress are influenced considerably by the thickness of the FRP plate. The interfacial stresses reduce as the thickness of adherend increases.

Fig. 6 Effect of Adherend thickness on interfacial stresses in strengthened beam

5

Conclusion

New theoretical interfacial stress analysis has been presented for simply supported RC beams bonded with a thin composite or steel plate and subjected to a uniformly distributed bending load. The salient features of the new analysis include the consideration of the adherend shear deformations by assuming a parabolic shear stress through the thickness of both the concrete beam and bonded plate. The solution methodology is general in nature and may be applicable to the analysis of other types of composite structures. By comparing with experimental results, this new solution provides satisfactory predictions to the interfacial stress in the plated beams. The classical solutions which neglect the adherend shear deformations over-estimate the non-uniformity of the adhesive stresses’ distributions and maximum interfacial stresses. In the final part of this paper, extensive parametric studies were undertaken by using the new solution for strengthened beams with various ratios of design parameters. Observations were made based on the numerical results concerning their possible implications to practical designs.

326

A. Rabahi et al.

References [Smith S.T 2001a] Smith, S.T., Teng, J.G.: FRP – strengthened RC beams. I: Review of debonding strength models. Engineering Structures 24(4), 385–395 (2002) [Smith S.T 2001b] Smith, S.T., Teng, J.G.: FRP – strengthened RC beams. II: Assessment of debonding strength models. Engineering Structures 24(4), 397–417 (2002) [Smith S.T 2001c] Smith, S.T., Teng, J.G.: Interfacial stresses in plated RC beams. Engineering Structures 23(7), 857–871 (2001) [Yu 1989] Yu, T.Q., Miao, X.S., Xiong, J.M., Jiang, H., Lee, H.: An orthotropic dam-age model for concrete at different temperatures. Engng. Fract. Mech. 32, 775–786 (1989) [Tounsi 2006] Tounsi, A.: Improved theoretical solution for interfacial stresses in con-crete beams strengthened with FRP plate. Int. J. Solids Struct. 43, 4154–4174 (2006)

A Sensitivity Analysis of Multi-objective Cooperative Planning Optimization Using NSGA-II Wafa Ben Yahia*, Omar Ayadi, and Faouzi Masmoudi Mechanics, Modelling and Production Research Laboratory, National Engineering School of Sfax (Route de Sokra B.P.1173 - 3038, Sfax, Tunisia ENIS), Sfax University, Tunisia [email protected], [email protected], [email protected]

Abstract. The Non-dominated Sorting Algorithm II (NSGA-II) is one of the most popular genetic algorithms (GA). It is characterized with a high optimization quality that is demonstrated for several multi-objective problems in various disciplines. During the optimization, several genetic parameters are involved and influence the solution quality. The purpose of this paper is to investigate the influence of the NSGA-II parameters on the optimization process, while solving a multi-objective planning model. Two cases, having opposite demand topology, are studied. Results show a considerable impact of NSGA-II parameters, especially the population size and the mutation operators, on the algorithm behaviour and the optimization process. This investigation offers to the partners several optimal production plans with different parameters combinations, and allows them to select the most influential parameter that provide several good solutions. Keywords: Multi-objective optimization, cooperative planning, NSGA-II, NSGAII parameters.

1

Introduction

The genetic algorithms (GA) have proven their robustness and powerful search mechanism to solve a variety of single as well as multi-objective problems. It features by the large search space and the multiple alternative trade-offs generated in a single run. NSGA-II is one of the most popular multi-objectives GAs that shows high performance and was successfully applied in various disciplines ((Murugan et al. 2009), (Kanagarajan et al. 2007), (Kannan et al. 2009), (Agrawal et al. 2007), (Bekele et al. 2007), (Atiquzzaman et al. 2006), (Hnaien et al. 2010), (Huang et al. *



327

328

W. Ben Yahia, O. Ayadi, and F. Masmoudi

2010)). Several GA parameters, like the population size and the mutation and crossover operators, are involved in the optimization process and interact in a difficult way. Many studies tried to give some guidelines for the choice of the GA parameters, like the population size or the probabilities operators ((Harik et al. 1999), (Goldberg et al. 1993), (Deb & Agrawal, 1999)...). However, most of these works made general observations about the parameters setting. According to Hart & Belew (1991) the parameters interaction mostly depends on the function being optimized. The problem arises especially in the choice of the adequate population size. Many researchers scruple about using small population sizes that may risk the possibility of finding good solutions, or using large population sizes which require more processing time, increase computational costs and may slow the performance. That is why in most of works, repeated simulations are required to select the suitable parameters values. Deb & Agrawal, (1999) investigated the effect of GA operators for solving simple and complex problems. The considered problems are Himmeblau's function, Rastrigin's function and deceptive function. They observed that mutation operator plays a significant role as well the crossover operator, in the case of simple problems. But, each operator has its working zone. In the case of complex problem, the most significant operator is the crossover operator, but this assertion necessitates the choice of the adequate population size. Pongcharoen et al. (2002) developed a factorial experiment to identify the most efficient genetic parameters (the probabilities of crossover and mutation, the population size and the generations’ number) to solve a scheduling problem that aims to minimize a total cost function. According to their study, the only significant parameters are the population size, the generation number and the mutation probability. Tran (2005) developed parameterless NSGA-II, to make NSGA-II easier. The developed algorithm performs quite well than the original NSGA-II, while solving a benchmark problem. Nevertheless, it is slower and has worse distribution of the solutions. According to Zeng et al. (2010), the performance of multi-objective evolutionary algorithms is limited by the suitability of their corresponding parameter settings. Thus, they develop a self-adaptive mechanism adjustment of the distribution index of the simulated binary crossover (SBX) operator to improve NSGA-II performance. Most of previous researches, consider benchmark functions in order to study the GA parameters. This paper focuses on cooperative planning problem, and considers the seven genetic parameters involved in the optimization process of NSGA-II. In Ben Yahia et al. (2013), the NSGA-II is adapted to solve the proposed multiobjective planning model. The model is a bi-objective one, which aims to minimize the total production cost and the average of inventory level of a cooperative supply chain (SC). While the proposed model is applied to experimental examples in order to generate corresponding optimal Pareto solutions, optimization process failed to generate Pareto of more than one solution and the sensitivity of the results to NSGA-II parameters variations is not addressed. Our interest here is to investigate and to analyze this sensitivity during the optimization process. The idea is to generate optimal production plans while varying the NSGA-II parameters and to explore their effects.

A Sensitivity Analysis of Multi-objective Cooperative Planning Optimization

329

The paper is organized as follows. A general description of the NSGA-II algorithm and its parameters is presented in section 2. The cooperative scheme is described and formulated in section 3, followed by the tests description in section 4. The effect of the parameters variation is explored in section 5. Finally, summary and potential research directions are pointed out in section 6.

2

The Non-dominated Sorting Algorithm II (NSGA-II)

The NSGA-II algorithm is detailed in (Deb & Agrawal, 2002). It is characterised by the use of the elitism and non-domination concepts. A solution dominates another if; first, it is not worse than the other in all the objective functions, second at least it is strictly better than the other in one objective function. Similarly, to any genetic algorithm, firstly, a random parent population is created. After that, it is sorted to provide different fronts composed of feasible solutions having the same rank. In fact, individuals are ranked based on the concept of domination. In addition, the parameter crowding distance is calculated for each individual to estimate the density of solutions surrounding a particular individual in the population. Then, the solution located in a lesser crowded region is selected. Tournament selection is used to select individuals for the next generation. In every generation, “N” new individuals (offspring) are generated from “N” parents using the Simulated Binary Crossover (SBX) and Polynomial mutation. Mainly, seven parameters are involved in the optimization process of the NSGAII. First, the generation number which is the maximum number of optimization iterations. The maximum number used here is equal to 1500 generations. Then, two parameters influencing the treated solutions, first the population size, which is the number of the treated solutions during the optimization, second the random seed influencing the initial population. The population size used for experiments varies between 25 and 400 and the random seed varies between 0 and 1. In addition, two crossover operators are involved in the optimization process. First, the distribution index for crossover “ηc” which relates the parent to their offspring by non-linear relationship. This parameter varies between 0.5 and 100. Second, the crossover probability “Pc” varies between 0.5 and 1. As well, two mutation operators are considered. First, the mutation distribution index “ηm” having also a non-linear relationship between parent and offspring, and it varies between 0.5 and 500. Finally, the mutation probability “P m” which varies between 0 and 1/n; where n is the number of decision variables.

3

Mathematical Formulation

The considered model is developed in Ben Yahia et al. (2013). It is a biobjective model, which aims to minimize simultaneously the total production cost and the average of inventory level, while considering a cooperative manufacturing supply chain scheme. The partners cooperate together in order to generate a global

330


optimal production plan for a whole manufacturing supply chain, within a fixed time horizon with a finite capacity of personnel and machines. The multi-objective optimization problem must satisfy the following assumptions: Several resources, with limited availabilities, can process several items. Raw materials are always available. The sequence of operations required to produce an item is fixed, and any alternative routing is prohibited. Inventories at the initial period are void. Items can be only produced if all their predecessor components are available. Periodic external demand of each item is known. Overtime is allowed to extend fixed capacity availabilities. Backlogging is not allowed. Inventory is calculated at the end of a time period. External demand has to be fully met in time and quantity. Setup time is neglected. The mathematical formulation considers the following notations. Indexes t j r

planning period, t= l,..., T. operation, j = 1,. . . , J. resource, r = 1,. . . , R.

Set Indexes T set of planning periods J set of operations R set of resources Sj set of direct successors of operation j Parameters cvj

unit cost of operation j

cfi cor Dj,t Cr,t Lj,t ar,j rj,k

setup cost of operation j unit cost of overtime (capacity expansion) for resource r (external) demand for operation j in period t Capacity of resource r in period t Large constant Unit requirement of resource r by operation j Unit requirement of operation j by successor operation k

Decision variables C Imoy xj,t

total production cost average of inventory level for all operations output level of operation j in period t (lot size)


331

inventory level of operation j at the end of period t ij,t yj,t setup variable of operation j in period t (yj,t =1 if product j is set up in period t; yj,t =0 otherwise ) or,t overtime of resource r in period t Formulation The formulation of the model (the signals disappeared):

M in ( C , Im oy ) St

C =

(1)

T

∑ ∑ [( cv . x j

t =1 j∈ J

T

j, t

) + ( cf j. y j , t )] + ∑ ∑ co r .o r , t

1 T ∑ ∑ ij , t T t =1 j ∈ J i j , t −1 + x j , t = D j , t + r j , k x k , t + i j , t ∀j ∈ J , ∀t ∀j ∈ J , ∀t, ∀r ∈ R ∑ j a r , j. x j , t ≤ C r , t + o r , t x j , t ≤ L j , t. y j , t ∀j ∈ J , ∀t x j , t ≥ 0, ij , t ≥ 0 ∀j ∈ J , ∀t ∀t, ∀r ∈ R or , t ≥ 0 y j , t ∈ {0,1} ∀j ∈ J , ∀t Im oy =

(2)

t =1 r ∈ R

(3) (4) (5) (6) (7) (8) (9)

The first objective function (2) consists of the minimization of operations, setup and overtime costs. The second objective (3) aims to minimize the average level of inventory with respect to the number of planning periods. The constraint (4) represents the flow balance between output, inventory and consumption by external demand or successor operations. The constraints (5) ensure the capacity restrictions in using the resources to produce the different items. Lot-sizing relationships and the choice of the items to be produced at each time-period and at each plant location are expressed in (6). The constraints (7), (8) and (9) specify the domains of the different variables.

4

Tests Description

The purpose of this study is to investigate the influence of the NSGA-II parameters on the previously proposed model. Two cases will be studied, where only the demand typology changes. In each test, customer demands are supposed to be

332


deterministic and e to be fulfilled during the planning horizon while facing finite resources capacities. The first case is labelled 2PU.2P.ID and is represented by a supply chain of two production units (2PU) planning its production activities over two periods (2P) to satisfy the external increasing demand (ID). As shown in table 1, the demand of the second period exceeds the main available capacity. The second case (2PU.2P.DD) has the same configuration, but the external demand has a decreasing trend (DD). This time, the demand of the first period exceeds the main available capacity. These two tests have the same number of decision variables and constraints. The search strategies of the both studied cases are completely different. First, the 2PU.2P.ID case has a larger solutions region and the conflict between considered objectives occurs especially in this case. In fact, for the increasing demand, in order minimize the total production cost the use of the overtime must be avoided. But like that, the production must be made in advance, which will increase the average of inventory level. However, in the 2PU.2P.DD case, the demand has a decreasing pattern, which does not exceed the main available capacity at the end of the planning period. So there is no need to produce or to storage. Therefore, during the optimization the algorithm has to find a balance between the use of the overtime and the inventory level. Ultimately, the choice of the most suitable production plan is made by the partners according to their preferences. Table 1 Customer demand features in the studied cases 2PU.2P.ID

2PU.2P.DD

Demand of

1st period

2nd period

1st period

2nd period

Product 1

20

140

90

15

Product 2

15

70

40

5

Product 3

10

70

50

5

The parameters are examined separately, i.e. changing the value for the given parameter while keeping the values of the rest of parameters at a constant level. During the investigation of the effect of each parameter is examined separately. In other words, when the considered parameter is varied, the other parameters keep constant values. The adopted constant values of each parameter are shown in table 2.


333

Table 2 Genetic parameters NSGA-II (parameters)

Parameter values

N, Population size

150

G, Generation number

2000

Pc, crossover probability

0.99

Pm, mutation probability

1/n (number of variables)

ηc, Crossover Index

50

ηm, Mutation Index

100

r, random seed

0.123

5

Result and Discussion

5.1

Generation Number Influence

Figure 1 shows the evolution of the optimization process while increasing the generation number. Both studied cases need more than 75 generations to find feasible solutions. One can note that for the 2PU.2P.ID case, the objective functions are optimized simultaneously and the algorithm is giving a Pareto of two optimal solutions only during the first 250 generations. However, for 2PU.2P.DD case, the objective functions are optimized simultaneously until generation 550. In addition, the Pareto of solutions contains up to five solutions. Besides, for both studied cases, the algorithm converges to a Pareto of unique solution. At convergence, the Pareto

Generation number generation number

1000

800

600

400

200

Inventory level

20

Inventory 30 40

50 60 000

40000

Total cost

Cost total

20 000

Fig. 1 Influence of the generation number on the objective functions

334


takes a regular pattern in other word the found solution does not improves while increasing the generation number. During the convergence to the optimal solution, the total production cost is minimized by 91% in the 2PU.2P.DD case and 96% in the 2PU.2P.ID case; while, the average of inventory level is minimized by 52% in the 2PU.2P.DD case and 15% in the 2PU.2P.ID case. Therefore, increasing the generation number improves the solution quality.

5.2

Other Parameters Influence

Figure 2 includes the graphics showing the influence of the NSGA-II parameters on optimal solutions, for the studied cases. Mostly, the solutions region swept by all the parameters is the same in the two cases. For the 2PU.2P.DD case, the solutions region swept for all the parameters is the same except some deviation while varying the mutation operators. Moreover, the solutions region is concentrated and many solutions are repeated with different NSGA-II parameters combinations. However, the solutions region of the 2PU.2P.ID case is very large, except when varying the mutation distribution index. Besides, only when varying the population size or the mutation probability the dispersion of the solutions is wider, otherwise, the browsed solutions region is the same. In addition, the lowest total production cost is found several times with different parameters combinations. While the lowest averages of inventory level are found only with the crossover and mutation distribution indexes. Whereas, in the case of decreasing demand, mostly a global optimal solution is found, except with the variation of the crossover probability where a Pareto of two optimal solutions is found. The best solution is found while the mutation distribution index is varied. Moreover, the optimal solutions found in each studied case do not correspond to the same parameters combination; and the objective functions are very sensitive to any variation of genetic parameters especially in the case of the increasing demand. For the 2PU.2P.ID case, although the lowest total production cost is found with small and large population size, the lowest average of inventory level is found only with large population size. While for the 2PU.2P.DD case, the best solution is found with large population size. Therefore, increasing the number of the treated individuals improves the solution quality. Nevertheless, the execution time will increase. Mostly, NSGA-II can obtain optimal solution in a reasonable amount of time. For instance, when varying the crossover operators and fixing the other parameters, in several times, 500 generations are enough to converge. Moreover, for our model, increasing the amount of crossover to its maximum value is not necessary to converge into optimal solutions. The same assertion is true for the mutation operators.


335

Inventory

Inventory

1

00

Inventory

1 00

Inventory 50

50

25 000 0.2

25000 100

20 000 0.4

15 000 0.6

Cost total

Population size

Total cost

10 000

15 000

200 population size

Cost total random seed Random seed

20 000

0.8

10 000 300

Total cost

5000

5000 400 1.0

a.

The random seed

b.

The population size

20

Inventory

40 Inventory

20 000

50

60 80

100

15 000

Cost total

Total cost 0.6

50

10 000

10 000

Cost total crossover distribution index

Pm c.

Total cost

ηc

0.8 crossover probability

5000

0

1.0

The crossover probability

d.

The crossover distribution index

80 60 100

Inventory

Inventory

100

Inventory

40 Inventory 20

50 Inventory

0.06

25000 20 000 15 000 totalCost

0.04 mutation probability

15 000

Pm

10 000 Cost total

Total cost

e.

Total cost

0

10000

0.02 5000

The mutation probability

200 distribution index mutation

ηm

f.

400

5000

The mutation distribution index

Fig. 2 Influence of the Genetic parameters on the objective functions

336

6


Conclusion

This paper proposes an investigation of NSGA-II parameters impact on the optimization process of a bi-objective cooperative planning model. The considered model aims at minimizing the total production cost and the average of inventory level of the SC cooperating partners. Two cases are tested. For each studied case, the NSGA-II parameters have a different behaviour, due to the randomization of the GA. In addition, while changing the GA parameters, a diversity of solutions is offered and the range of the solutions’ performance varies significantly. It is noticed that, the population size and the mutation operators have the most important and noticeable impact on the optimization process. Moreover, using large population sizes, with reasonable amount of crossover and mutation improve the solution quality. During optimization, in the increasing demand case, the lowest total production cost is found many times with different inventory level while using only the main capacity and a null inventory at the last period; but the lowest average of inventory level corresponds to a very high total cost because of the use of overtime. However, in the decreasing demand case, the lowest total production cost corresponds to the lowest average of inventory level with the use of the necessary overtime and a small reserve in inventory at the end of the planning period. The algorithm is giving various optimal solutions that are useful for multi-criteria group decision- making purposes, especially for decentralized systems.

References Agrawal, N., Rangaiah, G., Ray, A., Gupta, S.: Design stage optimization of an industrial low-density polyethylene tubular reactor for multiple objectives using NSGA-II and its jumping gene adaptations. Chemical Engineering Science 62(9), 2346–2365 (2007) Atiquzzaman, M., Liong, S., Yu, X.: Alternative decision making in water distribution network with NSGA-II. Journal of Water Resources Planning and Management 132(2), 122–126 (2006) Bekele, E.G., Nicklow, J.W.: Multi-objective automatic calibration of SWAT using NSGAII. Journal of Hydrology 341(3-4), 165–176 (2007) Ben Yahia, W., Cheikhrouhou, N., Ayadi, O., Masmoudi, F.: A Multi-objective Optimization for Multi-period Planning in Multi-item Cooperative Manufacturing Supply Chain. In: Haddar, M., Romdhane, L., Louati, J., Ben Amara, A. (eds.) Design and Modelling of Mechanical System, pp. 635–643. Springer, Heidelberg (2013) Deb, K., Agrawal, S.: Understanding interactions among genetic algorithm parameters. In: Foundations of Genetic Algorithms V, pp. 265–286. Morgan Kaufmann, San Mateo (1999) Deb, K., Agrawal, S.: A fast elitist non-dominated sorting genetic algorithm for multiobjective optimization: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2), 182–197 (2002) Harik, G., Cantú-Paz, E.: The gambler’s ruin problem, genetic algorithms, and the sizing of populations. Evolutionary Computation 7, 231–253 (1999)


337

Hart, W.E., Belew, R.K.: Optimizing an Arbitrary Function is Hard for the Genetic Algorithm. In: Proceedings of the Fourth International Conference on Genetic Algorithms, pp. 190–195 (1991) Hnaien, F., Delorme, X., Dolgui, A.: Multi-objective optimization for inventory control in two-level assembly systems under uncertainty of lead times. Computers & Operations Research 37(11), 1835–1843 (2010) Huang, B., Buckley, B., Kechadi, T.-M.: Multi-objective feature selection by using NSGAII for customer churn prediction in telecommunications. Expert Systems with Applications 37(5), 3638–3646 (2010) Kanagarajan, D., Karthikeyan, R., Palanikumar, K., Davim, J.P.: Optimization of electrical discharge machining characteristics of WC/Co composites using non-dominated sorting genetic algorithm (NSGA-II). The International Journal of Advanced Manufacturing Technology 36(11-12), 1124–1132 (2007) Kannan, S., Baskar, S., McCalley, J.D., Murugan, P.: Application of NSGA-II Algorithm to Generation Expansion Planning. IEEE Transactions on Power Systems 24(1), 454–461 (2009) Murugan, P., Kannan, S., Baskar, S.: NSGA-II algorithm for multi- objective generation expansion planning problem. Electric Power Systems Research 79(4), 622–628 (2009) Pongcharoen, P., Hicks, C., Braiden, P.M., Stewardson, D.J.: Determining optimum Genetic Algorithm parameters for scheduling the manufacturing and assembly of complex products. International Journal of Production Economics 78(3), 311–322 (2002) Tran, K.D.: Elitist Non-Dominated Sorting GA-II (NSGA-II) as a Parameter-Less MultiObjective Genetic Algorithm. In: Proceedings of the IEEE SoutheastCon 2005, pp. 359–367 (2005), doi:10.1109/SECON.2005.1423273 Zeng, F., Low, M., Decraene, J.: Self-adaptive mechanism for multi- objective evolutionary algorithms. In: Proceedings of the International MultiConference of Engineers and Computer Scientists, IMECS 2010, vol. I, pp. 7–12 (2010)

Transient Flow Analysis through Centrifugal Pumps Issa Chalghoum, Sami Elaoud, Mohsen Akrout, and Ezzeddine Hadj-Taïeb Research Laboratory "Applied Fluid Mechanics, Process Engineering and Environment" Engineering School of Sfax, BP'W'3038 Sfax, Tunisia {chalghoumissa,elaoudsa}@yahoo.fr

Abstract. A theoretical study has been carried out on the transient flow through a centrifugal pump during the starting period. The problem is governed by two hyperbolic partial differential equations which the continuity and the motion equations. The mathematical model is solved numerically by using the method of characteristics with specified time intervals. The comparison between the obtained numerical results and those obtained by experiment has shown a good concordance. In this study, the effect of the starting time, the impeller diameter, the number and the height of blades on the pressure increase were analyzed. The numerical results have shown that the pressure increase is inversely proportional to the starting time. However, during the starting period, these results have shown that the number of blades has no significant effect on this pressure increase. Keywords: centrifugal pump, transient flow, valve, starting period, method of characteristics.

1

Introduction

Studies of dynamic characteristics of centrifugal pumps have reached new importance in recent times In order to understand transient behavior of a centrifugal flow pump during the starting period, experimental and theoretical study has been performed (Tsukamoto et al. 1982). These experiments and analysis have shown that the impulsive pressure and lag in circulation formation around the impeller vanes is the main reasons for the difference between dynamic and quasi-steady characteristics. In his studies of the dynamic characteristics of a centrifugal pump during starting period, Saito (1986) has examined the transient characteristics of the pump with reference to three major factors disappear in the process when pump start up to normal steady-state performance, namely the mass of water in the pipeline, valve opening and starting time. Through experimental test, the dynamic behavior of a pump volute type during startup and stopping periods has been in© Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_35

339

340

I. Chalghoum et al.

vestigated (Thanapandi et al 1994).The obtained results were also compared to a mathematical model solved numerically by the method of characteristics. Dazin, et al. (2007) proposed a new method for predicting the transient behaviour of turbomachinery based on the angular momentum and energy equations. The results show that the transient behaviour of a pump impeller depends not only on the acceleration rate and flow rate but also on velocity profiles and their evolution during transient periods. In this paper we study transient flow through a centrifugal pump. The transient regime is created by considering a linear variation of the rotational speed. The governing partial equations are solved by the methods of characteristics. In this study, the effect of the starting time and the geometric properties of the impeller, on the pressure and the flow rate, are analyzed.

2

Mathematical Formulations

2.1

Governing Equations

The local continuity equation and the equation of motion in a relative movement in a reference frame rotating with the impeller passage having the same origin as the fixed reference are the following:

JJG dρ + ρ divW = 0 dt

ρ

(1)

G JJJG G G G G G G G G dW JJJJJG ∧ oM + Ω ∧ Ω ∧ oM ⎤ = grad p − p + divτ + ρ F − ρ ⎡ 2Ω ∧ W + Ω ⎣⎢ ⎦⎥ dt

(

)

(2)

G Where ρ the density of the fluid is, τ is the stress tensor of viscosity, F is the G G external body force, W is the relative velocity and Ω is the rotational speed. Figure 1 shows the control volume defined by two successive blades limited by suction and discharge sides of the impeller. The principle of the study of turbo machinery is based on the velocity diagram for determining the absolute velocity at any point of the blade. In particular, point 2 which represent the outlet of the G impeller, the absolute velocity V2 is given by the following equation:

G G G V2 = W2 + U 2

G where U 2 the entrainment velocity at that point.

(3)

Transient Flow Analysis through Centrifugal Pumps

341

In a centrifugal pump impeller, only the radial component of the absolute velocity which permits the passage of fluid from the impeller to the volute. According to the velocity diagram shown in Figure 1, this component is:

V2 r = V × cos α 2 = W × sin β 2

(4)

This relationship allows the passage from the rotating frame to the absolute frame. The moving frame is used as reference for the flow study through the impeller. Taking into account the assumption of radial flow through the different sections of thicknesses Δr of the impeller and the shear forces act in the radial direction, the equations of motion (1) and (2) are simplified [Analogy to unidirectional flow (Wylie et al, 1993; Thanapandi and Prasad, 1994)]:

∂H C 2 ∂Q + =0 ∂t gA ∂r

(5)

QQ 1 ∂Q ∂H +g +λ − r Ω2 = 0 ∂r A ∂t 2 Dh A2

(6)

where A is the impeller passage section, Q is the fluid discharge, H is the head, λ is the Darcy Weisbach friction coefficient, C is the pressure wave speed and Dh = 4lb 2(l + b) is the hydraulic diameter of the section A where b is the blade height and l is the width of the control volume.

2.2

Numerical Resolution by the Method of Characteristics

The method of characteristics (Wylie et al, 1993) is applied to the PDEs (5) and (6), and the partial differential terms associated with the flow velocity and the pressure are the reduced to ordinary differential ones compatible with two characteristics lines C + and C + (Figure.2).The ordinary differential equation of (5) and (6) are: ⎧ gA 2 ⎪ dH + dQ + JAdt − r Ω Adt = 0 C+ ⎨ C ⎪ dx = Cdt ⎩

(7)

⎧ gA dH + dQ + JAdt − r Ω 2 Adt = 0 ⎪− C ⎨ C ⎪dx = −Cdt ⎩

(8)

−

where J = λQ Q 2 DA2 is the head loss by unit of the impeller passage length.

342

I. Chalghoum et al.

Consider L = R2 − R1 the impeller total length subdivided in N equal reaches Δx = L N . If we start with known steady-state conditions at t = 0 , then we know Q and H at the N + 1 sections of the pipe. If we specify the time interval Δt defined as the Δt = Δx C , the characteristic lines from the sections i − 1 and i + 1 intersect at P , which is the section i (figure 2). In these conditions, the integration of equations (7) and (8) can be written in the following forms:

QPi − Qi −1 +

gAi −1 ( H Pi − Hi −1 ) + Ai −1 Ji −1Δt − ri −1Ω2 Ai −1Δt = 0 C

(9)

QPi − Qi +1 −

gAi +1 ( H Pi − H i +1 ) + Ai +1 J i +1Δt − ri +1Ω 2 Ai +1Δt = 0 C

(10)

Fig. 1 Forces in rotating impeller in the frame of reference

Fig. 2 Characteristic lines

The calculation of the discharge and head at any section i is obtained if we solve Eqs. (9) and (110) for the H Pi by eliminating QPi ,thus: H Pi =

1 ⎡C1H i −1 + C 2 H i +1 + ( Qi −1 − Qi +1 ) − G ⎤⎦ C1 + C 2 ⎣

(11)

where G = Δt (A i −1 J i −1 − Ai +1 J i +1 ) + Δt (ri −1 Ai −1Ω 2 − ri +1 Ai +1Ω 2 ) , C1 = gAi −1 C and C 2 = gAi +1 C By substituting equation (11) in equation (10), one can obtain the discharge at section i as follows:

Q pi = ⎡⎣Qi +1 + C 2 ( H Pi − H i +1 ) − Ai +1ΔtJ i +1 − ri +1 Ω 2 Ai +1Δ t) ⎤⎦

(12)

These equations permit to determine QPi and H Pi at any interior point of the pump. In order to get the solution until an elevated time level, it is necessary to introduce the appropriate boundary conditions. These conditions will be introduced in the next section.


3

343

Applications and Results

To study the transient flow through centrifugal pumps, we consider the hydraulic facility (TE-83), available at ENIS, represented by the simplified schematic (Figure 3). This installation has two identical centrifugal pumps permitting the study of a single pump, two pumps in parallel and two pumps in series. The test facility represented by figure 3 contains two digital pressure gages for measuring the pressure at the suction side and the discharge side of the pump. The rotational speed of the motors is controlled by potentiometers. A mechanical device is used to determine the engine torque. Figure 4 illustrates the centrifugal pump impeller. It is composed of 8 blades with a base suction diameter of 42.8 mm and a discharge diameter of 128 mm.

Fig. 3 Schematic description the hydraulic

Fig. 4 pump impeller TE-83

facility (TE-83)

3.1

Boundary Conditions

In this section, we are, mainly, interested in determining the boundary conditions through the valve (Figure 3).

Fig. 5 Boundary conditions at the discharge valve

When the valve is placed between the upstream end and the downstream end of the discharge pipe, the resolution by the method of characteristics requires that the main discharge pipe must be treated as if it is formed by two conduits (figure.5).

344

I. Chalghoum et al.

Compatibility equations that solve this problem are: equation (9) for the pipe 1 and equation (10) for the pipe 2. These equations are written for the pipes 1 and 2 respectively as follows: QP1, N 1+1 − Q1, N 1 +

QP 2,1 − Q2,2 −

gA ( H P1, N 1+1 − H1, N1 ) + AJ1, N 1Δt = 0 C

(13)

gA ( H P 2,1 − H 2,2 ) + AJ 2,2 Δt = 0 C

(14)

These equations can be written in the following form: QP1, N 1+1 = − K1 H P1, N 1+1 + K 2

(15)

QP 2,1 = K1 H P 2,1 + K 3

where K 2 = Q1, N 1 +

(16)

gA gA H1, N 1 − AJ1, N 1Δt , K 3 = Q2,2 − H 2,2 − AJ 2,2 Δt and C C

K1 = gA C

Based on the theory of the flows through the orifices, we get: QP1, N 1+1 = QP 2,1 = Cd AG H P1, N 1+1 − H P 2,1 ,

(17)

if the flow rate is positive, and

QP1, N 1+1 = QP 2,1 = Cd AG H P 2,1 − H P1, N 1+1

(18)

if the flow rate is negative In the above equations, AG is the open section of the valve and Cd = Cc × Cv is the flow coefficient, Cc is the coefficient of contraction since the actual flow is passing through a lower section AG and Cv = 1 1 + Kv is the velocity coefficient since the actual velocity is less than theoretical velocity due the singular pressure loss where K v is the singular pressure drop coefficient of the valve which depends of the valve opening. Tables 1 and 2 represent the coefficients K v and Cv respectively for DN40 gate valve (Same type of valve used in TE-83 test facility). Table 1 Contraction coefficient r/R

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Cc

0.611

0.612

0.616

0.622

0.633

0.644

0.662

0.687

0.722

0.781


345

Table 2 Coefficient K v for gate valve DN 40 Percentage of opening

100

75

50

25

0

Coefficient K v

0.15

0.26

2.1

17

∞

By substituting equations (15) and (16) in the sum of equations (11) and (12), we obtain the following quadratic equations:

K1

( Cd AG )

2

QP21, N1+1 − 2QP1, N1+1 + ( K1 + K2 ) = 0

K1

( Cd AG )

2

QP21, N1+1 + 2QP1, N1+1 − ( K1 + K2 ) = 0

(19)

(20)

Equation (19) is used to determine the positive flow rate while equation (20) is used to determine the negative flow rate.

3.2

Results by the Method of Characteristics

To numerically determine the characteristics of the hydraulic facility (TE-83) shown in figure 2, the impeller geometry is simplified. In this case, we considered an impeller having the same number of blade as the original one (8 blades) but a straight curve shape of the blades was adopted as illustrated in figure 6. Since the relative flow is assumed to be in the radial direction, this simplification of the blades shape can be considered as logical. r Ω

G W2

G β 2 V2 α2 G U2

Fig. 6 simplified impeller

Fig. 7 modelling the pump impeller

Furthermore, we consider an average blade height of 5 mm while the actual height of the blade is variable (8 mm at the inlet of the impeller and 3.2 mm at its outlet). In Figure 6, the angle β 2 = 90D is constant throughout the blade and particularly in the outlet point 2 of the centrifugal impeller. In this case, based on equation (4), the radial component of absolute velocity is equal to the relative velocity:

346

I. Chalghoum et al.

V2r = V × cosα2 = W × sin β2 = W

(21)

5

2,5

4

flow rate (l/s)

pressure (m)

When establishing the equations of motion, we model the impeller shown in figure 7 by 8 parallel pipes. By simulating the start-up phase, we assume that to reach the steady state, the engine’s rotational speed varies linearly during a startup time of 0.15 s. closed valve valve half open open valve

2,0

closed valve 0,5 valve open open valve

3

0,25 valve open 0,75 valve open

1,5 1,0

2

0,25 valve open 0,75 valve open

startup time 0.15 s - N = 1450 rpm

startup time 0.15 s/ N =1450 rpm

0,5

1

0,0

0 0,0

0,1

0,2

0,3

0,4

0,5

0,6 t (s) 0,7

0,0

0,1

0,2

0,3

0,4

0,5

0,6 t (s) 0,7

Fig. 8 the pressure difference versus time Fig. 9 Variation of flow rate versus time for for different valve opening positions different valve opening positions

Figure 9 shows the evolution of the pressure difference between inlet and outlet sides of the pump as a function of time and for the different valve opening positions. These curves are considered for a rapid starting period (≈ 0.15 s ) and a rotational speed of 1450 rpm. This figure shows that the pressure is inversely proportional to the valve opening percentage. Indeed, when the valve is fully open, the pressure drops to 0.2 m. This pressure drop is due to the reduction of the flow resistance caused by the friction at the suction and discharge circuits and mainly at the valve. This figure also shows an oscillating effect at the pressure curve when the valve is in the closed position. This phenomenon is due to the wave propagation along the pipes and its reflection at both ends of the hydraulic circuit. It can be seen also in figure 9 the presence of peak of pressure from the mid-open position up to complete valve opening. This peak is due to the pump start-up time and to the mass of water which increases when the resistance in the circuit decreases with the valve opening.

3.3

Validation of the Theoretical Results

Figure 11 shows a comparison between the pump characteristic curves of the experimental and the theoretical model for two rotational speeds 1450 rpm and 2900 rpm. This curve shows a slight shift between the experimental and theoretical curves. In fact, the characteristic curve of a pump depends on the following parameters: the rotational speed of the engine; the suction and discharge diameter of the impeller; the blade curve; the blade height; pressure losses within the pump (at the impeller and the volute) and the shape of the volute. This difference is mainly due to the approximation of shape and the averaged height of the blades. Nevertheless, the results can be considered as acceptable. The curves of Figure 11 show


347

that when is increased the rotational speed, the pressure range and the flow rate range is increased too. This result is logic since the centrifugal force which permits the flow through the impeller is proportional to the square root of the rotational speed. Figure 12 shows the effect of the diameter of the impeller on characteristic curve of the pump. It can be noted that the head range of the characteristic curve is proportional to the diameter of the impeller. This is due to the effect of centrifugal force which increases with the radius. For a constant impeller diameter and by changing the blades height, the flow rate range increases while the pressure remains unchanged as shown in Figure 13. 6

pressure (m)

pressure (m)

20

startup time 0.15 s - N = 1450 rpm

5

startup time 0.15 s N = 2900 rpm

16

4

12 Pump TE-83

3

Pump TE-83

8

2

theoretical model

theoretical m odel

4

1 0 0,00

0,25

0,50

0,75

1,00

1,25

0

Q(l/s) 1,50 1,75

0,00

0,50

1,00

1,50

2,00

2,50

Q(l/s)

3,00

3,50

Fig. 10 Comparison of theoretical and experimental curves for N = 1450 rpm and N = 2900 rpm 6 pressure(m)

pressure (m)

10 s tartup time 0.15 s - N =1450 rpm 8

R = 84 mm R = 74 mm

6

s tartup time 0.15 s - N =1450 rpm 4

h = 5 mm

R = 64 mm

h = 7 mm

4

2

2 0

0 0

0,4

0,8

1,2

1,6 Q (l/s )

2

0

1

2

3

4

Q (l/s)

5

Fig. 11 Effect of the impeller diameter on Fig. 12 Influence of the blades height on the the pump characteristics curve pump characteristics curve

3.4

Parameters Which Influence the Pressure During the Starting Phase

Saito (1982), in his research, has shown that during the starting period of the pump, the change of its dynamic characteristics is due to three factors, namely: the mass of water in the pipe; the opening position of the valve and the starting time. In fact, figure.14 shows that the peak of pressure during the starting period is largely reduced when the starting time is increased. Besides the statement of Saito which are well confirmed in this section and in the previous section, other parameters have also an effect during the starting period. In fact, it can be seen on figure 15 that the peak of pressure is reduced when the diameter of the impeller is reduced.

I. Chalghoum et al. 3

4

L2 = 1.5 m - N =1450 rpm startup time 0.15 s startup time 0.3 s startup time 0.6 s

2

pressure (m)

pressure (m)

348

startup time 0.15 s - N =1450 rpm

3 R = 0.64 mm R = 0.74 mm

2

R = 0.84 mm 1 1

0

0 0,0

0,1

0,2

0,3

0,4

0,5

0,6 t (s ) 0,7

0,0

0,1

0,2

0,3

0,4

0,5

0,6 t (s) 0,7

Fig. 13 Effect of starting time on the Fig. 14 Effect of the impeller diameter on pressure evolutions the pressure curve

4

Conclusion

In this paper, the transient flow inside a centrifugal pump has been conducted. The transient is created as a result of a linear variation of pump speed from rest to the permanent regime. This type of flow is governed by two partial differential equations: the momentum and the continuity equations. These equations are solved numerically by the use of the method of characteristics. The pump impeller was modelled as conduits connected in parallel. The obtained results have shown a good concordance with those obtained by experiment. In this study the effect of, the starting time, the geometric proprieties of the impeller and the valve openings, on the pressure evolution has been also analysed. These results have shown that the pressure increase during the starting period was influenced by the starting time and the valve openings. However, these parameters have no influence on the steady state pump characteristic curve. These results have shown that this curve is well influenced by the radius and the blade height of the impeller.

References Tsukamoto, H., Ohashi, H.: Transient characteristics of a centrifugal pump during starting period. ASME Journal of Fluids Engineering 104, 6–13 (1982) Dazin, A., Caignaert, G., Bois, G.: Transient behavior of turbomachineries: application to radial flow pumps startups. ASME Journal of Fluids Engineering 129, 1436–1444 (2007), doi:10.1115/1.2776963 Saito, S.: The transient characteristics of a pump during startup. Bulletin of JSME 25, 372–379 (1986) Thanapandi, P., Prasad, R.: Centrifugal pump transient characteristics and analysis using the method of characteristics. Int. J. Mech. Sci. 37, 77–89 (1995) Streeter, V.L., Wylie, E.B.: Hydraulic Transients. McGraw-Hill Book Company, New York (1967)

Vibro-Acoustic Analysis of Laminated Double-Wall: Finite Element Formulation and Reduced-Order Model Walid Larbi, Jean-François Deü, and Roger Ohayon Conservatoire National des Arts et Métiers (Cnam), Structural Mechanics and Coupled Systems Laboratory, 2 rue Conté, 75003 Paris, France [email protected]

Abstract. This paper presents a finite element model for sound transmission analysis through a double sandwich panels with viscoelastic core inserted in an infinite baffle. The proposed model is derived from a multi-field variational principle involving structural displacement of the panels and acoustic pressure inside the fluid cavity. To solve the vibro-acoustic problem, the plate displacements are expanded as a modal summation of the plate’s real eigenfunctions in vacuo. Similarly, the cavity pressure is expanded as a summation over the modes of the cavity with rigid boundaries. Then, an appropriate reduced-order model with mode acceleration method by adding quasi-static corrections is introduced. The structure is excited by a plane wave. The radiated sound power is calculated by means of a discrete solution of the Rayleigh Integral. Fluid loading is neglected. Various results are presented in order to validate and illustrate the efficiency of the proposed reduced finite element formulation. Keywords: double-wall, viscoelastic, vibroacoustic, finite element, modal reduction.

1 Introduction Double-wall structures are widely used in noise control due to their superiority over single-leaf structures in providing better acoustic insulation. Typical examples include double glazed windows, fuselage of airplanes and vehicles, etc. Various theoretical, experimental and numerical approaches have been investigated to predict the sound transmission through double walls [1, 2, 3]. By introducing a thin viscoelastic interlayer within the panels, a better acoustic insulation is obtained. In fact, sandwich structures with viscoelastic layer are commonly used in many systems for vibration damping and noise control. In such structures, the main energy loss mechanism is due to the transverse shear of the viscoelastic core [4, 5]. c Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_36

349

350

W. Larbi, J.-F. Deü, and R. Ohayon

In the first part of this paper, a non-symmetric finite element formulation of double-wall sandwich panels with viscoelastic core is derived from a variational principle involving structural displacement and acoustic pressure in the fluid cavity. Since the elasticity modulus of the viscoelastic core is complex and frequency dependent, this formulation is complex and nonlinear in frequency. Therefore, the direct solution of this problem can be considered only for small size models. This has severe limitations in attaining adequate accuracy and wider frequency ranges of interest. An original reduced order-model is then proposed to solve the problem at a lower cost. The proposed methodology, based on a normal mode expansion, requires the computation of the uncoupled structural and acoustic modes. The uncoupled structural modes are the real and undamped modes of the sandwich panels without fluid pressure loading at fluid-structure interface, whereas the uncoupled acoustic modes are the cavity modes with rigid wall boundary conditions at the fluid-structure interface. It is shown that the projection of the full-order coupled finite element model on the uncoupled bases, leads to a reduced order model in which the main parameters are the classical fluid-structure and residual stiffness complex coupling factors. Moreover, the effects of the higher modes of each subsystem is taken into account through an appropriate so-called static correction. Despite its reduced size, this model is proved to be very efficient for simulations of steady-state and frequency analyses of the coupled structural-acoustic system with viscoelastic damping and the computational effort is significantly reduced. In the last part, numerical examples are presented in order to validate and analyse results computed from the proposed formulation.

2 Finite Element Formulation of the Coupled Problem 2.1 Local Equations Consider a double-wall structure coupled to an acoustic enclosure shown in Fig. 1. Each wall occupies a domain ΩSi , i ∈ {1, 2} such that ΩS = (ΩS1 , ΩS2 ) is a partition of the whole structure domain. A prescribed surface force density Fd is applied to the external boundary Γt of ΩS and a prescribed displacement ud is applied on a part Γu of ΩS . The acoustic enclosure is filled with a compressible and

ΩF Σ Γt Ωs2

Fd u

d

Γu

Ωs1 ud

Sandwich panel with viscoelastic core

Fig. 1 Double sandwich wall structure coupled to an acoustic enclosure

Vibro-Acoustic Analysis of Laminated Double-Wall

351

inviscid fluid occupying the domain ΩF . The cavity walls are rigid except those in contact with the flexible wall structures noted Σ. The harmonic local equations of this structural-acoustic coupled problem can be written in terms of structure displacement u and fluid pressure field p [6] divσ(u) + ρS ω 2 u = 0

in ΩS

(1)

σ(u)nS = Fd

on Γt

(2)

σ(u)nS = pn

on Σ

(3)

on Γu

(4)

in ΩF

(5)

on Σ

(6)

u=u Δp +

d

2

ω p=0 c2F

∇p · n = ρF ω 2 u · n

where ω is the angular frequency, nS and n are the external unit normal to ΩS and ΩF ; ρS and ρF are the structure and fluid mass densities; cF is the speed of sound in the fluid; and σ is the structure stress tensor.

2.2 Constitutive Relation for Viscoelastic Core In order to provide better acoustic insulation, damped sandwich panels with a thin layer of viscoelastic core are used in this study (Fig. 1). When subjected to mechanical vibrations, the viscoelastic layer absorbs part of the vibratory energy in the form of heat. Another part of this energy is dissipated in the constrained core due to the shear motion. The constitutive relation for a viscoelastic material subjected to a sinusoidal strain is written in the following form: σ = C∗ (ω)ε

(7)

where ε denotes the strain tensor and C∗ (ω) is the complex elasticity tensor. It is generally complex and frequency dependent (∗ denotes complex quantities). It can be written as: (8) C∗ (ω) = C (ω) + iC (ω) √ where i = −1. Furthermore, for simplicity, a linear, homogeneous, viscoelastic constitutive equation will be used in this work. In the isotropic case, the viscoelastic material is defined by a complex and frequency dependence shear modulus in the form:

G∗ (ω) = G (ω) + iG (ω)

(9)

where G (ω) is know as shear storage modulus, as it is related to storing energy and G (ω) is the shear loss modulus, which represents the energy dissipation effects.

352


With these assumptions, the stress tensor of the sandwich structure is complex and frequency dependent.

2.3 Finite Element Equation After variational formulation using the test-function method [6, 7] and discretization by the finite element method, we obtain the following matrix system of the fluid/elastic structure with viscoelastic damping coupled problem: ∗ F Ku (ω) −Cup U 2 Mu 0 (10) = −ω CTup Mp 0 Kp 0 P where U and P are the vectors of nodal values of u and p respectively; Mu and K∗u (ω) are the mass and stiffness matrices of the structure; Mp and Kp are the mass and stiffness matrices of the fluid; Cup is the fluid-structure coupled matrix; F is the applied mechanical force vector. Note that since the elasticity modulus of the viscoelastic core of the sandwich panels is complex and frequency dependent, the stiffness matrix K∗u (ω) is also complex and frequency dependent.

3 Reduced Order Model In this section, we introduce a reduced-order formulation based on a normal mode expansion with an appropriate static correction.

3.1 Eigenmodes of the Structure in Vacuo In a first phase, the first Ns eigenmodes of the structure in vacuo are obtained from ∗ Ku (ω) − ω 2 Mu U = 0 (11) Due to the frequency dependent of the stiffness matrix, this eigenvalue problem is complex and nonlinear. It is assumed that vibrations of the damped structure can be represented in terms of the real modes of the associated undamped system if appropriate damping terms are inserted into the uncoupled modal equations of motion. Thus, the complex stiffness matrix is decomposed in the sum of two matrices: K∗u (ω) = Ku0 + δK∗u (ω)

(12)

where Ku0 = Re[K∗u (0)] is the real and frequency-independent stiffness matrix calculated with a constant Young module’s of the viscoelastic core and δK∗u (ω) is the residual stiffness matrix. The ith real eigenmode is obtained from the following equation 2 Ku0 − ωsi Mu Φsi = 0 for i ∈ {1, · · · , Ns } (13) where (ωsi , Φsi ) are the natural frequency and eigenvector for the ith structural mode.


353

3.2 Eigenmodes of the Internal Acoustic Cavity with Rigid Walls In this second phase, the first Nf eigenmodes of the acoustic cavity with rigid boundary conditions are obtained from the following equation (14) Kp − ωf2 i Mp Φf i = 0 for i ∈ {1, · · · , Nf } where (ωf i , Φf i ) are the natural frequency and eigenvector for the ith acoustic mode. It is important to note that the physical acoustic modes in a rigid fixed cavity are such that ΩF p dv = 0 which excludes the ωf i = 0, p = constant solution which is not physical but had to be introduced in the formulation of the coupled problem. Thus, the variational formulation of this kind of system, in order to be regularized for zero frequency situation, i.e. valid for a static problem, has to be modified by adding the following constraint ρF c2F Σ u.n ds + ΩF p dv = 0 (see [6] for details). When doing this, on one hand the static pressure is defined precisely by ρF c2F s u.n ds (15) p =− |ΩF | Σ and on the other hand, the reduced order formulation will be carried only by projection on the physical acoustic modes.

3.3 Modal Expansion of the General Problem By matrices Φs = [Φs1 · · · ΦsNs ] of size (Ms × Ns ) and Φf = introducing the Φf 1 · · · Φf Nf of size (Mf × Nf ) corresponding to the uncoupled bases (Ms and Mf are the total number of degrees of freedom in the finite elements model associated to the structure and the acoustic domains respectively), the displacement and pressure are sought as U = Φs qs (t) and P = Φf qf (t)

(16)

T where the vectors qs = [qs1 · · · qsNs ]T and qf = qf 1 · · · qf Nf are the modal amplitudes of the structure displacement and the fluid pressure respectively. Substituting these relations into Eq. (10) and pre-multiplying the first row by ΦTs and the second one by ΦTf , we obtain • Ns mechanical equations 2

−ω qsi +

Ns

k=1

∗ γik (ω)qsi

+

2 ωsi qsi

−

Nf

j=1

βij qf j = Fi

(17)

354


• Nf acoustic equations −ω 2 qf i + ωf2 i qf i − ω 2

Ns

βij qsj = 0

(18)

j=1

where Fi (t) = ΦTsi F is the mechanical excitation of the ith mode; βij = ΦTsi Cup Φf j ∗ is the fluid structure coupling coefficient and γik (ω) = ΦTsi δK∗u (ω)Φsk the reduced residual stiffness complex coefficient. At each frequency step, the reduced system (Eqs. (17) and (18)) is solved by ∗ (ω). After determining the complex amplitude vectors qsi and qf i , updating γik the displacement and pressure fields are reconstructed using the modal expansion (Eqs. (16)).

3.4 Static Corrections The process of mode truncation introduces some errors in the response that can be controlled or minimized by a modal truncation augmentation method. In this method, the effects of the truncated modes are considered by their static effect only. First the applied loading vector F is composed as: F=

L

αi (t)F0i

(19)

i=1

where F0i is the invariant spatial portion and αj (t) is the time varying portion. For each invariant spatial load, the static modal eigenvector Ψsi is given by: Ψsi = K−1 u0 F0i

(20)

The truncated basis containing the real and undamped structure modes is enriched by the static modal eigenvectors such that ¯ s = [Φs1 · · · ΦsN , Ψsi , · · · Ψsl ] = [Φs Ψs ] Φ s

(21)

The truncated fluid basis is enriched with the static pressure Ps computed from Eq. (15): ¯ f = [Φf Ps ] Φ (22) Thus, the displacement and pressure are sought as U = Φs qs (t) + Ψs q0s (t)

and P = Φf qf (t) + Ps qf0 (t)

(23)

where the vectors q0s and qf0 are the quasi-static modal amplitudes of the structure displacement and the fluid pressure respectively. Similar coupled differential equations than Eqs. (17) and (18) can of course be obtained after modal projection on ¯f. ¯ s and Φ the enriched bases Φ


355

4 Numerical Examples 4.1 Sound Transmission through an Elastic Double-Panel System In this section, the validation of the proposed coupled finite elements formulation for sound radiation is presented. The problem under consideration is shown in Fig. 2. A normal incidence plane wave excites a double-plate system filled with air (density ρF = 1.21 kg/m3 and speed of sound cF = 340 m/s). The plane wave has a pressure amplitude of 1 N/m2 and is applied to plate 1 as the only external force to the system. The plates are identical and simply supported with thicknesses of 1 mm. The density of the plates is 2814 kg/m3 , the Youngs modulus is 71 GPa, the Loss factor is 0.01 and Poisson ratio 0.33. The surrounding fluid is the air. This example was originally proposed by Panneton in [2].

Rigid baffle

b=0.22 m

Air Cav ity

Plate 2 Normal plane wave

Plate 1

a=0.35 m

4

76

m

.0

0 c=

Fig. 2 Double-plate system filled with air: geometric data

When the excitation is applied to the first plate, the second one vibrates and radiates sound caused by the coupling of air and plate 1. The normal incidence sound transmission loss is then computed using the Rayleigh’s integral method [1] which needs the finite element solution of surface velocities of plate 2. For this purpose, the resolution of the coupled system is done with a modal reduction approach using the first 10 in vacuo structural modes and the first 10 acoustic modes of the fluid in rigid cavity. The truncated bases are enriched by the static modal eigenvectors. Fig. 3 shows the normal incidence transmission loss through a simply supported plate (dashed line). Due to the modal behavior of the plate, dips in the transmission loss curve are observed at its resonance frequencies (modes (1, 1), (3, 1) and (1, 3)). When a second plate is used to form an airtight cavity (continuous line), an increase in the transmission loss is achieved except in the region of the so-called plate-cavity-plate resonance (mode (1, 1)*). At this frequency, the two plates move

356


80 Air−filled double wall system Simple panel

70 60 nSTL (dB)

50 40 30 20 10

mode (1, 3) mode (3, 1) mode (1, 1)*

mode (1, 1)

0 1

10

2

10 Frequency (Hz)

3

10

Fig. 3 Comparison of the normal incidence sound transmission (nSTL) through an air-filled double panel and a simple panel

out of phase with each other and the effect of the cavity on the plates is mostly one of added stiffness. This frequency is similar to the mass-air-mass resonance of unbounded double panels. In addition, the variation of the nSTL of an air-filled panels and a simple panel is in very good agreement with the published date from [2].

4.2 Sound Transmission through a Double Laminated Glazing Window The proposed reduced order finite-elements formulation is applied now to calculate the transmission loss factor of a double laminated glazing window. The system consists of two identical clamped laminated panels of glass separated by an air cavity of 12 mm thickness. Each laminated glass is composed of two glass plates bonded together by a Polyvinyl Butyral (PVB) interlayer. The thickness of outer and inner glass ply is h1 = h3 = 3 mm and those of the PVB interlayer is h2 = 1.14 mm. The glass ply is modeled as linear elastic material (density 2500 kg/m3 , Youngs modulus 72 GPa, and Poisson ratio 0.22). The material properties of the PVB are both thermal and frequency dependent. From dynamic and thermal tests, Havrillak and Negami have found an empirical law describing this dependence. The resulting complex frequency dependent shear modulus of the PVB is given at 20◦ C as [8]: −β0 G∗ (ω) = G∞ + (G0 − G∞ ) 1 + (iωτ0 )1−α0

(24)

where G∞ = 0.235 GPa, G0 = 0.479 Mpa, α0 = 0.46, β0 = 0.1946, τ0 = 0.3979. The Poisson ratio of the PVB is 0.4 and density is 999 kg/m3 . Concerning the excitation and the finite element discretization, we used the same ones as in the previous example.


357

Radiated Sound Power (dB) − ref 10

−12

100

80

60

40

20 Simple glass pane Laminted glass pane (3×1.14×3 mm)

0 0

500

1000 Frequency (Hz)

1500

2000

Fig. 4 Comparison of radiated sound power from a simple glass pane and a laminated glass with the same mass

A comparison between a simple glass and a laminated glass with PVB interlayer with an equivalent surface mass is shown in Fig. 4. Calculation was limited to 2000 Hz maximum. This comparison shows that laminated glass has a much lower acoustic radiation compared to conventional glass at resonance frequencies du to the effect of the viscoelastic layer. The reduction of sound radiation power is around 10 dB in lower frequencies and around 20 dB in higher frequencies. In fact, at low frequencies, the viscoelastic material is soft and the damping is small. At higher frequencies, the stiffness decreases rapidly and the damping is highest. Moreover, flexural vibrations causes shear strain in the viscoelastic core which dissipates energy and reduces vibration and noise radiation. Note that the thickness of the viscoelastic layer has a significant influence in terms of attenuation.

120 Modal reduction Direct method 100

nSTL (dB)

80 60 40 20 0 1

10

2

10 Frequency (Hz)

3

10

Fig. 5 nSTL through an air-filled double panel: comparison between the modal reduction approach and the direct nodal method

358


Fig. 5 shows a comparison between the nSTL of the coupled problem, obtained with the proposed accelerated modal reduction approach with a truncation on the first twenty structural modes (Ns = 20) and first twenty acoustic modes (Nf = 20) and the direct nodal method (Eq. (10)) where the displacement and pressure vectors are calculated for each frequency step. The structural modes are calculated from Eq. (13) using the constant shear storage modulus G∞ . As can be seen, a very good agreement between the two methods is proved. In this respect, it should be noted that the resulting reduction of the model size and the computational effort using the reduced order method are very significant compared to those of the direct approach.

5 Conclusions In this paper, a finite element formulation for sound transmission through double wall sandwich panels with viscoelastic core is presented. A reduced-order model, based on a normal mode expansion, is then developed. The proposed methodology requires the computation of the eigenmodes of the undamped structure, and the rigid acoustic cavity. Quasi-static corrections are introduced in order to accelerate the convergence. Despite its reduced size, this model is proved to be very efficient for simulations of steady-state analyses of structural-acoustic coupled systems with viscoelastic interlayers when appropriate damping terms are inserted into the modal equations of motion.

References 1. Fahy, F.: Sound and structural vibration, 1st edn. Academic Press, New York (1985) 2. Panneton, R.: Modélisation numérique tridimensionnelle par e´ léments finis des milieux poroélastiques: application au problème couplélasto-poro-acoustique. Ph.D., Université de Sherbrooke (1996) 3. Quirt, J.D.: Sound transmission through windows I. Single and double glazing. Journal of the Acoustical Society of America 72(3), 834–844 (1982) 4. Akrout, A., Hammami, L., Ben Tahar, M., Haddar, M.: Vibro-acoustic behaviour of laminated double glazing enclosing a viscothermal fluid cavity. Applied Acoustics 70(1), 82–96 (2009) 5. Basten, T.G.H.: Noise reduction by viscothermal acousto-elastic interaction in double wall panels. PhD-thesis, University of Twente, Enschede, The Netherlands (2001) 6. Morand, H.J.-P., Ohayon, R.: Fluid-structure interaction. John Wiley & Sons, New York (1995) 7. Larbi, W., Deü, J.-F., Ohayon, R.: Finite element formulation of smart piezoelectric composite plates coupled with acoustic fluid. Composite Structures 94(2), 501–509 (2012) 8. Havriliak, S., Negami, S.: A complex plane analysis of?-dispersions in some polymer systems. Journal of Polymer Science Part C: Polymer Symposia 14(1), 99–117 (1966)

Time Step Size Effect on the Hydrodynamic Structure around a Water Darrieus Rotor Ibrahim Mabrouki, Ahmed Samet, Zied Driss, and Mohamed Salah Abid Laboratory of Electro-Mechanic Systems (LASEM), National School of Engineers of Sfax (ENIS), University of Sfax (US), B.P. 1173, Road Soukra km 3.5, 3038 Sfax, TUNISIA [email protected], [email protected], [email protected]

Abstract. Turbulent flows are encountered in many hydraulic and water resources engineering problems. Their understanding is thus a critical prerequisite for designing stream and river restoration projects and a broad range of hydrodynamic structures. For this purpose, an unsteady Reynolds averaged Navier-Stokes (URANS) equations with a two-equations turbulence closure model is employed. The present paper aims to numerically explore the three-dimensional unsteady flow over a conventional Darrieus type rotor and to study the time step size effect on the hydrodynamic structure. Keywords: URANS, hydrodynamic, Darrieus rotor.

1

Introduction

Darrieus turbine was developed in France by Georges Darrieus in the 1920's to generate power from wind. It consists of a set of long, rectangular airfoils connected to a central rotating shaft. The airfoils may be curved to directly connect to the shaft or be straight and held parallel to the shaft by struts, arms or discs. These blades are there by oriented transversely to the fluid flow and parallel to the axis of rotation. Significant testing in the 1980's and 1990's demonstrated the utility of this turbine design. Several different models were successfully tested in the laboratory and various waterways. This kind of turbine considered for diffrent applications, such as tidal power, run-of-the-river hydroelectricity, and wave energy conversion (Shiono et al. (2000), Prabhu et al. (2013) and Maitre et al. (2013)). Hwang et al. (2009) studied the effect of number Blade on the performance of Darrieus Water Turbine. It’s one of the important parameter of Darrieus water turbine. These authors studied the influence of chord length of Darrieus water turbine. They show the evolution of maximum power coefficient and the TSR for a fixed number of blades. Attempts to improve the performance of H-Blade Darrieus rotor by using a duct placed outside the rotor have been documented. Shimokawa et al. (2010) measured the influence of the stream section with and without nozzle in the distribution of velocity vectors. In addition, Matsushita et al. (2008) studied the effect of draft tube on the efficiency in the case with narrow intake Sin /D=0.8. Malipeddi et al. (2012) studied the effect of changing position of the tur© Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_37

359

360

I. Mabrouki et al.

bine in the duct by performing simulation with the turbine at various locations in the duct, for a tip-speed ratio of 2. Rao et al. (2010) studied the performance of Darrieus rotor equipped with a symmetric two-foiled channeling device (Diffuser Annular Ring). They changed the channeling profile and, demonstrate their effect in the performance of the turbine. A ducted Darrieus turbine system as appropriate one for extra-low head hydropower has been proposed and developed by different authors. The design guidelines of Darrieus turbine and the characteristics have been clarified to obtain higher performance As the results of still continued investigation to simplify the structure for making cost effectiveness higher, it was recognized for small sized runner that there was no need to install a draft tube and the sidewalls of runner casing section in the case of runner with inlet nozzle. A numerical investigation was developed in the unsteady state using computational fluid dynamic (CFD) in order to characterize the hydrodynamic comportment of the flow around the rotor. We studied also the effect of time step size on the hydrodynamics characteristics. The used CFD code "Fluent" is based on the Unsteady Navier-Stokes equations by a finite volume discretization model. The numerical approach used is the sliding mesh (SM) model. In these condition, the test section equipped with 2 parts stationary domain and moving domain in which we have impose a rotational speed equal to ω = 25.61 rad.s-1. The turbulent flow is defined by the Reynolds number Re=3.8 106.

2

Geometrical System

The test section under investigation in this work is composed of a duct equipped with water Darrieus rotor. The duct have a prismatic shape with a height equal to b=0.2 m, a width equal to s=0.526 m, and a length equal to L=2.886 m .The Darrieus turbine, is composed by three straight blades and is placed inside the duct. The blades have profiles belonging to the symmetrical NACA 0018 four digit with a maximal thickness equal to 18% of the chord, a height equal to b=0.2 m and a diameter equal to D=0.39 m.

Fig. 1 Geometrical arrangement

The flow over Darrieus rotor is periodic and a proper time step selection is important in order to ensure a good results. The time step corresponds to one degree rotation of the rotor and is expressed as: Ts = Where: ω: Vitesse angulaire (rad/s)

/ (180 ω)

(1)

Time Step Size Effect on the Hydrodynamic Structure

361

In this study the used time steps correspond to 1.25° and 2.5° rotation and they are shown in Table 1. Table 1 Time steps used for time step effect

3

Time step (s)

Corresponding degree of rotation (degree)

0.1

2.5

0.05

1.25

Numerical Model

In the present work, the commercial CFD code Fluent is used to perform 3D simulations, solving the incompressible Unsteady Reynolds-Averaged Navier-Stokes (U-RANS) equations discretized by means of a finite volume approach. The implicit segregated version of the solver is employed. The pressure-velocity coupling is achieved by means of the SIMPLE algorithm. The convective terms are discretized using a second order accurate upwind scheme, and pressure and viscous terms are discretized by a second-order-accurate centered scheme. A first order implicit time formulation is also used. Turbulence closure is provided by shear stress transport k-ω turbulence model. The model has been validated, by comparing numerical results with available experimental data (Matsushita et al. 2008).

3.1

Meshing Size

A mesh with structured triangular grid has been created in the stationery area and, a structured quadrilateral grid in the moving area using "GAMBIT” software.

Total number of nodes

Total number of cells

239 538

956 054

Fig. 2 Overall grid (left) and near blade grid (right)

362

3.2

I. Mabrouki et al.

Boundary Conditions

Figure 3 illustrates all information about boundary conditions is required anywhere fluid enters or exits the system and can be set as a pressure, mass flow, volume flow or velocity. In the considered simulation, the velocity inlet, we will take as a value Vi=1 m.s-1 , and for the pressure outlet a value of 101325 Pa will be considered which means that at this opening the fluid exits the model to an area of static atmospheric pressure. Figure 3 illustrate all information. The rotating -1 domain is characteristic by an angular velocity equal to w = 25.61 rad.s .

Fig. 3 Boundary conditions

4

Numerical Results

In this study, we are interested to visualize the velocity field, the static pressure, the dynamic pressure, and the turbulent kinetic energy. Indeed, the variation of each result as function of time in different points located in the downstream of the rotor (figure 4). points

x (m)

y (m)

z (m)

Point 1

0.22

0

0

Point 2

0.22

0.15

0

Point 3

0.22

-0.15

0

Fig. 4 Visualization planes

4.1

Velocity Fields

Figure 5 shows the distribution of the velocity fields presented in the longitudinal plane containing the blade and defined by z=0 m. According to these results, it is clear that the velocity field is weak in the inlet of duct. Indeed, it is governed by the boundary condition value of the inlet velocity witch is equal to 1 m.s-1. While approaching to the water Darrieus rotor, the velocity field is affected by the geometric configuration. In fact, flow acceleration appears in the gap between the


363

rotor and the blades walls. In the downstream of the rotor, the velocity value keeps decreasing until the out of the test section. Then, a sharp increase has been noted through the divergent angle where the velocity values are recorded in the lateral walls of the duct. Globally, similar results are obtained with the different time step sizes. However, the difference is localized in the duct extremity and in the blades.

Fig. 5 Velocity fields distribution

4.2

Static Pressure

Figure 6 shows the distribution of the static pressure in the test section equipped by the water Darrieus rotor. According to these results, a considerable increase of the static pressure has been observed. In the blade upstream, a compression zone has been appeared. However, in the blade downstream depression zones have been observed. In the longitudinal section plane defined by z=0 m, it has been noted that the pressure increases slightly in the first part of the test section and it decreases rapidly in the water Darrieus rotor downstream. Globally, similar results are obtained with the different time step sizes. However, the difference is localized in the rotating are a around the blades. In fact, the compression zones are larger with Ts= 0.1 s. However, the depression zone are larger with Ts= 0.05 s.

Fig. 6 Static pressure distribution

364

I. Mabrouki et al.

Figure 7 shows the evolution of the static pressure in three different points (1), (2) and (3) localized in the downstream of the rotor for a period of 2 seconds. Overall, it has been noted a significant fluctuations of the static pressure curves. Particularly, it has been observed that the evolution of the static pressure is less important in point (1) than in the points (2) and (3). This fact due to the position of this point located in the middle of the duct. For the points (2) and (3) located in the lateral faces of duct, it has been noted a little difference between the results due to the effect of time steps sizes.

Fig. 7 Static pressure variation

4.3

Dynamic Pressure

Figure 8 illustrates the distribution of the dynamic pressure in the test section equipped by the water Darrieus rotor. According to these results, it has been noted that the dynamic pressure depends essentially on the fluid speed. A first view shows that the dynamic pressure is found weak in the inlet of duct. When it gets to the water turbine, the dynamic pressure keeps increasing in the upstream of the rotor and around it. In fact, flow acceleration appears in the gap between the rotor and the blades walls. This fact is confirmed in the longitudinal plane defined by z=0 m. In the rotor downstream of, the dynamic pressure value keeps decreasing until the out of the test section. Then, a sharp increase has been noted through the divergent angle where the velocity values are recorded in the lateral walls of the duct. This fact is confirmed in the transverse section plane defined by y=0 m. The maximum value is obtained in the gap between the rotor and the duct walls. Under these conditions, the dynamic pressure is equal to 2.36103 Pa at Ts= 0.1 s and 1.99103 Pa at Ts= 0.05 s. globally, similar results are obtained with the different time step sizes. However, the difference is localized in the extremity of the duct exactly in the gaps between the rotor and the blades walls.


365

Fig. 8 Dynamic pressure distribution

Figure 9 shows the evolution of the dynamic pressure indifferent in three points (1), (2) and (3) located in the downstream of the Darrieus rotor for a period of 2 s. Overall, it has been noted a significant fluctuations of the dynamic pressure curves. Indeed, it’s clear that the fluctuating of the dynamic pressure is less important in point (1) than the point (2) and (3). This fact is due to the position of this point located in the middle of the duct. Indeed, it has been noted a difference between the results due to the time steps effect.

Fig. 9 Dynamic pressure variation

4.4

Turbulent Kinetic Energy

Figure 10 shows the distribution of the turbulent kinetic energy in the test section equipped by the water turbine. From these results, it is clear that the turbulent kinetic energy is found to be very weak in the test section except in the area surrounding the rotor. The distribution of the turbulent kinetic energy shows the increase of the energy in the interior zone of the rotor. From a first glance, a small difference appears between the distributions of the turbulent kinetic energy for each time steps size. This difference is located in the rotating area, lacted around the blades. The maximum value of the turbulent kinetic energy is recorded around the blades as shown in the longitudinal and transverse planes defined by z=0 m and y=0 m. Under these conditions, the turbulent kinetic energy is equal to 0.222 m2.s-1withTs=0.1s and 0.196 m2.s-1 with Ts=0.05s.Globally, similar results are obtained with the different time step sizes. However, the difference is localized in the lateral faces of the duct exactly around the blades walls.

366

I. Mabrouki et al.

Fig. 10 Turbulent kinetic energy distribution

Figure 11 shows the evolution of the turbulent kinetic energy in different three points (1), (2) and (3) located in the downstream of the turbine for a period of 2 s. Overall, it has been noted a significant fluctuations of the turbulent kinetic energy curves. According to these results, it has been noted that the turbulent kinetic energy fluctuation is less important in point (1) than the point (2) and (3). This fact is due to the position of this point located in the middle of the duct. Indeed, it has been noted a difference between the two results, due to the effect of the time steps.

Fig. 11 Turbulent kinetic energy variation

5

Comparison with Experimental Results

To verify our numerical results, we have compared in figure 12 the dimensional axial velocity Va/Va value as a function of the 2y/D position obtained by our numerical model with the experimental value founded by Matsushita et al. (2008). Based on the numerical results, we found that the tow curves are close to the experimental data. Indeed, it has been noted that the curves with Ts=0.05 s close to the experimental. Thus, so we can conclude that when the time steps size decreases, the degree of rotation decrease, so we approach to the reality. According to these results, the relative gap calculated between the numerical and the experimental axial velocity values is equal to 8%. The good matching between these results indicates the validity of our computer method.


367

Fig. 12 Velocity profiles variation

6

Conclusion

In this paper, we have studied the numerical parameters effect in the unsteady state. Particularly, we have studied the time step sizes effect and we concluded that this parameter has an influence in the rotation degree of the water Darrius rotor. Then, we have concluded that the sliding mesh (SM) method can be used to model the water turbine characteristics. This study allows us to find the adequate numerical parameters enabling more detailed analyses of the flow around the Darrieus rotor.

References Shiono, M.I., Katsuyuki, S.U., Seiji, K.I.: An Experimental Study of the Characteristics of a Darrieus Turbine for Tidal Power Generation. Electrical Engineering in Japan 132(3) (2000) Prabhu, S.V., Vimal, P.A., Himanshu, C.H.: Performance Prediction of H-Type Darrieus Turbine by Single Stream Tube Model for Hydro Dynamic Application. International Journal of Engineering Research & Technology 2(3) (2013) Maitre, T., Amet, E., Pellone, C.: Modeling of flow in Darrieus water turbine: wall grid refinement analysis and comparison with experiments. Renewable Energy 51, 497–512 (2013) Huang, S.W., Tsai, Y.D., Liang, S.Y., Hsieh, C.H., Chen, S.J.: Wind-tunnel study on aerodynamic performance of small vertical-axis wind turbines (2009)

368

I. Mabrouki et al.

Shimokawa, K.A., Akinori, F.U., Kusuo, O.K., Daisuke, M.A.: Satoshi WA Side-wall effect of runner casing on performance of Darrieus-type hydro turbine with inlet nozzle for extra-low head utilization. Technological Sciences 53, 93–99 (2010) Matsushita, D.A., Okuma, K.U., Watanabe, S.A., Furukawa, A.K.: Simplified structure of ducted Darrieus-type hydro turbine with narrow intake for extra-low head hydropower utilization. Journal of Fluid Science and Technology 3, 387–397 (2008) Malipeddi, A.R., Chatterjee, D.: Influence of duct geometry on the performance of Darrieus hydro turbine. Renewable Energy 43, 292–300 (2012) Roa, A.M., Aumelas, V., Maître, T., Pellone, C.: Numerical and experimental analysis of a Darrieus-type cross flow water turbine in bare and shrouded configurations. In: 25th IAHR Symposium on Hydraulic Machinery and System (2010)

Simulation of the Dynamic Behavior of Multi-stage Geared Systems with Tooth Shape Deviations and External Excitations Hassen Fakhfakh1, Jérome Bruyère1, Philippe Velex1,*, and Samuel Becquerelle2 1 2

INSA de Lyon, LaMCoS, France Hispano-Suiza, France [email protected]

Abstract. In this paper, a torsional dynamic model of multi-stage idler spur and helical gears is presented which combines time-varying internal and external excitations such as time-varying external torques. Each contact line in the various base planes is discretized in elemental cells which are all attributed a time-varying mesh stiffness element and an initial separation to account for tooth shape deviations from ideal involute flanks. The corresponding non-linear differential system is solved by combining a Newmark’s numerical scheme and a normal contact algorithm. A number of simulation results are presented on the influence of the combined effect of errors and shape deviations along with external excitation sources on dynamic tooth loads. Keywords: Multi-stage idler gears, Spur and helical gears, Time-varying torques, Tooth shape deviations.

1

Introduction

In certain conditions, power circulation can influence vibrations as is the case for the dynamic behavior of multi-stage geared systems with the various possible combinations of resisting and driving torque positions. In this context, Kuria and Kihiu (2012) developed a numerical model to study the effect of three design variable on dynamic stress of multi-stage spur gear. Al-Shyyab and Karahman (2005) used a Harmonic Balance Method in conjunction with continuation technique to describe the periodical responses of multi mesh gear system. Velex and Saada (1991), Velex and Raclot (1999), Kubur and al. (2004) proposed FE models of double stage drives and investigated the influence of key design parameters on the system dynamics. Carbonelli (2014) used a Finite Difference approach accounting for both internal and external sources in the time domain such as clearance nonli*



369

370

H. Fakhfakh et al.

nearity. Yang (2013) studied the vibrational response in torsion of a two-stage gear submitted to deterministic and random excitations in the presence of backlash. This paper presents a torsional dynamic model of multi-stage idler spur or helical gears aimed at simulating the dynamic tooth loading in the presence of several time-varying mesh stiffness whose relative phases are determined based on the gear geometry and their relative positions. The model can moreover account for tooth shape modifications and constant or time-varying external loads which can strongly influence dynamic tooth forces.

2

Torsional Model of Multi-Stage Idler Gears

2.1

Mesh Stiffness Matrix

The normal deflection at any point of contact

M mij

(m is the stage number), is

equivalent to the normal approach δ mij with respect to rigid-body positions minus the initial separation

δ emij

possibly induced by tooth shape modifications:

Δ mij = δ mij − δ emij In what follows,

k mij

(1)

represents the discrete stiffness element associated with

each discrete cell on the contact lines which depends on load and shape deviations (Velex and Maatar 1996) and which is expressed as: k mij = kˆmij H ( Δ mij )

where

H ( Δ mij )

with

⎧ H = 0 if Δ ⎪ mij < 0 ⎨ ⎪ H =1 if Δ mij > 0 ⎩

(2)

is the Heaviside function used to simulate total or partial contact

losses. For a pinion-gear pair with one torsional degree of freedom each, the mesh stiffness matrix can be expressed as: JJJG JJJJGT ⎡ K (t )⎤ = ∑ k mij Vm Vm ⎣ m ⎦ mij

where

JJJG Vm = ζ Mm Rbm cos βb

gear geometry.

ζ Mm Rbm +1 cos βb

(3)

is a structural vector which accounts for

Simulation of the Dynamic Behavior of a Multi-stage Geared Systems

371

Extending the methodology to a system with one pinion, one idler gear and one output member, the global stiffness matrix reads: ⎡ 2 ⎢ k1 ( t ) Rb1 ⎢ ⎢k ( t ) R R b1 b 2 ⎢1 ⎡ K ( t )⎤ = cos 2 β ⎢ ⎣ ⎦ 0 b ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢⎣

( )

2.2

(

k1 ( t ) Rb1 Rb 2 k1 ( t ) + k2 ( t ) Rb22

)

k2 ( t ) Rb 2 Rb 3 0 0

⎤ ⎥ ⎥ ⎥ 0 0 k2 ( t ) Rb 2 Rb 3 ⎥ ⎥ 2 0 k2 ( t ) + k3 ( t ) Rb 3 " ⎥ ⎥ # ⎥ ⎥ 2 k N −1 ( t ) RbN + k h ⎥⎦ 0

(

0

0

)

(4)

Model and Equations of Motion

All the numerical results presented in this paper are related to the torsional model of N idler spur and helical gear shown in figure 1. The pinions are modelled as rigid discs connected by time-varying mesh stiffness functions (Kubo 1978, Gregory and al 1963, etc.) whose relative phasing are determined analytically. Time-varying loads are considered which can be caused by generators or pumps and fluctuating power input. In this paper, a periodic torque of the form Ct = C o + C s sin(k Ω t) will be considered where Ω is the angular frequency of the member submitted to torque Ct. At a given time t, the equivalent deviations are discretized on the base planes using the same elemental cells as those introduced for mesh stiffness elements. From an initial position of the contact lines, a time-step procedure reproduces the mesh stiffness and gives the instantaneous equivalent normal deviation em ( M )

at each contact point M. In these conditions, the equations of motion read:

[ M ] {q} + [C ] {q} + ⎡⎣K ( t ,q )⎤⎦ {q} = {F0( t )} + {F1( t ,q,δ em )} + {F2 ( t )}

(5)

where [M] and [K(t,q)] are the global mass and stiffness matrix, the damping matrix is expressed using a Rayleigh model such that [C]=a.[M]+b.[Kav] (Velex 1988) , {F0 ( t )} = Ct1 ( t ) Ct2 ( t ) ... Ct N ( t ) is the external torque vector,

{F1 (t , q ,δ em )} embodies the contributions of shape deviations and errors and its com-

ponents are:

{F1(t ,q,δ em )}

⎧ ⎫ k1 ( t )δ e1 ( M )ζ M 1Rb1 cos βb ⎪ ⎪ ⎪ ⎪ k1 ( t )δ e1 ( M )ζ M 1 + k2 ( t )δ e2 ( M )ζ M 2 Rb2 cos βb ⎪ ⎪ ⎪ ⎪ k2 ( t )δ e2 ( M )ζ M 2 + k3 ( t )δ e3 ( M )ζ M 3 Rb3 cos βb ⎪ ⎪ = ⎨ ⎬ # ⎪ ⎪ ⎪ ⎪ ⎪ k N − 2 ( t )δ eN − 2 ( M )ζ M N − 2 + k N −1 ( t )δ eN −1 ( M )ζ M N −1 RbN −1 cos βb ⎪ ⎪ ⎪ ⎪ ⎪ k N −1 ( t )δ eN −1 ( M )ζ M N −1RbN cos βb ⎭ ⎩

( (

(

) )

)

(6)

372

H. Fakhfakh et al.

and {F2 ( t )}=

− J1Ω 1

− J 2Ω 2

... − J N Ω N

is the inertial torque vector possibly gener-

ated by unsteady rotational speeds. The differential system is solved by a Newmark’s time step integration methods. The contact condition and constraints are directly inserted in the integration scheme so that the partial or total contact losses problem between the mating flanks can be solved at any time step.

Ω1(t) O1 K1: excitation due to mesh stiffness

e1(t): excitation due to geometrical errors c1

K2

O3

∆l2

c2 O2

e2(t) ∆l1: relative phase shift

kh ON

Fig. 1 Multi-stage idler gear with N meshes

3

Dynamic Results

The numerical applications were conducted on the 4-stage idler spur gear shown in Figure 2 where the system is represented with the pinion centres all aligned such that γ1 = γ2 = γ3 = 180°. The corresponding gear data are listed in Table 1.

Fig. 2 Idler gear system


373

Table 1 Gear data

Tooth number Face width(mm) Module (mm) Pressure angle (°) Helix angle (°) Addendum coefficient Profile shift coeff. Dedendum coefficient

3.1

Pinion 1

Pinion 2

50 25.4

39 25.4

1.0 0.0 1.4

Pinion 3

Pinion 4

47 51 25.4 46.5 3.18 20 0 (spur) or 20 (helical) 1.0 1.0 1.0 0.0 0.0 0.0 1.4 1.4 1.4

Pinion 5 28 25.4

1.0 0.0 1.4

Influence of an Oscillating Torque and Tooth Shape Modifications

The influence of a periodically varying torque is illustrated by comparing the dynamic tooth force curves versus speed for constant torque and those obtained when considering a fluctuating torque on pinion 1 (figure 3). In both cases, a damping factor ξ = 0.04 was used. Table 2 Tooth critical frequency W1 W2 W3 W4 W5

Tooth critical frequency (rad/s) 73.790 11814.744 25139.066 38266.122 34321.088

The critical speeds of the four stages appear simultaneously on the four curves which proves the significant inter-stages coupling so that each pinion is excited by all the mesh stiffness functions. The response curves for a constant load (figure 3.a) exhibit peaks at 775 rpm ≌ (W2/50)/3, 1075 rpm ≌ (W2/50)/2 and 1950 rpm corresponding to a natural frequency being excited by the mesh frequency and its harmonics. The introduction of a time-varying torque of frequency 20Ω1 generates mostly an additional response peak at 5650 ≌ W2/20 rpm. The system also exhibits contact losses between the teeth at some critical speeds on some of the reduction stages. The introduction of short profile corrections (linear relief of depth Pf=60μm at tooth tips and extents corresponding to 20% of the active profile) on all the pinions, leads to a substantial improvement of the dynamic behaviour below the main critical speed (≌775 rpm) as shown in figure 3.c.

374

H. Fakhfakh et al.

(a)

(b)

(c) Fig. 3 Influence of (a) constant torque and (b) time-varying torque (c) profile corrections on dynamic mesh forces


3.2

375

Influence of an Intermediate Oscillating Torque and Tooth Shape Modifications

This section deals with the influence of the power circulation combined with tooth profile relief on the dynamic behavior. To this end, the following loading conditions are investigated: i) a constant driving torque Ct1= 1200 Nm is applied to pinion 1, ii) a resisting torque either constant or periodic Ct3=-500 or -500+200.sin (20Ω1t) is imposed on pinion 3 and iii) a resisting torque on the output gear 5 (adjusted so that the total power is conserved).

(a)

(b) Fig. 4 Influence of (a) constant torque and (b) time-varying torque (c) profile corrections on dynamic mesh forces

376

H. Fakhfakh et al.

(c) Fig. 4 (continued)

The results in figure 4 show that although the critical rotational speeds are the same as in the previous case, the different power circulation modifies the dynamic amplifications. The introduction of the resisting torque on pinion 3 unloads the meshes downstream (between pinions 3, 4 and 5) over the whole speed range. When the periodic component is superimposed (Ct3=-500+200.sin (20Ω1t)), an additional tooth critical speed emerges for all the meshes amplifying contact losses especially on stage 3. The response spectra (not shown in this paper) exhibit significant modulation side-bands related to the torque frequency 20Ω1. The response curves for short profile corrections shown in figure 4.c illustrates tooth forces amplitude and contact losses decrease below the main critical speed (≌775 rpm).

4

Conclusion

In this paper, a simplified torsional dynamic model of N stage idler spur and helical gears has been presented. The time-varying mesh stiffness functions attached to every individual reduction stage are estimated from the classic formulae of Weber & Banaschek and their relative phases are determined analytically. The model can simulate the influence of periodic torques and power circulations (several outputs, time-varying resisting loads, etc.) on tooth load amplification and critical speeds. It has been shown that short profile corrections can improve the dynamic behavior for a range of rotational speeds. Research is under way to incorporate three-dimensional degrees of freedom, shafts, bearings and casings along with a variety of loading conditions. Acknowledgements. The authors would like to thank the support of Hispano-Suiza from the Safran Group.


377

Nomenclature Ωi ,Ω i

= angular velocity, acceleration of pinion i = base radius of pinion i

Rbi Ji

= polar moments of inertia of pinion i

βb

= base helix angle

ζ Mm = kh

= torsional stiffness added to node N ( k h = 10

2

Nm / rad

)

= average, oscillation amplitude of the periodic torque

c0 , cs

a ,b

⎡K ⎤ ⎣ av ⎦

sign of the driving torque of stage m

= constants that can be adjusted to get the desired damping factor ξ .

In this case a = 1, 12 ; b = average stiffness matrix

= 1, 2.10

−5

References Kuria, J., Kihiu, J.: Effect of Gear Design Variables on the Dynamic Stress of Multi stage Gears. Innovative Systems Design and Engineering (2012) ISSN 2222-1727 Al-Shyyab, A., Karahman, A.: Non-linear dynamic analysis of a multi-mesh gear train using multi-term harmonic balance method: sub-harmonic motions. J. of Sound and Vib. (2005), doi:10.1016/j.jsv.2003.11.029 Velex, P., Saada, A.: A model for the dynamic behavior of multi-stage geared system. In: Proc. of the 8th World Congress on Theory of Machines and Mechanisms, Prague, vol. 2, pp. 621–624 (1991) Raclot, J.-P., Velex, P.: Simulation of the dynamic behavior of single and multi-stage geared systems with shape deviations and mounting errors by using a spectral method. J. of Sound and Vib. 220(5), 861–903 (1999) Kubur, M., Kahraman, A., Zini, D., Kienzle, K.: Dynamic analysis of a multi-shaft helical gear transmission by finite elements – Model and experiments. J. Vib. Acous. 126(3), 398–406 (2004) Carbonelli, A.: Hammering noise modelling-Nonlinera dynamic of a multi-stage gear train. In: International Gear Conference, Lyon, pp. 447–456 (2014) Yang, J.: Vibration analysis on multi-mesh gear-trains under combined deterministic and random excitations. Mechanism and Machine Theory 59, 20–33 (2013), doi:10.1016/j.mechmachtheory.2012.08.005 Velex, P., Maatar, M.: A Mathematical Model for Analysing the Influence of Shape Deviations and Mounting Errors on Gear Dynamic Behaviour. J. of Sound and Vibration 191(5), 629–660 (1996), doi:10.1006/jsvi.1996.0148 Kubo, A.: Stress condition, vibrational exciting force, and contact pattern of helical gears with manufacturing and alignment error. J. Mech. Des. 100(1), 77–84 (1978), doi:10.1115/1.3453898

378

H. Fakhfakh et al.

Gregory, R.W., Harris, S.L., Munro, R.G.: Dynamic Behavior of Spur Gears. Proc. of the Institution of Mechanical Engineers 178(1), 207–218 (1963), doi:10.1177/002034836317800130 Velex, P.: Contribution à l’analyse du comportement dynamique de réducteurs à engrenages à axes parallèles. Thèse de Doctorat, INSA de Lyon.N°88 ISAL 0032 (1988)

Fuzzy Modeling and Control of Centrifugal Compressor Used in Gas Pipelines Systems Ahmed Hafaifa1, Guemana Mouloud1,2, and Belhadef Rachid3 1

Applied Automation and Industrial Diagnostic Laboratory, University of Djelfa 17000 DZ, Algeria [email protected] 2 Faculty of Science and Technology, University of Medea 26000 DZ, Algeria [email protected] 3 Faculty of Science and Technology, University of Sedik Ben yahia of Jijel, Algeria [email protected]

Abstract. Respond to changing technology industrial installations, this work propose solutions to the modeling and control problems in industrial processes with the use of new approaches. The objective of this work is the use of fuzzy techniques in modeling and control in the study of gas compression system instability. The obtained results show clearly how the main dynamic characteristics, in our examined compression system, are reproduced using the proposed fuzzy model, allowing better performance during its control synthesis operation. Keywords: Gas compression system, surge phenomena, centrifugal compressor, exploitation instability, fuzzy modeling, fuzzy control.

1

Introduction

The development emergence in new automation techniques led to significant changes in the control systems design. Indeed, in many industrial applications, the scientific vision takes in some technology areas, a particularly interesting turn in the direction of developing control theory and its applications [1, 18, 23 and 25]. It is therefore natural to seek a mechanism to show the face of intelligent features extensive changes in a more complex environment. In this concept, we are trying to improve some traditional techniques are unable to adapt to complex environments, uncertain or variants over time. To solve the problem of instability in compressors and in order to improve system performance, we propose in this work the use of new techniques. These new techniques may in some cases prevent the instability phenomenon, and give a certain approach to solving the problem of the permanent opening of the recirculation valve. Subject to integration of new algorithms that provide real-time regulation, this will recover the lost production to a certain extent while ensuring the functioning machine and protect the compressor from all dangers. © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_39

379

380

A. Hafaifa, G. Mouloud, and B. Rachid

In this work, we propose to examine and illustrate the application ability of fuzzy logic to the modeling of our examined gas compression system, and to involve this fuzzy model in the synthesis of control strategies appropriate for this type of system. Several studies have focused on the problem of its modeling [4, 10, 14], most of which have presented rigorous models, with a very complex mathematical structure limiting, therefore, the possibility of direct exploitation by the conventional control. These models are established in nominal conditions of work, making it difficult to control the compressor suction flow. Due to the precision, it uses control systems fairly robust, able to adapt to some degree of satisfaction to possible variations of process parameters and conditions of its operation. In fact, it seems more than necessary to be sufficiently accurate and complete knowledge of the process in order to satisfy the requirements of robustness and performance. It is in this subject we propose, a fuzzy model for our examined gas compression system to describing the dynamics of this system. Fuzzy logic provides performance benefits in modeling of the considered process.

2

Gas Compression System Modeling

In this work, we are interested by the compression system model developed by Tommy GRAVDAHL [4, 19]. Taking into account the maximum losses, in order to obtain a model, able to predict the actual characteristics of the examined gas compression system. The overall performance characteristics are identified at the point of maximum efficiency and normal operating conditions. The examined gas compression system, shown in figure 1, is composed by a turbine, compressor, pipes and an exhaust valve, the figure 2 show allows us to situate the different areas in the examined compressor. Yt 1.5

Anti Surge Valve

User

1

Zone de fonctionnement

Coefficient de débit

Mass flow coefficient

Normal normale operating area

N

Compressor

Turbine

Zone de Pompage Instability area

0.5

0

-0.5

-1

Gas Input Pa Ta

Fig. 1 Examined gas compression system

-1.5 -0.4

-0.2

0

0.2 0.4 Coefficient de Charge Load factor

0.6

0.8

1

Fig. 2 Centrifugal compressors characteristics

Fuzzy Modeling and Control of Centrifugal Compressor Used in Gas Pipelines Systems

381

The model developed by GRAVDAHL [4, 19] is based on the concepts of thermodynamics and fluid dynamics and it is written in the following form: 2

a PP = 01 ( m - m t ) VP = m

A1 LC

( P2 - PP )

(1)

1 (τ t -τ c ) N = 2Jπ

Where m is the mass flow of the compressor inlet, mt the output mass flow of the valve, Pp the output pressure, V p the output gas volume, P2 the pressure at the compressor outlet, a01 the sound speed at the compressor inlet, Lc the length of the compressor driving, A1 the goose surface of the impeller, J the compressor moment of inertia, τ c the compressor torque and τ t is the turbine torque. This model is similar to FINK AL model [7], while the two first equations of (1) are equivalent to Greitzer model [9, 10], where the speed is not included. The Pp equation is derived from the mass balance assuming the trial isentropic transfer. The third equation is derived from the second law of dynamics:

∑ M = J W

⇒

= 1 (τ -τ ) W t C J

(2)

With M is the moments applied to the compression system and W is the angular acceleration. The compressors characteristics can be defined by different methods, in this work we adopting the definitions suggested by Greitzer in [9, 10], namely: w V • The specific speed: η S = , Δ H 3/4 t/2 •

The mass flow coefficient: Φ =

•

The load factor: Ψ =

Δ H t/2 U 22

Vm U2

, D 2 is the outlet impeller diameter,

.

The trial is assumed isentropic, so we can write from a mass balance:

382


dPP dVP dρ =-k =k P PP ρP VP We have ρ =

(3)

1 dV dρ =which implies , where: V ρ V dP P P = k P ρ P = k R TP ρ P PP

(4)

The mass balance is given by the following relationship:

ρP = Where a s =

T 1 ( m - m t ) ⇒ P P = k R P ( m - m t ) = ( m - m t ) VP VP

(5)

K R TP is the speed of sound and the temperature ratio of the

compression close to unity

PP

ρP

= R TP , then the amount will be too near

P01

ρ 01

. So

we can conclude that: P PP = k 01 VP ( m - m t )

(6)

ρ 01

To complete the model (1) for give the dynamical model, we need to the output mass flow rate ( mt = k t

Pp - P01 ) with k t the proportional gain of the gas re-

circulation valve. The driving torque of the turbine is calculated by dividing the power of the turbine by the angular velocity, is given by :

τt =

Pt η t m tur C p , t Δ Ttur = w 2π N

(7)

Where: mtur is the steam flow at the inlet of the turbine and η is the mechanical efficiency of the turbine. Finally, we have the complete dynamical model given by the equations 8. 2 ⎧ a 01 ⎛⎜ m - k P Pp - P01 ⎞⎟ = ⎪ p t ⎝ ⎠ V p ⎪ ⎪ ⎪ ⎡ ⎪ A ⎢ ⎛ Δ h ideal ⎨ m = 1 ⎢ Pp ⎜ 1 + η i ( m , N ) ⎜ Lc ⎢ ⎝ C p T01 ⎪ ⎪ ⎣ ⎪ ⎪ = 1 τ -τ ⎪ N t c 2Jπ ⎩

(

)

⎞ ⎟ ⎟ ⎠

k −1 k

⎤ ⎥ - Pp ⎥ ⎥ ⎦

(8)


383

The previous study was conducted to centrifugal compressors in a single stage. The model of a multistage compressor is given by the equations (9).

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩

k P01 ⎛ P p = ⎜ m - kt ρ 01 Vp ⎝

Pp - P01 ⎞⎟ ⎠

⎡ ⎛ Δ h ideal A1 ⎢ ⎜ m= ⎢ P01 ⎜ 1 + η i (m, N) C p T01 Lc ⎢ ⎝ ⎣

⎞ ⎟ ⎟ ⎠

4 ( k -1 ) k

⎤ ⎥ - Pp ⎥ ⎥ ⎦

(9)

⎛ η m C Δ Ttur ⎞ = 1 ⎜ t tur p , t - 2 r22 σ π N m ⎟ N ⎟ 2 J π ⎜⎝ 2π N ⎠

The obtained model (9) is complex and difficult to exploit in control. Admittedly, it better describes the main physical characteristics of the system, but it is often necessary to simplify this complex model by linearization near a nominal operating point or by neglecting the less dominant dynamic process.

3

Fuzzy Modelling and Control of the Gas Compression System

Fuzzy logic offers a benefits modeling performance in industrial systems [3, 6, 8, 13, 21 and 22]. Let the differential system (9). In practice, the structure and parameters of this complex model are established under several simplifying assumptions. The linearization of the model given by the equations (9) about an operating point M ( Ppc 0 , mc 0 , u t 0 , ub 0 ) is given by : ⎡ Pˆ ⎤ ⎡ B m x = ⎢ pC ⎥ = ⎢ 1 ⎢⎣ mˆ c ⎥⎦ ⎢ B ⎣⎢

−B ⎤ ⎡ P ⎤ ⎡ 0 ⎤ 1 ⎥ ⎢ pC ⎥ + ⎢ uˆ b ⎥ m − V / B ⎥⎦ ⎣ c ⎦ ⎣ Bmte ⎦⎥

(10)

~ With t = twH and wH is the Helmholtz frequency defined by the following equation: wH = a

AC V P LC

with a = γRTa

G are defined as following: B =

and Bm =

B . The parameters B and G

Ut L A and G = t C , B is the Greitzer 2wH LC LC At

384


stability parameter [9, 10]. It follows that the linear model assuming (9) achieving status as follows: ⎧δx (t ) = A δx(t) + B δu(t) ⎪ ⎨ ⎪δy (t ) = C δx(t) + D δu(t) ⎩

(11)

and x 0 = [ 5.74 1.25 12.3] ,

With δx = x - x 0 , δu = u - u 0 , δy = y - y 0

T

y 0 = [ 2.23 6.6 0] , u 0 = [ 0.20 0.13 - 0.15 ] . The matrices A, B, C and D T

T

assessed in equilibrium are given by: ⎡ ⎢ ⎢ 0.2076 A = ⎢⎢ 3.7181 ⎢ ⎢ ⎢⎣- 9.6870 ⎡ ⎢ ⎢1 C = ⎢⎢ 0 ⎢ ⎢ ⎢⎣0

0.1323 - 0.4892 - 1.539

0 1 0

⎤ ⎥ 0⎥ ⎥, 0⎥ ⎥ ⎥ 1⎥⎦

⎤ ⎥ 0⎥ ⎥, 0⎥ ⎥ ⎥ 0⎥⎦

⎡ ⎢ ⎢ 0.2076 B = ⎢⎢ 0 ⎢ ⎢ ⎢⎣ 0

⎡ ⎢ 0 ⎢ ⎢ D=⎢ 0 ⎢ ⎢ ⎢⎣12.10740

⎤ ⎥ 0 ⎥ ⎥ - 2.312⎥ ⎥ ⎥ 0 ⎥⎦

1.2365 - 3.6891 - 3.8230

0 0 0.3582

⎤ ⎥ 0⎥ ⎥. 0⎥ ⎥ ⎥ 1⎥⎦

The first goal in centrifugal compressors modeling is to estimate losses, the first model called "Single-Zone Modeling" includes only global characteristics of the compressor. The proposed fuzzy model for the examined compression system is symbolically expressed by a set of IF-THEN rules of the form [5, 26]: IF Z 1 (t ) est Fi1 et ... .. et Z n est Fin THEN

⎧ xˆ (t ) = A i xˆ(t) + B i u(t) + K i ( y (t ) - yˆ(t) ) ⎪ ⎨ ⎪ yˆ (t ) = C xˆ(t) , i = 1, 2 ...... , r i ⎩ i

(12)

With: Rule i, i = 1, 2 ..... r We are interested by fuzzy Takagi-Sugeno modelling [2, 20, 26], defined by:

Fuzzy Modeling and Control of Centrifugal Compressor Used in Gas Pipelines Systems r ⎧ ⎪ x (t) = ∑ h i ( z(t) ) A i x(t) + B i u(t) i= ⎪ ⎨ r ⎪ ⎪ y (t) = ∑ h i ( z(t) ) C i x (t) i= ⎩

[

385

] (13)

The obtained results using fuzzy controller, shown in figure 3 are better than those obtained with conventional control algorithms. The proposed controller is a system based on special knowledge, using a limited depth of reasoning in a forward chaining procedure rules (rule firing by the premises). Compression system output

Compression system input

Knowledge Base

Decision mechanism Premises treatment

Inference

Fuzzification

Normalization

Aggregation rules

Defuzzification

System control

Denormalizatio

Compression system

Fig. 3 Proposed fuzzy controller structure

Several stages in the processing rules are distinguishing in the proposed fuzzy controller structure. The variables are the output quantities of gas compression system and capture decisive action to the dynamic evolution in the examined system. The knowledge base is composed of a database and a database of rules. For the different cases studied, the symbolic representation of the dynamics of the compression system, we have adopted consists of nine rules type If _Then.

386


The membership functions associated with various linguistic variables are represented. Each partition consists of five fuzzy sets (N, NM, Z, PM, P) is uniformly distributed over an interval normalized [-1 1]. Matrices ( A j , B j , C j , D j ), j = 1, ..., 9, are determined from the combination of the operating point considered in the rule j. The dynamic behavior was studied for various cases, varying the Greitzer stability parameter B for a variation of the flow and then to a pressure variation (load factor and the mass flow coefficient). The figures 4, 5 illustrate the fuzzy model of the mass flow coefficient for B = 1.25 and B = 1. Comparison Comparison 0.35 0

0.3

-0.05

Fuzzy model Linearized model Complex model

0.25

Débit d'as pération

M a ss flow co effic ien t

Débit en P om page

Mass flow coefficient

-0.1

-0.15

-0.2

-0.25

0.2

Fuzzy model Linearized model Complex model

0.15

0.1

-0.3

0.05 -0.35 0

1.6

3.2

time(s)

4.8

6.4

8

0 0

1.6

3.2

4.8

6.4

8

time(s)

Fig. 4 Fuzzy model of the mass flow coeffi- Fig. 5 Fuzzy model of the mass flow coefcient for B = 1.25 ficient for B = 1

From the obtund results it is clear that the proposed fuzzy model is more accurate than the linearized model. In the various cases considered, the characteristics of the compression system described by the model complex are perfectly reproduced by the proposed fuzzy model. To validate the proposed model, we used a fuzzy controller that allows distributed seven fuzzy sets on the universe of discourse of each input variable and output (NG, NM, NP, EZ, PP, PM, PG). Increasing the number of fuzzy sets to seven (NG, NM, NP, EZ, PP, PM, PG) requires the processing of 49 rules, so that the distribution of these sets on the universe of discourse of each variable becomes a choice. As regards the matrix inference, the sets of the output variable (increment) are always derived from the Mac-Vicar matrix. By analyzing the responses of fuzzy controllers we see the smooth continuation of fuzzy controllers to variations of the operating. The figure 6 shows the evolution of load factor, we see that it is properly oriented and stabilizes reference and the figure 7 shows the response of proposed fuzzy controller applied to the load factor.


Pression B =d’aspiration 1.25 / f

= 1 d’aspiration Résidu sur la B pression 0

1

45

-0.1

40

factorflou sortieLoad du régulateur 2.5

2

-0.2 35

387

1.5

-0.3 1

30

0

100

200

300

-0.4

0

100

200

300 0.5

B =d’aspiration 0.25 / f Pression

B = 0.50

Résidu sur la pression d’aspiration -0.1

2

45

0

-0.2

40

-0.5

-0.3 35

30

-1

-0.4

0

100

200

300

-0.5

0

100

200

300

-1.5 0

200

400

600 800 temps Time

1000

1200

1400

Fig. 6 Evolution of load factor in the Fig. 7 Response of proposed fuzzy controller examined system

The increase of the fuzzy sets increasing linguistic rules, therefore the computation time, the fuzzy controller with five sets in this investigated case represents a good choice that satisfies the tradeoff between the computation time and the requirement of performance specifications.

4

Conclusion

The obtund results confirm that the proposed fuzzy model is more accurate than the linearized model. In the various cases considered, the characteristics of the compression system described by the model complex are perfectly reproduced by the proposed fuzzy model. Also, the fuzzy controller used appears to have multiple benefits. It does not require a mathematical programming is difficult and easy. It should be noted that it requires a relatively slow computation time, while traditional algorithms often result in very short programs. It may seem at first glance that the fuzzy controller requires very little knowledge of the system to settle. The controller is very sensitive to variations of system parameters, so its rugged good looks. In reality should not conceal a certain number of disadvantages of this type of controller.

References Adamy, J., Schwung, A.: Qualitative modeling of dynamical systems employing continuous-time recurrent fuzzy systems. Fuzzy Sets and Systems 161(23), 3026–3043 (2010)

388


Fiordaliso, A.: A nonlinear forecasts combination method based on Takagi–Sugeno fuzzy systems. International Journal of Forecasting 14(3), 367–379 (1998) Esogbue, A.O.: Fuzzy sets modeling and optimization for disaster control systems planning. Fuzzy Sets and Systems 81(1), 169–183 (1996) Bøhagen, B., Gravdahl, J.T.: Active surge control of compression system using drive torque. Automatica 44(4), 1135–1140 (2008) Bortolet, P., Merlet, E., Boverie, S.: Fuzzy modeling and control of an engine air inlet with exhaust gas recirculation. Control Engineering Practice 7(10), 1269–1277 (1999) Wong, C.-C., Lin, N.-S.: Rule extraction for fuzzy modeling. Fuzzy Sets and Systems 88(1), 23–30 (1997) Fink, et al.: Surge dynamics in free-spool centrifugal compressor system. Journal of Turbomachinery 114, 321–332 (1992) Panoutsos, G., Mahfouf, M.: A neural-fuzzy modelling framework based on granular computing: Concepts and applications. Fuzzy Sets and Systems 161(21), 2808–2830 (2010) Greitzer, E.M.: Surge and rotating stall in axial flow compressors, part I: Theoretical compression system model. Journal of Engineering for Power 98, 190–198 (1976) Greitzer, E.M., Moore, F.K.: A Theory of Post-Stall Transients in Axial Compression Systems: Part II – Applications. American Society of Mechanical Engineers, paper 85-GT172 (1985) Ahmed, H., Daoudi, A., Guemana, M.: SCADA for Surge Control: Using a SCADA network to handle surge control in gas suppression systems in pipelines. Control Global | Process Automation Technologies Journal, ISA Transactions 24(3), 69–71 (2011) Ahmed, H., Daoudi, A., Laroussi, K.: Application of fuzzy diagnosis in fault detection and isolation to the compression system protection. Control and Intelligent Systems 39(3), 151–158 (2011) Ahmed, H., Laaouad, F., Laroussi, K.: A Numerical structural approach to surge detection and isolation in compression systems using fuzzy logic controller. International Journal of Control, Automation, and Systems, IJCAS 9(1), 69–79 (2011) Ahmed, H., Daoudi, A., Laroussi, K.: Modelling and control of surge in centrifugal compression based on fuzzy rule system. Studies in Informatics and Control (SIC) 19(4), 347–356 (2010) Ahmed, H., Laroussi, K., Laaouad, F.: Robust fuzzy fault detection and isolation approach applied to surge in centrifugal compressor modeling and control. Fuzzy Information and Engineering Journal 02(01), 49–73 (2010) Ahmed, H., Laaouad, F., Laroussi, K.: Fuzzy logic approach applied to the surge detection and isolation in centrifugal compressor. Automatic Control and Computer Sciences 44(01), 53–59 (2010) Ahmed, H., Guemana, M., Daoudi, A.: Spectral analysis approach applied to the vibrations detection and isolation in centrifugal pump used in petroleum industry. In: 5th International Congress on Technical Diagnostics, Didactics Center, AGH Univeristy of Science and Technology, Kraków, Poland, September 3-5 (2012) Hafaifa, A., Laaouad, F., Laroussi, K.: Fuzzy modeling and control for detection and isolation of surge in industrial centrifugal compressors. Automatic Control Journal of the University of Belgrade 19(01), 19–26 (2009) Gravdahl, J.T., Egeland, O., Vatland, S.O.: Drive torque actuation in active surge control of centrifugal compressors. Automatica 38(11), 1881–1893 (2002) Guelton, K., Bouarar, T., Manamanni, N.: Robust dynamic output feedback fuzzy Lyapunov stabilization of Takagi–Sugeno systems—A descriptor redundancy approach. Fuzzy Sets and Systems 160(19), 2796–2811 (2009)


389

Wang, L.-X.: Modeling and control of hierarchical systems with fuzzy systems. Automatica 33(6), 1041–1053 (1997) Mei, F., Man, Z., Nguyen, T.: Fuzzy modelling and tracking control of nonlinear systems. Mathematical and Computer Modelling 33(6-7), 759–770 (2001) Bernal, M., Guerra, T.M., Kruszewski, A.: A membership-function-dependent approach for stability analysis and controller synthesis of Takagi–Sugeno models. Fuzzy Sets and Systems 160(19), 2776–2795 (2009) Guemana, M., Aissani, S., Hafaifa, A.: Use a new calibration method for gas pipelines: An advanced method improves calibrating orifice flowmeters while reducing maintenance costs. Hydrocarbon Processing Journal 90(8), 63–68 (2011) Guemana, M., Aissani, S., Hafaifa, A.: Flow measurement and control in gas pipeline system using intelligent sonic nozzle sensor. Studies in Informatics and Control (SIC) 20(2), 85–96 (2011) Sugeno, M., Kang, G.T.: Fuzzy modelling and control of multilayer incinerator. Fuzzy Sets and Systems 18(3), 329–345 (1986)

Burst Test and J-Integral Crack Growth Criterion in High Density Poly-Ethylene Pipe Subjected to Internal Pressure Mohamed Amine Guidara1,2, Mohamed Ali Bouaziz1,2, Christian Schmitt2, Julien Capelle2, Ezzeddine Hadj-Taïeb1, Zitouni Azari2, and Said Hariri3 1

Laboratoire de Mécanique des Fluides Appliqués, Génie des Procédés et Environnement, Ecole Nationale d’Ingénieurs de Sfax, Université de Sfax, BP 1173, 3038, Sfax, Tunisie 2 Laboratoire de Biomécanique, Polymères et Structures (LaBPS), Ecole Nationale d’Ingénieurs de Metz, 57070 Metz, France 3 Département Technologie des Polymères et Composites and Ingénierie Mécanique, Ecole des Mines de Douai, 59508 Douai, France [email protected], [email protected]

Abstract. In the present work we are interested on the analysis of the severity of crack defects created by a disc cutter and to study the behavior of a high density polyethylene pipe (HDPE pipe) when subjected to an internal pressure, either in the absence or presence of a pre-crack. In order to do this, experimental tests was performed to measure the toughness and to determine the mechanical behavior of HDPE. These features were used to perform numerical simulations using ABAQUS on pipe solicited by an increase in internal pressure. This allows to compare with burst tests of cracked pipes and to determine the fracture energy that will be compared to the toughness. The results show that the crack is initiated in the radial direction. And the size of the crack has a great influence on the energy of rupture and consequently on the ultimate pressure. There is a good agreement between experimental and numerical results. Keywords: J integral, burst test, FEM model, crack propagation, HDPE.

1

Introduction

High density polyethylene (HDPE) pipe provides the lowest life cycle cost when compared to other systems due to significantly reduced or no leakage, increased billable dollars, water conservation, fewer new water-treatment plants, reduced maintenance crews, reduced seasonal water-main breaks, and its viscoelastic behavior makes it a good resistant to water hammer (CSIRO 2008).


391

392

M.A. Guidara et al.

In the present study, two part investigation will be made for HDPE pipes, with superficial defect, using the well-known approach “J-integral” of the fracture mechanics (Rice 1968). In the first part, experimental tests will be made to determine mechanical behavior (stress-strain curve) and failure analysis (JIc) of high density polyethylene. In the next part, FEM cracked tube model was performed on ABAQUS for calculating J-integral around the crack type. Placed into the J integral curve JIc allow to know the internal pressure which leads to the crack initiation. To validate FEM Model, experimental burst tests are performed on cracked HDPE pipes. The measurement results will be presented as a curve of pressure versus volume of the added water.

2

Mechanical Behavior and Failure Analysis of High Density Polyethylene

2.1

Tensile Testing

Tensile test were conducted on a standard traction-compression machine piloted in displacement mode using an extensimeter to measure the deformation of the -3 -3 specimen were carried out at three different test strain rates ( ε ) of 10 , 5.10 and -2 -1 10 s . Stress- strain curves for each strain rate are represented in figure1. 30

V=0.001(1/s)

25

V=0.005(1/s) V=0.01(1/s)

Stress (MPa)

20

15

10

5

0 0

100

200

300

400

Strain (%)

Fig. 1 Stress-strain curves for the HDPE

500

600

700

Burst Test and J-Integral Crack Growth Criterion

2.2

393

Determination of Fracture Toughness JIC

Several methods have been developed specifically for determining the fracture toughness of polymeric materials. ASTM D 6068 describes a method for measuring J-R curves (a measure of elastic-plastic fracture toughness) for polymer specimens that are not large enough to experience conditions of plane strain during loading. Standard compact tension (CT) specimen according to the recommendations of ASTM designation D6068 is used (figure 2). All tests were carried out at room temperature using partial unloading compliance method on a servo-hydraulic testing machine. The tests followed the guide-lines of the ASTM D 6068 standard for the fracture toughness determination.

W

= 41.7

±0.2

width

L

= 52.5

±0.2

overall length

(1,25 w ± 0,01 w)

transverse width

(1,2 w ± 0,01 w)

±0.2

d1

= 50

R

= 5+0.1

Radius

(0,125 w ± 0,005 w)

b

= 8+0.1

Thickness

(0,4 w < b < 0,6 w or b < w/2)

a

= 18

d2

= 11.5

±0.1 ±0.1

(0,45 w < a < 0,55 w)

crack length distance

between

the

center

of

two

holes,

(0,55 w ± 0,005 w)

balanced/Plans crack ± 0.005w

Fig. 2 CT specimen

The crack growth was followed by an optical microscope on specimen cracked surfaces to calculate the average physical crack extension Δap. The energy required to extend the crack, U, is used to calculate J. The total energy, UT, determined from the area under the load versus load-point displacement curve obtained for each specimen is the sum of U and Ui, the indentation energy. And J is calculated according to the following relation:

J=

ηU

b(W − a0 )

(1)

Figure 3 shows a representative J-Δa curve obtained from the 8 mm thick CT specimen. The intersection point between the regression line and 0.2 mm offset line gives a candidate value JQ which becomes JIC provided that the validity requirements are satisfied. In the case of HDPE PE100, JIc=7.69 KJ/m².

394

M.A. Guidara et al. 100 90 80 70

J (KJ/m 2 )

60

data points 50

regression line

40 30 20 10

J0.2=7.69 0 0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

2,2

2,4

2,6

2,8

3

3,2

3,4

3,6

Cra ck extention Δa (mm)

Fig. 3 Fracture Resistance vs Crack Growth

3

Burst Tests

3.1

Experimental Study

After the experimental determination of HDPE’s fracture toughness, the experimental investigation on this material was carried on by burst testing. The specimens for these tests were a 400 mm length of pipe section with different precracks depths (figure 4). These pre-cracks are machined by disc cutter that has a plate thickness of 0.5 mm and an outer diameter of 80 mm. The depth of the precrack (a) is due to the disc cutters penetration depth into the pipe wall.

2 Fig. 4 Pre-crack geometry

a (pre-crack depth in mm) 4 6

8


395

All tests were performed on a sample size of 3 specimens per design type of the same age and service history (new pipes). And these tests were made according to the following instructions - Setting up the specimen and fill it with water until completely filled and bled from air. - All specimens have been loaded with a continuous pressure rate of 10 bar per minute until failure (figures 5a and 5b). 60

Pressure (bar)

50 40 30 20 10 0 0

1

2

3

4

5

6

7

Time (min)

(a)

(b)

Fig. 5 Burt test

Unlike metallic structure, the use of strain gauge on HDPE pipes to obtain the strain-internal pressure curve needs a complicated and delicate pre-treatment and results will not accurate enough. So a more reliable solution should be adopted The measurement results will be presented as a curve of pressure versus volume of water added. Comparison between experimental data and numerical simulation results will be made to validate the FEM model.

3.2

Finite Element Modeling

Finite element simulation was performed using the code Warp3D (non-mechanical linear Out). The numerical simulation of crack propagation in three dimensions is known tricky for reasons related to the mesh near the pre-crack, especially when it comes to polymer materials. Figure 6 shows a three dimensional representation (3-D FE) of the finite element mesh. Taking into account of the symmetry of loading and geometry, only a quarter of the model is studied in order to reduce the computation time. The mesh was created using HyperMesh in three dimensions with quadratic elements C3D20. It has been sufficiently refined in the vicinity of crack with special elements to increase the accuracy in local area and predict correctly the fracture energy into the crack. Overall, the FE model contains 150000 elements and 600000 nodes. The calculations are made with the geometrically nonlinear ABAQUS software (large displacements) by taking into account of the variation of the strain rate.

396

M.A. Guidara et al.

Fig. 6 3D Crack Mesh

Thermoplastic materials such as PE 100 have a viscoelastic behavior and a high degree of ductility. Normally, the fracture behavior (the initiation and propagation of cracks) must be analyzed using the concept of viscoelastic-plastic fracture mechanics (Brostow et al. 1991) and (Favier et al. 2002). However, it has been shown (Benhamena et al. 2010, 2011) that elastic–plastic approaches can approximate with acceptable accuracy the critical J integral of the polyethylene.

3.3


The figure 7 shows the variation of the internal pressure as a function of volume of water added to each specimen during burst test. The ultimate pressure is greatly affected by the size of the initial crack. 60

Without crack a=2mm a=4mm a=6mm a=8mm

50

Pressure (bar)

40

30

20

10

0 0,0

0,2

0,4

0,6

0,8

Added volume(l)

Fig. 7 Internal pressure vs Added volume (Burst test)

1,0

1,2

1,4


397

58 56

55,44

Burst pressure (bar)

54

53,8 52,46

52 50 48

46,8 46 44

43,1 42 40 0

1

2

3

4

5

6

7

8

9

pre-crack depth (mm)

Fig. 8 Burst pressure variation with pre-crack depth

Burst pressures for different cracked pipes are deduced from figure 7 and represented on figure 8 as a function of the pre-crack’s depth. These results show a sharp drop in the burst pressure when the depth of crack is more than 4mm. The figure 9 shows the evolution of the fracture energy in different contours in the normal plane to the crack front. They converge to the same value. The curve of the fracture energy versus pressure and the value of the critical 2 failure energy (JIC=7.69KJ/m ) allow to determine the internal pressure (Pa=46bar) which lead to the crack initiation (figure 9). It is less than the ultimate pressure (Pu=52.5 bar) (figure 10). 14

contour2 contour3

fracture resistance J (KJ/m2)

12

contour4 contour5

10

contour6 contour7

8 JIC=7.69 6 4 2 0 -2 0

10

20

30

40

Pa=46

Pressure (bar)

Fig. 9 Variation of the fracture toughness J versus Pressure (a=4mm)

50

60

398

M.A. Guidara et al.

In the numerical application, the behavior of PE 100 was considered elasticplastic and the damage criterion is not studied. This explains the error found between the experimental and numerical calculations (figure 10). For any pressure, the volume of water added during the burst test is greater than that found experimentally. It can also be caused by experimental errors (measurement, leak ...). 60 Pu=52.2 50 Pa=46 Numerical test

Pressure (bar)

40

Experimental test

30

20

10

0 0

0,05

0,1

0,15

0,2 0,25 Added Volume(l)

0,3

0,35

0,4

Fig. 10 Pressure vs Added volume (a=4mm)

In the case of a cracked pipe with a 4 mm crack’s depth, the ultimate pressure and maximum admissible pressure are calculated from the burst test using fracture toughness JIC experimentally obtained. The result is quite logic since Pa < Pu. This proves the accuracy of the value of JIC. Numerical calculation of admissible pressure for the other cracked pipe will be made soon.

4

Conclusion and Perspectives

In this study, the mechanical propriety of high density polyethylene (PE100) pipe such as behavior law and fracture toughness JIC were experimentally obtained. These parameters were used to study the effect of the crack’s depth on the ultimate pressure of a PE100 pipe.


399

Results show that the crack is initiated in the radial direction. And the size of the crack has a great influence on the energy of rupture and consequently on the ultimate pressure. There is a good agreement between experimental and numerical results. In the upcoming study some improvement will be made like: -

Use a concept of visco elastic-plastic fracture mechanics to analyze the fraction behavior Introduce a damage criterion in the numerical simulation Perform a dynamic testing to study the effect of transient flow in cracked pipes

References CSIRO, Life Cycle Analysis of Water Networks. presented at Plastics Pipe XIV, Budapest (2008) Rice, J.R.: A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks. Journal of Applied Mechanics 35, 379–386 (1968) ASTM D 6068 Standard Test Method for Determining J-R Curves of Plastic Materials WARP3D. Release 15.0 manual, civil engineering, reports no UILU-ENG-95-2012. Urbana, University of Illinois (2004) Benhamena, A., Bouiadjra, B., Amrouche, A., Mesmacque, G., Benseddiq, N., Benguediab, M.: Three finite element analysis of semi-elliptical crack in high density poly-ethylene pipe subjected to internal pressure. Materials and Design 31, 3038–3043 (2010) Benhamena, A., Aminallah, L., Bouiadjra, B., Benguediab, M., Amrouche, A., Benseddiq, N.: J integral solution for semi-elliptical surface crack in high density poly-ethylene pipe under bending. Materials and Design 32, 2561–2569 (2011) Favier, V., Giroud, T., Strijko, E., Hiver, J.M., G’Sell, C., Hellinckx, S., et al.: Slow crack propagation in polyethylene under fatigue at controlled stress intensity. Polymer 43, 1375–1382 (2002) Brostow, W., Fleissner, M., Müller, W.: Slow crack propagation in polyethylene: determination and prediction. Polymer 3, 419–425 (1999)

Solving the Three-Dimensional Time-Harmonic Maxwell Equations by Discontinuous Galerkin Methods Coupled to an Integral Representation Nabil Gmati1, Stéphane Lanteri2, and Anis Mohamed3 1

National Engineering School of Tunis, ENIT-LAMSIN BP 37, 1002 Tunis, LR 95–ES–20 Tunisia [email protected] 2 INRIA Sophia Antipolis Méditerranée 2004 route des lucioles-BP 93 06902, Sophia Antipolis Nice cedex France [email protected] 3 National Engineering School of Tunis, ENIT-LAMSIN BP 37, 1002 Tunis, LR 95–ES–20 Tunisia [email protected]

Abstract. In this paper, we present a numerical study of three-dimensional timeharmonic Maxwell equations. We use a finite element discontinuous Galerkin method coupled with an integral representation. This study was completed by several numerical examples to test the efficiency of the proposed approach. The numerical simulation was down by an iterative solver implemented in FORTRAN. Keywords: time-harmonic, maxwell equations, integral representation, discontinuous galerkin method.

1

Introduction

Electromagnetic phenomena are generally described by the electric and magnetic fields E and H which are related by the following Maxwell equations:

-ε∂ t E + curl H = 0 -μ∂ t H + curl E = 0

(1) (2)

Where ε and μ are the complex-valued relative dielectric permittivity and the relative magnetic permeability, respectively. We are interested in time-harmonic solutions which satisfy the following equations: curl E - iωμ H =0

(3)

curl H - iωε E =0

(4)

These two equations are posed in an unlimited domain. Or approach consists of limiting the obstacle D by a fictitious boundary Γ a where we impose the absorbent of Silver-Muller condition. On the boundary Γ m of D, we take the perfect conductor condition.


401

402

N. Gmati, S. Lanteri, and A. Mohamed

Fig. 1 Diffraction of an electromagnetic wave in the presence of an obstacle D

2

Formulation of the Problem

Our problem is the following: Find E,H such that: curl E - iωμ H =0 curl H - iωε E =0 E ∧ n = − E inc ∧ n

In Ω

(5)

In Ω On Γ m

(6) (7)

E ∧ n − Z ( H ∧ n ) ∧ n = ℜ( E ) ∧ n − Z ( ℜ( H ) ∧ n ) ∧ n

On

Γa

(8)

Where: • ω : The angular frequency of the problem, n is a normal vector to the boundary Γ a .

μ . ε • ℜ( E ) , ℜ( H ) are the expression of the electric and magnetic fields E and H , respectively, on Γ a . They are given by the following integral representation: 1 ℜ( E ) = curl x ∫ n( y ) ∧ E ( y )G ( x, y )∂σ y − curlx curlx ∫ n( y ) ∧ H ( y )G ( x, y )∂σ y Γa Γa iωμ •

Z=

ℜ( H ) = curlx ∫ n( y ) ∧ H ( y )G ( x, y )∂σ y − Γa

Let ( e1 , e2 , e3 ) fined as follows:

1

curlx curlx ∫ n( y ) ∧ E ( y )G ( x, y )∂σ y Γa iωε 3 be the canonical basis of \ . We consider the matrix Gl de-

Solving the Three-Dimensional Time-Harmonic Maxwell Equations

403

⎛ 03×3 N el ⎞ Gl = ⎜ t , for l ∈ {1, 2,3} , ⎜ N e 03×3 ⎟⎟ ⎝ l ⎠ ⎛ v1 ⎞ ⎜ ⎟ where for a given vector v = ⎜ v2 ⎟ , the matrix N is: ⎜v ⎟ ⎝ 3⎠ v3 −v2 ⎞ ⎛ 0 ⎜ ⎟ N v = ⎜ −v3 0 v1 ⎟ . ⎜v ⎟ ⎝ 2 −v1 0 ⎠ ⎛E ⎞ It is easy to verify that Gn = G1n1 + G2 n2 + G3 n3 . Let W = ⎜ ⎟ , therefore the ⎝H ⎠ system of equations (5), (6), (7) and (8) can be rewritten in the following conservative form: iωQW + G1∂ xW + G2 ∂ yW + G3 ∂ zW = 0

In Ω

(9)

( M Γm − Gn )(W − W inc ) = 0

On Γ m

(10)

( M Γa − Gn )(W − ℜ(W )) = 0

On Γ a

(11)

Where: • •

⎛ ε .I 3 03×3 ⎞ ⎛ 03×3 N n ⎞ ⎛ ℜ( E ) ⎞ ℜ(W ) = ⎜ ⎟ , and M Γa = ⎜ t ⎟. ⎟, Q =⎜ ⎝ ℜ( H ) ⎠ ⎝ 03×3 μ .I 3 ⎠ ⎝ N n 03×3 ⎠ M Γa = Gn = Gn+ − Gn− Where Gn+ and Gn− are the positive and negative parts of Gn , respectively.

3

Discretization

We discretize the domain Ω into N tetrahedral cells such that: 

N

Ωh =

∪τ

K∈

K

h



P mi [τ i ] = {The space of polynomials of degree ≤ mi defined on τ i }.



Vh = W ∈ ( L2 (Ω))3 / W\τ i ∈ P mi [τ i ]

{

3

}

Wh = ⎡⎣ E1h , E2h , E3h , H1h , H 2h , H 3h ⎤⎦ the projection of W on ( P mi [τ i ])6 Also we note: i , Γm = Γ0 = K ∩K K ∩ Γ and Γ a = K ∩Γ t



∪

i ∈τ K ,K h

∪

K ∈τ h

m

∪

K ∈τ h

a

404


If we multiply the equation (5) by a test function V and we integrate on Ω h , we then get:

∫

Ωh

(\

Ωh

F (W ) = (G1W , G2W , G3W ) is

where

×\ ×\

6

(iωQWh )t .Vdx + ∫ (∇.( F (W ))h )t .Vdx = 0 ,

6

6

a

linear

application

\6

from

to

)

Using Green formula we obtain:

∫

Ωh

∑τ ( ∫ ( ( F (W ) ) ) .∇Vdx + ∫ ( F (W ) .n ) .V ∂σ ) = 0 t

(iωQWh )t .Vdx −

K∈

(12)

h

∂K

h

K

h

Using the same technics adopted by Ern and Guermond [1,2], and by adding the terms of the integral representation , we obtain the following formulation: ∀V ∈ Vh × Vh and K ∈ τ h ; Find Wh ∈ Vh × Vh such that:

∫

Ωh

+∑

F ∈Γ0

(iωQWh )t .Vdx −

∑τ ∫ ( ( F (W )) ) .∇Vdx t

K∈

h

K

h

∫ ⎡⎢⎣( S .aW b) .cdeV fgh − (G .aW b) .{V }⎤⎥⎦ ∂σ t

t

F

F

h

nF

h

t t ⎡ ⎛1 ⎤ ⎞ ⎛1 ⎞ + ∑ ⎢ ∫ ⎜ M F , K − I FK GnF Wh ⎟ .V ∂σ − ∫ ⎜ M F , K − I FK GnF ℜ (W ) ⎟ .V ∂σ ⎥ F F ⎠ ⎝2 ⎠ ⎢ ⎝2 F ∈Γa ⎣ ⎦⎥

(

)

(

)

t

t

⎛1 ⎞ ⎛1 ⎞ + ∑ ∫ ⎜ M F , K − I FK GnF Wh ⎟ .V ∂σ = ∑ ∫ ⎜ M F , K − I FK GnF Whinc ⎟ .V ∂σ F 2 F 2 m m ⎝ ⎠ ⎝ ⎠ F ∈Γ F ∈Γ Where: ℜ (W )( x ) = ∫ m K ( x, y ) W ( y )dy such that K : \ 6 × \ 6 → M 6 ( ^ ) is a Green

(

)

(

)

Γ

kernel. We also define the jump and average of a vector V ∈ Vh × Vh , i: respectively, on the face F shared with two elements K and K

(

)

1 V\ + V\ Ki such as I FK represents the 2 K incidence matrix between oriented faces and elements and they are given by:

aV b = I FK .V\

K

+ I F Ki .V\ Ki and

{V } =

⎧1 if F ∈ K and their orientations match, ⎪ I FK = ⎨-1 if F ∈ K and their orientations do not match, ⎪0 if F ∉ K . ⎩

We mention here that for the terms on the boundary, we used two classical numerical flux.


3.1

405

Centered Flux

In this case, for all the faces S F = 0 and for the boundary faces, we have:

M F ,K

⎧ ⎛ 03×3 N nF ⎞ m ⎪⎪ I FK ⎜ ⎟ if F ∈ Γ t = ⎨ ⎜⎝ − N nF 03×3 ⎟⎠ ⎪ a ⎪⎩ GnF if F ∈ Γ

3.2

An Upwind Flux

In this case: ⎛ α N nF N SF = ⎜ ⎜ 03×3 ⎝ E F

⎞ ⎟ , M F ,K H t α F N nF N nF ⎟⎠

t nF

03×3

⎧ ⎛η F N nF N nt F I FK N nF ⎪⎪ I FK ⎜⎜ I N t 03×3 =⎨ ⎝ FK nF ⎪ a ⎪⎩ GnF if F ∈ Γ

For homogeneous media, η F = α FE = α FH =

4

⎞ ⎟ if F ∈ Γ m ⎟ ⎠

1 2

Linear System

The problem is reduced to the linear algebraic system:

( A − C ) .X

= b such that

A and C are square size matrices N = 6 × di × N c and where: • d i : number of degrees of freedom. • N c : number of cells. A is a block defined matrix of size 6.d i × 6.d i such that:



For i, j =1,… ,N c ; A ( i , j ) ∈ M 6.di ( ^ ) •

A ( i , i ) = Di1 − Di2 + DiΓ × δ ij + DiΓ × δ Γm + DiΓ × δ Γa

•

A ( j , i ) = B × δ ij

0

m

a

i

i

Γ0 ij

δij : Kronecker symbol C is a block defined matrix of size 6.di × 6.d j such that:



For i, j =1,… ,N c ; C ( i , j ) ∈ M 6.di ×6.d j (  ) •

C ( i , j) = −Cij × δ Γa × δ Γm i

1 i

2 i

Γ0 i

Where D , D , D extra-diagonal block.

j

Γm i

, D

and DiΓ

a

0

are diagonal blocks, BijΓ defines the

406


Cij The Coupling matrix defined as follows: Cij =

5

1 ( I3 ⊗ Ψ ia ) .Ki ij .( I 3 ⊗ Ψ im ) 2

Numerical Results

The numerical implementation, we used the FORTRAN code developed by Hugo Fol & Stéphane Lanteri with a simple modification based on the Bi-Conjugate Gradient method. We have illustrated the numerical solutions obtained with a transparent provided CT (Discontinuous Galerkin coupled to an integral representation) and the exact solution in the following figures: (in these two cases, the inner radius is equal to 1) We choose in these tests a single layer enter the boundary Γ m and the fictitious boundary Γ a . 

Test No. 1

Table 1 The first test data Parameters

Value

Outer radius

1.18

Frequency(Ghz) 0.3 H max

0.30

Number of cells 3119

Fig. 2 The contour lines of the computed solution


Fig. 3 The contour lines of the Exact Solution



Test No. 2

Table 2 The second test data Parameters

Value

Outer radius

1.1

Frequency(Ghz) 0.3 H max

0.16

Number of cells 10512

Fig. 4 The contour lines of the computed solution

407

408


Fig. 5 The contour lines of the Exact Solution

According to figures we can see that this method leads to a solution which is very similar to the exact one. Acknowledgements. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.

References [1] Ern, A., Guermond, J.-L.: Discontinuous Galerkin methods for Friedrichs systems I. General theory. SIAM J. Numer. Anal. 44(2) (2006) [2] Ern, A., Guermond, J.-L.: Discontinuous Galerkin methods for Friedrichs systems II. Second-order elliptic PDEâs. SIAM J. Numer. Anal. 44(6) (2006) [3] El Bouajaji, M., Gmati, N., Lanteri, S., Salhi, J.: Coupling of an exact transparent boundary condition with a DG method for the solution of the time-harmonic Maxwell equations, vol. 95, pp. 237–247. Springer (2014) [4] Ben Belgacem, F., Fournie, M., Gmati, N., Jelassi, F.: On the Schwartz algorithms for the elliptic exterior boundary value problems. ESAIM. Math. Model. Numer. Anal. 39, 639–714 (2005) [5] Colton, D., Kress, R.: Inverse acoustic and electromagnetic scattering theory. Springer (June 1997)

Iterative Methods for Steady State Looped Network Analysis Zahreddine Hafsi, Sami Elaoud, Mohsen Akrout, and Ezzeddine Hadj-Taïeb Laboratory of Applied Fluids Mechanics Process and Environment Engineering, ENIS P.O. Box, W, Sfax, 3038, Tunisia [email protected], [email protected]

Abstract. The aim of this paper is to numerically study the steady state of water flows in looped networks. This study will be performed by the use of Hardy Cross Method and Newton-Raphson algorithm. The comparison of the numerically obtained results by these two methods to those obtained by the use the commercial software Pipe Flow Expert has shown a good concordance between them. The numerical analysis has shown that the convergence of Newton-Raphson method is more rapid than that of the classic Hardy Cross Method. Keywords: steady flow, pipe network, Newton-Raphson method, Hardy-Cross method.

1

Introduction

Fluid transport from production sites to the exploitation sites is a major concern for manufacturers. Many studies have shown that the most effective method is to transport via apparent or underground pipelines. These pipelines, connected together, form what called a "hydraulic network" that can be either branched or looped or even adopting a hybrid topology. To analyze the steady state, hydraulic networks flows are to be balanced. For a branched network, by knowing the flow demand at each terminal, it’s easy to determine the flow rate at each conduit of the network since the flow direction is evident. The problem arises when we start facing a purely looped or hybrid network (containing loops and branches). Several methods and techniques are used to balance flows in these types of network. Between these methods, one can cite: the Hardy Cross Method (HCM), Node-Loop Method, Modified Node Method and Andrijašev Method. A comparative study between the previous techniques has been carried out (Dejan 2011). In this study, the number of iterations needed to achieve the network equilibrium is taken as criterion to put in evidence the efficiency of these methods. Dejan proved that, for complex networks, Node Loop Method is more efficient as it does not require complex numerical scheme for algebraic addition of corrections in each of iterations. © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_42

409

410

Z. Hafsi et al.

These techniques, and as are iterative methods, require an initial guess of flow rates through pipelines. This guess has to satisfy the first Kirchhoff’s law: nodes law. In many studies the initial guess is introduced manually. In this paper, Hardy Cross and Newton-Raphson methods are used to solve a water looped network. A comparative study between the two methods in terms of efficiency, elapsed time, and precision will be conducted. In this study, the initial guess imposed by these iterative methods will be generated automatically.

2

Mathematical Formulations

In an electric network, Kirchhoff’s second law reposes on relationship between nodal variable (electric potential ) and traversing variable (current ) given by Ohm’s law ( is the electrical resistance). In a hydraulic circuit the traversing variable is the flow-rate and the nodal variable is the head . Basing in theory of analogy, a relationship between and is written as follows: (1) is the hydraulic resistance. where The values to be assigned to and depend on the equation being used. In literature, for a single-phase fluid flow, several relations in the above form can be found. A common used relation is that of Darcy-Weisbach (Giles 1962; Fox 1992; Walski 2001) given by: (2) where is the pipe length, is its diameter, is the gravitational acceleration and is the friction factor. The friction factor is given by Colebrook-White equation (Giles 1962; Fox 1992; Shames 1989; White 1994): √

2

. .

where e is the internal roughness of the pipe and by the following expression:

√

(3) is the Reynolds number given

(4) where is the kinematic viscosity of the fluid. Equation (3) is an implicit form in term of by the use of an iterative process.

which can be solved numerically

Iterative Methods for Steady State Looped Network Analysis

3

411

Hardy Cross Method

The Hardy Cross method is used to balance flows through looped networks. In this method, an initial guess of flow per conduits has to be assigned, and in that way chosen values are to be used for first iteration. Then, an iterative process is conducted with correction of the flow-rate for each loop. The following assumptions must be satisfied for Hardy Cross calculation: - Algebraic sum of flows per each node must be exactly equal to zero (first Kirchhoff”s law: continuity of flow). - Algebraic sum of head losses per each loop must be approximately zero at the end of iterative procedure (second Kirchhoff’s law: continuity of potential). The loop flow correction is given by the following equation: ∑ ∑

(5)

⁄

where ∑ is the algebraic head loss sum of all pipes in the considered loop, is the flow-rate through each pipe of the loop and is depending on the used formula for head loss expression. In general, the iterative correction process is continued until the integrated head loss around all loops in the network is reduced to a specified limiting value .

4

Newton Raphson Method

The Newton Raphson method is based on the resolution of non linear equation 0 by successive iterations using the following expression:

;

0,1, …

(6)

where is an initial guess of the solution. In each step we have to verify the convergence criterion of the algorithm, for | , where is a tolerance value. example | To solve non linear system of equations 0 written under matrix form, same principle can be applied. In that case equation 6 is written:

;

5

0,1, …

(7)

Applications and Results

The following assumptions will be considered: the variation of the potential energy is neglected; one-dimensional flow is admitted; the pipe is supposed to be ) and the head loss through each pipe of the rigid (its section network is calculated using Darcy-Weisbach formula (equation 2).

412

5.1

Z. Hafsi et al.

Branched Network Study

In a branched network having pipes, the number of nodes is 1. Only nodes law is needed to analyze such network. Figure 1 and Table 1 show respectively the case study and the input data needed to simulate the network.

Fig. 1 Example of a branched network Table 1 Data describing the network topology Pipe number Upstream node Downstream node

1 1 2

2 2 3

3 3 4

4 3 5

5 2 6

6 2 7

For a branched network we have to solve a linear system formed by nodes equations. These equations can be written under the following matrix form: (8) where is commonly known as incidence matrix, represents the vector of the unknowns flow-rates in the conduits and is the consumptions (or demands) vector. The terms of the matrix are determined by respecting the following rules: 1 1

(9)

0 The dimensions of the matrix 1, the obtained matrix is:

are

. By respecting equation (9) and table


1 1 0 0 0 0 0

0 1 1 0 0 0 0

0 0 1 0 1 0 0

0 0 1 1 0 0 0

0 1 0 0 0 1 0

413

0 1 0 0 0 0 1

(10)

300 ; 0 ; 0 ; 70 ; 150 ; 0 ; 80

(11)

Vector C is written as follows:

In this vector, the minus sign preceding the first flow-rate indicates that the first node is a supply node (the node that feeds the network). In a hydraulic network, we can encounter one or more supply nodes. Respecting the mass continuity principle, the sum of vector elements has to be equal to zero. To solve the system of equations (8), matrix must be non singular. However, its first row can be written ∑

(12)

This equality has a physical signification since the first row concerns the first node (the supply node) and the other rows concern consumption nodes. To ensure that matrix is non singular, its first row is removed and subsequently the first element of the vector . By taking into account the previous considerations, the resolution of system (8) permits to determine the vector : ⁄

300 ; 220 ; 70 ; 150 ; 0 ; 80

(13)

All the described steps to solve the branched network equations, in this section, will be integrated to balance the looped network flows.

5.2

Looped Network Study

For a looped network containing pipes and nodes, the flow in each pipe should be determined. By applying Kirchhoff’s laws, independent equations are obtained. Since we have nodes, nodes equations can be written. Nevertheless, as mentioned in section 4.1, only 1 are available. It remains to find 1 other independent equations that correspond to the loops equations. As an application, we consider a looped network composed by 12 pipes and 9 nodes as shown in figure 2. Tables 2 and 3 show respectively the geometric description of the pipes and the fluid properties of the looped network shown in figure 2. Table 4 resumes the studied looped network topology.

414

Z. Hafsi et al.

Fig. 2 Example of a looped network

Fig. 3 Branched generated network

Table 2 Tubular pipes material properties: steel Sch 80 pipe (ANSI standards) Designation Internal roughness Nominal Pipe Size Internal diameter

Value

Unit

0.0459 500 455.625

Temperature Density Kinematic viscosity

5.2.1

Value

Upstream node

Downstream node

Length (m)

1

1 2

2

900

5 5

1200 900

6 3

1200 900

4 4 8

1200 900 1200

8 9

900 1200

7 7

1200 900

4 5 6 7 8

Unit

25°

°

997

/

0.89

Pipe number 2 3

Table 3 Fluid properties: water Designation

Table 4 Studied looped network characteristics

.

9 10 .

11 12

6 1 2 3 5 5 9 6 4 8

Network Balancing by the Use of Hardy Cross Method

The initial guess required by the HCM will be generated automatically. As a first step, a branched network is derived from the initial looped network. As mentioned in section 5.1 the number of pipes in a branched network is equal to 1. As consequence, the number of pipes to be removed from a looped network containing pipes and nodes is : 1 one from each loop. In our case study, the number of pipes to be removed is equal to 4. Figure 3 shows the obtained branched network. Then, an incidence matrix 1 of the eliminated pipes and 2 of the branched network are built respecting the principles of equation (9) and a vector solution for the eliminated pipes is generated. This vector can be introduced randomly or, simply, chosen as null vector. For the flow-rate vector of the branched network, we have:


415

1

(14)

2 1

2

1

(15)

By combining and , a vector of flow-rates through the looped network respecting the nodes law is obtained. This vector is the starting guess needed to apply the HCM principle developed in section 3. To apply the loops law properly, A Direction Matrix is introduced. For each pipe from each loop the initial flow-rate is multiplied by the correspondent element in (1 if the supposed flow direction is clockwise or 1 if it’s anticlockwise). For this case study, the matrix is written as follows: 1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

(16)

For pipes that belong to two loops, the correction factor to apply is where is the correction factor of the concerned loop and is that of the other one. In each iteration of HCM algorithm, the friction factor of each pipe has to be evaluated by solving equation 3 using an iterative method. Respecting the above steps, the following matrix of flow-rates through each pipe in each loop is obtained: 208.3183 110.9658 83.1999 78.9862

/

97.3525 70.9658 81.5662 49.9520

83.1999 78.9862 28.4818 30.0480

191.6817 97.3525 108.4818 81.5662

(17)

It is noticed that the elements of the flow-rates matrix have the same signs of their correspondents in the direction matrix , so, the real flow direction through each pipe is the same we had supposed. In case we have a difference of sign, the considered flow is actually in the opposite direction than we supposed. 200 Pipe 2 belonging to the loop 1 Pipe 2 belonging to the loop 2

150

Flow -rate (l/s)

100

50

0

-50

-100

-150

-200

2

4

6

8

10

12

14

16

18

20

22

Iteration number

Fig. 4 Variation of flow-rate through pipe 2 Fig. 5 Pipe Flow Expert simulation result

416

Z. Hafsi et al.

Figure 4 illustrates the correction of flow-rate through pipe 2 from the initial supposed value to the real one. This figure shows sign respect while applying the correction procedure, so that the flow-rate value of a common pipe between two loops is considered positive through one loop negative through the other. 5.2.2

Network Balancing by the Use of Newton-Raphson Method

Respecting the Newton-Raphson Method’s principle mentioned in section 4, the studied looped network is analyzed by adopting a matrix representation. The system of equations to solve is formed by independent loops and nodes equations taking in consideration the matrix. The flow rate vector Q is the unknown to evaluate. Starting from an initial guess and by fixing a predefined value for , the solution is evaluated in each iteration until verifying the | : convergence criterion | As in HCM, in each iteration, the friction factor λ of each pipe has to be calculated using an iterative process (Newton Raphson principle was used to evaluate in both algorithms) and the initial guess can be assigned using a generated branched network as detailed in section 5.2.1. The final solution is: ⁄

208; 97; 83 ; 191 ; 110 ; 70 ; 78; 81 ; 28; 108; 49; 30

(18)

The obtained vector contains no negative values. So, the real flow direction is the same initially supposed (from the upstream to the downstream node).

5.3

Comparison and Validation of Results

It’s important to mention that when conducting a comparative study between iterative methods, we have to ensure that the same convergence criterion is Table 5 Obtained flow-rates values

/

by the used methods

Pipe number

Hardy Cross

Newton-Raphson

Pipe Flow Expert

1

208.3183

208.3191

208.325

2

97.3525

97.3523

97.375

3

83.1999

83.2003

83.671

4

191.6817

191.6809

191.675

5

110.9658

110.9669

110.950

6

70.9658

70.9669

70.950

7

78.9862

78.9866

79.308

8

81.5662

81.5659

81.738

9

28.4818

28.4806

28.004

10

108.4818

108.4806

108.004

11

49.9520

49.9535

50.258

12

30.0480

30.0465

29.742


417

considered. In our case, and as mentioned above for both HCM and Newton Raphson Method, the solution is reached when the difference between two successive obtained results ( and ) is less than the chosen limiting value for HCM and for Newton Raphson Method. To evaluate the efficiency of the used methods of analysis, a simulation of the studied network was performed by the use of the commercial software Pipe Flow Expert. The obtained results by that software are shown in figure 5. Table 5 shows the flow distribution in each conduit of the network obtained by the three used methods under the same limiting value ( =10 ). The comparison of Hardy Cross and Newton-Raphson methods results show a good concordance. Added to that, our numerical results compare very favorably with those obtained by “Pipe Flow Expert” software.

5.4

Comparison of Resolution Methods in Term of Efficiency

Figure 6 illustrates the convergence of HCM and Newton-Raphson Methods. 220

400

Hardy Cross Method

Pipe 4 Pipe 6 Pipe 9 Pipe 12

350 300

Newton Raphson Method

200 180

Pipe Pipe Pipe Pipe

250

Flow-rate (l/s)

Flow-rate (l/s)

160

200 150

140

4 6 9 12

120 100 80

100

60 50

40 0

10

15

20

25

30

35

Time (s)

40

45

50

55

60

20

6

8

10

12

14

16

18

20

22

Time (s)

Fig. 6 Convergence of Hardy Cross and Newton-Raphson Methods

For 4 pipes of the network, this figure shows the variations of flow-rate during the correction process from the initial guess to the final result. The correction phase has required more than 53 seconds to achieve the actual solution for HCM and just 14 seconds for Newton-Raphson Method. Added to that the total elapsed time for the HCM is about 60 seconds while it’s about 21 seconds for the NewtonRaphson Method. This difference in elapsed time is explained by the quick correction phase for the second method. Indeed the flow rate through each pipe is evaluated once in each iteration, while in the HCM method, we have to correct the flow-rate value through each loop that means pipes belonging to two loops have to be calculated twice (in clockwise and anticlockwise directions). In term of iterations number required to achieve final result, figure 7 shows that Newton-Raphson Method converges after about 9 iterations and reaches the desired solution while HCM requires about 21 iterations to find that solution. Both used methods are efficient to analyze a looped network but in term of rapidity convergence and simplicity to implement the algorithm, Newton-Raphson Method is more adequate mainly that the elapsed time will be so important to reduce when analyzing more complex networks.

418

Z. Hafsi et al. 180

Hardy Cross Method Newton Raphson Method

160

Flow-rate (l/s)

140 120 100 80

Pipe 2 Final flow- rate: 97.35 l/s

60 40 20 0

2

4

6

8

10

12

14

16

18

20

22

Iteration number

Fig. 7 Comparison of convergence in term of iterations number

6

Conclusion

A numerical study of looped water networks is presented in this paper. The network flow equilibrium is performed by the use of Hardy Cross and NewtonRaphson methods. The obtained results of each method are compared to those of the other method and confirmed by a “Pipe Flow Expert” simulation. This study also shows that the introduction of Newton-Raphson Method to analyze such problems is very useful mainly with the evolution of computing tools as that method fit non linear problems and even in convergence efficiency it’s very favorable comparing to traditional methods like HCM.

References Cross, H.: Analysis of flow in networks of conduits or conductors (1936) Brkic, D.: Iterative Methods for Looped Network Pipeline Calculation. Springer Science+Business Media B.V. (2011) Giles, R.V.: Theory and Problems of Fluid Mechanics and Hydraulics, p. 274. Schaum Publishing Co., New York (1962) Fox, R.W., McDonald, A.T.: Introduction to Fluid Mechanics. Wiley, New York (1992) Shames, I.H.: Mechanics of Fluids, p. 692. McGraw-Hill Book Co., Singapore (1989) Walski, T.M., Chase, D.V., Savic, D.A.: Water Distribution Modeling. Haestad Press, Waterbury (2001) White, F.M.: Fluid Mechanics, p. 736. McGraw-Hill Book Co., Singapore (1994)

Investigation and Modeling of Surface Roughness of Hard Turned AISI 52100 Steel: Tool Vibration Consideration Ikhlas Meddour, Mohamed Athmane Yallese, and Hamdi Aouici Mechanics and Structures Research Laboratory (LMS), May 8th 1945 University, P.O. Box 401, Guelma 24000, Algeria [email protected]

Abstract. Since the great interest of the achieved parts quality, this experimental study focuses on the modeling and the investigation of the surface roughness of hard turned AISI 52100 steel. The tool vibration was taking into consideration beside cutting speed, depth of cut, feed rate and tool nose radius. The response surface methodology (RSM) was employed for modeling process. Models reliability was established by conducting confirmation tests. Slight divergence between the experimental and their corresponding predicted values were observed. The significance of the different factors on surface roughness was established by applying analysis of variance (ANOVA). The results revealed that the best surface roughness is obtained by using small feed rate and large nose radius. Furthermore, a correlation between the surface roughness and tool vibrations was established. Keywords: Hard turning, ANOVA, RMS, Surface roughness, tool vibration.

1

Introduction

A numerous researches were carried out to study the effect of cutting conditions on surface roughness. (Yallese et al. 2005) examined the effect of cutting parameters, i.e. cutting speed, depth of cut, feed rate on surface roughness in hard machining of X200Cr12 steel. The results showed that large feed rate deteriorates surface roughness while depth of cut did not show a remarkable effect. So, they advised to machine with a large depth of cut in order to increase the material removal rate. The mixed ceramic tool was used by (Aslan et al. 2007) in their experimental investigation regarding the hard turning of AISI 4140 steel. They indicated that the surface roughness was mainly affected by cutting speed-feed rate and feed rate-depth of cut interactions. They modeled the surface roughness by multiple linear regressions. (Deruja et al. 2009) observed the best surface roughness at a small feed rate and high cutting speed in hard turning of AISI H11 steel with coated mixed ceramic inserts. The effect of cutting parameters on the © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_43

419

420

I. Meddour, M.A. Yallese, and H. Aouici

surface roughness was also investigated by (Lalwani et al. 2008) in finish hard turning of MDN250 steel using coated ceramic tool. The employment of ANOVA demonstrated that the feed rate is the prevailing parameter whereas cutting speed had no significant effect on surface roughness. In hard turning of AISI 52100 steel with CBN tool, (Bartarya and Choudhuryb 2012) used the multiple linear regression to develop a surface roughness model prediction based on cutting parameters. Other experimental studies, as the work of (Horng et al. 2008), considered the tool nose radius as variable along with the cutting parameters. The ANOVA revealed that the cutting speed and the tool nose radius influenced significantly the surface roughness of Hadfield steel hard turned with mixed ceramic insert. Also they linked the inspected increase of surface roughness when using a large tool nose radius to the excessive thermal crack and flank wear. In their point of view it is due to the enlargement of the contact area. The same number of variables was also adopted by (Saini S et al. 2012) in the case of hard turning of AISI H11 steel with ceramic tool, the ANOVA results showed that the feed rate influences the surface roughness most significantly followed by cutting speed and nose radius. Thus, according to the author the best surface quality was attainable by decreasing feed rate and increasing cutting speed. In hard turning AISI H11 with CBN tool, (Aouici et al. 2012) observed that feed rate had a statistical significance on the surface roughness, because its increase generates helicoids furrows the result of tool shape and helicoids movement tool-workpiece. They developed mathematical models for surface roughness by using response surface methodology (RSM). Other experimental researches were interested by the influence of the vibration generated during the process on the machined products. This vibration could result in the instability of the system, known as chattering, if an inadequate selection of the cutting parameters is done. (Tlusty 1999), consider that this phenomenon occurs when the uncut chip thickness is great regarding the dynamic stiffness of the system. (Thomas et al. 1996) related the difference between the measured and the theoretical surface roughness to the vibrations. Based on this concept, (Jang et al. 1996) developed a real-time monitoring algorithm to calculate the surface roughness. The algorithm superpose the theoretical profile and the oscillatory profile determined by the relative vibration between the cutting edge and the workpiece. The radial acceleration of the tool was taking into account beside the cutting parameters by (K.A Risbood et al. 2003), when they developed a neural net work models to predict surface roughness and dimensional deviation. (Kirby et al. 2004) developed a prediction surface roughness model based on feed rate and the measured vibration in three directions. (Abouelatta and Madl 2001) disclosed that the mathematical models of surface roughness including both machining conditions and tool vibrations were more efficient than those depending on machining conditions only. (Hessainia et al 2013) correlated between the surface roughness and tool nose displacement. They developed roughness models containing all cutting parameters, tool-nose displacements, spindle and machine tool frame.

Investigation and Modeling of Surface Roughness of Hard Turned AISI 52100 Steel

421

The aim of this work is the investigation of the effects of cutting conditions (cutting speed, depth of cut, feed rate and tool nose radius) and tool vibration on the surface roughness of hard turned AISI 52100 steel. This, by applying the analysis of variance (ANOVA). Also, the surface roughness is modeled by employing the response surface methodology (RSM), first by using only cutting conditions, then, the tool vibration is included in the model and both models are compared.

2

Experimental Equipments and Methods

2.1

Equipments

Figure 1 shows the experimental procedure and the different equipments used in this study. The experimentation was performed under dry conditions using AISI 52100 steel as workpiece material. The hardness was raised by quenching and tempering treatment followed by checking measurement with a digital Micron Hardness Tester DM2-D390. The average of measured values was 59 HRC. The workpiece is mounted on a universal lathe, model SN40C, spindle power 6.6 KW. The ceramic insert tool is a mix of 70% Al2O3 and 30% TiC. This type of inserts is commonly called CC650, and its ISO designation is SNGN 1204(r) T01020 where

Fig. 1 Experimental procedure

422


(r) is the tool nose radius ranged from 0.8 to 1.6 mm in this work, indicating a 20° chamfer angle over 0.1mm width. Tool holders are codified as PSBNR2525M12 with a common active part tool geometry described by: cutting edge angle χr =+75°, clearance angle α =+6°, rake angle γ =-6° and cutting edge inclination angle λ =-6°. New cutting edge is used for each test in order to reduce the wear influence on measured responses. The vibrometer (SmartBalancer2) is used for the measurement of the acceleration amplitude of the tool vibration in both tangential and radial directions. The sampling frequency is 12800 Hz. Regarding the measurement of the surface roughness, a Mitutoyo Surftest–201 (cut-off = 0.8) is applied after each test in three different positions by going around the workpiece each 120°. Afterward, the average of the three measurements is estimated. In addition, the three-dimensional topographic maps of the machined surfaces are produced using the optical platform of metrology modular Altisurf 500.

2.2

Plan of Experiments

Four cutting parameters are investigated in this study, which are: cutting speed (Vc), feed rate (f), cutting depth (ap) and tool noise radius (r). In order to minimize the cost and the time of the experiments, the parameters are varied according to a central composite design CCD, in particular the commonly called face-centred. It counts thirty runs composed of sixteen factorial points, six center points, and eight axial points located at cube faces which mean α = 1. Thus, each factor is varied at three levels which are presented in Table 1. The responses are: acceleration amplitude in radial direction (AMPy), acceleration amplitude in tangential direction (AMPz), and the arithmetic mean of roughness (Ra). The principal of the RMS is to model the experimental data. The central composite design allows the evaluation of the linear effects, and also to the eventual quadratic interaction effects between the different variables. We used the quadratic model with interactions as follow: Y = a0 +

∑

k bX i =1 i i

+

∑

k b X Xj i , j ij i

+

∑

k b X2 i =1 ii i

(1)

where i, j, k =1, 2... n a0, bi, bij and bii are the regression coefficients of the model and Xi, Xj, are the explicative variables. The analysis of variance ANOVA was employed in order to determine the significance of the independent variables on the output responses. The results of the experimental tests are presented in Table 2. Table 1 Factors variation level

Cutting speed (m/min)

Feed rate (mm/rev)

Depth of cut (mm)

Tool nose radius (mm)

-1

100

0.08

0.05

0.8

0

150

0.11

0.15

1.2

+1

200

0.14

0.25

1.6


Table 2 Experimental run and results Input variables

Output responses

run

Vc m/min

ap mm

f mm/rev

r mm

AMPy m/s²

AMPz m/s²

Ra ȝm

1

200

0.25

0.14

0.8

0.53

0.56

0.83

2

150

0.15

0.11

1.2

0.55

0.62

0.59

3

100

0.05

0.08

0.8

0.75

0.69

0.64

4

100

0.05

0.14

0.8

0.39

0.56

1.09

5

100

0.25

0.08

1.6

0.22

0.42

0.33

6

150

0.15

0.11

1.2

0.63

0.8

0.54

7

100

0.25

0.08

0.8

0.14

0.48

0.35

8

100

0.25

0.14

1.6

0.17

0.38

0.62

9

100

0.05

0.14

1.6

0.71

0.52

0.45

10

200

0.25

0.14

1.6

0.53

0.46

0.44

11

200

0.25

0.08

0.8

0.47

0.39

0.37

12

200

0.05

0.08

0.8

0.8

0.63

0.65

13

100

0.25

0.14

0.8

0.12

0.46

0.79

14

200

0.05

0.14

0.8

1.41

1.24

0.98

15

150

0.15

0.11

1.2

0.63

0.74

0.53

16

200

0.05

0.08

1.6

0.69

0.56

0.34

17

200

0.25

0.08

1.6

0.27

0.37

0.30

18

150

0.15

0.11

1.2

0.99

0.71

0.53

19

200

0.05

0.14

1.6

1.04

0.73

0.37

20

100

0.05

0.08

1.6

0.39

0.46

0.31

21

150

0.15

0.14

1.2

1.04

0.67

0.61

22

150

0.15

0.11

1.2

0.6

0.63

0.48

23

150

0.25

0.11

1.2

0.4

0.59

0.63

24

100

0.15

0.11

1.2

0.86

0.82

0.45

25

150

0.15

0.11

1.2

0.54

0.53

0.56

26

150

0.05

0.11

1.2

0.62

0.9

0.59

27

150

0.15

0.11

1.6

0.63

0.6

0.41

28

200

0.15

0.11

1.2

1.09

1.42

0.54

29

150

0.15

0.08

1.2

0.63

0.37

0.37

30

150

0.15

0.11

0.8

0.63

0.48

0.61

423

424


3


3.1

Statistical Analysis and Modelling

The analysis of variance is performed to establish the statistical significance of the regression models, model coefficients and lack of fit. It is done by comparing “Prob > F” to 0.05 or in other words at 95% of confidence. The proportion of contribution of each model coefficient was calculated. The regression equations are generated by the statistical software Minitab 16. Those equations describe the statistical relationship between the factors and the responses and could predict new runs 3.1.1

Surface Roughness

The adopted model for surface roughness including cutting parameters is the linear with interaction. The results given by ANOVA presented in Table 3 shows that the terms r and f are the most significant with the respective contribution (41.98, 35.51) %. The term ap has a small contribution with 3.23 %, whereas cutting speed does not affect the surface roughness. A similar result was obtained by (Aslan et al. 2007) when turning AISI 4140 steel (63 HRC) with mixed ceramic tool. The interactions which show significant effect on surface roughness are ap×r and f×r with the respective contributions (9.67 and 7.34) %. The determination coefficient model’s R² is 0.94. The final equation in terms of real factors is given as follow: Ra = −0.049 + 2.242 × 103Vc − 3.618ap + 12.479 f + 0.051r − 0.015Vc × f − 6.875 × 10 −4 Vc × r + 7.917 ap × f

(2)

+ 1.938ap × r − 5.625 f × r

The ANOVA results for surface roughness model including cutting parameters and tool vibrations in both radial and tangential directions are presented in Table 4. It shows that in addition to the terms cited above, the surface roughness is influenced by the tool vibration in the radial direction AMPy with 3.71 % of contribution. The determination coefficient model’s R² is improved to 97.10 %. The final equation in terms of real factors is given as follow: Ra = −0.895964 − 2.32358 ap + 16.7492 f + 0.528119 r + 1.19221AMPz + 0.937027 AMPy + 0.0128869 Vc × ap − 0.0190204Vc × f-7.89172ap × f + 0.841952ap × r1.64463ap × AMPz − 1.41237ap × AMPy

(3)

− 4.99520 f × r + 0.589721 f × AMPz-4.94334 f × AMPy-0.538542r × AMPz-0.486322r × AMPy-0.988144 AMPz × AMPy

3.1.2

Vibration Analysis

According to the Table 5, the significant terms of acceleration amplitude model in radial direction (AMPy) were Vc, ap with the respective contribution (24.79,


425

39.11) % and f with the lower contribution of 6.27 %. The term ap² is also significant (14.97 %). The lack of fit P-value of 0.561 was greater than 0.05. Therefore, the regression model does not fail to adequately explain the functional relationship between the experimental factors and the response. Lack of fit may occur if important terms from the model such as interactions or quadratic terms are not included. The correlation coefficient R² of about 84.47 is considered as good. The final equation in terms of real factors acceleration amplitude in radial direction (AMPy) is given as follow: AMPy = 1.5015 - 0.0224Vc + 7.6269 ap - 20.6830 f + 2.0932r - 0.0069Vc × ap + 0.0578Vc × f - 0.0024Vc × r (4) - 13.9167ap × f + 0.7063 ap × r + 3.0625 f × r + 7.7846×10-5Vc2 - 27.0386 ap2 + 60.6823 f 2 - 0.9399r 2

The results given by the ANOVA analysis of the acceleration amplitude in tangential direction (AMPz) presented in Table 6, show that the most significant model terms are Vc and ap with the respective contributions (10.29 and 19.85) %, whereas f and r do not affect the tool vibration in tangential direction. The quadratic term f² is significant with 21.71 % of contribution. The coefficient R² of about 81.22 is considered good. The final equation in terms of real factors is given as follow: AMPz = -1.6633 - 0.0428Vc + 1.2938ap + 60.0136 f + 3.5225r - 0.0111Vc × ap + 0.0488Vc × f - 9.0625 × 10- 04Vc × r - 10.625ap × f + 0.9219ap × r - 1.8229 f × r + 0.0001Vc 2 - 2.5790ap 2 - 278.655 f 2 - 1.4424r

2

Table 3 ANOVA for Ra vs cutting parameters Source

DF

Seq SS

Adj SS

Adj MS

F-value

P-value

Regression

10

0.99358

0.99358

0.09936

29.09

0.000

Vc

1

0.00245

0.00245

0.00245

0.72

0.408

0.25

ap

1

0.03209

0.03209

0.03209

9.40

0.006

3.23

f

1

0.35280

0.35280

0.35280

103.30

0.000

35.51

r

1

0.41709

0.41709

0.41709

122.13

0.000

41.98

Vc×ap

1

0.00000

0.00000

0.00000

0.00

1.000

0.00

Vc×f

1

0.00810

0.00810

0.00810

2.37

0.140

0.82

Vc× r

1

0.00302

0.00303

0.00303

0.89

0.358

0.30

ap× f

1

0.00902

0.00903

0.00903

2.64

0.121

0.91

ap× r

1

0.09610

0.09610

0.09610

28.14

0.000

9.67

21.35

0.000

7.34

3.11

0.108

f× r

1

0.07290

0.07290

0.07290

Residual Error

19

0.06489

0.06489

0.00342

Lack-of-Fit

14

0.05821

0.05821

0.00416

Pure Error

5

0.00668

0.00668

0.00134

Total

29

1.05847

Cont.%

(5)

426


Table 4 ANOVA for Ra vs cutting parameters and tool vibrations Source

DF

Seq SS

Adj SS

Adj MS

F-value

P-value

Cont.%

Regression

21

1.02779

Vc

1

0.00245

1.02779

0.04894

12.76

0.000

0.00479

0.00479

1.25

0.296

ap

1

0.03209

0.00017

0.24

0.00017

0.04

0.840

3.12

f

1

0.3528

0.00258

0.00258

0.67

0.436

34.33

r

1

0.41709

0.02122

0.02122

5.53

0.047

40.58

AMPz

1

0.00067

0.00179

0.00179

0.47

0.514

0.07

AMPy

1

0.03813

0.0105

0.01050

2.74

0.137

3.71

Vc×ap

1

0.00003

0.00634

0.00634

1.65

0.234

0.00

Vc×f

1

0.00371

0.00298

0.00298

0.78

0.404

0.36

Vc×r

1

0.00753

0.00049

0.00049

0.13

0.729

0.73

Vc×AMPz

1

0.00262

0.00000

0.00000

0.00

0.996

0.25

Vc×AMPy

1

0.00000

0.00607

0.00607

1.58

0.244

0.00

ap×f

1

0.00801

0.0009

0.00090

0.24

0.640

0.78 9.40

ap×r

1

0.09658

0.00158

0.00158

0.41

0.539

ap×AMPz

1

0.00064

0.00175

0.00175

0.46

0.518

0.06

ap×AMPy

1

0.00011

0.00253

0.00253

0.66

0.440

0.01 5.19

f×r

1

0.05331

0.01917

0.01917

5.00

0.056

f×AMPz

1

0.00005

0.00001

0.00001

0.00

0.961

0.00

f×AMPy

1

0.00166

0.00426

0.00426

1.11

0.323

0.16

r×AMPz

1

0.00173

0.00139

0.00139

0.36

0.564

0.17

r×AMPy

1

0.00017

0.00176

0.00176

0.46

0.517

0.02

2.19

0.177

0.82

Cont.%

AMPz×AMPy

1

0.00840

0.0084

0.00840

Residual Error

8

0.03068

0.03068

0.00384

Total

29

1.05847

Table 5 ANOVA for AMPy Source

DF

Seq SS

Adj SS

Adj MS

F-value

P-value

Regression

14

2.21852

2.21852

0.15847

5.83

0.001

Vc

1

0.52771

0.52771

0.52771

19.4

0.001

23.79

ap

1

0.86768

0.86768

0.86768

31.91

0.000

39.11

f

1

0.13904

0.13904

0.13904

5.11

0.039

6.27

r

1

0.01921

0.01921

0.01921

0.71

0.414

0.87

Vc×Vc

1

0.03511

0.09813

0.09813

3.61

0.077

1.58

ap×ap

1

0.33218

0.18942

0.18942

6.97

0.019

14.97

f×f

1

0.00029

0.00773

0.00773

0.28

0.602

0.01

r×r

1

0.0586

0.0586

0.05860

2.15

0.163

2.64

Vc×ap

1

0.01877

0.01877

0.01877

0.69

0.419

0.85

Vc×f

1

0.12041

0.12041

0.12041

4.43

0.053

5.43

Vc×r

1

0.03725

0.03725

0.03725

1.37

0.260

1.68

ap×f

1

0.02789

0.02789

0.02789

1.03

0.327

1.26

ap×r

1

0.01277

0.01277

0.01277

0.47

0.504

0.58

f×r

1

0.02161

0.02161

0.02161

0.79

0.387

0.97

Residual Error

15

0.40792

0.40792

0.02720

Lack-of-Fit

10

0.26719

0.26719

0.02672

0.95

0.561

Pure Error

5

0.14073

0.14073

0.02815

Total

29

2.62644 R²

84.47


427

Table 6 ANOVA for AMPz Source

DF

Seq SS

Adj SS

Adj MS

F-value

P-value

Regression

14

1.33035

1.33035

0.09503

4.63

0.003

Cont.%

Vc

1

0.13694

0.13694

0.13694

6.68

0.021

10.29

ap

1

0.26402

0.26402

0.26402

12.88

0.003

19.85

f

1

0.08134

0.08134

0.08134

3.97

0.065

6.11

r

1

0.05445

0.05445

0.05445

2.66

0.124

4.09

Vc×Vc

1

0.00214

0.31596

0.31596

15.41

0.001

0.16

ap×ap

1

0.17865

0.00172

0.00172

0.08

0.776

13.43

f×f

1

0.28883

0.16296

0.16296

7.95

0.013

21.71

r×r

1

0.13800

0.13800

0.13800

6.73

0.02

10.37

Vc×ap

1

0.04951

0.04951

0.04951

2.41

0.141

3.72

Vc×f

1

0.08556

0.08556

0.08556

4.17

0.059

6.43

Vc×r

1

0.00526

0.00526

0.00526

0.26

0.62

0.40

ap×f

1

0.01626

0.01626

0.01626

0.79

0.387

1.22

ap×r

1

0.02176

0.02176

0.02176

1.06

0.319

1.64

f×r

1

0.00766

0.00766

0.00766

0.37

0.55

0.58

Residual Error

15

0.30754

0.30754

0.02050

Lack-of-Fit

10

0.26046

0.26046

0.02605

2.77

0.136

Pure Error

5

0.04708

0.04708

0.00942

Total

29

1.63790 R²

3.2

Graphics Analysis

3.1.1

Surface Roughness

81.22

It is well seen in Figure 2(a) that the use of small feed rate and large nose radius results in the lowest surface roughness. Whereas, the use of large feed rate and small tool nose radius results in highest surface roughness. Because at large feed rate the distance between peaks and valleys of the feed marks is much more important. The ANOVA revealed that cutting speed does not influence the surface roughness. The depth of cut is found to be statistically significant. The effect of tool nose radius and depth of cut on surface roughness is depicted in Figure 3(b). We can observe that using a small depth of cut gives worst surface roughness especially with tool nose radius of 0.8 mm. We could attribute this to the material side flow phenomenon, due to temperature elevation and specific cutting pressure in the cutting zone when using small undeformed chip thickness. It consists of ploughing and squeezing of the machined material by the tool causing its deterioration (Kishawy and Elbestawi 1999). Figures 3(a) and 3(b) show the topography of machined surfaces using the lower feed rate and with both depth of cut (0.05 and 0.25) mm respectively. We can observe that the valleys of surface (a) are distant above those of surface (b) due to the material accumulation. This

428


additional volume of squeezed material presses on the tool flank face which can explain the increase of acceleration amplitude in the radial direction (AMPy) at small depth of cut (Figure 5(a)). The same trend is observed for the acceleration amplitude in tangential direction (AMPz) (Figure 5(b)). This figure shows that the tangential vibration (AMPz) increases with the increase in tool nose radius. Then, it decreases by using tool nose radius of 1.6 mm. We can explain this by the stabilization of cutting process when using large nose radius leading to the improvement of surface roughness.

b)

a)

Fig. 2 Effect of cutting conditions on surface roughness

a)

Vc = 200 m/min, ap = 0.05 mm, f = 0.08 mm/rev and r = 0.8 mm

b) Vc=200 m/min, ap = 0.25 mm, f = 0.08 mm/rev and r = 0.8 mm

Fig. 3 Topography of AISI 52100 steel hard turned with mixed ceramic insert


a)

429

b)

Fig. 4 Effect of Effect of cutting conditions on the tool vibration

4

Models Confirmation

The reliability of the developed mathematical model is investigated by performing three new runs. The confirmation runs conditions were set within the range of the factors levels already defined. Table 6 shows the percentage error between the experimental values and their corresponding predicted values obtained from the Eqs. (2). slight values of divergence were observed surface roughness. Consequently, the mathematical models can be used to effectively predicting surface roughness values for any combination of cutting speed, feed rate, depth of cut and tool nose radius within the limits of the actual experimentation. Table 7 Comparison between experimental and predicted values Vc

ap

f

r

1

115.4

0.05

0.09

2

131.9

0.20

3

160.9

0.10

run

5

exp Ra

pred Ra

error (%)

1.60

0.34

0.33

- 2.78

0.09

0.80

0.48

0.48

0.68

0.12

1.20

0.64

0.60

- 5.78

Conclusion

Through the results obtained above, the following conclusions could be drawn: 1. The best surface roughness is obtained by using small feed rate and large nose radius. The use of depth of cut of 0.05 mm with tool nose radius of 0.8 mm gives the worst surface roughness due to the material side flow phenomenon. 2. The ANOVA results show that depth of cut and cutting speed influence considerably the tool vibrations on both tangential and radial direction, whereas the feed rate has a significant effect only on acceleration amplitude in radial direction (AMPy). It is also revealed that the tool nose radius has

430


only a slight statistical significance on acceleration amplitude in tangential direction (AMPy). 3. When using a small undeformed chip thickness, the material side flow phenomenon could be detected by the increase of the tool vibrations. It’s due to the exerted pressure of the squeezed material on the tool flank face. 4. The optical platform of metrology modular was an important tool in the investigation of the surface roughness, through the produced threedimensional topographic maps of the machined surfaces.

References Ozel, T., Karpat, Y., Figueira, L.: Predictive modeling of surface roughness and tool wear in hard turning using regression and neural networks. Int. J. Mach. Tools Manuf. 45, 467–479 (2005) Yallese, M.A., Rigal, J.-F., Chaoui, K., Boulanouar, L.: The effects of cutting conditions on mixed ceramic and cubic boron nitride tool wear and on surface roughness during machining of X200Cr12 steel (60 HRC). Proc. IMechE, Part B: J. Eng. Manuf. 219 (2005) Aslan, E., Camuscu, N., Birgoren, B.: Design optimization of cutting parameters when turning hardened AISI 4140 steel (63 HRC) with Al2O3 + TiCN mixed ceramic tool. Mater. Design 28, 1618–1622 (2007) Dureja, J.S., Gupta, V.K., Sharma, V.S., Dogra, M.: Design optimization of cutting conditions and analysis of their effect on tool wear and surface roughness during hard turning of AISI-H11 steel with a coated–mixed ceramic tool. J. Eng. Manuf. 223(B), 1441–1453 (2009) Lalwani, D.I., Mehta, N.K., Jain, P.K.: Experimental investigations of cutting parameters influence on cutting forces and surface roughness in finish hard turning of MDN250 steel. J. Mater. Process. Technol. 206, 167–179 (2008) Bartarya, G., Choudhuryb, S.K.: Effect of cutting parameters on cutting force and surface roughness during finish hard turning AISI52100 grade steel. Procedia CIRP 1, 651–656 (2012) Horng, J.-T., Liu, N.-M., Chiang, K.-T.: Investigating the machinability evaluation of Hadfield steel in the hard turning with Al2O3/TiC mixed ceramic tool based on the response surface methodology. J. Mater. Process. Technol. (2008) Saini, S., Ahuja, I.S., Sharma, V.S.: Influence of Cutting Parameters on Tool Wear and Surface Roughness in Hard Turning of AISI H11 Tool Steel using Ceramic Tools. Int. J. Precision Eng. Manuf. 13(8), 1295–1302 (2012) Aouici, H., Yallese, M.A., Chaoui, K., Mabrouki, T., Rigal, J.F.: Analysis of surface roughness and cutting force components in hard turning with CBN tool: prediction model and cutting conditions optimization. Measurement 45, 344–353 (2012) Tlusty, J.: Manuf. Process. Equipment. Prentice Hall (1999) Thomas, M., Beauchamp, Y., Youssef, A.Y., Masounave, J.: Effect of tool vibration on surface roughness during lathe dry turning process. Comput. Ind. Eng. 31(3-4), 637–644 (1996) Jang, D.Y., Choi, Y.G., Kim, H.G., Hsiao, A.: Study of the correlation between surface roughness and cutting vibrations to develop an on-line roughness measuring technique in hard turning. Int. J. Mach. Tools Manuf. 36, 453–464 (1996)


431

Risbood, K.A., Dixit, U.S., Sahasrabudhe, A.D.: Prediction of surface roughness and dimensional deviation by measuring citting forces and vibrations in turning process. J. Mater. Process. Technol. 132, 203–214 (2003) Kirby, E.D., Zhang, Z., Chen, J.C.: Development of An Accelerometer based surface roughness prediction system in turning operation using multiple regression techniques. J. Ind. Technol. 4(20), 1–8 (2004) Abouelatta, O.B., Madl, J.: Surface roughness prediction based on cutting parameters and tool vibrations in turning operations. J. Mater. Process. Technol. 118, 269–277 (2001) Hessainia, Z., Kribes, N., Yalles, M.A., Mabrouki, T., Ouelaa, N., Rigal, J.-F.: Turning roughness model based on tool-nose displacements. Mechanika 19(1), 112–119 (2013) ISSN 1392-1207 Kishawy, H.A., Elbestawi, M.A.: Effects of process parameters on material side flow during hard turning. Int. J. Mach. Tools Manuf. 39, 1017–1030 (1999)

Finite Element Simulation of Fatigue Damage Accumulation for Repaired Component by Cold Expansion Method Abdelkrim Aid1, Mostefa Bendouba1, Mohamed Benguediab2, and Abdewahab Amrouche3 1

Laboratoire LPQ3M, B.P. 305, Universite de Mascara, Algeria [email protected], [email protected] 2 Département de Génie Mécanique, Université de Djilali Liabès, Sidi Bel Abèss, Algeria [email protected] 3 Laboratoire de Génie Civil et Géo-Environnement LGCgE, EA 4515, Faculté des Sciences Appliquées FSA Béthune, Université d’Artois, France [email protected]

Abstract. This manuscript investigates the effectiveness of applying the cold expansion process to extend the fatigue life of mechanical structures. During the cold expansion process compressive residual stresses around the expanded hole are generated. The enhancement of fatigue life and the crack initiation and growth behavior of a holed specimen were investigated by using the 6082 Aluminum alloy. The present study suggests a simple technical method for enhancement of fatigue life by a cold expansion hole of pre-cracked specimen. This technique produces beneficial high compressive residual stresses which have been predicted by means of finite element models, both 3D for proper assessment of thickness effects. Finite element models have been developed to increase their complexity, Fatigue damage accumulation of cold expanded hole in aluminum alloy which is widely used in transportation and in aeronautics was analyzed. Experimental tests were carried out using pre-cracked SENT specimens. Tests were performed in two and four block loading under constant amplitude. These tests were performed by using two and four blocks under uniaxial constant amplitude loading. The experimental results were compared to the damage calculated by the Miner’s rule and a new simple fatigue damage indicator based on an energy criterion. This comparison shows that the ‘energy criterion model’, which takes into account the loading history, yields a good estimation according to the experimental results. Keywords: cold expansion, compressive residual stress, finite element method, energies criterion, fatigue damage accumulation.

1

Introduction

Over the last 25 years, the cold expansion process has been commonly used to improve the fatigue life of components containing fastener holes. Cold expansion © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_44

433

434

A. Aid et al.

is employed usually in components that are exposed only to service conditions at ambient temperature. The cold-worked process introduces beneficial residual circumferential stresses into an annular region around the hole, and the presence of this compressive residual stress inhibits the growth and propagation of cracks (Lacarac et al. 2001, Pasta 2007). In order to increase the service life of components some methods have been developed. One of these methods, which were used in this study, is to form a controlled compressive residual stress field around the hole (Chandawanich and Sharpe 1979; Su and Yan 1986). However, the improvement in fatigue life is difficult to quantify. The residual stresses resulting from the cold expansion process are not uniform through the thickness (Pavier et al. 1997). Earlier work on cold expansion has used both analytical and numerical two-dimensional models to predict residual stress (Poussard et al. 1995). Furthermore, combining the residual stress distribution with a fatigue crack growth rate calculation is difficult due to the three-dimensional nature of the problem and complicating phenomena such as crack closure (Lacarac et al. 2004). In (Bernard et al. 1995; Ghfiri et al. 2000; Semari et al. 2013; Aid et al. 2013), the authors show the sensitivity of the degree of expansion. In a first phase the residual life increases but beyond a critical value of remaining life decreases sharply and growth is extremely detrimental. In service conditions, the components or structures are subjected to random or variable block loading. Different relationships (Lacarac et al. 2004) have been proposed to calculate the effect of variable amplitude loading conditions. However, these procedures need the identification of many parameters. In literature, in the particular case of block loading, the analysis for this phenomenon is oriented only to two loading steps. Miner’s rule (Miner 1945) is very much used to evaluate the fatigue damage accumulation when the components or structures are subjected to variable block loading. In this work, the results of a study on fatigue damage accumulation of cold expanded hole in aluminum alloys subjected to block loading are presented. Tests were carried out using pre-cracked SENT specimens and inserting an expanded hole at the crack tip. The degree of the cold expansion was chosen equal to 4.3%. Tests were performed in two and four block loading under constant amplitude. The experimental results were compared to the damage calculated by the Miner’s rule and a new simple fatigue damage indicator (Aid 2006a; Aid 2011; Amrouche 2008; Aid 2006b) and a new version of this damage indicator based on energy approach (Bendouba et al. 2011; Djebli et al. 2013).

2

General Principal of the Method

In this study, cold expansion is achieved by inserting an oversized rigid ball from one side (entry face) of the holed plate and removing it from the other side (exit face). The degree of cold expansion DCE is defined by the relation:

Finite Element Simulation of Fatigue Damage Accumulation

DCE % =

(D − d ) × 100 d

435

(1)

A cold expansion of the hole introduces compressive residual stresses that are beneficial in terms of lifetime until a critical degree of expansion where it becomes not beneficial for structures. Both aspects are shown in Figure 1.

Fig. 1 The curve of crack initiation according to the numbers of the cycles

Where d is the diameter of the hole drilling and D is the diameter of the rigid ball. Different technique can be used to repair a cracked component. The cold working expansion process was realized by forcing a hard steel ball of 6 mm inside a predrilled hole (the initial diameters of the hole are: 5.9, 5.8, 5.75, 5.6 and 5.5 mm for aluminium alloy and 5.8, 5.75 for steel), this process is illustrated in (Aid 2006) .

3

Materials and Specimens

The material used for this study was aluminum alloy Al 6082 T6, some of the mechanical properties are given in (Aid 2006). A difference gap of 10% between the characteristics of the uncracked specimens and the batch of specimens with hole was noticed. The specimens used for this investigation were conforms to ASTM standards (ASTM 1997). The geometry of the fatigue test specimen cut in the longitudinal direction (Aid 2006). For getting specimens with an expanded hole of 6 mm in diameter we drilled a hole of Ø5.75 mm at the pre-crack tip and then a cold-working expansion process was conducted by forcing a steel ball of 6 mm diameter. The fatigue tests were carried out using a 100 kN capacity Instron hydraulic machine. The loading frequency was 30 Hz and a stress ratio R of 0.57. During

436

A. Aid et al.

fatigue testing, a video camera with scale of 0.1 mm was used to determine the crack initiation in the entry and exit faces of specimen.

3.1

3D Finite Element Modeling

3-D FEM simulations using MSC.Marc have been used to evaluate the rigid ball during the cold expansion process. In this case the axial movement of the rigid ball through the specimen has been introduced in the model to obtain the distribution of the residual stresses through the thickness. Because of the symmetry, half of the lug is representative of the useful of the specimen. The quadrilateral element topology QUAD4 has been adopted for the analysis. Globally, the finite element model consists of 3886 elements and 5721 nodes. (see figure 2).

Fig. 2 3D FEM mesh, with rigid ball for cold-expansion and fatigue critical section

The contact between the surfaces of the steel ball and the hole is simulated by using the contact elements. Residual stress is obtained at the three surfaces (entry face, middle and exit face) to analyze the influence of cold expansion direction on the fatigue crack initiation life. During the simulation, the rigid ball was moved through the hole incrementally. The boundary conditions imposed are as following: The displacements of the nodes located on the X–Y plane are constrained in the z direction. The displacements of the nodes located on the X–Z plane are constrained in the y direction within a rectangular surface on the exit face.

4

Results and Discussions of the Simulation

The figure 3 illustrates the hoop stresses at the entry, middle and exit expansion step. Portrays the finite element prediction of the tangential residual stress variation at different through thickness positions from the entrance face, this figure reveals the large variation in hoop residual stress in the region of influence.


entry face

middle face

437

xit face

Fig. 3 Circumferential residual stress distribution in the 3D model

Low hoop residual stresses on entrance and exit faces are caused by the shear stress from the axial movement of the rigid ball. The compressive residual stresses into the annular region around the hole reduce the origin of cracks. During fatigue testing, this residual stresses act to change the effective stress intensity factor at the crack tip, i.e. the crack growth rates are lower than those in absence of residual stress. The residual stress profiles are in agreement with other studies presented in previous literature (Poussard et al.1985; Pavier et al. 1999 ; De Matos et al. 2005). Figure 4 shows the FE prediction of hoop residual stresses through the thickness of the plate from the entrance face to the exit face. A significant difference in the magnitude of tangential stress is recorded between the entry side and exit side. The magnitude of hoop residual stresses on the entrance and the exit faces are lower than the stresses on the mid-thickness. These results are in accordance with recent research (Yan et al. 2012 ; Chakherlou 2012).

5

Damage Accumulation Based an Energy Parameters

An energy approach is presented in this paper, which forms the basis for a new damage model. The strain energy density is proposed as a parameter of the fatigue analysis. The models do not include a division of the strain energy density into elastic and plastic parts, like in all the cases of the parameters proposed by Smith– Watson–Topper (SWT) (Smith et al. 1970), Hoffman and Seeger (Hoffman and Seeger 1989), Bergman and Seeger (Bergman and Seeger 1979). In the elastic range, energy is calculated from

1 W = σε 2

(2)

For uniaxial fatigue loading, authors (Bendouba et al. 2011; Djebli et al. 2013) introduce a new damage parameter, Di, defined as the ratio of the increment of energy due to stress damage over the difference between the energy due to ultimate stress and the applied stress. The damage indicator is defined by:

438

A. Aid et al.

Hoop residual stress (MPa)

200 150 100 50 0 -50 0 -100 -150 -200 -250 -300

5

10

15

20

Entry… Exit face

Distance from hole edge (mm) Fig. 4 Finite element predictions of residual hoop stress at entry face and exit face

D=

W edi −W i W u −W i

(3)

where: Wedi : energy is due to stress damage. Wi : energy is due to the applied stress. Wu : energy is due to the ultimate stress of the material.

6

Fatigue Testes

The experimental conditions are given in (Aid 2006). Eight tests were carried out for increasing loading conditions and as much for decreasing loading. To evaluate the effect between these loading conditions, the Miner’s Rule was considered for damage accumulation. Endurance curves are shown in (Aid 2006). These are based on constant amplitude test and the failure was considered at the crack initiation (Aid 2006). Two and four cyclic stress levels were considered and two different sequences were applied (Aid 2006). The aim of this set of tests is to determine the influence of increasing or decreasing loading conditions on lifetime and to prove that the proposed model takes into account the history of blocks loading and the nonlinearity of the accumulated damage unlike the Miner’s rule. Figure 5 demonstrates the comparison between the prediction models (Miner’ Rule, DSM model and the energy approach) with experimental results for different loading conditions (two and four blocks with increasing and decreasing loads) for specimens pre cracked and repaired by the technical process of cold expansion. This figure confirms that the estimates results by the DSM model and the energy approach used in this investigation are in good agreement with the experimental results (Aid 2006).


439

Figure 5 demonstrates the comparison between the prediction models (Miner’ Rule, DSM model and the energy approach) with experimental results for different loading conditions (two and four blocks with increasing and decreasing loads) for specimens pre cracked and repaired by the technical process of cold expansion. This figure confirms that the estimates results by the DSM model and the energy approach used in this investigation are in good agreement with the experimental results.

Fig. 5 Comparison between theoretical results and experimental results for different loading conditions

7

Conclusion

At first, a modelization study on cold-expansion process by finite element method has been made. Different FE models have been defined to evaluate different characteristics as plastic strain representation, contacts simulation and material characterization has been used. Residual stresses after the cold-expansion process have been evaluated by means of FEM models using MSC Marc software. Secondly, this investigation is also set to study fatigue damage accumulation in arresting crack holes with cold expansion process was studied. In total, 24 specimens were tested, eight specimens were used to obtain the fatigue endurance curve and 16 were subjected to increasing or decreasing blocks loading. Using the Miner’s rule to calculate the cumulative life time. We found that in both cases, increasing and decreasing blocks loading, the experimental results were below prediction. The results obtained by the energy approach and model of the damage stress are compared with the experimental results and a good agreement has been found. The experimental results show that the load sequence has no significant effect on the crack reinitiating.

440

A. Aid et al.

It seems that in the case of drilling with a cold expansion, the combination of the geometrical and mechanical effect attributed to the stress concentration factor associated with the compressive residual stresses predominate in the life time. In the other hand the load sequence has a minor effect. In this investigation, the compressive residual stresses at the edge of the hole are around of the yielding stress. The effective local applied stress is lower than the residual stress; this observation can explain the raison why there is no significant influence of the sequence loading. Currently, we achieve tests with more important loading in order to evaluate the sequence loading effect.

References Lacarac, V.D., Smith, D.J., Pavier, M.J.: The effect of cold expansion on fatigue crack growth from open holes at room and high temperature. Int. J. Fatigue 23, 161–170 (2001), doi:10.1016/S0142-1123(01)00125-6 Pasta, S.: Fatigue crack propagation from a cold-worked hole. Engng. Frac. Mech. 74, 1525–1538 (2007), doi:10.1016/j.engfracmech.2006.08.006 Chandawanich, N., Sharpe Jr., W.N.: An experimental study of fatigue crack initiation and growth from cold worked holes. Engng. Frac. Mech. 11, 609–620 (1979), doi:10.1016/0013-7944(79)90122-X Su, X., Gu, M., Yan, M.: A simplified residual stress model for predict-ing fatigue crack growth behavior at cold worked fastener holes. Fat. Frac. Engn. Mat. & Str. 9(1), 57–64 (1986), doi:10.1016/0013-7944(79)90122-X Pavier, M.J., Poussard, C.G.C., Smith, D.J.: A finite element simulation of the cold working process for fastener holes. J. Strain Analysis 32, 287–300 (1997) Poussard, C., Pavier, M.J., Smith, D.J.: Analytical and finite element pre-dictions of residual stresses in cold-worked fastener holes. J. Strain Analysis 30(4), 291–304 (1995) Lacarac, V.D., Garcia-Granada, A.A., Smith, D.J., Pavier, M.J.: Prediction of the growth rate for fatigue cracks emanating from cold expanded holes. Int. J. Fatigue 26(6), 585– 595 (2004), doi:10.1016/j.ijfatigue.2003.10.015 Bernard, M., Bui-Quoc, T., Burlat, M.: Effect of re-cold working on fatigue life enhancement of a fastener hole. Fat. Fra. Eng Mat. Structures 18(17-18), 765–775 (1995), doi:10.1111/j.1460-2695.1995.tb00902.x Ghfiri, R., Shi, H.J., Guo, R., Mesmacque, G.: Effect of Expanded and non expanded Hole on the Delay of Arresting Crack Propagation for Aluminum Alloys. Mat. Sci. Engineering A 286, 244–249 (2000), doi:10.1016/S0921-5093(00)00805-4 Semari, Z., Aid, A., Benhamena, A., Amrouche, A., Benguediab, M., Sadok, A., Benseddiq, N.: Effect of residual stresses induced by cold expansion on the crack growth in 6082 aluminum alloy. Eng. Fra. Mechanics 99, 159–168 (2013), doi:10.1016/j.engfracmech.2012.12.003 Aid, A., Semari, Z., Benguediab, M.: Cold expansion effect on the fatigue crack growth of Al 6082: numerical investigation. Structural Engineering and Mechanics 49(2), 225–235 (2014), doi:http://dx.doi.org/10.12989/sem.2014.49.2.225 Miner, M.A.: Cumulative damage in fatigue. J. App. Mech. 67, A159–A164 (1945) Aid, A., Semari, Z., Amrouche, A., Mesmacque, G., Benguediab, M.: Non linear damage cumulative criterion on fatigue under random loading. Alg. J. Adv. Materials (2006)


441

Aid, A., Amrouche, A., Bachir Bouiadjra, B., Benguediab, M., Mesmacque, G.: Fatigue life prediction under variable loading based on a new damage model. Mat. Design 32, 183–191 (2011), doi:10.1016/j.matdes.2010.06.010 Amrouche, A., Su, M., Aid, A., Mesmacque, G.: Numerical study of the op-timum degree of cold expansion: Application for the pre-cracked specimen with the expanded hole at the crack tip. J. Mat. Pro. Technology 197, 250–254 (2008), doi:10.1016/j.jmatprotec.2007.06.030 Aid, A.: Cumul d’endommagement en fatigue multiaxiale sous sollicitations variables. [Thesis], p. 195. Universitè de Sidi-Bel Abbes, Algérie (2006) Bendouba, M., Aid, A., Zengah, S., Benguediab, M.: Modèle énergétique de cumul du dommage et prevision de la dureé de vie en fatigue sous chargements aléatoires. In: Proceedings of the Cong. Alg. Mécanique, Guelma, Algeria (2011) Djebli, A., Aid, A., Bendouba, M., Amrouche, A., Benguediab, M., Benseddiq, N.: A non-linear energy model of fatigue damage accumulation and its verification for Al-2024 aluminum alloy. Int. J. Non-Linear Mechanics (2013), doi:10.1016 /j.ijnonlinmec.2013.01.007 ASTM E1049-85, Standard practices for cycle counting in fatigue analysis. Annual Book of ASTM Standards, Philadelphia, vol. 03(01), 614–620 (1997, 1999) Poussard, G.C., Pavier, M.J., Smith, D.J.: Analytical and finite element predictions of residual stresses in cold-worked fastener holes. J. Strain Anal. 32(4), 287–300 (1985) Pavier, M.J., Poussard, G.C., Smith, D.J.: Effect of residual stress around cold worked holes on fracture under superimposed mechanical load. J. Strain Anal. 63, 751–773 (1999) de Matos, P.F.P., Moreira, P.M.G.P., Camanho, P.P., de Castro, P.M.S.T.: Reconstitution of fatigue crack growth in Al-alloy 2024-T3 open-hole specimens using microfractographic techniques. Engng. Frac. Mech. 72, 2232–2246 (2005), doi:10.1016/j.engfracmech.2005.02.005 Yan, W.Z., Wang, X.S., Gao, H.S., Yue, Z.F.: Effect of split sleeve cold ex-pansion on cracking behaviors of titanium alloy TC4 holes. Eng. Fra. Mechanics 88, 79–89 (2012), doi:10.1016/j.engfracmech.2012.04.008 Chakherlou, T.N., Shakouri, M., Aghdam, A.B., Akbari, A.: Effect of cold expansion and bolt clamping on fretting fatigue behavior of Al 2024-T3 in double shear lap joints. Mat., Design 33, 185–196 (2012), doi:10.1016/j.engfailanal.2012.04.008 Smith, K.N., Waston, P., Topper, T.H.: A stress–strain functions for the fatigue on materials. J. Mater. 5, 767–776 (1970) Hoffman, H., Seeger, T.: Stress-strain analysis and life predictions of a notched shaft under multiaxial loading. In: Leese, G.E., Socie, D. (eds.) Multi-axial Fatigue: Analysis and Experiments AE-14, pp. 81–99. Society of Automative Engineers, Inc., Warrendale (1989) Bergman, J., Seeger, T.: On the influence of cyclic stress–strain curves, damage parameters and various evaluation concepts on the life prediction by the local approach. In: Proceedings of the 2nd European Conference on Fracture, Darmstadt, Germany. VDIReport of Progress, vol. 18(6) (1979)

Geometrically Non-linear Free Vibration of Fully Clamped Symmetrically Laminated Composite Skew Plates Hanane Moulay Abdelali1, Bilal Harras2, and Rhali Benamar1 1

Université Mohammed V Agdal-Ecole Mohammadia D’ingénieurs, Avenue Ibn Sina, B.P.765, Agdal, Rabat, Morocco [email protected], [email protected] 2 FST B.P.2202- Route D’Imouzzer Fes, Morocco [email protected]

Abstract. The present work concerns the geometrically non-linear free vibration of fully clamped symmetrically laminated composite skew plates. The theoretical model based on Hamilton’s principle and spectral analysis previously applied to obtain the non-linear mode shapes and resonance frequencies of thin straight structures, such as beams, plates and shells is used. A convergence study has been performed and has shown that 18 plate functions should be taken into account. Results are given for the linear and non-linear fundamental mode shape of fully clamped symmetrically laminated composite skew plates, considering different parameters such as the skew angle, the number of layers, the fiber orientation, the vibration amplitudes and plate aspect ratio. It was found that the non-linear frequencies increase with increasing the amplitude of vibration and increasing the skew angle, which corresponds to the hardening type effect, expected in similar cases, due to the membrane forces induced by the large vibration amplitudes. Keywords: Non-linear vibrations, Composite laminated skew plate.

1

Introduction

Composite materials are used in many industrial fields. Resulting composite material present the best qualities of their constituents and some properties can be improved by forming a composite material including reduction in structural weight, reduction in fatigue and corrosion problems, that’s why it’s present interested use in aircraft structures and space vehicles. For the potential advantages listed above, the fundamental carbon fiber reinforced plastic (CFRP) materials have been used as structural members, taking the place of traditional metals. The use of composite materials requires complex analytical methods in order to predict accurately their response to external loading, especially in severe environments, © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_45

443

444

H.M. Abdelali, B. Harras, and R. Benamar

which may induce geometrically non-linear behaviour. The geometrically nonlinear analysis of composite skew plates exhibits specific difficulties due to the anisotropic material behavior, and to the higher non-linearity induced by a higher stiffness, inducing tensile mid-plane forces in skew plates higher than that observed with conventional homogeneous materials. The problem of non-linear dynamic response of plates at large vibration amplitudes was investigated by many authors and subjected to some important works and methodological approaches during the last few decades. In the monograph of (Leissa 1969), a summary of the basic knowledge on linear dynamic isotropic plate problems has been presented. In the series of studies carried out by Benamar and his co-workers (Benamar and Bennouna 1990), a wide programme of experimental work was concentrated on the understanding of the dynamic behaviour of homogeneous and CFRP beams and plates. In reference (Chia 1988), a survey was presented of the literature on the geometric non-linear analysis of laminated composite elastic plates. (Han and Petyt 1996), (Ribeiro and Petyt 1999) have been presented dealing with the geometrically non-linear dynamic behavior of symmetrically laminated plates by using the hierarchical finite element method (HFEM). (Harras and Benamar 2001, 2002) investigated theoretical and experimental of the non-linear behavior of various fully clamped rectangular composite panels at large vibration amplitudes. Good agreement has been found in each case with previous published works. (Ray et al. 1995) have analytically studied the non-linear vibration behavior of isotropic skew plates. (Han and Dickinson 1997) applied Ritz approach for analyzing the vibration of symmetrically laminated composite skew plates. Using a differential quadrature (DQ) method, large amplitude free vibration analysis of laminated composite skew thin plates have been considered by (Malekzadeh 2007). Nonlinear vibration of symmetrically laminated composite skew plates by Finite element method have been considered by (Singha an Daripa 2007). Recently, large deformation flexural behavior of laminated composite skew plates using an analytical approach have been considered by (Upadhyay and Shukla 2012). The aim of this paper is to apply the theoretical model developed by (Benamar and Bennouna 1990) to analyze the geometrical non-linear free dynamic response of CFRP symmetrically laminated plates in order to investigate the effect of non-linearity on the non-linear resonance frequencies and the non-linear fundamental mode shape at large vibration amplitudes. The general formulation of the model for nonlinear vibration of laminated plates at large vibration amplitudes is presented. Fully clamped boundaries have been considered here and Periodic displacement was assumed. Results were compared to the previous ones in the literatures. The first non-linear mode shape is examined. The relationships between the non-linear resonance frequency ratio ωnl/ωl and the vibration amplitudes for the first non-linear mode shape, for various plate skew angle, fiber orientation and aspect ratios are discussed.

Geometrically Non-linear Free Vibration

445

2

Geometrically Non-linear Free Vibration of Fully Clamped Symmetrically Laminated Composite Skew Plates

2.1

Constitutive Equation of a Laminated Skew Plate at Large Deflections

Consider the skew plate with a skew angle θ shown in Fig.1. For the large vibration amplitudes formulation developed here, it is assumed that the material of the plate is elastic, isotropic and homogeneous. The thickness of the plate is considered to be sufficiently small so as to avoid the effects of shear deformation. The skew plate has the following characteristics: a, b, S: length, width and area of the plate; x-y: plate co-ordinates in the length and the width directions; ξ-η, H: Skew plate co-ordinates and plate thickness; E, ν: Young’s modulus and Poisson’s ratio; D, ρ: plate bending stiffness and mass per unit volume.

η

y

θ

b

x, ξ

a Fig. 1 Skew plate in x-y and ξ-η co-ordinate system

For the classical plate laminated theory, the strain-displacement relationship for large deflections are given by:

λ In which

,

and

λ

are given by:

(1)

446

H.M. Abdelali, B. Harras, and R. Benamar U

W

V

ε U

; k V

k k k

W Y

W

λ λ

; λ

λ

W

W W

(2)

W

U, V and W are displacements of the plate mid-plane, in the x, y and z directions respectively. For the laminated plate having n layers, the stress in the Kth layer can be expressed in terms of the laminated middle surface strains and curvatures as: Q

σ

ε

(3)

and terms of the matrix can be obtained by In which the relationships given in reference (Timoshenko 1975). The in-plane forces and bending moments in a plate are given by: N M

A B B D

λ

ε

(4)

k

A, B and D are the symmetric matrices given by the following Equation 5. 0 for symmetrically laminated plates (Mallick 1993). H

A ,B ,D

H

Q

1, z, z

dz

(5)

are the reduced stiffness coefficients of the kth layer in the plate Here the co-ordinates. The transverse displacement function W may be written as in reference (Ribeiro and Petyt 1999) in the form of a double series:

W

A

T

W sinkωt

(6)

a ,a ,…,a is the matrix of coefficients corresponding Where A T w ,w ,…,w is the basic spatial functions to the kth harmonic, W T matrix, k is the number of harmonics taken in to account, and the usual summation convention on the repeated index k is used. As in reference (Benamar, Bennouna and White 1993), only the term corresponding to k=1 has been taken into account, which has led to the displacement function series reduced, to only one harmonic: i.e., W

a w x, y sinωt

(7)


447

Here the usual summation convention for the repeated indexes i is used. i is summed over the range 1 to n, with n being the number of basic functions considered. The expression for the bending strain energy Vb, axial strain energy Va and kinetic energy T are given in reference (Harras 2001) in the rectangular coordinate (x,y). The skew co-ordinates are related to the rectangular co-ordinate (ξ,η) by: ξ=x-y tanθ ; η=y/cosθ. So, the strain energy due to bending Vb, axial strain energy Va and kinetic energy T are given in the ξ-η co-ordinate system. In the above expressions, the assumption of neglecting the in plane displacements U and V in the energy expressions has been made as for the fully clamped rectangular isotropic plates analysis considered in reference (Benamar, Bennouna and White 1993). Discretization of the strain and kinetic energy expressions can be carried out leading to: V

sin ωt a a k ; V

sin ωt a a a a b

ω cos ωt a a m

;T

(8)

In which mij, kij and bijkl are the mass tensor, the rigidity tensor and the geometrical non-linearity tensor respectively. Non-dimensional formulation of the nonlinear vibration problem has been carried out as follows. w ξ, η

Hw

ξ η

Hw ξ , η

,

Where ξ and η are non-dimensional co-ordinates ξ obtains:

k

H E

k

H E

;b

b

;m

(9) ξ

and η

ρH abm

η

one then

(10)

Where the non-dimensional tensors m*ij, k*ij and b*ijkl are given in terms of integrals of the non-dimensional basic function wi*, non-dimensionnal extensional and bending stiffness coefficient A*ij and D*ij , skew angle θ and aspect ratio α. Upon neglecting energy dissipation, the equation of motion derived from Hamilton’s principle is:

δ

V

T

0

(11)

Where V=Va+Vb. Insertion of Equations 8 into Equation 11, and derivation with respect to the unknown constants ai, leads to the following set of non-linear algebraic equations: 2a k

3a a a b

2ω a m

0, r=1…n.

(12)

These have to be solved numerically. To complete the formulation, the procedure developed in (Harras 2001) is adopted to obtain the first non-linear mode. As

448


no dissipation is considered here, a supplementary equation can be obtained by applying the principle of conservation of energy, leads to the equation: /

ω

(13)

This expression for ω*2 is substituted into Equation 12 to obtain a system of n non-linear algebraic equations leading to the contribution coefficients ai, i=1 to n. ω and ω* are the non-linear frequency and non-dimensional non-linear frequency parameters related by:

ω

D ρ

ω

(14)

To obtain the first non-linear mode shape of the skew plate considered, the contribution of the first basic function is first fixed and the other basic functions contributions are calculated via the numerical solutions of the remaining (n-1) non-linear algebraic equations.

3


The aim of this section is to apply the theoretical model presented above to analyze the geometrical non-linear free dynamic response of skew fully clamped symmetrically laminated plates in order to investigate the effect of non-linearity on the non-linear resonance frequencies and non-linear fundamental mode shape at large vibration amplitudes. Convergence studies are carried out, and the results are compared with those available from the literature through a few examples of laminated composite skew thin clamped plates with different fiber orientation and aspect ratio. The material properties, used in the present analysis are: (1) Plate 1, 5-layers symmetrical angle-ply (45°,-45°, 45°,-45°, 45°); EL/ET = 40.0; GLT/ET = 0.6; νLT = 0.25. (2) Plate 2, 8-layers symmetrically laminated plate (90°,45°,45°,0°)sym ; EL=120.5 Gpa; ET=9.63 Gpa; GLT=3.58 Gpa; νLT = 0.32. (3) Plate 3, 16-layers symmetrically laminated plate (45°,-45°,0°,-45°,45°,-45°,0°,45°)sym ; EL=173Gpa; ET=7.2 Gpa; GLT=3.76 Gpa; νLT = 0.29. Where E, G and ν are Young’s modulus, shear modulus and Poisson’s ratio. Subscripts L and T represent the longitudinal and transverse directions respectively with respect to the fibers. All the layers are of equal thickness. Calculation was made by using 18 functions corresponding to three symmetric beam functions in the ξ direction and three symmetric beam functions in the η direction, and three anti-symmetric beam functions in the ξ direction and three anti-symmetric beam functions in the η direction. Table 1 shows the linear results for a fully clamped laminated skew composite plate 2 for skew angle θ=30° and aspect ratio α=0.6 obtained using only 18 well-chosen plate functions. It can be seen a good convergence with results presented in reference (Harras 2001). The variation of non-dimensional nonlinear


449

Table 1 First linear mode shape of a fully clamped laminated skew composite plate 2 (Skew angle =30°, α=0.6): typical numerical results obtained with 18 basic functions ωl* a11 a13 a15 a22 a24 a26 a31 a33 a35

1 2 3 4 5 6 7 8 9

9.4343 1.05 -1.549667E-02 -1.665776E-03 -1.689070E-01 -4.314628E-03 -3.644569E-04 -2.185097E-02 2.017160E-02 1.995894E-03

10 11 12 13 14 15 16 17 18

a42 a44 a46 a51 a53 a55 a62 a64 a66

-2.591686E-02 -2.517736E-03 -3.854766E-04 -1.109130E-02 1.227799E-02 1.692331E-03 1.826523E-03 -3.078395E-03 -1.112399E-03

Table 2 Comparison of non-linear frequency ratio ωnl*/ ωl* of plate 2 for θ= 0°, α=2/3, between results obtained in references (Han and Petyt 1997) and (Harras 2001) associated with the fundamental non-linear mode shape and present results Am/H 0.2 0.4 0.6 0.8 1.0 1.2

results (Han and Petyt 1997) 1.0058 1.0232 1.0516 1.0903 1.1382 1.1941

results (Harras 2001) 1.0054 1.0214 1.0475 1.0826 1.1267 1.1759

Present results 1.0054 1.0214 1.0466 1.0821 1.1271 1.1743

Error (%) 0,020 0.088 0.281 0.400 0.472 0.903

Table 3 Convergence and accuracy of the nonlinear frequency ratios (ωnl/ωl) of clamped 5-layered angle-ply [45°/-45°/45°/−45°/45°] skew plate 1 (b/a = 1) Skew Angle θ° 0°

15°

30°

45°

Wmax/h

Present (Singha, Daripa 2007) Error% Present (Singha, Daripa 2007) Error% Present (Singha, Daripa 2007) Error% Present (Singha, Daripa 2007) Error%

0.2 1.0074 1.0068 0.06% 1.0077 1.0076 0.01% 1.0086 1.0086 0.0016% 1.0093 1.0098 0.0519%

0.4 1.0286 1.0271 0.14% 1.0299 1.0300 0.01% 1.0342 1.0340 0.01% 1.0379 1.0387 0.08%

0.6 1.0641 1.0599 0.39% 1.0675 1.0663 0.11% 1.0741 1.0749 0.08% 1.0814 1.0849 0.32%

450


Table 4 (continued) Skew Angle θ° 0°

15°

30°

45°

Wmax/h

Present (Singha, Daripa 2007) Error% Present (Singha, Daripa 2007) Error% Present (Singha, Daripa 2007) Error% Present (Singha, Daripa 2007) Error%

0.8 1.1076 1.1041 0.32% 1.1139 1.1149 0.10% 1.1275 1.1297 0.19% 1.1401 1.1458 0.50%

1 1.1609 1.1582 0.24% 1.1710 1.1744 0.29% 1.1944 1.1963 0.15% 1.2096 1.2215 0.98%

1.2 1.2189 1.2214 0.20% 1.2372 1.2431 0.48% 1.2607 1.2729 0.96% 1.2935 1.3076 1.08%

frequency ratio ωnl/ωl with respect to non dimensional maximum amplitude wmax/h is evaluated for clamped square plate and is shown in Tables 2, along with published results. It is observed that present results are in close agreement with those of assumed hierarchical finite element method (Han and Petyt 1997) and Energy balance method (Harras 2001). Next, the nonlinear free vibration behaviors of fully clamped angle-ply [45°/-45°/45°/-45°/45°] thin square and skew plates are examined in Table 3. It is observed that the results obtained here closely matches with those from (Singha, Daripa 2007) for the case of symmetrically angle ply laminated composite skew plates. It is observed that the nonlinear frequency ratio increases with the increase in amplitude of vibration, indicating hardening type of nonlinear behavior. Furthermore, for the chosen amplitude, it can be noted that, with the increase in skew angle, the degree of nonlinearity is high.

Fig. 2 Comparison of backbone curve of plate 1, 2, 3 and isotropic skew plate for θ=30°


451

In figure 2 comparison is make between non-linear frequency of the skew symmetrically angle ply laminated plate 1, the skew symmetrically laminate plate 2 and plate 3 and the skew isotropic plate for a skew angle θ=30°. It can be seen that plate 1 has a hardening effect compared to the isotropic one. Also the plate 2 and plate 3 are less hard than the isotropic skew plate.

4

Conclusion

The theoretical model established and applied to beams, plates and shells ((Benamar, Bennouna and White 1993) has been successfully applied to calculate the first non-linear mode shape of fully clamped skew symmetrically laminated plates for various aspect ratios. The present formulation has been verified with the available analytical and finite element results. Limited numerical studies are conducted to examine the effect of skew angle, fiber orientation, and aspect ratio on the large-amplitude frequency of composite skew plates. As has been shown in this work, the fundamental non-linear mode shape of the fully clamped skew symmetrically laminated plate can be obtained with sufficient accuracy by using 18 basic plate functions. This provides a more accurate description of the non-linear mode shape, compared with previous results based on the use of nine symmetric basic plate functions, since it allows the non-symmetry induced by the fiber orientation to be taken into account. Present study reveals hardening type of nonlinearity and the nonlinear frequency ratio in general increases with the increase in skew angle. Further work is needed to investigate the behaviour of higher modes, the effects of fiber orientation, and forced response of fully clamped symmetrically laminated skew plates.

References Leissa, A.W.: Vibration of Plates. US Government Printing Office, NASA SP-160, Washington, DC (1969) Benamar, R.: Non-linear dynamic behaviour of fully clamped beams and rectangular isotropic and laminated plates. Ph.D. Thesis, Institute of Sound and Vibration Research (1990) Chia, C.Y.: Geometrically nonlinear behavior of composite plates: a review. Applied Mechanical Reviews 41, 439–451 (1988) Han, W., Petyt, M.: Linear vibration analysis of laminated rectangular plates using the hierarchical finite element method-I: free vibration analysis. Computers and Structures 61, 705–712 (1996) Ribeiro, P., Petyt, M.: Multi-modal geometrical non-linear free vibration of fully clamped composite laminated plates. Journal of Sound and Vibration 225, 127–152 (1999) Harras, B.: Theoretical and Experimental Investigation of the non-Linear Behaviour of various Fully Clamped Rectangular Composite Panels at large Vibration Amplitudes. Ph.D. thesis, University Mohammed V-Agdal (2001)

452


Harras, B., Benamar, R., White, R.G.: Investigation of non-linear free vibrations of fully clamped symmetrically laminated carbon-fibre-reinforced PEEK (AS4/APC2) rectangular composite panels. Composites Sciences and Technology 62, 719–727 (2002) Han, W., Dickinson, S.M.: Free vibration of symmetrically laminated skew plates. Journal of Sound and Vibration 208, 367–390 (1997) Malekzadeh, P.: A differential quadrature nonlinear free vibration analysis of laminated composite skew thin plates. Journal of Thin-Walled Structures 45, 237–250 (2007) Singha, M.K., Daripa, R.: Nonlinear vibration of symmetrically laminated composite skew plates by finite element method. Int. J. Non-linear Mech. 42, 1144–1152 (2007) Ray, A.K., Banerjee, B., Bhattacharjee, B.: Large amplitude free vibrations of skew plates including transverse shear deformation and rotary inertia-a new approach. J. Sound Vib. 180, 669–681 (1995) Upadhyay, A.K., Shukla, K.K.: Large deformation flexural behaviour of laminated composite skew plates: An analytical approach. Composite Structures 94, 3722–3735 (2012) Benamar, R., Bennouna, M.M.K., White, R.G.: The effects of Large Vibration Amplitudes on the Fundamental Mode Shape of a Fully clamped, Symmetrically laminated rectangular plate. In: Fourth International Conference of Recent Advances in Structural Dynamics, pp. 749–760. Elsevier Applied Science (July 1990) Mallick, P.K.: Fibre-reinforced composites: Materials, Manufacturing and Design, vol. 152. Marcel Dekker, Inc., New York (1993); 2nd edn. revised and expanded Benamar, R., Bennouna, M.M.K., White, R.G.: The effects of large vibration amplitudes on the fundamental mode shape of thin elastic structures. Part II fully clamped rectangular isotropic plates. Journal of Sound and Vibration 164, 295–316 (1993) Han, W., Petyt, M.: Geometrically Nonlinear vibration analysis of thin, rectangular plates using the hierarchical finite element method-II: 1st mode of laminated plates and higher modes of isotropic and laminated plates. Computers and Structures 63, 309–318 (1997) Timoshenko, S., Weinsowsky-Krieger, S., Jones, R.M.: Mechanics of Composite Materials. International Student Edition, vol. 51. McGraw-Hill Kogakusha, Ltd., Tokyo (1975)

Scratch Adhesion Characteristics of PVD Cr/CrN Deposited on High Speed Steel and Silicon Substrates Kaouthar Khlifi, Hafedh Dhifelaoui, and Ahmed Ben Cheikh Larbi Ecole Nationale Supérieure des Ingénieurs de Tunis, Université de Tunis, 5 Rue Taha Hussein, 1008 Tunis, Tunisia [email protected], {dhafedh,ahmed.cheikhlarbi}@gmail.com

Abstract. The substrate is one of the significant parameters of technology controlling the mechanical properties of PVD coatings. The quality of the coating is highly influenced by the adhesion. An improvement of these properties can open new areas of applications. In this paper, the adhesion results for PVD Cr/CrN on high speed steel and silicon substrates are presented. The adherence of the coatings to the steel substrate was evaluated using a scratch test. In the scratch test, the critical force as the beginning of spalling or delaminating of the coating was determined. All the coatings show critical values higher than 7 N. The critical load depends on the film thickness and the substrate composition. Keywords: PVD thin films, CrN, scratch test, damage mechanisms.

1

Introduction

Today PVD thin films are widely used in order to increase the machining performance because PVD coated tools have significantly contributed to higher production rates and higher tool life's. Several parameters can be controlled to improve mechanical behavior of thin films. The substrate material is one of the significant parameters of technology controlling the mechanical properties of PVD coatings. In order to develop new PVD coatings, it is important to characterize the mechanical properties of the coating/substrate system and the deformation mechanisms in a tribological contact. Several methods are available to examine the efficiency of this system [1]. But the study of adhesion is still essential to examine coated systems interface. In this context scratch test is probably the most common and used test to understand coating failure modes of a hard coating substrate system [2]. During the conventional scratch test a diamond indenter Rockwell C flattened the surface of the coated material under an increasing normal load until failure occurs or the damage of the film is noted. The identification the coating failure mechanisms can be made by the following terms: © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_46

453

454

K. Khlifi, H. Dhifelaoui, and A. Ben Cheikh Larbi

LC1, the lower critical load, which is defined as the load where the first cracks occurred (cohesive failure), LC2, the upper critical load at which the first delaminating [3] and LC3 the critical load when the damage of the film exceeds 50%. Critical normal loads can be controlled by the coating thickness, coating hardness and substrate hardness .The present study focuses on the adhesion of Cr single layer and CrN multilayer deposited on 100C6 Steel substrate and Silicon substrate by magnetron sputtering system.

2

Experimental Details

Substrates were used in this study is 100C6 steel (AISI 52100) and Si. Cr/Si single layer, CrN/Cr/Si multilayer and CrN/Cr/100C6 coatings were deposited with a magnetron sputter unit with a pure chromium target. Thicknesses of thin films studied were illustrated in table 1. The morphological and microstructural characterizations were evaluated using an atomic force microscopy (AFM) and a Scanning electron microscopy (SEM). The scratching response of the coating– substrate composites were evaluated using the CSM Instruments with Rockwell C diamond styli with radius 100μm. A normal load range of 3–28N, a scratch length of 3 mm and a scratching velocity of 6mm/min were used in the experiments. Hardness of 100C6 steel and Silicon substrate are respectively 11GPa and 8GPa. Young's modules of both substrates are respectively 200 GPa and 130 GPa. Table 1 Thickness and roughness of Cr/Si, CrN/Cr/Si and CrN/Cr/100C6 coatings

Coatings Cr/Si CrN/Cr/Si CrN/Cr/100C6

Thickness (μm) 1,6 2 3,2

Roughness (nm) 10 4 14

3


3.1

Morphological and Microstructured Examination

Results obtained by AFM observation are illustrated in Figure 1. The AFM examination revealed the presence of domes and craters which are uniformly distributed over the surface. However, we have noticed a difference in the size of the domes and craters that characterize the morphology. The passage from coating pure chromium based to another nitrogen rich, a less rough surface is observed which was characterized by bigger and larger grains. With the introducing of nitrogen in the plasma roughness pass from 10 for Cr to 4 nm for CrN. Referring to the structure of zones described by Thornton [4], we go from the T-zone (less dense Zone) to a zone of type II with peaks pyramidal columns. Coating de-posited on steel appears rougher due to the high substrate surface

Scratch Adhesion Characteristics of PVD Cr/CrN Deposited on High Speed Steel

455

roughness when comparing with Si surface roughness. Cross-section SEM of all coatings showed a columnar structure as shown in Figure 2 similar to that proposed by Thornton [4] in his model of thin film growth.

Fig. 1 Three-dimensional AFM images of Cr/Si single layer, CrN/Cr/Si and CrN/Cr/100C6 multilayer

Fig. 2 SEM micrographs CrN/Cr/Si single layer

3.2

Scratch Test

Scratch methodology was used to evaluate the relative adhesive and cohesive failure. Progression of scratching is accompanied by successive degradations described by: cohesive failure signaled by micro cracks at the edge, local chipping of the coating from its substrate and, finally, a total loss of adhesion between the two constituents. In this study, the corresponding critical forces are summarized in table 2.

456


Table 2 Critical loads and friction coefficients of CrN/Cr/100Cr6, CrN/Cr/Si and Cr/Si coatings

Coatings LC1 LC2 LC3

CrN/Cr/100C6 5,68 12,79 13,54

CrN/Cr/Si 3,23 7,42 12,34

Cr/Si 4,27 9,16 13,16

Figures 3-5 shows the tracks were corresponding to the Cr/Si, CrN/Cr/Si and CrN/Cr/100C6 coating after scratch test.

Fig. 3 Optical micrographs of the scratch track of Cr/Si coating

Fig. 4 Optical micrographs of the scratch track of CrN/Cr/Si coating

Scratch Adhesion Characteristics of PVD Cr/CrN Deposited on High Speed Steel

457

Fig. 5 Optical micrographs of the scratch track of CrN/Cr/100C6 coating

Under the same loading conditions, the coatings behavior is different. In the single layer structure, with the increase of the scratch load, cracks have been developed throughout the cross section of the film and have spread rapidly. In this type of coating, adhesive failure occurred for low load comparing with the other coatings and we observed a total chipping of the film and exposure of substrate at higher load. The critical load is the normal load at the first coating failure as detected. This load was determined with combining observations of scratches by optical microscopy with measurements of acoustic emission, of normal and tangential forces and of the residual depth during scratch testing. Several authors [5,6] have linked the appearance of the first crack (cohesive failure) at a first critical load noted by LC1 and the upper critical load at which the first delaminating at the edge of the scratch track occurred (adhesion failure) by LC2. The critical loads LC1, LC2 and LC3 values were presented in figure 3-5. Higher critical loads of adhesion were obtained for CrN/Cr/100Cr6. In multilayer structure coatings, cracks develop mostly in the vicinity of the upper surface, and the layers interfaces can change the initial crack orientation.

4

Conclusion

In this work, PVD thin films were deposited on different substrate materials, 100C6 (AISI 52100) steel and Si by PVD magnetron sputtering system. The morphological and microstructural proprieties were evaluated using an atomic force microscopy (AFM) and a Scanning electron microscopy (SEM). Adhesion quality was studied with scratch tests. Damages mechanisms that were occurred during scratching were further discussed. The AFM examination revealed the presence of domes and craters which are uniformly distributed over the surface. CrN/Cr/Si has a less rough surface because of a small grains size. SEM cross section of all coatings showed a columnar structure

458


The results obtained in the present study show that the coating/substrate damage mechanisms during scratching and the critical normal loads obtained are strongly dependent on the mechanical properties of substrate material. The results obtained in the present study show that the coating/substrate damage mechanisms during scratching and the critical normal loads obtained are strongly dependent on the mechanical properties of substrate material. Acknowledgements. This work is partially supported by BALZER FRANCE. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.

References [1] Sveen, S., Andersson, J.M., M’Saoubi, R., Olsson, M.: Scratch adhesion characteristics of PVD TiAlN deposited on high speed steel, cemented carbideand PCBN substrates. Wear 308, 133–141 (2013) [2] Holmberg, K., Laukkanen, A., Ronkainen, H., Wallin, K., Varjus, S.: A model for stresses, crack generation and fracture toughness calculation in scratched TiN- coated steel surfaces. Wear 254, 278–291 (2003) [3] Khlifi, K., Ben Cheikh Larbi, A.: Investigation of the adhesion of PVD coatings using various approaches. Surface Engineering 29, 555–560 (2013) [4] Thornton, J.A.: J. Vac. Sci. Technol. 11, 666–670 (1974) [5] Yıldız, F., Alsaran, A., Celik, A., Efeoglu, I.: Surf. Eng. 26(6), 578–583 (2010) [6] Kataria, S., Kumar, N., Dash, S., Ramaseshan, R., Tyagi, A.K.: Surf. Coat. Technol. 205, 922–927 (2010)

Load Sharing Behavior in Planetary Gear Set Ahmed Hammami1,2, Miguel Iglesias Santamaria2, Alfonso Fernandez Del Rincon2, Fakher Chaari1, Fernando Viadero Rueda2, and Mohamed Haddar1 1

Laboratory of Mechanics, Modeling and Production (LA2MP), National School of Engineers of sfax, BP1173 – 3038 – Sfax – Tunisia [email protected], [email protected], [email protected] 2 Department of Structural and Mechanical Engineering, Faculty of Industrial and Telecommunications Engineering, University of Cantabria, Avda de los Castros s/n 39005 – Santander - Spain [email protected], [email protected], [email protected]

Abstract. The objective of this paper is to study the effects of meshing phase between planets, the effects of the gravity of carrier and the planet position error on the load sharing behavior in planetary gear set. These effects will be studied numerically under the stationary condition and will be validated experimentally by a back-to-back planetary gear test bench. In this test bench, strain gauges are installed on each planet’s pin hole in order to compare strains in the pinhole of each planet. Keywords: Planetary gear, load sharing, meshing phase, gravity, planet position error.

1

Introduction

Planetary gear can transmit higher power because they use multiple power paths formed by each planet branches. This allows the input torque to be divided between the n planet paths, reducing the force transmitted by each gear mesh. Under ideal conditions, each planet path carries an equal load. Nevertheless, planetary gears have inevitable manufacturing and assembly errors. So, the load is not equally shared amongst the different sun-planet and planet-ring paths, which can be a problem in terms of both dynamic behavior and durability. Many researchers have done significant works on the subject of planetary gear load sharing. Their works were based on transmission modeling and assessed by experimental tests. Kahraman (Kahraman, 1994) used a discrete model to study the influence of carrier pin hole and planet run-out errors on planet load-sharing characteristics of a four-planet system under dynamic conditions. He employed later (Kahraman, 1999) a planet load-sharing model to determine the static planet © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_47

459

460

A. Hammami et al.

load sharing of four-planet systems and presented experimental data for validation of the model predictions. Iglesias (Iglesias et al, 2013) studied the effect of planet position error on the load sharing and transmission error. Singh (Singh, 2005) found that the tangential pin position error has a greater effect on the load sharing than the radial pin position error. Ligata (Ligata et al, 2008) proved experimentally that for the same amount of error, the degree of inequality in the planet loadsharing behavior increases with the number of planets in the system. Guo and Keller (Guo and Keller 2012) presented a three-dimensional dynamic model which take into account to the addressing gravity, bending moments, fluctuating mesh stiffness, nonlinear tooth contact, and bearing clearance. They validated this model against the experimental data. In this paper, the effects of meshing phase, the gravity and the error position of the pin hole of planets on the load sharing characteristics are studied numerically and validated experimentally with back-to-back planetary gear.

2

Description of the Test Bench

The test bench is composed of two identical planetary gear sets with the same gear ratio (Fig. 1). The first planetary gear is a “test gear set” and the second planetary gear is a “reaction gear set” having a free ring. An arm is fixed on this ring and allowing the introduction of external load ((Hammami et al 2015) and (Hammami et al 2014a)).

Fig. 1 Back-to-back planetary gear test bench

Load Sharing Behavior in Planetary Gear Set

461

The two planetary gears are connected back-to-back in order to provide a mechanical power circulation: the sun gears of both planetary gear sets are connected through a common shaft and the carriers of both planetary gear sets are connected to each other through a rigid hollow shaft (Hammami et al 2014b). In order to compare the load sharing between the tests planets, three strain gauges are installed in the pin holes of each planet in the tangential direction of the test carrier (Fig. 2).

Fig. 2 Strain gauges mounted in each pin hole of the test carrier planets

The wires from the strain gauges are connected to the acquisition system through a hollow slip ring (HBM SK5/95) which is installed with the hollow shaft that connects the carriers (Fig.3). Strains gauges are used in quarter bridge configuration.

Fig. 3 Instrumentation layout

462

A. Hammami et al.

Strain signals registered by strain gauges will be acquired by Programmable Quad Bridge Amplifier module (PQBA) of the acquisition system “LMS SCADAS 316 system”. This module can support four channels of strain transducers, piezo-resistive or variable capacitor sensors. Every channel is connected by 6 poles LEMO connector. The data will be processed with the software “LMS Test.Lab” to visualize time history of strains. Additionally, an optic tachometer (Compact VLS7) is combined with pulse tapes along the axe in order to measure its instantaneous angular velocity. It was placed along the hollow shaft of carriers.

3

Numerical Results

The effect of meshing phase, the gravity of planets carrier and the position error of planets on the load sharing behavior is studied in this part. First of all, we define the planet load sharing ratio (LSR) as the ratio of the meshing torque due to sun-planet(i) and ring-planet(i) meshes of planet (i) by the meshing torque of all planets. L Pi =

Tmesh ( Pi )

(1)

n

∑ Tmesh ( Pi )

i =1

3.1

Effect of Meshing Phase

For the case of equally spaced planets and in phase meshes gear (sun planets and ring planets), the planet load sharing factor is equal to 1/N (N: number of planets). In our case, planets are equally spaced and gear meshes functions are sequentially phased. Z jψ i 2π

N

≠ n and ∑ Z jψ i = m π i =1

( j = r , s)

(2)

Z j is the number of tooth of the gear (j). ψ i is the angle position of the planet

(i). n and m are integer. The gear meshes stiffness of ring-planets gear pairs (Fig. 4) and sun-planets gear pairs are modeled using the finite element model (Fernandez Del Rincon et al 2013). The dynamic response is computed according to the procedure given in (Kahraman, 1994) and the load sharing ratio is computed according to equation (1). Fig.5 shows the planet load sharing ratio for the nominal position of planets (faultless system) for 100 N.m of input torque and for speed of motor 165 rpm.


463

8

7.5

x 10

ring-planet1 ring-planet2 ring-planet1

Gear mesh stiffness (N/m)

7 6.5 6 5.5 5 4.5 4 0

0.02

0.04

0.06

0.08

0.1 Time (s)

0.12

0.14

0.16

0.18

0.2

Fig. 4 Gear mesh stiffness ring-planets

0.355 Planet1 Planet2 Planet3

0.35 0.345

LSR

0.34 0.335 0.33 0.325 0.32 0.315 0.31 0

0.02

0.04

0.06

0.08

0.1 Time (s)

0.12

0.14

0.16

0.18

0.2

Fig. 5 Planet load sharing ratio for the nominal position of planets

In this case, the LSR for all planets fluctuates slightly around the 1/N value (N=3: number of planets) because the number of tooth in contact changes. The fluctuation of LSR of each planet is the same with a phase shift of 2π/3. This phase is induced by the fact that gear mesh sun-planets and ring-planets are sequentially phased.

3.2

Effect of Gravity of Carrier

The gravity of carriers has effect on the distances between ring-planets and sunplanets so that they are variable during the running of gearbox. Then, the values of gear mesh stiffness decrease as the distance between gears (sun-planets and ring-planets) increase and vice versa. Fig.6 shows the evolution of gear mesh stiffness ring-planet1 during one period of rotation of carrier with effects of gravity of carrier. This period can be divided into four subperiods:

464

A. Hammami et al.

- The first subperiod : where the planet 1 is in the upper position. The values of gear mesh stiffness are low. - The second subperiod where the planet 1 is in an intermediate position in (the left). The values of gear mesh stiffness are medium. - The third subperiod where the planet 1 is in a lower position. The values of gear mesh stiffness are high. - The fourth subperiod where the planet 1 is again in an intermediate position (the right). The values of gear mesh stiffness are medium.

8

7

x 10

ring-planet1

Gear mesh stiffness (N/m)

6.5

6

5.5

5

4.5

4 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (s)

Fig. 6 Evolution of gear mesh stiffness ring-planets with gravity effects

0.5

0.45

3

2

1

3

Planet1 Planet2 Planet3

LSR

0.4

0.35

0.3

0.25 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (s)

Fig. 7 Planet load sharing ratio with gravity effects

The gravity of carrier will lead to a periodic fluctuation of the maximum amplitude of gear mesh stiffness with a period corresponding to that of the rotation of the carrier Tc. The higher values of the gear mesh stiffness of ring-planets are successively on the planet 3 then planet 2 then planet 1.


465

Fig.7 shows the evolution of LSR during one period of rotation of carrier. This period is divided into three sub-periods of Tc/3. For each sub-period, the LSR of each planet does not oscillate with the same amplitude; there is always a planet that will include the maximum (planet1then planet2 then planet3).

3.3

Effect of Planet Position Error

If a planet has an error “e” on the position of its pin hole, and all other planets are at their ideal position, then the force due to this error is given by (Singh, 2010):

Fe = K eff .e

(4)

Keff is the cumulative stiffness due to meshing stiffness of the contact at the sun–planet Kps and planet–ring Kpr, and the planet bearing stiffness Kb. Keff is defined as (Ligata et al, 2009):

1 1 1 = + K eff K b K ps + K pr

(5)

In our case, the planet 1 has an error “e1=62μm” and the planet 2 has an error “e2=-60μm”. The LSR in this case is represented in Fig.8. 0.7 0.6 Planet 1 Planet 2 Planet 3

0.5

LSR

0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (s)

Fig. 8 Planet load sharing ratio with planets position errors

The position errors of planets 1 and 2 have an important effect in the LSR. Planet 1 which has a positive error anticipates the contact, being preloaded before planets 2 and 3 begin to transmit load; whereas planet 3 which has a negative error is preloaded after planet 2 and 1.This defects are in tangential direction and they have an effect more important in the LSR than the effect of gravity of carrier which is a variable radial error (Bodas and Kahraman 2004).

466

4

A. Hammami et al.

Correlation with Experimental Results

Strain-time histories for three planets system having errors “e1=62μm” and “e2=60μm” are shown in Fig.9. It is clear that the positioning error has an important effect in the strain of each pin hole of planet. 170.00

1.00

Amplitude

Time Planet1 Time Planet2 Time Planet3

uE Amplitude

F F F

10.00

0.00 0.00

s

1.84

Fig. 9 Variation of the measured planets strains for the speed motor 165 rpm 1

Planet1(Experiments) Planet2(Experiments) Planet3(Experiments) Planet1(Numerical) Planet2(Numerical) Planet3(Numerical)

0.9 0.8 0.7

LSR

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (s)

Fig. 10 Evolution of measured and calculated LSR

The computational results of the load sharing ratio are compared to the measured LSR (Fig. 10). In general, the calculated load sharing agrees with the measured data. In fact, the planet position errors have an important effect. In addition, a slight deviation of the load sharing evolution due to the effect of gravity of carrier is observed on the numerical and experimental results whereas the effect of meshing phase is observed only in the numerical results because the recorded signals presents noise due to the contact between brushes and slip ring. So, a signal processing was necessary.


5

467

Conclusion

In this paper, the planets load sharing behaviour is studied and correlated with those obtained experimentally using strain gages on planets of a back-to-back planetary gear test bench. The effect that planets are sequentially phased is included to the model and as results; the load sharing ratio for all planets fluctuates slightly around the 1/N value. The time evolution of LSR of each planet bends a little bit every Tc/3 by the effect of gravity of carrier. However, the planets position errors have an important effect on the LSR of each planet. The final numerical results with the three effects agree with the measured strains on planets. Future investigation will be focused mainly on the planets load sharing under non-stationary condition (run-up and run-down regimes). Acknowledgements. This paper was financially supported by the Tunisian-Spanish Joint Project N° A1/037038/11. The authors would like also to acknowledge project “Development of methodologies for the simulation and improvement of the dynamic behavior of planetary transmissions DPI2013-44860” funded by the Spanish Ministry of Science and Technology. Acknowledgment to the University of Cantabria cooperation project for doctoral training of University of Sfax’s students.

References Bodas, A., Kahraman, A.: Influence of carrier and manufacturing errors on the static lad sharing behavior of planetary gear sets. Bulletin of the Japan Society of Mechanical Engineers 47(3), 908–915 (2004) Fernandez del Rincon, A., Viadero, F., Iglesias, M., García, P., de-Juan, A., Sancibrian, R.: A model for the study of meshing stiffness in spur gear transmissions. Mechanism and Machine Theory 61(2013), 30–58 (2013) Guo, Y., Keller, J.: Combined effects of gravity, bending moment, bearing clearance, and input torque on wind turbine planetary gear load sharing. American Gear Manufacturers Association technical paper, 12FTM05 (2012) Hammami, A., Fernández, A., Viadero, F., Chaari, F., Haddar, M.: Modal analysis of backto-back planetary gear: experiments and correlation against parameter model. Journal of Theoretical and Accepted Applied Mechanics 53(1) (2015) Hammami, A., Fernández, A., Viadero, F., Chaari, F., Haddar, M.: Dynamic behaviour of back-to-back planetary gear in run up and run down transient regimes. Journal of Mechanics (2014a), doi:10.1017/jmech.2014.95 Hammami, A., Fernández, A., Chaari, F., Viadero, F., Haddar, M.: Dynamic behaviour of two stages planetary gearbox in non-stationary operations. Mechatronic System: Theory and Application, 23–35 (2014b) Iglesias, M., Fernandez, A., De-Juan, A., Sancibrian, R., Garcia, P.: Planet position errors in planetary transmission: Effect on load sharing and transmission error. Frontiers Mechanical Engineering 8(1), 80–87 (2013) Kahraman, A.: Load Sharing Characteristics of Planetary Transmissions. Mechanism and Machine Theory 29, 1151–1165 (1994)

468

A. Hammami et al.

Kahraman, A.: Static Load Sharing Characteristics of Transmission Planetary Gear Sets: Model and Experiment. SAE Paper No. 1999–01–1050 (1999) Ligata, H., Kahraman, A., Singh, A.: An experimental study of the influence of manufacturing errors on the planetary gear stresses and planet load sharing. Journal of Mechanical Design 130 (2008) Ligata, H., Kahraman, A., Singh, A.: A closed-form planet load sharing formulation for planetary gear sets using a translational analogy. Journal of Mechanical Design 131, 021007 (2009) Singh, A.: Application of a system level model to study the planetary load sharing behavior. Journal of Mechanical Design 127, 469–476 (2005) Singh, A.: Load sharing behavior in epicyclic gears: Physical explanation and generalized formulation. Mechanism and Machine Theory 45, 511–530 (2010)

Low Velocity Impact Behavior of Glass FibreReinforced Polyamide Jamel Mars1, Mondher Wali1, Remi Delille2, and Fakhreddine Dammak1 1

Mechanical Modelisation and Manufacturing Laboratory (LA2MP), National Engineering School of Sfax, B.P W3038, Sfax, University of Sfax, Tunisia {Jamelmars,mondherwali}@yahoo.fr, [email protected] 2 Laboratory of Industrial and Human Automation control (LAMIH / C2S), University of Valenciennes [email protected]

Abstract. The low velocity impact behavior of composites made of polyamide (PA) as matrix and glass fibre as reinforcement has been investigated. The assessment of the impact behavior has driven the need to perform tensile tests to determine the elasto-plastic behavior of the composites. The specimens were manufactured by injection molding techniques for the experimental tensile testing. ABAQUS/EXPLICIT for finite element modeling is employed in order to predict the impact behavior of glass fibre-reinforced polyamide. The determinations of the impact force history and elasto-plastic structure deflection are the most important objectives in impact engineering structures design. Keywords: Mechanical Composite, PA66, mechanical characteristics.

1

Introduction

Fibre-reinforced composites are widely used in many engineering applications such as automobile and aerospace due to their light weight, high strength and stiffness, resistance to chemical and environmental agents, design freedom and manufacturing advantages. During their manufacture, maintenance and service life, this Fibre-reinforced composites may suffer different damage, of which low velocity impact is considered as one of the most important and dangerous, because it can induce internal damage in the form of delamination, matrix cracking, local permanent deformation, debonding and fiber breakage, leading to a significant strength reduction of the structure. Numerous investigations dedicated to the problem of the impact behavior of composite materials are performed by several authors such as (Wali et al. 2011), (Dhakal et al. 2012) and (Yang et al. 2013). The published results show that the impact resistance of composite structures depends, in a complex way, on the properties of composite structures (material, thickness, laminate stacking sequence in © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_48

469

470

J. Mars et al.

case of laminated composites), properties of impactor (mass, velocity, energy, shape of impacting head) and experimental setup (clamping conditions of the test specimen) (Giangiacomo et al. 2012). In addition, research has shown that composites are capable of absorbing energy and dissipating it by various fracture and elastic processes when subjected to a low velocity impact (Cantwell and Morton 1991). A number of investigations were conducted to assess the impact on thermoplastic composites reported in the works of (Xu et al. 2011) and (Simeoli et al. 2014). Glass fiber-reinforced polyamide (PA66) is one of the most widely used thermoplastic composites in the engineering industry and the security imposes to perform impact behavior. Therefore, it becomes necessary to know the behavior of this composite material which is very sensitive to elasto-plastic impact promoting high levels of strain, leading to failure and damage. Despite increased use in engineering applications, few published work on the impact behavior of polyamide short fiber-reinforced thermoplastic composites is found. The fracture toughness and impact behavior of a range of glass-fibrereinforced polyamides of high fiber content, which were promoted as potential metal replacement materials, are examined by (Akay et al. 1995). The compressive behavior of Nylon 6 and Nylon 66, on wide range of strain-rates, was investigated by (Benaceur et al. 2008) using Hopkinson bar tests. A localized low velocity impact experiments and simulations were conducted by (Mouti et al. 2010, 2013) on glass fiber-reinforced polyamide automotive product to investigate typical flying stones impact scenarios. The purpose of this paper is to predict the behavior of an elasto-plastic glass fibre-reinforced polyamide structures under low velocity impact. The mechanical elasto-plastic model of glass fibre-reinforced polyamide materials for different glass fibre volume fractions is then modeled basing on experimental tensile tests. The model is implemented into the commercial finite element code ABAQUS/Explicit to predict the Low velocity impact behavior of circular glass fibre-reinforced polyamide. Numerical results, including contact force and displacement of the composite plate subjected to low-velocity impact, are calculated.

2

Experimental

2.1

Materials

Two commercial polyamide 66 with 30 wt.% of short glass fibre (HERAMID A NAT with 30 wt.% glass fibre, provided by RADICI GROUP) and natural polyamide 66 (A HERAMID NAT) were used to produce molded glass fibre-reinforced composites with 00, 10, 20 and 30 wt.% .They were designed as PA66_00, PA66_10, PA66_20 and PA66_30, respectively. The specimens were made by injection molding (Figure 1).

Low Velocity Impact Behavior of Glass Fibre-Reinforced Polyamide

4771

Fig. 1 Molded samples

2.2

Tensile Tests

Mechanical testing consisted of tensile tests. The tensile tests have been carrieed out using an Instron Electtropuls tensile testing machine with crosshead speed off 1 mm/min (Figure 2). Thee jaws used for attachment of the test specimens arre

Fig. 2 Instron Electropuls ten nsile testing machine

Fig. 3 Shape and dimensionss of the specimen, according to ISO 527-2 standard

472

J. Mars et aal.

controlled pneumatically, 4 bars. Three specimens of each composition werre tested and the average vallue reported. Strains were measured with a RUDLPH laaser extensometer. Markerrs (white color and width 3 mm) are placed in each teest specimen for measuring the t deformation in the active area during the tensile tesst. Specimens had the shapee and dimensions reported in Figure 3, according to ISO 527-2 standard. Tensile testing t was carried out in air at 20°C and 50% relativve humidity.

2.3

Analysis of fibrre Length Distributions

The fibre orientation distrribution in a material sample was examined by using thhe X-Ray microtomography machine (Skyscan 1172). The local variation of the annisotropy from the shell to the core of the molded tensile sample is captured and innformation about the local average fibre orientation angle is observed. Figure 4 shows that th he orientation of fibers in assemblies is moderately influuenced by the processes ussed to manufacture the samples. Fibre content-images foor specimens used in tensilee tests is examined and illustrated in figure 5 to evaluaate the global anisotropy .Fig gure 5 shows that, the amount of homogeneously distrributed fibers is greater.

PA66_10

PA66_20

PA66_30

Fig. 4 Microtomography of reeinforced PA66 (sagittal scanning in the test specimen middle)

Fig. 5 Fiber content-images for specimens PA66_10, PA66_20, PA66_30


3


3.1

Quasi-static Characterization

473

Tensile test data was captured for different weight fractions: 0, 10, 20 and 30 wt%. For each material tested, three specimens were used. The degree of repeatability was found excellent in all cases. For example, in tensile tests of PA_00 at a strain rate of 1 mm/min, this repeatability is illustrated in Figure 6. The medium curve of three tests is chosen for all polyamides tested. Tensile test data was then used to plot stress strain relations. Data captured from the testing machine and the laser extensometer was plotted in Figure 7. Figure 7 shows Stress-strain curves of natural PA66_00, PA66_10, PA66_20 and PA66_30. These results are in good agreement with literature (Ghorbel et al. 2011). 50

Stress (MPa)

40 30 Test 1 Test 2 Test 3

20 10 0 0

10

20 Strain %

30

40

Fig. 6 Repeatability of simple tensile test of PA66_00

120

PA66_00 PA66_10 PA66_20 PA66_30

Stress (MPa)

100 80 60 40 20 0 0

10

20 Strain %

30

40

Fig. 7 Uniaxial stress-strain curves of PA66_00, PA66_10 , PA66_20 and PA66_30

474

J. Mars et al.

The increasing of the fiber content has the effect: • An improvement in elastic modulus • An increase in tensile strength • A reduction in fracture strain. The materials behavior changes from ductile to brittle behavior.

3.2

Identification of the Hardening Law

The mechanical elasto-plastic model of glass fibre-reinforced polyamide composites for different glass fibre volume fractions must be modeled to accurately predict the low velocity impact responses. However, the least-squares method was used to fit the Stress/strain curves to the experimental tensile data points and the mechanical parameters of glass fibre-reinforced polyamide material are determined. The elastic-plastic material model was implemented in the commercial nonlinear finite element code ABAQUS/EXPLICIT for studying the impact behavior of polyamide. The mechanical elasto-plastic model used during the identification phase is defined by

σ = σ e + Q (1 − e

− β .ε p

) + K .ε p

(1)

where σ e is the yield stress, Q, β and K are the mechanical parameters of the elasto-plastic model and ε p is the plastic deformation. Finally, the comparison of the numerical approach with the experimental data shows the accuracy of the mechanical elasto-plastic model. Figure 8 shows that the numerical approach is in good argument with the experimental result for PA66_20 material. This approached law is used in next section when studying the impact behavior of glass fibre-reinforced polyamide. Table 2 shows the values of the mechanical parameters for the glass fibrereinforced polyamide during the identification phase. Table 1 Identified mechanical parameters of the reinforced polyamide

Yield stress E (MPa) Q (MPa) β (MPa) K (MPa)

e (MPa)

PA66_00 15 1100 23.5 30 72

PA66_10 25 2350 35.1 144 83

PA66_20 30 4200 47 198 103

PA66_30 34 6000 72.6 198 135


475

80 Stress (MPa)

60 40

Experience

20

Num Approach

0 0

0.02 Strain0.04

0.06

Fig. 8 True Stress/strain curves for PA66_20

3.3

Finite Element Modeling and Validation of Impact Model  Finite element modeling

Various procedures can be used in ABAQUS for modeling the elasto-plastic impact phenomena. The accurate modeling requires an appropriate selection of contact modeling, element type, resolution method and number of element in plate thickness. The determinations of the impact force history and elasto-plastic structure deflection are the most important objectives in impact engineering structures design. For studying the impact ABAQUS 6.10 version is used. A clamped circular plate impacted at its center by a cylindrical impactor with hemi-spherical nose is considered. The plate with radius R=60mm and thickness h=1mm is made of aluminum alloy. The initial velocity of impactor is V0 (Figure 9). Proj Rp h

V0

R

Fig. 9 Cylindrical projectile impacted circular plate

The presence of the rotational symmetry axis, can simplify the threedimensional impact problem in an axisymmetric problem. The problem can thus be analysed using 2D axisymmetric model. The plate was meshed using the structured meshing technique and 6000 4-nodes per elements with reduced integration (CAX4R in ABAQUS). The impactor mesh consisted of 433 4-nodes per elements (CAX4R in ABAQUS).

476

J. Mars et al.

 Validation of the contact model

Given that the impact model needs reliable data for simulation, we need to validate the mechanical properties and the contact law introduced into numerical model. For that, a comparative study between the numerical predictions and experimental response found in literature is established. The plate material is assumed to be an isotropic and homogeneous, and the elastic, perfectly plastic material and the Von Mises yield criterion, J2, is employed same as in the work of (Chen et al. 2007). In this section, the deformable impactor is assumed to be elastic. For the validation, the mechanical properties of both impactor and plate are illustrated in Table 2. The impactor velocity predicted by the present numerical model and the result obtained by experimental data tested through LDA technique (Chen et al. 2007) are plotted in Figure 10. One can note from this figure that the impactor velocity is in good agreement with the experimental test. 40

Velocity (m/s)

30 20 10 0 Chen et al. 2007 Present Axi-model

-10 -20 0

0.5 Time (ms)

1

Fig. 10 Validation: impactor velocity time responses (mp = 54.4 g, Rp=6.35mm, V0 = 29.9 m/s) Table 2 Mechanical properties of impactor and circular plate Parameters

Material

Young modulus Poisson’s ratio ν E ( GPa)

(

Density

ρ Kg / m3

)

Yield stress

σ e ( MPa)

Circular Plate

Aluminum

69

0.3

2600

290

Impactor

Steel

200

0.3

7800

-

3.4

Impact Behavior of Glass Fiber Reinforced Polyamide

The dynamic behavior of a clamped circular plate, of natural PA66_00, PA66_10, PA66_20 and PA66_30, subjected to impact by a projectile is discussed in the following section. The target circular plate with radius R = 60mm and thickness


477

h = 4mm whose density, Young’s modulus, Poisson ratio, yield stress and σe are already mentioned in the previous paragraph. The impactor has the same characteristics described above except that the mass and speed become respectively 1.369 Kg and V0 = 3.418 m / s. The evolution of plate center displacement according to the time is presented in Figure 11 for incidental impact energy of 8J for PA66_00, PA66_10, PA66_20 and PA66_30.

0

2

4

Time (ms) 6 8

10

12

14

Displacement (m)

0.000 -0.002 -0.004 -0.006 PA66_00

-0.008

PA66_10

-0.010

PA66_20

-0.012

PA66_30

Fig. 11 Center plate displacement time responses for impact energy of 8J

Force MAX (KN)

3

PA66_00 PA66_10 PA66_20

2

PA66_30

1

0 0

2

4

6 Time (ms)

8

10

12

Fig. 12 Impact force versus time curves for an impact energy of 8J

Figure 12 presents examples of force-times curves obtained for impact energy of 8 Joules. One can note that an increase in values of the maximal impact forces when varying the glass fibre volume fractions introduced in polyamide, but the impact time decrease greatly by increasing the glass fibre volume fractions.

478

J. Mars et al.

According to the displacement response and impact force as shown in Figures 11 and 12, it can be noted that the central deflection keeps decreasing until the peak force appears. Also, it can be noticed that the circular plate will rebound when the projectile is completely separated, then the circular plate will be in a free vibration mode around the permanent deflection position.

4

Conclusion

Impact behaviors at low velocity of glass fibre-reinforced polyamide are investigated for different glass fibre volume fractions. The assessment of the impact behavior has driven the need to perform tensile tests to determine the elasto-plastic model of the polyamides. The X-Ray microtomography machine is used to check the glass fibre distribution. The mechanical elasto-plastic model used can predict the stress-strain behavior of the studied materials. The elastic-plastic model was implemented in ABAQUS/EXPLICIT for studying the behavior of the composites under low velocity impact. The results have shown that the impact force increase with increasing fibre volume fractions.

References Akay, M., O’Regan, D.F., Bailey, R.S.: Fracture Toughness and impact behaviour of glassfibre-reinforced polyamide 6,6 injection mouldings. Composites Science and Technology 55, 109–118 (1995) Benaceur, I., Othman, R., Guegan, P., Dhieb, A., Damek, F.: Sensitivity of the flow stress of Nylon 6 and Nylon 66 to strain-rate. International Journal of Modern Physics B 22, 1249–1254 (2008) Cantwell, W.J., Morton, J.: The impact resistance of composite materials-a review. Composites 22, 347–362 (1991) Dhakal, H.N., Zhang, Z.Y., Bennett, N., Reis, P.N.B.: The low-velocity impact response of non-woven hemp fibre reinforced unsaturated polyester composites: Influence of impactor geometry and impact velocity. Composite Structures 94, 2756–2763 (2012) Chen, L.B., Xi, F., Yang, J.L.: Elastic-plastic contact force history and response characteristics of circular plate subjected to impact by a projectile. Acta Mech. Sin. 23, 415–425 (2007) Ghorbel, A., Saintier, N., Dhiab, A., Dammak, F.: Etude du comportement mécanique d’un polyamide 66 chargé de fibres de verre courtes. Mécanique & Industries 12, 333–342 (2011) Minak, G., Kostic, S.C., Šoškic, Z., Radovic, N.: Equivalent stiffness as measure of low velocity impact damage of complex composite structures. Facta Universitatis: Mechanical Engineering 10, 91–104 (2012) Mouti, Z., Westwood, K., Kayvantash, K., Njuguna, J.: Low Velocity Impact Behavior of Glass Filled Fiber-Reinforced Thermoplastic Engine Components. Materials 3, 2463–2473 (2010)


479

Mouti, Z., Westwood, K., Long, D., Njuguna, J.: An experimental investigation into localised low-velocity impact loading on glass fibre-reinforced polyamide automotive product. Composite Structures 104, 43–53 (2013) Simeoli, G., Acierno, D., Meola, C., Sorrentino, L., Iannace, S., Russo, P.: The role of interface strength on the low velocity impact behaviour of PP/glass fibre laminates. Composites 62, 88–96 (2014) Wali, M., Abdennadher, M., Fakhfakh, T., Haddar, M.: Dynamic analysis of an elastoplastic sandwich subjected to low velocity impact. Multidiscipline Modeling in Materials and Structures 7, 184–206 (2011) Wali, M., Abdennadher, M., Fakhfakh, T., Haddar, M.: Vibration response of sandwich plate under Low-velocity impact loading. WSEAS Transactions on Applied and Theoretical Mechanics 6, 27–36 (2011) Xu, J., Li, Y., Ge, D., Liu, B., Zhu, M.: Experimental investigation on constitutive behavior of PVB under impact loading. International Journal of Impact Engineering 38, 106–114 (2011) Yang, L., Yan, Y., Kuang, N.: Experimental and numerical investigation of aramid fibre reinforced laminates subjected to low velocity impact. Polymer Testing 32, 1163–1173 (2013)

Burst Pressure Estimation of Corroded Pipeline Using Damage Mechanics Djebbara Benzerga Department of Mines and Metallurgy, LSCMI, Faculty of Mechanical, University of Science and Technology of Oran Mohamed Boudiaf, B.P. 1505 El M’naouer 31000 Oran, Algeria [email protected]

Abstract. Pipelines are being widely employed worldwide as means of conveyance of crude oil and its derivatives. Especially in the south and in the north of Algeria many pipelines connect oil fields to oil refineries. Nevertheless, a considerable distance is covered crossing hills, in which landslides could change not only the pipelines alignment but also the stresses. Besides, landslides may cause cracks in the pipes. Furthermore, both the close contact with soil and the action of weather can provide the corrosion of the pipes, which will reduce the cross section area, allowing the formation of disturbed flow areas, and also will develop stress concentrated regions on the pipe wall. Generally, the main cause of high-pressure gas and oil pipeline ruptures is metal loss in a pipe wall from corrosion. Particularly, SONATRACH company data show that corroded defects (general corrosion and pitting corrosion) are the primary causes of accidents. Corrosion is one of the most common causes of accidents involving pipelines. To avoid these undesirable situations, computational models are playing an important role, as they are able to predict the behaviour of pipelines in several ways. . The computational simulation through Finite Element Method (FEM) is one of the most efficient tools to quantify reliably the remaining strength of corroded pipes. This work presents a new method based on the concept of Continuum Damage Mechanics (CDM) which currently has reached a stage of maturity enabling it to model any type of degradation. The value of the critical pressure, i.e., the allowable operating pressure, of a corroded pipeline is obtained by using a post processor based upon damage mechanics. This post processor allows the calculation of the crack initiation conditions from the history of strain components taken as the output of the ANSYS Software. The ANSYS code is used for physical and geometrical non-linear analysis to obtain the critical point where at any time the damage equivalent stress is maximum. This method was validated by comparing the results of numerical simulations with experimental and ASME/ B31G results available in the literature. Keywords: Burst pressure, pipe, corrosion, damage mechanics, damage, finite element. © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_49

481

482

1

D. Benzerga

Introduction

In most applications, the damage is very localized in such a way that the damaged material occupies a volume small in comparison to the macroscale of the structural component and even to the mesoscale of the representative volume element RVE. This is due to the high sensitivity of damage to stress concentrations at the macroscale and to defects at the microscale. This allows us to consider that the effect of the damage on the state of stress and strain occurs only in very small damaged regions. In other words, the coupling between damage and strains may be neglected everywhere in the structure except in the RVE(s) where the damage develops. This is the principle of the locally coupled analysis [Lemaitre and Doghri] where the procedure may be split into the following two steps as shown in Fig. 1: - a classical structure calculation in elasticity or elastoplasticity by the finite element method (FEM) to obtain the fields of strain and stress; - a local analysis at the critical point only dealing with the elasto-plastic constitutive equations coupled with the kinetic law of damage evolution, that is a set of differential equations. This method is much simpler and saves a lot of computer time in comparison to the fully coupled analysis, which takes into account the coupling between damage and strain in the whole structure. The fully coupled method must be used when the damage is not localized but diffused in a large region [Lemaitre and Doghri]. Our main motivation in this study is to show that this method is a contribution to the rehabilitation of corroded pipes. The study and analysis of the damage at critical point where at any time the damage equivalent stress is maximum in the vicinity of the corrosion defect of a steel pipe X65. The method developed here is to construct a numerical tool to determine the new maximum pressure that could support a corroded pipe. The Modeling of the damage phenomenon is based

Fig. 1 Locally coupled analysis of crack initiation

Burst Pressure Estimation of Corroded Pipeline Using Damage Mechanics

483

upon Damage Mechanics, which currently has reached a stage of maturity enabling it to model any type of degradation. Using the programmable ANSYS, a subroutine was developed and implemented in the main code for the determination of the critical point (M*) in the vicinity of corrosion defect where potential crack may occurs. In a second step, a post-processor based on the iterative Newton method was applied to this critical point (M*) for the determination of the maximum internal pressure that a corroded pipe could support in the case of a longitudinal corrosion defect. The maximum pressure is the pressure value corresponding to the critical value of the DC damage (crack initiation). This method was validated by comparing the results of numerical simulations with experimental and ASME/ B31G results available in the literature.

2

Modeling

To model the pipeline we used the ANSYS code with two scenarios, a model without a cavity (no corrosion, Figure 2) and a model with cavity (corroded pipeline, Figure 3). Taking into account the symmetry of the pipeline, we modeled only one-fourth of cylinder. The dimensions of the structure are length, inner radius and thickness. To do this, we have created a rectangular area, which has the same length as the pipeline, and a width equivalent to the thickness of the pipeline. By a 90° rotation of this surface around the axis X, a volume was obtained in the form of a quarter cylinder. For meshing the structure, we used volume elements with 20 nodes Solid95. These elements have compatible displacement shapes and are well suited to model curved boundaries. The mesh of the volume is made to obtain a concentration of elements around the side where the cavity is represented (Figure 2). For boundary conditions, we have respected the conditions of symmetry and the experience. On the corroded pipe (model with cavity), we kept the same data used in the model without cavity, and created a cavity that represents the corrosion defects on the pipeline. To do this we created a parabolic surface to the thickness of one quarter of the cylinder with a length equivalent to the defect length and width equal to the depth of the defect. The volume of the cavity is obtained by a rotation of 90 ° around the upper line of the pipeline wall. This volume is subtracted from the volume of the non-corroded pipeline (Figure 2). The same meshing is used with a mesh filtering around the cavity. We varied the values of the depth and extension of the cavity (corrosion defects, see Figure 2). The burst pressure is determined according to the geometrical characteristics of each cavity. The critical pressure or the maximum pressure that will support a corroded pipe is that corresponding to the critical value of the damage Dc (this value depends on the material and temperature).

484

D. Benzerga

Fig. 2 Pipe without cavity (pipe not corroded)

Fig. 3 Model with cavity

Using the programmable language ANSYS, a subroutine was developed and implemented in the main code for the determination of critical point (M *) in the zone of corrosion defect (see figure below) where microcracks may grow.


MEF Calculations

Post-Processor

Dangerous area

Critical point (M*)

485

Next, the constitutive law of the critical point (M, *) where the equivalent stress σ * is maximum, obtained using the ANSYS code, is implanted in the postprocessor based on the iterative Newton method [Benallal et al, 1988.]:

σ* (M ) = Sup⎛⎜ σ* ⎞⎟ ⎝ ⎠ with

1 σ* = σ eq Rv 2

⎛ σH ⎞ 2 ⎟ RV = (1 + ν ) + 3(1 − 2ν )⎜⎜ ⎟ 3 σ eq ⎠ ⎝

(1)

2

In most cases, this criterion is satisfied in areas of high stress concentration with a high coefficient of triaxiality σH / σeq. The final step is to determine the evolution of damage in solving the following constitutive laws: e +Ep E ij = E ij ij 1 + ν σ ij υ σ kk − δ ij E 1− D E 1− D ~D 3 σ ij p P Si f = f = 0 E ij = 2 σ eq 2 σ eq D= Si p ≥ p0 Rv P 2ES e = E ij

(2)

Using a stepwise method simultaneously with the method (scheme) implicit Newton, which ensures good convergence [Benallal et al, 1988].

486

D. Benzerga

The method developed above gives versus of the internal pressure of the pipe, the value of the damage, the accumulated plastic strain and stress components at each instant until a macroscopic crack initiation at the defect corrosion. This allows determining the new maximum pressure that will support a pipe with a corrosion defect.

3

Validation of the Methodology

In order to validate our model, we compared our results with ASME/B31G standard and experimental results obtained by TA Netto [Netto, 2005] [3] (see table 1). For the model without cavity, we took the analytical value P = 50.89 MPa. Figure 4 illustrates the results obtained by three methods, the experimental, ASME / B31G and our proposed method. Table 1 Maximum predicted pressure

N°

d (mm)

L (mm)

Burst Pressure tests (MPa)

Burst Pressure(MPa) ASME/B31G

T1 T2 T3 T4 T5 T6 T7

1.58 1.59 1.87 1.91 2.13 2.14

42 21 42 21 42 21

57.33 37.02 44.65 32.47 41.28 26.76 34.55

50.89 35.05 42.53 30.85 38.93 24.50 34.86

60 56

53.4 36 44.7 32.5 42.5 29.5 36.8

Burst pressure versus depth for L=41 mm

55

54 52

50

50

Burst pressure MPa

Burst pressure MPa

6.85 2.75 0.11 0.099 2.95 10.23 6.51

60

Burst pressure versus depth for L=21 mm

58

Predicted Burst Erreur pressure(MPa) en %

48 46 44

Simulation ASME/B31G Expérimental

42 40 38 36 34

45

Simulation ASME/B31G Experimental

40

35

30

32 30

25

28 0.0

0.5

1.0

1.5

corrosion defect depth mm

2.0

2.5

0.0

0.5

1.0

1.5

2.0

2.5

Corrosion defect depth mm

Fig. 4 Burst pressure depending on the depth of corrosion defects for two extensions L = 21mm and L = 41 mm.


4

487

Analysis and Interpretation of Results

We note that the numerical results are more realistic (experimental results). These numerical results do not exceed a 10% difference. Comparing simulation results and ASME/B31G, we note that the average error is smaller in numerical case. This is due to a factor of safety used by B31G standard. The ASME/B31G uses the projection of corrosion defect, while the numerically exact corrosion surface defect is used. The difference between the numerical and experimental results depends on several factors: It was considered a perfectly continuous material, while in reality there are defects, discontinuities and micro-voids in the material, which can be potential sources of damage. There are always residual stresses after machining that we have ignored in the laws of behavior considered in this study. The shape of the cavity is not identical for all specimens. It was considered perfect symmetry, which is not the case in reality For the boundary conditions, it was considered a locking surface, there are small displacements, which cause deformations, and the structure of the material is changed. The mesh is not fine enough or sufficiently homogeneous. S values (damage coefficient) εpD (plastic deformation damage threshold) and DC (critical damage) of X65 material taken approximately, true values are determined by load-unload tests. The yield condition was considered without hardening. We considered that damage only occurs when strain hardening is saturated

5

Conclusion

In this paper, we presented a new method based upon damage mechanics, which is now in its stage of maturity. Our main motivation in this work was to show that this method is a contribution to the rehabilitation of corroded pipes. In spite of some differences between proposed method results and the experimental results, our method have had a good performance in predicting the failure pressure of a pipeline containing corrosion defects. The ASME/B31G results were more conservative than proposed method; however, our method provided more precise results. Our models proved to be capable of simulating the corroded pipe burst tests adequately.

488

D. Benzerga

References Benallal, A., Billardon, R., Doghri, I.: An integration algorithm and the coresponding consistent tangent operator for fully coupled elastoplastic and damage equations. Comm. Appl. Numer. Methods 4, 731–740 (1988) Lemaitre, J., Doghri, I.: Damage 90: a post processor for crack initiation. Comput. Methods Appl. Mech. Eng. 115, 197–232 (1994) Netto, T.A., Ferraz, U.S., Estefen, S.F.: The effect of corrosion defects on the burst pressure of pipeline. J. Constr. Steel Res. 61, 1185–1204 (2005)

Numerical Simulation of Incremental Sheet Metal Forming Process Lotfi Ben Said, Mondher Wali, and Fakhreddine Dammak Mechanical Modelisation and Manufacturing Laboratory (LA2MP), National Engineering School of Sfax, B.P W3038, Sfax, University of Sfax, Tunisia {bensaid_rmq,mondherwali}@yahoo.fr, [email protected]

Abstract. This paper presents a numerical simulation of the incremental sheet metal forming (ISF) process, type single point incremental forming (SPIF). A finite element model (FEM) is developed by using the commercial FE code ABAQUS. An elasto-plastic constitutive model with quadratic yield criterion of Hill and isotropic hardening behavior has been adopted during ISF operation. Results including thickness variation of sheet metal and forming force along Z-axis are presented. Keywords: Sheet forming, Incremental forming, Numerical simulation.

1

Introduction

Incremental Sheet Forming (ISF) is an innovative process to manufacture sheet metal with numeric increments on CNC machines through plastic deformation. This process is relatively slow, allows the manufacture of prototype parts with complex shapes and high precisions. Two ISF types can be distinguished: the Single Point Incremental Forming (SPIF) and the Two Points Incremental Forming (TPIF). These tow variants of ISF are presented in Figure 1.

Fig. 1 The two variants of ISF (Azaouzi and Lebaal 2012)


489

490

L. Ben Said, M. Wali, and F. Dammak

The experiences realized with the ISF process are concentrated in the SPIF process using a CNC machines with a forming tool with specific shape. This tool replaces the milling tool and it uses the same mounting elements in CN machine. Many trajectories are used in incremental forming with numerical increments by the use of CN program in 3 axis or 5 axis machines. The incremental forming tool trajectories are specific to this process and materials of sheets manufactured in ISF. The literature on the subject of ISF: the integrity of surfaces during and after the incremental forming, the geometry of the specimen manufactured, forming efforts and optimization of forming time are investigated by the majority of researches. The majority of manufactured parts, mentioned in the articles, are not industrial parts but they are test pieces with cone or pyramidal shape where a hemispherical punch is used in the sheet forming process, with a manner close to the operation of a pocket in a CN milling process. So the study of an industrial piece will be significant with a good experiment results. Several studies have been made to understand ISF process taking into account different material behavior and various methods have been proposed for assessing the formability in ISF. (Robert et al. 2012) made comparison between incremental deformation theory and flow rule to simulate sheet metal forming processes. They used the material user subroutine (VUMAT) to implement two material behavior laws (incremental deformation theory and classical flow rule plasticity) in their simulation. At the aim to analyze the stress state and thickness distribution of sheet metal studied. But in this study they did not take into account kinematic hardening in the material behavior for a best of prediction of spring back in sheet metal incremental forming. (Ben Ayed et al. 2014) presented a simplified numerical approach to simulate ISF with efficient CPU time. A shell element DKT12 was implemented and coupled with an elasto-plastic model based on a classical flow rule, isotropic hardening has been considered in their simulations. (Fang et al. 2014) build an investigation on deformation mechanism and fracture behavior in single point incremental forming. In their analytic model they make on consideration two parameters, bending effect and strain hardening, to describe the localized deformation mechanism. By numerical and experimental investigation they demonstrate that the deformation occurs not only in the contact zone, but also in the neighboring wall which has been already formed in the vicinity of the contact zone. And that the fracture tends to appear at the transitional zone between the contact area and the formed wall. They conclude also that the strain hardening has a positive effect on the material formability. (Belchior et al. 2014) developed an approach applied in robotized ISF process, this approach consists in coupling a finite element analysis of the ISF (using to predict efforts in this process) with an elastic model of a robot structure where the punch is mounted. The result of finite element simulation was used like an input data for the elastic model at the aim to optimize the punch path and to make correction in robotized ISF when the punch makes deviation. The effect of the hardening law (Ludwick or Voce) has been evaluated.

Numerical Simulation of Incremental Sheet Metal Forming Process

491

(Kurra and Regalla 2014) analyzed the formability and thickness distribution in ISF of extra-deep drawing steel sheet which is a widely used material in automotive applications involving simple and complex parts requiring high formability. (Seong et al. 2014) concluded after their research that the stress states in top, middle and bottom surfaces exceed the forming limit curve at different times and each layer has different stress state in terms of their deformation history to suppress necking. (Behera et al. 2014) propose an analysis methodology using topological conceptual graphs to capture the effects of different phenomena on the final accuracy of a sheet metal part manufactured by SPIF. So they create an algorithm creating partial tool paths when the shape of the sheet formed is to complex and demand sum accuracy. (Han et al. 2013) studied the springback at the aim to predict and control accurately this factor to design an accurate tool path for ISF. A three-dimensional elasto-plastic finite element model was implemented to simulate ISF this model is based on the particle swarm optimization neural network and it has shown good results in the prediction of springback. The present investigation is a contribution of work that started using an incremental sheet metal forming (ISF) process by taking into account the anisotropic effect presented in sheet metal combined with isotropic hardening behavior. Results including thickness distribution and forming forces can be estimated from the numerical model.

2

Elasto-plastic Constitutive Equations

An extensive description for formulating constitutive elasto-plastic rate equations in order to model anisotropic yielding, and non-linear isotropic hardening can be found elsewhere, and we will not elaborate on these details; see, for example, (Lemaitre and Chaboche 1990). A summary of the resulting set of equations considered in this paper - Partition of the total strain

ε = ε +ε

p

(1)

- Hooke’s law

σ = D: ε

e

(2)

- Yield function

f =

e

ϕ (σ ) = σ p =σY + R

,

3 2

ϕ (σ ) −σ p ≤ 0

σ Pσ T

(

R =Q 1− e

(3)

(4) − βκ

)

(5)

492


p ε = γ

- Flow rule

- Isotropic hardening

∂f ∂σ

=

3 2

γ n

,

n=

κ = γ

1

ϕ

Pσ

(6)

(7)

where ε denotes the elastic strain tensor and ε is the plastic strain tensor, σ is the stress tensor, D is the general elastic operator, Q and β are material parameters, γ is the plastic multiplier and P is a fourth order tensor which define e

p

the yield criterion. This yield function includes the classical J2 plasticity yield condition and the quadratic Hill criterion as special cases. The Hill yield criterion, in three-dimensional cases, is obtained by taking

P=

2 3

H

⎡H + G −H ⎢ H+F ⎢ ⎢ , [H ]= ⎢ ⎢ ⎢ Sym ⎢ ⎣

−G

0

0

−F

0

0

F +G

0

0

2N

0 2M

⎤ 0⎥ ⎥ 0⎥ ⎥ 0 ⎥ 0⎥ ⎥ 2L⎦ 0

(8)

where F , G , H , N , M and L are material constants obtained by tests of the material in different orientations. The J2 plasticity yield criterion is recovered using Eq. (9) and setting F = G = H = 0.5

3

,

N = M = L = 1.5

(9)


In this section, square box application of ISF was investigated and the results carried out with the use of numerical treatment based on ABAQUS/Explicit code. The material is an aluminum alloy (AA5754-O). The mechanical properties of the AA5754-O material are given in Table 1. The isotropic hardening behavior is modeled by the exponential law (Eq. 5), and the Hardening parameters of the AA5754-O material are presented in Table 2.


493

The initial dimension of the sheet is 200 x 200 x 1.5 mm, the punch tool has a diameter of 10 mm. The punch tool is assumed as a rigid body and no property is assigned. The frustum of the square box has a depth of 10 mm and a base area of 80x80 mm. The tool path and the dimensions of the desired shape are shown in Figure 2. The friction coefficient between punch tool and sheet metal is assumed to be 0.1. The sheet is meshed using 2728 triangular elements (S3R in Abaqus) with five integration points through the thickness. Table 1 Mechanical properties of the aluminum AA5754-O material

Elastic Prop.

Hill 48 Coefficients

E(GPa)

ν

F

G

H

N

70

0.33

0.748

0.572

0.403

1.467

Table 2 Hardening parameters of the AA5754-O material

Parameter

Unit

IH

σY

MPa

95

Q

MPa

159

β

--

9

Fig. 2 Tool path and the final shape of sheet incremental formed (Robert et al. 2012)

494


Numerical predictions carried out using Abaqus are shown in Figures 3, 4 and 5. The thickness variation along the Y = 0 cut is presented in Fig. 3. According to the thickness evolution, significant deformations of square box are located principally in the vicinity of punch path. The evolution of the vertical forming force during the ISF process of the studied sheet is illustrated in Figure 4. The vertical forming force due to the contact pressure reaches the maximum value in the last pass of the punch when incremental forming is in a depth of 10 mm. the value of this effort is in the same range of efforts recognized in the majority of articles established by many research group. The distribution of the von Mises stress in the deformed zone under the action of the punch during the ISF process is as shown in Figure 5. According to this Figure the major Von Mises stress is localized under the punch tool, located principally along the punch path.

Thickness (mm)

1.55 1.5 1.45 1.4 1.35 1.3 1.25 0

0.05

0.1

0.15

X direction (m) Fig. 3 Thickness variation in the cut plane Y = 0

Fig. 4 Evolution of the forming force Fz (N)

0.2


495

Fig. 5 Distribution of Von Mises Stress

4

Conclusion

In this paper, finite element model is developed, based on Hill yield criterion and isotropic hardening behavior, simulating an example of single point incremental sheet forming. The material model is implemented in the commercial finite element code ABAQUS/Explicit to predict ISF process. The results show significant deformations have been located along the contour of the square box base but it should be noted that there are also minor deformations on the borders near to the clamped shape.

References Azaouzi, M., Lebaal, N.: Tool path optimization for single point incremental sheet forming using response surface method. Simulation Modelling Practice and Theory 24, 49–58 (2012) Behera, A.K., Lauwers, B., Duflou, J.R.: Tool path generation framework for accurate manufacture of complex 3D sheet metal parts using single point incremental forming. Comput. Ind. 65, 563–584 (2014) Belchior, J., Leotoing, L., Guines, D., Courteille, E., Maurine, P.: A Proc-ess/Machine coupling approach: Application to Robotized Incremental Sheet Forming. J. Mater. Process. Tech. 214, 1605–1616 (2014) Ben Ayed, L., Robert, C., Delamézière, A., Nouari, M., Batoz, J.L.: Simplified nu-merical approach for incremental sheet metal forming process. Eng. Struct. 62-63, 75–86 (2014) Fang, Y., Lua, B., Chena, J., Xua, D.K., Oub, H.: Analytical and experimental investigations on deformation mechanism and fracture behavior in single point incremental forming. J. Mater. Process. Tech. 214, 1503–1515 (2014) Han, F., Mo, J., Qi, H., et al.: Springback prediction for incremental sheet forming based on FEM-PSONN technology. Trans. Nonferrous Met. Soc. China 23, 1061–1071 (2013) Kurra, S., Regalla, S.P.: Experimental and numerical studies on formability of extra-deep drawing steel in incremental sheet metal forming. Journal of Materials Research and Technology 3, 158–171 (2014)

496


Lemaitre, J., Chaboche, J.L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1990) Robert, C., Delamézière, A., Dal Santoa, P., Batoz, J.L.: Comparison between in-cremental deformation theory and flow rule to simulate sheet-metal forming processes. J. Mater. Process. Tech. 212, 1123–1131 (2012) Seong, D.Y., Haque, M.Z., Kimb, J.B., Stoughton, T.B., Yoon, J.W.: Suppression of necking in incremental sheet forming. Int. J. Solids Struct. 51, 2840–2849 (2014)

A Higher Order Shear Strain Enhanced SolidShell Element for Laminated Composites Structures Analysis Abdessalem Hajlaoui1, Abdessalem Jarraya2, Mondher Wali1, and Fakhreddine Dammak1 1

2

Mechanical Modelisation and Manufacturing Laboratory (LA2MP), National Engineering School of Sfax, B.P W3038, Sfax, University of Sfax, Tunisia

Faculty of Engineering, King Abdulaziz University, North Jeddah, Saudi Arabia [email protected], jarraya_abdessalem,mondherwali}@yahoo.fr, [email protected]

Abstract. This paper presents a free from locking higher order solid-shell element based on the Enhanced Assumed Strain (EAS) for laminated composite structures analysis. The transverse shear strain is divided into two parts: the first one is independent of the thickness coordinate and formulated by the Assumed Natural Strain (ANS) method; the second part is an enhancing part which ensures a quadratic distribution through the thickness. This permit to remove the shear correction factors and improves the accuracy of transverse shear stresses. Also, volumetric locking is completely avoided by using the optimal parameters in the EAS method. Comparisons of numerical results with those extracted from literature show the performance of the developed finite element. Keywords: Laminated composites, Solid-shell, Higher Order Shear Deformation.

1

Introduction

In the last decades the use of composite materials is becoming more widely used since they provide many advantages to structural designers. The stress and the strain field in these structures is very complex, therefore numerical methods have been developed to give an accurate prediction. Several methods have been proposed in the literature namely: The Assumed Natural Strain method (ANS) used to solve the shear locking problems in the case of thin plate and shell structures, we can cite (MacNeal 1982), (Bathe and Dvorkin 1985), (Bucalem and Bathe 1993) among many others. (Hauptmann and Schweizerhof 2000) assumed a linear distribution of the strain in thickness direction and developed a solid-shell element formulation. (Sze and Yao 2000) used this method to obtain a locking-free solid-shell element. © Springer International Publishing Switzerland 2015 M. Haddar et al. (eds.), Multiphysics Modelling and Simulation for Systems Design and Monitoring, Applied Condition Monitoring 2, DOI: 10.1007/978-3-319-14532-7_51

497

498

A. Hajlaoui et al.

The Enhanced Assumed Strain method (EAS) was proposed by (Simo and Rifai 1990). It allowed avoiding the thickness locking caused by coupling of the inplane and transverse normal stress and normal strain responses (Doll et al. 2000). Based on this method, (Alves de Sousa et al. 2003) proposed a locking-free solidshell element formulation. The application of the EAS method in shell analysis was presented in the work of (Andelfinger and Ramm 1993) with linear elastic range. The extension to non-linear aspects was carried out by (Büchter et al. 1994) and (Bischoff and Ramm 1997) among others. The use of a single method cannot entirely solve the locking problems in solidshell elements. Therefore many studies have combined them for a better prediction of the structure behavior. (Klinkel et al. 1999) and (Vu-Quoc and Tan 2003) developed three-dimensional locking free solid-shell elements by using the EAS and ANS methods. (Hajlaoui et al. 2012) presented a solid-shell finite element based on the classical ANS method to compute the shear locking effect and a nine parameters EAS method which avoided completely the volumetric locking to study the bucking behavior of laminated composite plate with delamination. In the literature, there is no solid-shell formulation with high-order transverse shear enhancement. The only exception is the work of (Quy and Matzenmiller 2008), where the authors developed a finite element based on the theoretical foundations of high-order shear deformation theories. In this works authors decomposed the transverse shear strains into two parts: the first one is compatible and continuous, which can present a shear locking in thin limit structures, the second part is based on the EAS with parabolic function in terms of natural thickness coordinate. The membrane part in this work is not enhanced, which can present a numerical locking in the case of in-plane bending and in the nearly incompressible elasticity. However, in the case of plates and classical shell, the higher order shear deformation theory (HOSDT) has been widely used. (Liu and Reddy 1985), (Rastgaar Aagaah et al. 2003) and (Reddy and Lee 2004), among others, used this theory to improve the transverse shear strain and stress distribution. In a recent work, (Wali et al. 2014) presents an efficiency three dimensional double directors shell element for the functionally graded material shell structures analysis. The vanishing of transverse shear strains on top and bottom faces is considered in a discrete form inspired form the work of (Dammak et al. 2005). Thus, the third-order shear deformation plate theory (TSDT) is a particular case of the developed formulation. Nevertheless, the treatment of delamination with this HOSDT is complicated which is not the case with the solid-shell elements. This paper is an improvement of the solid-shell formulation developed by (Hajlaoui et al. 2012); the transverse shear strain is composed by two parts: a compatible part based on the ANS formulation of (Bathe and Dvorkin 1985), and a second part based on EAS method with parabolic function in terms of natural thickness coordinate as proposed in (Quy and Matzenmiller 2008). In addition, the enhancements of membrane part and the transverse strain were used to avoid the in-plane bending and volumetric locking.

A Higher Order Shear Strain Enhanced Solid-Shell Element

2

499

Solid-Shell Finite Element Formulation

The strain field is enhanced with the introduction of internal variables (Simo and Rifai, 1990), as following: c E = E + E

(1)

Where E and E are respectively the compatible part and the enhanced part of the Green-Lagrange strain tensor. c

2.1

Compatible Strains

The compatible part is arranged in (6x1) column matrix as follows: E =⎡ ⎣ E11 , E22 , E33 , 2 E12 , 2 E13 , 2 E23 ⎤⎦ c

c

c

c

c

c

c

T

c

(2) c

The compatible transverse shear strains E13 and E23 and the transverse thickc

ness strains E33 are developed on the base of ANS method proposed by (Bathe and Dvorkin 1985). This formulation is detailed in the author’s previous work, c

c

Hajlaoui et al. (2012). The compatible transverse shear strains E13 and E23 are evaluated at four midpoints of the reference element mid-surface A = (-1,0,0), B = c

(0,-1,0), C = (1,0,0), D = (0,1,0) and the transverse thickness strains E33 is evaluated at the following four points of the reference element A1=(-1,-1,0), A2= (1,1,0), A3= (1,1,0), A4= (-1,1,0).

⎡ 2E13c ⎤ ⎡(1 − η ) E13B + (1 + η ) E13D ⎤ ⎢2 E c ⎥ = ⎢ 1 − ξ E A + 1 + ξ E C ⎥ ) 13 ( ) 13 ⎦ ⎣ 23 ⎦ ⎣(

(3)

4

E33 = c

1 ∑ 4 (1 + ξξ )(1 + ηη ) E A

A

A 33

(4)

A =1

Then the compatible part of the Green-Lagrange strain tensor takes the following form

500

A. Hajlaoui et al. 1 ( g11 − G11 ) ⎡ ⎤ 2 ⎢ ⎥ 1 ( g 22 − G22 ) 2 ⎢ ⎥ 4 ⎢ ⎥ A A 1 1 1 + ξ A ξ )(1 + η Aη ) 2 ( g 33 − G33 ) ⎥ ∑ 4 ( c −T ⎢ E =T ⎢ A =1 ⎥ ( g12 − G12 ) ⎢ ⎥ ⎢ 1 ⎡(1 − η ) ( g B − G B ) + (1 + η ) ( g D − G D ) ⎤ ⎥ 13 13 13 13 ⎦ ⎢2⎣ ⎥ A A C 1 ⎢⎣ 2 ⎡⎣(1 − ξ ) ( g 23 − G23 ) + (1 + ξ ) ( g 23 − G23C ) ⎤⎦ ⎥⎦

(5)

Where Gij = Gi .G j and g ij = gi .g j are the components of the metric tensors at reference and actual configurations. T is the transformation matrix of the strain tensor from parametric coordinates to the local Cartesian coordinates. The variation and the increment of the total compatible strain tensor can be written as follows:

δ E = B δUn c

,

Δ E = B ΔU n c

(6)

Where Un is the displacement nodal vector and B is the strain interpolation matrix, associated with node ( I ) and denoted BI is given by: T g1 N I ,1 ⎡ ⎢ T g1 N I ,1 ⎢ 4 ⎢ T 1 1 + ξ A ξ )(1 + η Aη ) g3 N I ,3 ∑ ⎢ 4 ( −T A =1 BI = T ⎢ T T g2 N I ,1 + g1 N I ,2 ⎢ ⎢1 B B B B D D D D ⎢ 2 ⎡⎣(1 − η ) g3 N I ,1 + g1 N I ,3 + (1 + η ) g3 N I ,1 + g1 N I ,3 ⎢ ⎢⎣ 12 ⎡⎣(1 − ξ ) g3A N IA,1 + g1A N IA,3 + (1 + ξ ) g3C N IC,1 + g1C N IC,3

( (

2.2

T

T

T

T

) )

( (

T

T

T

T

) )

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤⎥ ⎦ ⎥ ⎤⎥ ⎦⎦

(7)

Enhanced Strain

The enhanced part is related to the vector of the internal strain parameters α as

α , δE = M δα E = M

Δα , ΔE = M

(8)


501

is the interpolation function matrix for the enhanced assumed strain where M field. The interpolation is first defined in the natural coordinate and must be transferred to the global coordinates as follows

= det J 0 T − T M M ξηζ 0 det J

(9)

Where the subscript ‘ 0 ’ means evaluation at the center of the element in the natural coordinates,

J = [G1 , G2 , G3 ] is the Jacobian matrix. The interpolation

, in equation (9), is expression in term of the parametric coordinates matrix M ξηζ

( ξ ,η ,ζ ) .

Three choices of matrix M will be considered with, 7, 9 and 11 ξηζ

parameters.

7

M ξηζ

9

M ξηζ

⎛ξ 0 0 0 0 ⎜0 η 0 0 0 ⎜ ⎜0 0 ζ 0 0 ⎜ 0 0 0 ξ η =⎜ ⎜ ⎜0 0 0 0 0 ⎜ ⎜ ⎜0 0 0 0 0 ⎝

⎛ξ 0 0 0 0 ⎜0 η 0 0 0 ⎜ ⎜0 0 ζ 0 0 ⎜ 0 0 0 ξ η =⎜ ⎜ ⎜0 0 0 0 0 ⎜ ⎜ ⎜0 0 0 0 0 ⎝

1 5

⎞ ⎟ 0 ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 1 2 ⎟ −ζ ⎟ ⎠ 5

0

0

0 0 0 1 5

−ζ

2

0

0

0

0

0

0

0

0

0

ξζ

0

0

0

0

0

−ζ 0

2

1 5

−ζ

2

0

(10.a)

0 ⎞

⎟ ⎟ ηζ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎠ 0

(10.b)

502

A. Hajlaoui et al.

11

M ξηζ

⎛ξ ⎜0 ⎜ ⎜0 ⎜ 0 =⎜ ⎜ ⎜0 ⎜ ⎜ ⎜0 ⎝

0

0

0

0

0

0

0

0

ξη

η

0

0

0

0

0

0

0

0

0

ζ

0

0

0

0

ξζ

ηζ

0

0

0

ξ

η

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

5 0

0

0

−ζ

0

2

1

0

−ζ

5

2

⎞ ξη ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎠ 0

(10.c)

Since the shear strain interpolation in thickness direction is parabolic, at least 2x2x3 Gauss quadrature integration rule is used. After including interpolation functions for enhanced strain fields, we obtain the solid-shell element enhanced with 7, 9 and 11 incompatible modes.

3

Weak Form and Linearization

The weak form of the proposed solid-shell formulation is G( u , E ) =

∫

V

(

)

c S : δ E + δ E dV −

∫ F .δ udV − ∫ V

V

∂Vf

FS .δ udA = 0

(11)

where S is the second Piola–Kirchhoff stress tensors and FS and FV are surface and volumetric distributed load respectively. This equation will be solved by the Newton-Raphson method and must be linearized. The strain parameters are then eliminated at the element level which leads to the following tangent stiffness matrix and the residual vector −1

KT = K e + K G − L H L , T

where K e , L , H , f Ke = f

int

=

∫

Ve

T

∫

Ve

int

,f

ext

T

ext

=

∫

Ve

T

ext

−f

int

(12)

and h are given by the following expressions

B  B dVe , L =

B S dVe ; f

−1

Re = L H h + f

∫

Ve

T B dV , M e

N FV dVe + T

∫

∂V fe

H=

∫

Ve

T M dV M e

N FS dA; h = T

∫

Ve

T S dV M e

(13)

(14)

where,  is the 6 × 6 three directional isotropic elasticity tensor and K G is given in (Hajlaoui et al. 2012).

n Enhanced Solid-Shell Element A Higher Order Shear Strain

4

5003

amples Numerical Exa

In this section we presentt two numerical simulations in order to illustrate efficienncy of the proposed higherr order shear strain enhanced solid-shell element. Resullts well-known formulations. A listing of elements and thhe are compared to other w ntify them is given in Table 1. abbreviations used to iden ments Table 1 Listing of shell elem Description

Name Q8A3E5

(Klinkel eet al. (1999), ANS with

Q8A3E7

4.1

E33 , E13 E23 and 5 parameters

(Vu-Quocc and Tan 2003), ANS with

E33 , E13 E23 and 7 parameters

Q8S12

(Quy and Matzenmiller 2008), with 12 parameters

C3D8C11

Present ellement with 11 parameters

der with End Diaphragms Pinched Cylind

To investigate the capaability of the developed elements to model the innextensional bending and complex membrane states of stress, we consider one oof the more severe tests for sshell and solid-shell formulations. 1.8 Normalized displacement (x10-5)

1.6 1.4 1.2 1

Q8A3E5

0.8

Q8A3E7,C3D8C11

0.6

Q8S12

0.4 0.2 0 0

10 20 Number of Elements per Side

30

Fig. 1 Description and resultts of pinched cylinder with end diaphragms

A short cylinder, witth two pinching vertical forces at the middle section, ms at the end, is modeled using one octant with approoand two rigid diaphragm priate symmetry boundarry conditions, (figure 1). The length of the cylinder is

504

A. Hajlaoui et al.

L=600mm, the radius is R=300mm, and the thickness is h=3mm. The material

properties are: E = 3.0 × 10 MPa, ν = 0.49 . The results are shown in Figure 1. In comparison to other solid-shell formulations in the literature listed in Table 1, our developed formulation and those obtained with (Vu-Quoc and Tan 2003) shows superior convergence behavior. This proves the accuracy and efficiency of the present solid-shell formulation for the static analysis of shells. 6

4.2

Buckling of Laminated Plate with Delamination

To study the influence of the delamination size on the buckling behavior of a simply supported symmetric laminated plate, we consider a cross ply square laminated plate [0/90/90//0]. It is subjected to a uniform in plan force with throughthe-width délamination between the two last layers, as shown in figure 2.

Fig. 2 Laminated plate with delamination

The material properties are: E11 = E22 = open , E22 = E33 , G23 = 0.5E22 ,

G12 = 0.5G13 = 0.6 E22 and ν 12 = ν 13 = ν 23 = 0.25 . Figure 3 illustrates the variation of the buckling load versus the delamination size for symmetric cross ply laminated square plates. It is shown that the largest buckling load is obtained for the case of symmetric laminated [0/90/90//0], because the fiber orientation at the layer with the delamination is parallel to the in-plane loads. From the plots it can be concluded that the influence of the delamination is more important for the symmetric laminated [0/90/90//0].


Critical buckling load(N/mm²)

180

505

a/b=1 et b/h=20 E11/E22=30

160 140

[90/0/0//90]

120

[0/90/90//0]

100 80 60 40 20 0 0

0.2

0.4 D/a

0.6

0.8

Fig. 3 Description and results of Pinched cylinder with end diphragrams

5

Conclusions

In this study a new solid-shell finite element based on the partition of shear strain is developed: a first part is independent of the thickness coordinate and formulated by the ANS method which avoid the shear locking in thin limit structure; a second part, enhancing part, ensures a quadratic distribution of the transverse shear strain. This permit to remove shear correction factors and improves the accuracy of transverse shear stresses. The finite element formulation is completely free from the volumetric locking phenomena which occur in the treatment of nearly incompressible elasticity. The developed finite element model is validated by comparing numerical results with other results from literature. The buckling behavior of composite laminates is investigated based on the use of solid-shell element.

References Alves de Sousa, R.J., Natal Jorge, R.M., Valente, R.A.F., César de Sá, J.M.A.: A new volumetric and shear locking-free 3D enhanced strain element. Eng. Computation 20, 896–925 (2003) Andelfinger, U., Ramm, E.: EAS-Elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements. Int. J. Numer. Meth. Eng. 36, 1311–1337 (1993) Bathe, K.J., Dvorkin, E.N.: A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. Int. J. Numer. Meth. Eng. 2, 367–383 (1985) Bischoff, M., Ramm, E.: Shear Deformable shell elements for large strains and rotations. Int. J. Numer. Meth. Eng. 40, 4427–4449 (1997) Bucalem, M.L., Bathe, K.J.: Higher-order MITC general shell elements. Int. J. Numer. Meth. Eng. 36, 3729–3754 (1993)

506

A. Hajlaoui et al.

Büchter, N., Ramm, E., Roehl, D.: Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept. Int. J. Numer. Meth. Eng. 37, 2551–2568 (1994) Dammak, F., Abid, S., Gakwaya, A., Dhatt, G.: A formulation of the non linear dis-crete Kirchhoff quadrilateral shell element with finite rotations and enhanced strains. European Journal of Computational Mech. 14, 1–26 (2005) Doll, S., Schweizerhof, K., Hauptmann, R., Freischlager, C.: On volumetric locking of loworder solid and solid-shell elements for finite elastoviscoplastic deformations and selective reduced integration. Eng. Computation 17, 874–902 (2000) Hajlaoui, A., Jarraya, A., Kallel-Kamoun, I., Dammak, F.: Buckling analysis of a laminated composite plate with delaminations using the enhanced assumed strain solid shell element. J. Mech. Sci. Technol. 26, 3213–3221 (2012) Hauptmann, R., Schweizerhof, K., Doll, S.: Extension of the ‘solid-shell’ concept for application to large elastic and large elastoplastic deformations. Int. J. Numer. Meth. Eng. 49, 1121–1141 (2000) Klinkel, S., Gruttmann, F., Wagner, W.: A continuum based three-dimensional shell element for laminated structures. Comput. Struct. 71, 43–62 (1999) Liu, L.F., Reddy, I.N.: A higher-order shear deformation theory of laminated elastic shells. Int. J. Engng. Sci. 23, 319–330 (1985) MacNeal, R.H.: Derivation of element stiffness matrices by assumed strain distributions. Nucl. Eng. Des. 70, 3–12 (1982) Noor, A.K.: Stability of multilayered composite plates. Fibre Sci. Technol. 8, 81–88 (1975) Quy, N.D., Matzenmiller, A.: A solid-shell element with enhanced assumed strains for higher order shear deformations in laminates. Tech. Mech. 28, 334–355 (2008) Rastgaar Aagaah, M., Mahinfalah, M., Nakhaie Jazar, G.: Linear static analysis and fi-nite element modeling for laminated composite plates using third order shear theory. Compos. Struct. 62, 27–39 (2003) Reddy, J.N., Lee, S.J.: Vibration suppression of laminated shell structures investigated using higher order shear deformation theory. Smart Mater. Struct. 13, 1176–1194 (2004) Simo, J.C., Rifai, M.S.: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Meth. Eng. 29, 1595–1638 (1990) Sze, K.Y., Yao, L.Q.: A hybrid stress ANS solid-shell element and its generalization for smart structure modeling. Part I—solid-shell element formulation. Int. J. Numer. Meth. Eng. 48, 545–564 (2000) Vu-Quoc, L., Tan, X.G.: Optimal solid shells for non-linear analyses of multilayer composites. I. Statics. Comput. Methods Appl. Mech. Engrg. 192, 975–1016 (2003) Wali, M., Hajlaoui, A., Dammak, F.: Discrete double directors shell element for the functionally graded material shell structures analysis. Comput. Methods Appl. Mech. Engrg. 278, 388–403 (2014)

New Approch of High Cycle Fatigue Behaviour of Defective Material under Multiaxial Loading in 1045 Steel Hassine Wannes1, Anouar Nasr2, and Chokri Bouraoui1 1

LGM, université de Monastir ENIM, Avenue Ibn Eljazzar, Monastir 5019 [email protected], [email protected] 2 LGM, EPEIM, Avenue Ibn Eljazzar, Monastir 5019 anouar.nasr @hotmail.fr

Abstract. This paper presents a new approach to predicting the HCF behaviour of defective material submitted to Multiaxial loading .Defects resulting from the casting processes are simplified to semi-spherical pore at surface of specimen. Finite Element (FE) method was used to determine stress distribution around defect. Papadoupolos criterion was used to evaluate stress equivalent around defect. A definition of affected area was given, which is the area close to surface defect where papadoupolos criterion was violated .The evolution of the affected area, with the amplitude of loading and defect size, allows us to determine fatigue limit for different defect sizes. Results of the new approach are in good agreement with experimental results and show that affected area is a good parameter to characterise the influence of a defect on fatigue behaviour.

1

Introduction

The fatigue limit of components containing inherent defect is of great importance for the industrial application and always depends on morphology defect and the loading mode. The designer needs to compromise between the fatigue resistance of the component and the allowable defect size due to the process. Consequently, the influence of the defect behaviour must be characterized to correlate the fatigue limit into size of defects. The objective of this paper is to study the triaxiality repartition of stress gradient around defect and correlate fatigue limit of defective material into defect size. Several ways are explored and different authors have proposed fatigue limit calculation based on the size of defect .Murakami [1] proposed an interesting approch based on experimental results .A geometrical parameter, √ , which is the square root of the projected area of defect on the plane perpendicular to the direction of the maximal principal stress, is used to model the defect size. This approch suggests a practical relationship between the fatigue limit, hardness and this parameter of size.


507

508

H. Wannes, A. Nasr, and C. Bouraoui

Endo [2] has extended Murakami’s criterion to proportional biaxial loading. According to this criterion, the ratio between torsion and tension limits is supposed constant whatever the size of the defect. Using numerical tools other approach have been proposed as The CDM (Critical Distance Method) criterion which is based on measurement of a criterion at a given distance from the surface of defect[3 ]. The last criterion leads to very good results in the case of small defects as shown by Leopold and Nadot [4 ]. Recently, Nadot and Billaudeau [5] proposed an approch to determine the influence of defect size and geometry on fatigue behaviour .They use the gradient of the maximal hydrostatic stress, which was taken into account to propose the HCF criterion for defective material . In the present paper we propose a new approch to correlate fatigue limit to defect size. Triaxiality of stress gradient around defect was taken into account. In the critical plane, perpendicular to the maximal principal stress, we define the affected area, which is the area close to defect where the Papadoupolos criterion was violated .The evolution of affected area versus amplitude of loading and defect size helps us to calculate the fatigue limit for each defect size .The approach was applied for fully reversed tension and torsion for1045 steel. Results are in good agreement with experimental and show that affected area is a good parameter to characterise the influence of a defect on the fatigue behaviour.

2

Material and Experimental Database

This study was carried on the 1045 carbon steel. It has long been used in aircraft and automotive industry. An experimental database containing Multiaxial fatigue results for 1045 steel with induced defects has been previously published [6].

3

The 3D Simulation of Stress Gradient around Defect

3.1

F E Analysis

Defects are always the preferential sites for crack initiation .So we need to characterise the stress distribution around the defect. Finite Element simulations have been conducted for each size of defect and loading mode. Symmetry and boundary conditions are applied . Plasticity occurs around defect so that all simulations are conducted with an elastic- plastic Prandtl-Reuss type model with a linear isotropic hardening law. Meshing is refined and optimized in the critical zone around the defect. The cyclic loading is included in the cyclic stabilised experimental stress-strain curve used to identify the model .

3.2

Stress Distribution around Defect

The plane perpendicular to maximum stress direction is the highest loaded plane (HLP) .Experimental works provides that the crack initiated always at the tip of

New Approch of High Cycle Fatigue Behaviour of Defective Material

5009

the defect in the maximum m shear plane [7] .And the macroscopic crack that leadds to failure of the sample propagate p in HLP. So that we will be interested in thhe distribution of stress on th his plane (Fig.5).

(a)

(b)

ound defect on HLP (S22 maximal principal stress) (a) Tensioon Fig. 1 Stress distribution aro loading

= 180MPa (b) to orsion loading

=150 MPa

A calculation of Papad doupolos equivalent stress plane HLP allows us to deeduce the following results: − −

in the highest loadeed

is almostt constant along the equator of defect (AB) with an erroor that doesn’t exceed d 4% . The distribution off Papadoupolos equivalent stress is independent of thhe chosen direction.

As first approximation to o interpolate the variation of following equations:

in HPL we can propose thhe

• Fully reversed tensio on: =

*

=

*(

/

+1)

(1)

• Fully reversed torsio on /

+1)

Where:

R r

: : : : : :

Defect freee fatigue limit fully reversed tension. Defect freee fatigue limit fully reversed torsion. Amplitudee of tension applying load Amplitudee of torsion applying load Radius of defect. d Distance a from defect centre to a considered point on HLP

These equations, (1) an nd (2), give good results with error less than 5% .

(22)

510


4


4.1

Methodology

From the results of numerical simulations, we can conclude that the variation of is independent of the chosen direction. In fact, on a centred outline at the centre of defect is constant .Our task is to study the variation of in HLP. In this plane, there is an area, close to the defect, where the Papadoupolos criterion violated even with a loading less than the endurance limit of the defective device. Consequently we are led to introduce the following definition: Affected area The affected area by the gradient of stress for a given defect and an applied load is the area where the Papadoupolos equivalent stress is greater than or equal to ( = =169MPa)

Fig. 2 Affected area in HLP

The approach that we suggest consists in following the evolution of the affected area in relation to the loading and the size of the defect. When the loading grows up, the affected area evolves. To the loading of endurance limit corresponds an affected area limit. Thus, to a given defect under a given loading; if affected area affected area limit, there is no rupture before 10 cycles. At the endurance limit we have: affected area = affected area limit.


511

Using equations (1) and (2) the affected area as defined above is determined by the following equations: • Fully reversed tension . affected = ( D 1 √area (3) • Fully reversed torsion: Affected = (

D

1

√area

(4)

Table 1 Experimental results [6]

√area m 170 400 900 √area 300 400 900

σD MPa 195 152 130 τD 157 145 128

Fully reversed tension Affected area 15630 25140 35280 Fully reversed torsion Affected area 20642 42105 57857

MPa. 24378 24100 24418

31000 29700 30788

Using experimental data base table 2.we note that at fatigue limit we have a relationship between affected area limit and the fatigue limit as bellow: * = ct = 24000 MPaμm in case of reversed fully tension. * =ct = 30000 MPaμm in case of fully reversed torsion. Affected area limit was calculated by equations (3) and (4) using the fatigue limit value in each cases Hypothesis For fully reversed tension and torsion the product of fatigue limit by the square root of is almost constant and independent of defect size . The proposed approach consists on the following points: ♦At first we plot curves : = f( equations (3) and (4) for many defect sizes. or ♦ Secondly, from a medium size of a defect we deuce ♦Then at the same graphic we plot the curve: = or ♦Finally, the crossing points allow as to determinate the fatigue limit for each defect size.

512


To summarize necessary steps for the proposed approach, we can use the algorithm fig.3.

Fig. 3 Algorithm to simulate Kitagawa diagram

4.2 4.2.1

Applications Alternative Tension

In alternative tension case the affected area was determined by (3). From an experimental fatigue limit of a medium defect size we calculate the parameter .In this study, a defect size √ = 400μm corresponding to a fatigue = 152 MPa. Using hypothesis 2 we deduce =24100 MPa.um. strength In the same graphic we plot the curves: ♦ Affected area versus loading amplitude, for many defect sizes. ♦ Affected area limit versus loading amplitude: Affected area = . We obtain the graphic Fig.4

Affectedareaum2


513

40000

Affectedarealimit

30000

size:50um size:100um

H 20000

size:200um size:300um size:500um

10000

size:700um size:900um 0 120

140

160

180

200 220 240 Loadingamplitude(MPa)

Fig. 4 Application of proposed approach: fully reversed tension

Fatiguelimt(MPa)

From the crossing point H we deduce that a defect size √ = 300 μm 165 . Referring to experimental results corresponding to fatigue limit where =167MPa.Using all crossing points, for each defect size we can deduce Kitagawa diagram in case of fully reversed tension for 1045 steel fig.6

240

Experimentalresults Proposedapproach

220 200 180 160 140 120 100 0

200

400

Fig. 5 Kitagawa diagram tension loading

600

800 Defectsize(um)

1000

= -1. Material 1045 steel

As shown, obtained results are in good agreement with experimental results and the approach treats well the issue of small defects.

514


4.2.2

Alternative Torsion

In alternative torsion case the affected area was determinate by (4).From an experimental fatigue limit of a medium defect size we calculate the parameter .In this study, a defect size √ = 400μm corresponding to a fatigue strength = 145 MPa. Using hypothesis 2 we deduce =29700 MPa.um. In the same graphic we plot the curves: ♦ Affected area versus loading amplitude, for many defect sizes. ♦ Affected area limit versus loading amplitude: = . We obtain the graphic Fig.6

Affectedareaum2

70000 60000 50000

Affectedarealimit size:170

40000

N

size:200

30000

size:300

20000

size:400 size:600

10000

size:900

0 120

130

140

150

160

170

180

Loadingamplitude(Mpa)

Fig. 6 Application of proposed approach: fully reversed torsion

From the crossing point N we deduce that a defect size √ = 300 μm corresponding to fatigue limit 155 . Referring to experimental results where =155MPa. Using all crossing points, for each defect size we can deduce Kitagawa diagram in case of fully reversed torsion for 1045 steel fig.14


515

Fatiguelimit(Mpa)

180 170

__

160

Experimentalresults Proposedapproach

150 140 130 120 110 100 0

200

400

600

800

1000

dfectsize(um)

Fig. 7 Kitagawa diagram torsion loading

= -1.Material 1045 steel

As shown, obtained results are in good agreement with experimental results but the approach overestimates fatigue limit for small defects (less than 200μm).

5

Conclusions

An experimental database containing multiaxial fatigue results for 1045 carbon steel [7] was used in this study. Defects are simplified as spherical pores at surface of specimen. A simplified model was taken. Load and boundary conditions are applied. Around defect the mesh is refined. After simulation in each point the stress tensor is deduced. Papadoupolos criterion is used to calculate the equivalent stress ( ). The plane perpendicular to the maximum principal stress is the highest loaded was studied in HLP. plane (HLP).The repartition of From this study the following conclusions and remarks can be done: is constant along the equator ♦ For tension and torsion cases the of defect and independent of the chosen direction in the HLP in HLP. ♦ In each case an equation is proposed to interpolate ♦ A definition of affeted area is introduced, which is the area close to defect where Papadoupolos criterion is violated. .which depends on ♦ At fatigue limit we obtain affected area defect size and loading mode. is introduced, which is the product of fatigue limit ♦ A parameter and supposed constant for all defect by the square root of affected area sizes (hypothesis 1) ♦ In each case of loading mode an experimental fatigue limit of medium size is used to evaluate

516


♦ The following of affeted area evolution versus the amplitude of load and defect size allows us to determine fatigue limit for each defect size. ♦ The proposed approch adopted in this study provides encouraging result in alternative tension for 1045 steel and treats well the issue of small defects . ♦ The representation of the equivalent stress gradient in HLP by a quadratic form, equations (1) and (2) is better representative than using the average gradient of hydrostatic pressure or equivalent stress.

References [1] Murakami, Y., Endo, M.: The parameter model for small defects and non metallic inclusion in fatigue strength: experimental evidences and application. In: Blom, A.F., Beevers, C.J. (eds.) Actes de “ Theoretical Concepts and Numerical Analysis of Fatigue”, pp. 51–71. EMAS, Birmingham (1992) [2] Endo, M.: Effects of small defects on the fatigue strength of steel and ductile iron under combined axial/torsional loading. In: Ravichandran, K.S., Ritchie, R.O., Murakami, Y. (eds.) Proceeding of Small Fatigue Cracks: Mechanics, Mechanisms and Applications, Hawaii, pp. 375–387 (1999) [3] Susmel, L., Taylor, D.: Two methods for predicting the Multiaxial fatigue limits of sharp notches. Far. Frac. Eng. Mater. Struct. 26, 821–833 (2003) [4] Leopold, G., Nadot, Y.: Fatigue from an induced defect: experiments and application of different Multiaxial fatigue approach. J. ASTM Int., 7 (2007) [5] Nadot, Y., Billaudeau, T.: Multiaxial fatigue limit criterion for defective materials. Engineering Fracture Mechanics (2005) [6] Billaudeau, T., Nadot, Y., And Bezine, G.: Multiaxial fatigue limit for defective materials: mechanisms and experiments. Acta Mater. 52, 3911–3920 (2004)

Determination of Stress Concentration for Orthotropic and Isotropic Materials Using Digital Image Correlation (DIC) Mhalla Mohamed Makki and Bouraoui Chokri Laboratory of Mechanical Engineering, University of Monastir Tunisia [email protected], [email protected]

Abstract. The objective of this study was to perform a tensile test on isotropic and orthotropic plates with holes and measured deformations around their hole by using the method of digital image correlation (DIC). The results of this measure were used to determine the local stress at the edge of the hole and measured the net and global stress concentration factor for isotropic and orthotropic materials and see the difference between the two types of materials. This paper presents an analysis of the net and global stress concentration factor for different hole diameters and their influence on the isotropic and orthotropic material using a camera DIC. Keywords: orthotropic, isotropic, plate with hole, stress concentration factor, digital image correlation DIC.

1

Introduction

The stress concentration is a local phenomenon that increases the stresses in an area having a geometrical modification of the part. This discontinuity appears in a part of the structure or with the presence of a cut (hole, notch ...). The stress concentration area is often the site of initiation failure, but may also be the cause of a sharp break in the case of a brittle material. Heywood [1] Peterson [2] and Pilkey [3] studied several forms of stress concentration for isotropic materials with a wide range of holes. Howland [4] determined the solution of the rectangular isotropic plate problem with a hole in the center with a tensile load. Peterson and Heywood introduced different equations for a finite plate with different aperture shapes. Lekhnitskii [5] and Tan [6] introduced different formulations for the stress concentration in orthotropic materials. Lekhnitskii derived an equation for an infinite plate with circular holes. And for a finite orthotropic plate Tan has a various equations.


517

518

2

M.M. Makki and B. Chokri

Digital Image Correlation (DIC)

The camera allows the use of in situ recording of a burst frame. These are used to measure the displacement fields and local deformation. The principle of image correlation is to compare local travel in the plane perpendicular to the camera between two different images. Comparing a said deformed image and the reference image allows the measure of local movements (Figure 1). The calculations do not take into account the movement of solid body. From these measurements, the local deformations are deducted in the directions of the study design. To determine the displacement at point of the specimen, it is divided into small images and is squared. The grid represents the space between two thumbnail images, they may intersect. The user can therefore play on two adjustable parameters are the size of the thumbnail images and size of the grid. Although it appears to have the perfect pattern thinnest and finest grid in order to gain accurate measurements, this configuration is not always the most suitable. When the grid is too fine, the correlation has a noise significant measurement.

Fig. 1 Principle of image correlation: determining an optimal field of displacement by image analysis. Reference image and images solicited are devisees of series thumbnail images (set of pixels) [7]

This method is implemented on plate specimens with hole than is covered with a white paint before the test. Is then carried out by spraying an artificial marking paint with a speckle pattern on the surface of the material, via a black spray. Thereafter, the image correlation is performed using the standard software ICASOFT® [Icasoft] developed by Fabrice Morestin (Mguil-Touchal and al.1996) [7] to LaMCoS laboratory of INSA Lyon (Example Icasoft image with Figure 2). The principle is used to match two (original and distorted) images by comparison pixel by pixel with using a criterion based on the grid level in order to evaluate the displacement field and the corresponding strain.

Determination of Stress Concentration for Orthotropic and Isotropic Materials

519

. Fig. 2 Example of an image with camera DIC

3

Experimental Study

The equipment used for the tests of uniaxial tension is: traction machine EZ20 with computer controlled device and image acquisition, including a lamp for lighting. The tests summers Films has captured with the camera positioned facing the test, all using software (Icasoft, INSA Lyon, LAMCOS-MSE) to determine displacement fields and strain. See Figure 3: Traction machine Lamp

Camera DIC PC Software Icasoft

Fig. 3 Diagram of the mechanical acquisition data and digital images with a uniaxial tensile test.

Tensile tests were carried out in a universal testing range of the EASY TEST Machine combine flexibility with exceptional usability, EZ20 model with transverse hydraulically setting with constant velocity about 2 mm / min, with ambient temperature. Tensile tests were performed according to standard ASTM D3039 [8] (Figure 4), using a minimum of twelve specimens (The most current specimens to ASTM D3039 has a constant rectangular cross-sectional dimensions: 250 mm long x 25 mm wide x 2 mm thick test .A typical speed for standard test samples of 2 mm / min) for each family of laminate.

520


Fig. 4 Standard ASTM D3039 [20]

The specimens were made from carbon / epoxy with 0.22 mm thick. The tapes were strength carbon fibers (T700) high strength unidirectional of M21 Hexcel epoxy resin (35% resin). The plate contains eight unidirectional layers [0/90] 2s laminate with a total thickness of 1.76 mm. These tacking sequences have helped to get an overview of the evaluation with different types of damage in notched specimens under uniaxial loading using DIC. The specimens were prepared by bonding the end of glass fiber / epoxy laminate legs. This procedure resulted in the available 150 mm2 of the sample for the tensile test surface. Figure below shows the laminated specimens with different diameters:

Fig. 5 specimens of plate

4

Results

With an ambient temperature, it was two families of tests: steel and composite plates with different hole in the center with the same size and speed of loading (Figure 6) .on notice the break in the steel is still in hole and perpendicular to the loading, against composite the breakage is around the hole but the orientation of crack always followed the direction of tissue .In our example we have a orientation tissue [0.90]2s. So the crack almost perpendicular to the load.


521

Fig. 6 Tensile test for a composite plate with hole (a=4mm)

Thereafter we can plot the stress-time curve for the composite plate (Figure7) such that we can see the first damage and then the fiber break with the total damage:

Fiber break First dommage

Fig. 7 Stress-Time curve for a composite plate with hole (a=4mm) 400 ı

d=3

300

d=5 d=7

200

400 ı

d=4

300

d=5.5

200

d=7

d=9 100

100

0

İ 0

0,01

0,02

0,03

İ

0

0,04

Ͳ100

0

0,005 0,01 0,015 0,02 0,025

Fig. 8 Stress-Strain curve for a steel plate Fig. 9 Stress-Strain curve for a composite with different diameters plate with different diameters

522


failure stress 400 300 200 100 0 3

4

5

6

7

8

9

a 10

Fig. 10 Variation of failure stress with a different hole diameter

Fig. 11 Image of damaged Steel plate

Fig. 12 Image of damaged steel plate with camera DIC

Fig. 13 Image of damaged composite plate

Fig. 14 Image of damaged composite plate with camera DIC


523

Before beginning the analysis of the diameters influence on the intensity of stress concentration, it is necessary to present the general appearance of the stress respect to strain (Figures 8-9). In addition, it is important to know the failure stress of each hole diameter to estimate its influence on the resistance plate (Figure 10). Note that the failure stress down with increasing of notch diameter .We conclude that the hole size has a very important role in the resistance of plaque rupture. For this we used the method of image correlation with DIC camera that gives travel and local deformations to determine the local stresses at geometric discontinuity. Thereafter we can see in Figures (11-18) images of holes in steel and composite plates given by DIC camera: Figure (15) shows the variation of the longitudinal strain (ε11) along the y axis and passing through the center hole, this variation measured from DIC camera. In These planar structures, the state of stress is fairly well represented by a plane stress tensor, that:

İ0,12 11 0,1 0,08 0,06 Isotropic material

0,04

Orthotropic material

0,02 0

Y pixel

1,38E+01 1,58E+01 1,78E+01 1,98E+01 2,18E+01 2,38E+01 Ͳ0,02

Fig. 15 Variation of the longitudinal strain (ε11) along the y axis

Subsequently, the longitudinal stress around the hole can be calculated from the deformation field (ε11 and ε22) obtained from DIC:

Finally we can plot the stress concentration (Kt = σ /σ avec σ σ ) with a different hole diameters for steel and composite (carbon / epoxy) materials.

524

5

M.M. Makki and B. Chokkri

Theoretical Co oefficient of Global Stress Concentration

Theoretical coefficient off stress concentration is defined as the ratio of the actuual maximum stress in the zo one of discontinuity (notch, hole, for example) to the noominal stress in the section n: Kt = σ

/σ

: Is calculable by numerical n methods such as finite element method or bby σ analytical methods for siimple geometries. It is also measured by the techniccal analysis of experimental constraints c such as photo-elasticity or DIC methods. σ : Is calculable using the formulas of strength materials, considering thhe specimen like a bar or plaate without account the geometric discontinuity.

σ

=

With F: traction, uniforrm on edges of plate. w: width of the plate. h : plate thickness.

6

Evolution of th he Global Stress Concentration with Hole Diameter

6.1

Case of Isotrop pic Material

For a rectangular plate wiith a circular hole in the middle and subjected to a tensiile loading in one direction.. It occurs around the hole increasing the value of thhe stress that is characterized d by the stress concentration factor KTg Howland [9]:

6.2

Case of Orthotrropic Material (Composite)

For a thin composite platee (assumed orthotropic) infinite (w >> r) with a central hole. The coefficient Kt iss obtained by Lekhnitskii [5]: = =1+ 2

/

/

Determination of Stress Con ncentration for Orthotropic and Isotropic Materials

5225

In our case, the study focuses on an infinite orthotropic plate 3D is made oof carbon /epoxy (HR [0/90] 2s) .According of Table 1 Kt equal to 5.24 for an inffinite orthotropic plate. According to the inveestigation of Heywood [1] on a rectangular orthotroppic plate with center hole und der the action of unidirectional axial load, the factor is inntroduced in the form of th he following equation:

With

7

The Theoretica al Value of the Net Stress Concentration

Another way to determinee the stress concentration is the application of the averagge normal stress across the net n section instead of the hole width of the plate (global surface).

σ

=

This concentration is know wn as the net concentration of stresses K T can be related to the globaal stress concentration factor K T by:

K T = K T (1-a/w)

[9, 1] and

526


8

Validation by Comparison with the Results of the FE Analysis

8.1

Modelisation with Finite Elements

To validate our work can be compared with the results found by finite elements analysis and with literature. A simulation under "ABAQUS 6.12" software was performed with a mesh-like continuum Shell hexagonal, progressive and refined around the hole. Two systems of different materials are considered: Table 1 Material Properties orthotropic carbon / epoxy (T700 / M21 [0 90] 2s) Table 2 E11

E22 (GPa)

G12 (GPa)

ν12

σ11T (MPa)

σ11c (MPa)

τ12 (MPa)

7.8

4.5

0.35

2375

1465

95

(GPa) 148

Table 2 Properties of Isotropic Steel Type Elastic modulus E(GPa) 210

Poisson ratio v 0.3

mass density ρ (kg / m3) 7800

A tensile load of 119 MPa at one and a lock on the opposite. The fineness of the mesh is optimized in order to have a steady value of maximum stress. To reduce the computation time while maintaining good accuracy, only a quarter of the plate is meshed with progressive mesh designed to produce a very fine mesh around the hole and larger elsewhere (Figure 16).

Fig. 16 gradual meshing of the plate


8.2

527

Evolution of the Stress Concentration with Diameter Hole (Isotropic Material):

The results found after an experiment method with camera DIC and with ABAQUS simulation are presented in Table 3:

Table 3 Isotrope Material a

a/w

0,2 2 4 6 8 10

0,02 0,1 0,2 0,3 0,4 0,5

Kt infinite 3 3 3 3 3 3

Ktg DIC 3,0048 3,0582 3,1900 3,4097 3,7543 4,2937

Ktg FEA 3,004 3,04 3,15 3,3 3,6 4,1

Ktn DIC 2,82 2,61 2,34 2,22 2,12 2,06

Ktn FEA 2,78 2,53 2,21 2,07 1,91 1,81

For a given width, this ratio Kt equal to 3 for an infinitely thin plate with a circular hole and thereafter the global stress concentration factor Ktg increases if the hole diameter increases against the net stress concentration factor Ktn decreases.(Figure 17 and 18):

Ktg 11,5

DIC

10,5

FEA

9,5

Heywood

8,5

Howland

7,5 6,5 5,5 4,5 3,5 a/w

2,5 0

0,2

Fig. 17 Variation Ktg isotropic with ratio a/w

0,4

0,6

a/w

528


Ktn 3,5

DIC

3

FEA

2,5

Heywood

2

Howland

1,5 1 0,5 a/w

0 0

0,2

0,4

0,6

Fig. 18 Variation Ktn isotropic with ratio a/w

For isotropic material we can notes that the results of finite element analysis are very close to the results of DIC camera but for the stress concentration Ktg there is a difference from a / w = 0.4 compared with Heywood and Howland.

8.3

Evolution of the Stress Concentration with Diameter Hole (Orthotropic Material):

Table 4 compares the stress concentration between the finite element analysis and image correlation method: Table 4 Orthotropic Material a

a/w

0,2 2 4 6 8 10

0,01 0,1 0,2 0,3 0,4 0,5

Kt infinite 5,24 5,24 5,24 5,24 5,24 5,24

Ktg (DIC) 3,4 3,9 4,52 5,26 6,43 7,12

Ktg (FEM) 5,41 6,02 6,32 6,62 7,02 7,22

Ktn (DIC) 2,9 2,72 2,47 2,21 1,82 1,62

Ktn (FEM) 4,81 4,12 3,2 2,8 2,1 1,7


Ktg 14

529

DIC

12

FEA

10 Heywood 8 6 4 2 a/w

0 0

0,2

0,4

0,6

Fig. 19 Variation Ktg orthotropic with ratio a/w

On observed that a / w is increased when the edges of the plate is closer to the hole. Consequently, the lines of force will be more compressed and Ktg increases.

6Ktn

DIC

5

FEA

4 Heywood 3 2 1 a/w

0 0

0,2

0,4

Fig. 20 Variation Ktn orthotropic with ratio a/w

0,6

530


Note that the net stress concentration decreases if the ratio a / w increases and unlike different with global stress concentration factor Ktg. It has almost the same variation of KTn respect to a/ w for isotropic and orthotropic material.

9

Conclusion

In this work we have used the image correlation method with DIC camera to measure the tensile stress concentration and which gives consistent results with the finite element analysis and thereafter it is found that the DIC method given a best results for a planar structure isotropic or orthotropic. The results obtained show that the presence of a geometrical discontinuity in a flat plate has a significant effect on the stress concentration to an isotropic and orthotropic material. In addition we see the evolution of global and net stress concentration respect to different hole diameters for composite and steel. Finally, the DIC approach don't provide good results for composite structures because this method can be show only the surface damage against the composite material having many damage inside the structure (delamination ,cracking matrix….). Acknowledgements. I want to thank deeply Mr. Jury members who have done me the honor to judge my work and have improved my presentation with helpful comments.

References [1] Heywood, R.B.: Designing by photoelasticity. Chapman and Hall (1952) [2] Peterson, R.E.: Stress concentration factors. John Wiley & Sons (1974) [3] Pilkey, W.D.: Peterson’s stress concentration factors, 2nd edn. John Wiley & Sons Inc., New York (1997) [4] Howland, R.C.J.: On the stresses in the neighborhood of circular hole in a strip under tension. Phil. Trans. Roj. Soc. 229, 49–86 (1929) [5] Lekhnitskii, S.G.: Theory of Elasticity of an Anisotropic Elastic Body, 1st edn., p. 4. Holden-Day, San Francisco (1963) [6] Tan, S.C.: Finite-width correction factors for anisotropic plate containing a central opening. J. of Composite Materials 22, 1080–1097 (1988) [7] Mguil-Touchal, S.: Une technique de corrélation directe d’images numériques. Thèse INSA Lyon (1996) [8] ASTM D3039/D3039M-00. Standard test method for tensile properties of polymer matrix composite materi-als. American Society for Testing Materials [CD-ROM] (2004) [9] Howland, R.: On the stresses in the neighborhood of a circular hole in a strip under tension. Philosophical Transactions of the Royal Society A 229, 67 (1929-1930)

The Extended Finite Element Method for Cracked Incompressible Hyperelastic Structures Analysis Mehrez Zaafouri1, Mondher Wali1, Said Abid1, Mohammad Jamal2, and Fakhreddine Dammak1 1

Mechanical Modelisation and Manufacturing Laboratory (LA2MP), National Engineering School of Sfax, B.P W3038, Sfax, University of Sfax, Tunisia {zaafourimehrez,mondherwali}@yahoo.fr, [email protected], [email protected] 2 Faculté des Sciences Ben M’Sik, Laboratoire dÊIngénierie et Matériaux (LIMAT), Université Hassan II de Casablanca, B.P. 7955, Sidi Othman, Casablanca, Morocco [email protected]

Abstract. This paper aims to examine the contribution of the extended finite element method (XFEM) in finite strain fracture mechanics problems. A generalized neo-Hookean hyperelastic material is considered in an incompressible plane stress approximation. The accuracy of the implementation is demonstrated by a series of numerical tests. Keywords: XFEM, Hyperelasticity, Crack Tip fields, Finite strain.

1

Introduction

The numerical simulation of the cracked structures represents an important issue for many industrial sectors. Among the numerical tools developed to simulate cracked structures one finds the finite element methods (FEM). A number of approaches have been developed within the FM framework in over the years, which makes it as a most suited method to investigate the asymptotic stress and displacement fields at the crack tip. However, FEM requires that the crack surface coincide with the finite elements boundary and a conformal mesh is needed. Therefore, frequent remeshing is necessary for crack growth modeling. To overcome these difficulties, several approaches were proposed, the most recent is the extended finite element method (XFEM) introduced by (Moës et al. 1999) and ameliorated by (Dolbow et al. 2000). It is an extension of the FEM dedicated to the modeling of geometrical discontinuities The principal characteristic of the XFEM is the decoupling between discontinuity and the grid: the crack is


531

532

M. Zaafouri et al.

represented independently of the grid by introducing enrichment functions into the finite elements approximation base taking into account the displacement jump along the crack as well as the asymptotic displacement field near the crack tip. And since, several problems of fracture mechanics were solved by the mentioned approach (Fries and Belytschko 2010) and (Mohammadi 2008). Although this alternative is promising, it remains often held for the linear elastic problems of fracture mechanics. The extension of the XFEM method to non-linear fracture mechanics raises major difficulties and a few works are developed for this subject. (Dolbow and Devan 2004) present a geometrically non-linear assumed strain method that allows for the presence of arbitrary intra-finite element discontinuities. In (Legrain et al, 2005), the others used the XFEM method in incompressible plane stress problems with classical Neo-Hook materials. Later, (Karoui et al. 2014) present an extension of XFEM method to large deformation of cracked hyperelastic bodies in plane strain approximation. The major problem in XFEM method in to large strain of cracked hyperelastic bodies is the singular enrichment witch concern nodes whose support contains the crack tip. In linear elasticity, the asymptotic displacement field near the crack tip is used as singular enrichment. In case of hyperelasticity, some authors obtained the analytical expression of the asymptotic fields within the simplified framework of plane elasticity. Theoretical results were established by (Knowles and Sternberg 1973, 1974) for compressible Blatz–Ko materials in plane strain, by (Stephenson 1982) for incompressible materials in plane strain, by (Knowles and Sternberg 1983) for neo-hookean material in plane stress and by (Geubelle and Knauss 1994) for incompressible generalized neo-Hookean material in plane stress. (Long et al. 2011) presents a new method to determine the asymptotic stress and deformation fields near the tip of a Mode-I traction free plane stress crack. There analysis is based on the fully nonlinear equilibrium theory of incompressible hyperelastic solids. Two types of soft materials are used: generalized neo- Hookean solids and a solid that hardens exponentially. For the generalized neo-Hookean solids, this method is able to resolve a difficulty in the work by (Geubelle and Knauss 1994). In this paper, a Neo-Hook material is considered in incompressible plane stress approximation. This paper is organized as follows: in the next section, the constitutive equations of the problem are presented. Weak form and linearization are given in section three. In section four, technical aspects of the implementation of the XFEM in non-linear elasticity are presented. Numerical examples of this implementation, which show the performance of the method, are given in section five. Finally, same concluding remarks and possible extensions to this work are given.

The Extended Finite Element Method

2

533

Constitutive Equation

Let us consider the deformation movement defined by the function 3 φ ( X ,t ) : B × R → R and F ( X ,t ) is the deformation gradient with X ∈ R

3

indicate the position of a point in the reference configuration. F =

∂φ

, J = det ( F )

∂X

(1)

The symmetric right Cauchy-Green tensor C is defined by

C=F F T

(2)

The hyperelasticity implies the existence of an energy function dependant on the tensor C. In the case of isotropic hyperelasticity, the energy function is a function of C as follows

ψ = ψ ( I1 ,I 2 ,I3 )

(3)

where I1 , I 2 and I 3 are the three invariants of C tensor

I1 = tr ( C ) , I 2 =

(I 2

1

2 1

− trC

2

)

,

I 3 = det ( C ) = J

2

(4)

In the case of incompressibility constraint, the third invariant of the right Cauchy-Green tensor is equal to unit. Therefore, the principal invariant I1 and I 2 are the only independent deformation variables.

ψ = ψ ( I1 ,I 2 )

J =1

,

(5)

To determine the constitutive equations for isotropic hyperelastic materials in term of strain invariants, we consider the differentiation of ψ with respect to tensor C. By means of the chain rule of differentiation we find, (Dammak et al. 2007 and Jarraya et al. 2011), the most general form of the second Piola-Kirchhoff stress tensor in the incompressible case 2

S = 2∑ψ ,i i =1

∂I i ∂C

= α I +α C +α C 1

2

3

−1

(6)

534

M. Zaafouri et al.

Where I , C and C

−1

are in plane tensors. However, the most general form

of the elasticity tensor  in terms of the principal invariants is given by the following expression

∂S

 =2  = δ I ⊗ I +δ 1

δ C ⊗C +δ 4

2

∂C

−1

(7)

∂C ∂C

(I ⊗C +C ⊗ I)+δ

(C ⊗ C

5

∂ ψ ( I1 , I 2 ) 2

=4

3

)

(I ⊗C

−1

)

−1

+C ⊗I +

+ C ⊗ C + δ C ⊗ C + δ I C + δ I ijkl −1

−1

6

−1

7

8

(8)

−1

where the following notations are used

(I ) C

−1

=

ijkl

(C

1 2

( )i =1,3

The terms α

−1 ik

C

−1 jl

+C

( )

i and δ iso

i

−1 il

i =1,8

C

−1 jk

)

I ijkl =

,

1 2

(δ

ik

δ jl + δ il δ jk )

(9)

are given in (Dammak et al. 2007 and Jar-

raya et al. 2011).

3

Weak Form and Linearization

The numerical solution with the finite element method is based on the weak form of equilibrium equations. The three dimensional form of the latter in the total Lagrangian formulation is given as G=

∫

V

S : δ E dV − Gext = 0

(10)

where Gext is the virtual work of the external forces, S is the second PiolaKirchhoff stress tenser and δE represent the virtual Green-Lagrange strain tensor. E =

δE =

1 2

δC =

1 2

(C − I )

( δF 2

1

T

(11)

F + F δF T

)

(12)


535

If the following kinematics is adopted

φ= x = X +u

(13)

then, in 2D, F and δE can be written as F = [ x ,1

x ,2 ]

,

(14)

with

δE = B δu

where

,

T x ,1 ( .) ,1 ⎡ ⎤ ⎢ ⎥ T B= x ,2 ( .) ,2 ⎢ ⎥ ⎢⎣ x ,T1 ( .) ,2 + x ,T2 ( .) ,1 ⎥⎦

(15)

( .),i is a material derivative. With these matrix notations, the weak form

(10) can be written in the following form

G=

∫ ( Bδ u )

T

V

S dV − Gext ( δ u ) = 0

(16)

In the optics of a resolution with the Newton method, its linearization is developed in two parts: material and geometrical DG ⋅ Δu =

∫ ( B δ u )  ( B Δu ) dV + ∫ ∇ (δ u ) :∇ ( Δu ) SdV T

V

V

(17)

where  is the elasticity material tensor given in equation (8).

4

Extended Finite Element Formulation

Within the Xfem framework, the crack is taken into account by two different types of enrichment: discontinuous enrichment and singular enrichment (Fig. 1). The energy function of generalized neo-Hookean materials is

μ ⎛⎡

n ⎞ ⎤ ψ = ⎜ 1 + ( I − 3) − 1 ⎟ ⎢ ⎥ 2b ⎝ ⎣ n ⎦ ⎠

b

(18)

536

M. Zaafouri et al.

As in (Legrain at al. 2005), the singular enrichment function that will be used is

•



Discontinuous enrichment Singular enrichment

Fig. 1 Xfem enrichment

{

F= r

}

sin (θ / 2 )

1/ 2

(19)

The numerical implementation of the presented formulation is based upon a four node non-linear plane stress element. Initial geometry and displacement vectors are approximated by

X=

∑N X i

(20)

i

i∈I n

u=

∑ N u + ∑ N (H − H )a + ∑ N (F − F )b i

i

i

i

i ∈I n

i

i

i∈ I H

i

i

δ u = ∑ N δ ui + ∑ N ( H − H i ) δ ai + ∑ N ( F − Fi ) δ bi i

i

i ∈I n

(21)

i ∈I F

i

i ∈I H

(22)

i ∈I F

Matrix B , in equation (15), will be at node (i):

B = ⎡⎣ Bu i

T i x ,1 N ,1 ⎡ ⎤ ⎢ ⎥ i T i Bu = x ,2 N ,2 ⎢ ⎥ ⎢⎣ x ,1T N ,i2 + x ,T2 N ,1i ⎥⎦

i

i

Ba

,

Bb ⎦⎤ i

T i x ,1 M ,1 ⎡ ⎤ ⎢ ⎥ i T i Ba = x ,2 M ,2 ⎢ ⎥ ⎢⎣ x ,1T M ,i2 + x ,T2 M ,i1 ⎥⎦

(23)

(24)


537

T i x ,1 L,1 ⎡ ⎤ ⎢ ⎥ i T i Bb = x ,2 L,2 ⎢ ⎥ ⎢⎣ x ,1T Li,2 + x ,T2 Li,1 ⎥⎦

M =N i

With

i

(H − H )

(25)

L =N i

,

i

i

(F − F )

(26)

i

( ⋅) ,

, i = 1,3 are materials derivatives. Finally, the element residual and material and geometric tangent operator are expressed as:

and

i

Gin = i

KG = ∫ ij

S

∫

i

T

B SdV

⎡ f ( Ni ,N j ) I ⎢ i j ⎢ f (M ,N ) I ⎢⎣ f ( Li , N j ) I

∫

ΚM = ij

,

V

V

( )I f ( M ,M ) I f ( L ,M ) I i

j

i

j

f N ,M

i

B  B dV i

T

j

(27)

( ) ⎥ f ( M , L ) I dS ⎥ f ( L , L ) I ⎥⎦ f N ,L I ⎤

j

i

i

j

i

j

(28)

j

where:

(

i

f N ,M

5

j

) = N, (S i

1

11

)

(

M ,1 + S12 M ,2 + N ,2 S 21 M ,1 + S 22 M ,2 j

j

i

j

j

)

(29)

Numerical Example

In order to illustrate the accuracy and versatility of the extended finite element method X-FEM in nonlinear fracture mechanical problems for large displacements whose behavior is isotropic hyper-elastic kind, a numerical example is presented in this section. The example is solved using both FEM and X-FEM techniques, and the results are compared. For the purpose to perform a real comparison, the number of elements in FEM and X-FEM meshes are taken almost equal. For XFEM a uniform meshes elements are used independent of the shape of crack. While in FEM analyses, the crack is aligned with element edges and refined mesh is applied near the crack region. All examples are simulated with the plane stress approximation. The example shown in Figure 2 illustrate a plate subjected to an imposed displacement on its top edge equal to 4mm and its bottom edge is fixed.

538

M. Zaafouri et aal.

h

(a)

(b)

(c)

Fig. 2 (a) Plate with lateral ccrack under imposed displacement (b) X-FEM mesh, (c) FEM m mesh (Abaqus)

Fig. 3 (a) XFEM deformed configuration c (b) Abaqus deformed configuration

Plate geometry and maaterial properties are adopted from (Legrain et al, 20055). The width and height of this plate are respectively w = 2 mm and h = 6 mm. A crack of length a = 1mm is inserted at mid-height. The material is assumed to bbe Neo-Hookean with μ = 0.4225.


539

The deformed configurations obtained by X-FEM and FEM methods are presented in figure 3. We notes that the XFEM field displacement discontinuity near the crack is well represented and conformed to Abaqus results. U2-XFEM

U2 : Displacement

3,5

U2-Abaqus

3 2,5 2 1,5 1 0,5 0

0,5 Distance X

1

1,5

Fig. 4 Comparison of crack U2-displacement

In order to make a comparison between two techniques, the deformed shape of the crack obtained by the vertical displacements of the nodes situated on the edges of crack are plotted in figure 4. The results are in perfect agreement proving the advantage of the X-FEM method versus FEM.

6

Conclusions

In this paper, the extended finite element method (XFEM) is used to analyze the finite strain fracture mechanics problems with a neo-Hookean hyperelastic material with an incompressible plane stress approximation. In order to demonstrate the efficiency of the X-FEM method, a numerical example was considered showing displacement field near the crack and the deformed shape. A comparison of XFEM and FEM results was carried showing the reliability of XFEM method to nonlinear fracture mechanic. Acknowledgment. The authors gratefully acknowledged and express their gratitude to the CMPTM (Tunisian-Moroccan Joint Standing Committee) under reference 13/TM42 for their support and technical assistance of this research.

References Dammak, F., Regaieg, A., Kallel Kammoun, I., Dhiab, A.: Modeling the law of transversely isotropic hyperelastic behavior of elastomers. European Journal of Computational Mechanics 16(1), 103–126 (2007)

540

M. Zaafouri et al.

Dolbow, J., Moes, N., Belytschko, T.: Modeling fracture in Mindlin-Reissner plates with the eXtended finite element method. Int. J. Solids Structures 37, 7161–7183 (2000) Dolbow, J.E., Devan, A.: Enrichment of enhanced assumed strain approximations for representing strong discontinuities: Addressing volumetric incompressibility and the discontinuous patch test. International Journal for Numerical Methods in Engineering 59, 47–67 (2004) Fries, T.P., Belytschko, T.: The extended/generalized finite element method: An overview of the method and its applications. Int. J. Numer. Meth. Engng. 84, 253–304 (2010) Geubelle, P.H., Knauss, W.G.: Finite strains at the tip of a crack in a sheet of hyperelastic material: I. Homogeneous case. J. Elasticity 35, 61–98 (1994) Jarraya, A., Kallel-Kammoun, I., Dammak, F.: Theory and finite element implementa-tion of orthotropic and transversely isotropic incompessible hyperelastic membrane. Multidiscipline Modeling in Materials and Structures (MMMS) 7(4), 424–439 (2011) Karoui, A., Mansouri, K., Renard, Y., Arfaoui, M.: The extended Finite Element Method for cracked hyperelastic materials: a convergence study. Int. J. Numer. Meth. Engng. 100(3), 222–242 (2014) Knowles, J.K., Sternberg, E.: An asymptotic finite-deformation analysis of the elastostatic field near the tip of a crack. Journal of Elasticity 3, 67–107 (1973) Knowles, J.K., Sternberg, E.: Finite deformation analysis of the elastostatic field near the tip of a crack: reconsideration of higher order results. Journal of Elasticity 4, 201–233 (1974) Knowles, J.K., Sternberg, E.: Large deformation near the tip of an interface crack between two neo-Hookean sheets. J. Elasticity 13, 257–293 (1983) Legrain, G., Moës, N., Verron, E.: Stress analysis around crack tips in finite strain problems using the eXtended finite element method. Int. J. Numer. Meth. Engng. 63, 290–314 (2005) Long, R., Krishnan, V.R., Hui, C.Y.: Finite strain analysis of crack tip fields in incompressible hyperelastic solids loaded in plane stress. Journal of the Mechanics and Physics of Solids 59, 672–695 (2011) Möes, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 46, 131–150 (1999) Mohammadi, S.: Extended finite element method. Blackwell Publishing (2008) Stephenson, R.A.: The equilibrium field near the tip of a crack for finite plane strain of incompressible elastic materials. Journal of Elasticity 12, 65–99 (1982)

Displacement Influence on Frequencies and Modal Deformations of a Sandwich Beam Idris Chenini1, Youssef Abdelli2, and Rachid Nasri3 1

Assistant to the ESIER: elkef road, 5 km Medjez elbâb, 9070, Beja [email protected] 2 Aassociate professor at the military academy, Address: Fondek ejjdid Grombalia, Nabeul [email protected] 3 Professor at the ENIT, Address: 37 B.P. Le Belvédère, 1002 Tunis [email protected] 1, 2,3 Research laboratory on Applied Mechanics and Engineering (LR-MAI-ENIT)

Abstract. Composite structures are used in the aerospace, stars and automotive especially the structures made of composite sandwich panels which are subject to vibration harmful sources of noise and mechanical failures. In this paper, we propose to analyze the influence of the shapes of displacement fields on the vibration behavior of a sandwich beam. For this energy method based on the minimum energy is used to achieve the equations frequencies and a sandwich beam modes. The kinetic and potential energies of the skins are, in turn, derived from the classical laminate theory. Several fields of polynomial movements are tested. Other parts will study the effects of rotational inertia, taking into account the bending energy of the body of sandwich NIDA. We analyzed the frequencies and modes based on different parameters. The experimental data are obtained for recessed-free conditions limits exciting near underrun using an impact hammer. The vibrational response is measured with a laser vibrometer. The natural frequencies are obtained experimentally by modal analysis. Numerical simulations complete this work for two types of sandwich Nomex paper and aluminum. The natural frequencies obtained from the theoretical formulation for numerical solution of the system are compared with experimental results and the results of numerical simulation. The very good agreement between the results shows that the model is correct. Keywords: Displacement, energy, sandwich, Eigen frequency, modal deformation.

1

Introduction

Energy methods are among the most important methods to study the vibration behavior of beams. The problem with these methods is that they require the


541

542

I. Chenini, Y. Abdelli, and R. Nasri

introduction of a field of kinematic ally admissible displacement or statically admissible stress or both at once. The accuracy of the method will therefore depend on the right choice of these fields. In the literature, there are several choices of the field selection. But it is difficult to decide on their preferences depending on their application. Mindlin [1] proposed a model to study the transverse shear of a thick sandwich plate isotope; this model is known by the theory of Mindlin. Yarlagadda Lesieutre and [2] have developed an analytical method based on the theories of Rayleigh-Ritz to determine the influence of the change of bends orientation directions, temperature and thickness of the laminated plates on the Frequencies and the damping of the bending vibration of a composite plate. Wang [3] compared the classical theories of thin plates with the theories of Mindlin thick plates then he studied the bending vibrations of a rectangular face and core isotropic sandwich panel. Meunier and Shenoi [4] introduced the mechanical properties of the elements which form the sandwich plate in the analytical equations of elastic-viscoelastic model and the results of the differential equations give their frequencies and the factors influencing them. Soula et al. [5] have studied the influence of the vibration on the kinematic behavior of laminated plates. They used three theories plates (Kirchhoff, Mindlin and Reddy) to estimate the natural frequencies and modal damping of symmetric and antisymmetric plates. Maheri and Adams [6] extended the tests used for measuring the damping of stratified monolithic laminated beams in bending vibration for NIDA sandwiches beams. The contribution due to the damping components, in particular the fiber orientation of the skins, was considered. They compared their experimental results with theoretical study. Banerjee [7] applied the Timoshenko theory on a beam using the coupling of bending and torsion and taking into consideration the rotational inertia. The characteristics of the composite are taken experimentally as if the material is homogeneous. Nilsson et al [8] used equivalent homogeneous characteristics to study the bending vibrations of a sandwich beam from the Hamilton integral taking into consideration the transverse shear, the rotational inertia and distortion. They write the dispersion equation giving the wave number for the three modes according to the frequency for some boundary conditions and then they find the modal deformations and dynamic stiffness, the formulation is then validated by a test. Blevins [9] presented a detailed list of formulas for determining the natural frequencies and their modes corresponding to various structure shapes. On this paper, we study the influence of the displacement fields shape under different boundary conditions of a sandwich beam on the natural frequencies and modal deformations. This study which is applied on two different beams shall give us more ideas about the influence of various fields on the modeling accuracy.

Displacement Influence on Frequencies and Modal Deformations

543

An experimental study of two sandwich beams also allows us to understand more this field by comparing the results of tests and calculations on the one hand and various tests according to the excitation and measuring points on the other.

2

Mathematical Formulation

We choose an n order polynomial transversal displacement depending of the constants (1..n = i-1) and the kinetic and potential energies are determined as follows. The potential energy of a sandwich beam is the sum of the bending elastic deformation energy of and traction of both upper and lower faces as well as the shear and bending energy of the body.

Y

c

Z

X

b

a

Fig. 1 Nida Sandwich beam

The elastic deformation of the lower side can be written as follows:

ε xB = ( z + ε ' xB = ( z − c − VBF =

f l ∂ 2W c )( ) 2 ∂x 2

(1)

f u ∂ 2W c )( ) 2 ∂x 2

(2)

a 0 E dz (ε xB ) 2 dx 2(1 − ν 2 ) ∫− fl ∫0

(3)

With the same method we determine the energy of elastic bending deformation of the upper layer:

V ' BF =

c+ fu a E dz ∫ (ε ' xB ) 2 dx 2 ∫c 0 2(1 − ν )

(4)

To determine the tensile energy of one of the faces we must first determine the normal force to this surface and its extension in the parallel axis to the face. N FL : The normal Force at the lower face and N FU : the normal force at the upper face

544

I. Chenini, Y. Abdelli, and R. Nasri 0

N FL = ∫ σ f dz − fl

The Formula of

(5)

N FU = ∫

And

c + fu

c

σf

σ f dz

(6)

which is the stress in an arbitrary face of the fiber is as

follows:

c ∂ 2W c ) 2 ∂x 2

σ f = − E ( z − )(

(7)

Substitute (7) in (5) and (6) we obtain: 0 c ∂ 2W c )dz N FL = − ∫ E ( z − )( − fl 2 ∂x 2

(8)

N FU = − ∫

And

c+ fu

c

c ∂ 2W E ( z − )( 2 c )dz 2 ∂x

(9)

Using the integral we determine the normal forces at the both the upper and the lower faces:

N FL = E

fl ∂ 2W c (c + f l )( ) 2 ∂x 2

(10)

N FU = − E

fu ∂ 2W c (c + f u )( ) 2 ∂x 2

(11)

VNF : is the tensile energy of the lower layer of the normal force of this face.

ε xL : The elastic deformation in the x direction of the lower face. ε xU : The elastic deformation in the x direction of the upper face. V NF =

1 a N FL ε xL dx 2 ∫0

(12)

1 a (− N FL ).(−ε xL )dx 2 ∫0

V ' NF =

(13)

The bending elastic deformation energy of the bottom layer per unit width is as the following:

1 2

ε xL = (c + f l )(

∂ 2W c ) ∂x 2

2 (14) and ε xU = − 1 (c + f u )( ∂ W c ) 2 ∂x 2

(15)

Substituting (10) and (14) in (12) we obtain:

VNF =

E. f l .(c + f l ) 2 8

∫

a

0

∂ 2W c 2 ) dx ∂x 2

(16)

∂ 2W c 2 ) dx ∂x 2

(17)

(

Substituting (11) and (15) in (13) we obtain:

V ' NF =

E. f u .(c + f u ) 2 8

∫

a

0

(

The body Shear energy (Vc) par unit width is as following:

Vc =

a 1 c 2 dz ∫ G XZ ⋅ (γ XZ ) dx ∫ 0 2 0

(18)


With: γ XZ = 1 ⋅ τ XZ

545

and τ XZ dx = −

(19)

G XZ

∂ N FL dx ∂x

(20)

The body bending energy is as following: 2

VFC

⎛ ∂ 2W ⎞ 1 a = ∫ Ec .I .⎜⎜ 2 c ⎟⎟ dx 2 0 ⎝ ∂x ⎠

With

I=

c 3 .b 12

(21)

EC : Young Modulus of core

From the previous formula we determine the total potential energy which is:

V = VCC + VFC + V NF + V ' NF + VBF + V ' BF

(22)

The kinetic energy of the vibration of the sandwich panel is equal to the sum of kinetic energy of translational and rotational kinetic energy.

T = ECT + ECR

(23)

2

Where: E = 1 ρ a ⎛⎜ ∂W ⎞⎟ dx CT ∫ 2

0

⎝ ∂t ⎠

(24)

And

ECR =

1 a ⎛ ∂ ⎛ ∂W I⎜ ⎜ 2 ∫0 ⎜⎝ ∂t ⎝ ∂x

The vibration system is considered as conservative where

2

⎞⎞ ⎟ ⎟⎟ dx ⎠⎠

(25)

Ai (i =1..n-1) are

constants. The resolve of the obtained system allows us to determine (n-1) natural frequencies and their shape mode.

3

The Beam Results

First, we presented the mechanical and geometrical characteristic of test specimens analyzed Table 1 Geometrical and mechanical characteristics of used specimen NIDA in Al

NIDA in NOMEX

Length (a) in mm

250

250

Width (b) in mm

53

53

Thickness core (c) in mm

5

8

0.75

0.5

573

221

Ec in Pa

130.106

2.5 .106

E p in Pa

70.109

8.63 .106

G XZ in Pa ν

5600.106

70.109

0.33

0.33

Each layer thickness f u Density

ρ in Kg/m3

= fl

in mm

546


Second, we take a beam where field checking the no translation and not the non-rotation on the fixed end section x=0 is: We present only the results where the differences between fields are observable. We mention that the fields are the following: Field 1: order 2 polynomial Field 3: order 4 polynomial Field 5: order 6 polynomial

Field 2: order 3 polynomial Field 4: order 5 polynomial

4000

377

sans prise en compte de l’inertie rotationnelle et de la flexion dans le corps

sans prise en compte de l’inertie rotationnelle et de la flexion dans le corps Avec flexions corps

Avec flexions corps

Avec effet d'inertie rotationnelles

Avec effet d'inertie rotationnelles

375

avec prise en compte de l’inertie rotationnelle et flexion dans le corps

3500

avec prise en compte de l’inertie rotationnelle et flexion dans le corps

373

3000

371

2500 369

2000 367

365 Champs 2

Fileds 1

Champs 3

Champs 4

Filds 2

Filds 3

Champs 5

Filds 4

Fig. 2 First natural frequency depending of several parameters

1500 Champs 2

Fileds 1

Champs 3

Filds 2

Champs 4

Filds 3

Champs 5

Filds 4

Fig. 3 Second natural frequency depending of several parameters

The first pulse from the first displacement fields (order 2 polynomial) with only one variable is widely superior to other values which tend to the same region limit 370 rad/s if we consider both the effect of of rotational inertia and the bending body. The value of the 2nd natural frequency tends to a limit close to 2300 rad/s without the effect of the rotational inertia and converges to 2200 rad / s from the third field (order 4 polynomial). The value of the third natural frequency converges only from the fourth field (order 5 polynomial).

------ Fileds 1 ∗ Filds 2 + Filds 4

-------

*Fileds 5

Filds 3

- - -

Fig. 4 first modal deformation comparison between different fields

∗ Filds 2 ------- Filds 3 *Fileds 1

+ Filds 4

- - - Numerical

Fig. 5 second modal deformation comparison between different fields


-------

Filds 3

+ Filds 4

*Fileds 5

- - - Numerical simulation Fig. 6 Third modal deformation comparison between different fields

+ Filds 4

547

*Fileds 5

- - - Numerical simulation Fig. 7 Fourth modal deformation comparison between different fields

It can be concluded that the high order frequencies require more order to stabilize and more the mode order rises more we need a higher order polynomial. We notice that the first modal deformation obtained from the field (1) is not close to other deformed because of the function that characterizes the displacement field does not define the shear energy of the body (∂ 3W / ∂x 3 = 0) . Even the first obtained natural frequency is clearly superior to the others results, therefore this result will be eliminated from our comparative study. When we increase the number of parameters that characterize the displacement fields we notice that the natural frequencies and modal deformed curves converge to a common solution. Each time we increase the number of parameters that characterize the displacement fields, the calculations become more and more difficult and sometimes we can‘t solve the systems of the obtained equations where we find unusable results (problems with the field (5)), that’ s why we will consider the results obtained from the field (4) as a reference. From figure 5, 6 and 7 we remark that the deformed reach their max and min in the same x-axis and also get canceled at the same points.

4

Experimental Study

4.1 4.1.1

Test Presentation Used Material Presentation

The test rig, shown in Figure 8, consists of a massive steel structure for embedding beams and insulation from external vibrations. The beams are excited by an instrumented hammer connected to the acquisition system. The vibrational response of the beam is itself measured with a laser vibrometer.

548


Fig. 8 Test rig with vibrometer laser

Performing a test of static bending with considering the effect of shear force (the thickness of the specimen is relatively large) allows us to determine the static stiffness and EIz. In these calculations the specimen is assumed to be elastic homogeneous isotropic and linear. The last two assumptions are verified in the used loading area. 4.1.2

Experimental Device

The specimen which is fixed at one end and free on the other is suspended vertically to avoid the effect of static deflection. The excitation hammer (Figure 10) with rubber tip and incorporated force sensor excites the bar horizontally with a transient load of small amplitude. The bi- channels spectrum analyzer calculates the transfer function averaged over several samples. The coherence function is checked at each test and is close to one.

Fig. 9 Experimental device- Hammer shock sensor

Fig. 10 Experimental device- Hammer shock sensor


4.2

549

Test Results

During the tests we used a single excitation point (at x = a) is the free end and the acceleration is measured in a sufficiently close to the fixed end (x = a / 6). The results of the transfer functions in amplitude and phase are shown in the following section. The choice of the frequency band is imposed by the bandwidth of the accelerometer concerning the adhesive assembly.

Fig. 11 Transfer function magnitude and phase of the Al beam

5

Fig. 12 Transfer function magnitude and phase of the Nomex

Numerical Simulation

Fig. 13 First modal deformation of flexion in Fig. 14 First modal deformation of flexion plan (o,x,y) in plan (o,x,z)

550


Fig. 15 second modal deformation of flexion Fig. 16 First modal deformation of torsion in plan (o,x,y)

Fig. 17 Third modal deformation of flexion Fig. 18 Fourth modal deformation of flexin plan (o,x,y) ion in plan (o,x,z)

6

Comparative Analysis

Comparing results among them, we see a slight variation between the natural frequencies and modal deformation determined digital simulation compared with analytical and experimental results this variation may be due to the not taken into account the effect of the glue between the layers has a significant effect known vibration damping.

7

Conclusion

The shape of the displacement field has a great influence on the natural frequency, the right choice is essential to reach a common solution. The second value of the first and the third natural frequency converge respectively from a displacement field of 3, 4 and 5 order polynomial.


551

For each form of the displacement field we notice the relationship between the coefficients of the matrix which may be written as follows For n order polynomial: i = 1....(n − 1) M ij = K ij − L ij Avec j = 1....(n − 1)

Lij can be in a general form.

Where L ij =

1 2 ρω 2 a ( 3 + i + j) 2 ( 3 + i + j)

K 1 j = 4 (1 + j) a j ⋅ C

K 2 j = N 2 j + (12 ⋅ j) ⋅ a ( j+1) ⋅ C

With relations between the N ij coefficients. We can suggest to increase the degree of polynomial that characterizes the displacement field or to modify its shape in order to reach at a general relationship. We can also choose a field of displacement according to the three direction (x, y, z) wherein for better vibration behavior of our beam while taking account of the torsion and bending in both horizontal and vertical plane.

References [1] Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Appl. Mech., 1831–1838 (1951) [2] Yarlagadda, S., Lesieutre, G.: Fiber contribution to modal damping of polymer matrix composite panels. Journal of Spacecraft and Rockets 32(5), 825–831 (1995) [3] Wang Deducing, C.M.: thick plate solutions from classical thin plate solutions. Struct. Eng. Mech. 11(1), 89–104 (2001) [4] Meunier, M., Shenoi, R.A.: Dynamic analysis of composite sandwich plates with damping modeled using higher-order shear deformation theory. Composite Structures 54, 243–254 (2001) [5] Soula, M., Nasri, R., Ghazel, M.A., Chevalier, Y.: The effects of kine-matic model approximations on natural frequencies and modal damping of laminated composite plates. Journal of Sound and Vibration 297, 315–328 (2006) [6] Maheri, M.R., Adams, R.D.: Steady-state flexural vibration damping of honeycomb sandwich beams. Composites Science and Technology 52(3), 333–347 (1994) [7] Banerjee Frequency, J.R.: equation and modal deformation formulae for composite Timoshenko beams. Composite Structures 51, 381–388 (2001) [8] Nilsson, E., Nilsson Prediction, A.C.: Measurement of some dynamic properties of sandwich structures with honeycomb and foam cores. Journal of Sound and Vibration 251(3), 409–430 (2002) [9] Blevins, R.D.: Formulas for Natural Frequency and Modal deformation. Krieger Publishing Company, Florida (2001)

Author Index

Abbes, Mohamed Slim 189 Abdallah, Bouabidi 209 Abdelali, Hanane Moulay 443 Abdelli, Youssef 541 Abid, Mohamed Salah 91, 101, 111, 359 Abid, Said 531 Adim, Belkacem 317 Affi, Zouhaier 47 Agred, Souhila 9 Ahmed Benyahia, Ali 1, 169 Aid, Abdelkrim 433 Ait-Sghir, Khalid 69 Akrout, Mohsen 339, 409 Amara, Idriss 161 Amira, Bilel Ben 91 Amouri, Ammar 141 Amrouche, Abdewahab 433 Aouici, Hamdi 419 Ayadi, Omar 289, 327 Azari, Zitouni 81, 391 Bacetti, Abdelmoumen 229 Baganna, Moez 219 Becquerelle, Samuel 369 Belhadri, Mansour 27 Belkadi, Abdelghani 1 Bellagi, Ahmed 307 Ben Said, Lotfi 489 Bennacer, Hamza 179 Benamar, Rhali 443 Benarous, Abdallah 9 Bendouba, Mostefa 433 Benguediab, Mohamed 433 Benguesmia, Hani 269

Ben Mohamed, Ahmed 219 Bentaleb, Fayçal 161 Ben Yahia, Wafa 327 Benzerga, Djebbara 481 Berrah, S. 179 Bolaers, Fabrice 69 Bouaziz, Mohamed Ali 391 Boubakeur, Ahmed 269 Boudjemai, Abdelmadjid 229 Boudjenane, Nasr-Eddine 27 Boukharouba, Taoufik 281 Boukortt, A. 179 Boulahia, Ramdane 281 Bouraoui, Chokri 507 Bruyère, Jérome 369 Capasso, Clemente 249 Capelle, Julien 391 Chaari, Fakher 459 Chalghoum, Issa 339 Chargui, Selma 317 Chaterbache, Omar 59 Chehaibi, Kaouther 239 Chelbi, Tarek 111 Chen, Mian 299 Chenini, Idris 541 Chokri, Bouraoui 517 Choley, Jean-Yves 189 Dalleli, Manel 81 Dammak, Fakhreddine 469, 489, 497, 531 Daouadji, Tahar Hassaine 317 Delille, Remi 469

554

Author Index

Del Rincon, Alfonso Fernandez 459 Deü, Jean-François 349 Dhifelaoui, Hafedh 453 Driss, Dorra 101 Driss, Slah 131 Driss, Zied 91, 101, 111, 131, 209, 359 Dron, Jean Paul 69 Elaoud, Sami

339, 409

Fakhfakh, Hassen 369 Felfel, Houssem 289 Fudym, Olivier 307 Ghanmi, Samir 37 Gloaguen, Jean-Michel 281 Gmati, Nabil 401 Graa, Mortadha 47 Guedri, Mohamed 37 Guidara, Mohamed Amine 391 Guizani, Amir 189 Haddar, Mohamed 189, 459 Hadj-Taïeb, Ezzeddine 81, 339, 391, 409 Hafaifa, Ahmed 379 Hafsi, Zahreddine 409 Hajlaoui, Abdessalem 497 Hammadi, L. 27 Hammadi, Moncef 189, 249 Hammami, Ahmed 459 Hammami, Omar 299 Hariri, Said 391 Harras, Bilal 443 Hocine, Rachida 229 Houidi, Ajmi 47 Issam Ziane, Mohamed 179 Ichchou, Mohamed 37 Jamel, Mohammed 531 Jarraya, Abdessalem 497 Kaddeche, Slim 121 Kaffel, Ahmed 111 Kammoun, Imen Kallel Karmed, Djamel 9 Khlifi, Kaouthar 453 Lanteri, Stéphane Lanzotti, Antonio

401 199

131

Larbi, Ahmed Ben Cheikh Larbi, Walid 349 Loukarfi, Larbi 9

453

Maaloul, Makram 101 Mabrouki, Ibrahim 359 Mahfoudi, Chawki 141 Makki, Mhalla Mohamed 517 Mars, Jamel 469 Masmoudi, Faouzi 289, 327 Meddour, Ikhlas 419 Mejri, Sabrine 307 Merzoug, Mustapha 69 Meziani, S. 19 Miloudi, Abdelhamid 59, 69, 151 Mlayeh, Olfa 101 Mohamed, Anis 401 Mouloud, Guemana 379 Moussaoui, M. 19 Mrabet, Elyes 37 Mrad, Charfeddine 239, 259 M’Ziou, Nassima 269 Nasr, Anouar 507 Nasr, Aymen 259 Nasri, Rachid 219, 239, 259, 541 Neder, Mahmoud 151 Nejlaoui, Mohamed 47 Ohayon, Roger 349 Ouali, Nourdine 1, 169 Patalano, Stanislao

199, 249

Rabahi, Abderezak 317 Rachid, Belhadef 379 Renaud, Ruixian 249 Romdhane, Lotfi 47 Rueda, Fernando Viadero

459

Saidi, Mohamed 121 Salah, Abid Mohamed 209 Samet, Ahmed 359 Santamaria, Miguel Iglesias 459 Schmitt, Christian 81, 391 Sghaier, Jalila 307 Skander, Achraf 121 Soriano, Thierry 189 Soula, Mohamed 37

Author Index Velex, Philippe 369 Veneri, Ottorino 249 Vitolo, Ferdinando 199 Wali, Mondher 469, 489, 497, 531 Wannes, Hassine 507

555 Yallese, Mohamed Athmane Zaafouri, Mehrez 531 Zaatri, Abdelouahab 141 Zafrane, Mohammed Amine Znaidi, Amna 219

419

229

ACM 2 - Multiphysics Modelling and Simulation for ...

ACM 2 - Multiphysics Modelling and Simulation for ...

Suggest Documents

multiphysics modelling for electronics design - CiteSeerX

Coupling Schemes for Multiphysics Reactor Simulation

Multiphysics welding simulation model - CiteSeerX

Multiphysics Modeling and Simulation of Nanoscale ...

Multiphysics Modeling and Simulation of the Behavior

Multiphysics Coupled Modelling in HVDC GILs

Introduction to comsol multiphysics - Numerical Modelling Laboratory

Introduction to comsol multiphysics - Numerical Modelling Laboratory

Multiphysics simulation of thermoelectric systems - ECT2008

Multiphysics Simulation using high resolution FSI

A Multiphysics Simulation approach to Optimizing ... - cloudfront.net

Multiphysics simulation of corona discharge ... - Semantic Scholar

Modelling and Simulation

Mathematical Modelling and Simulation

MICROGRID MODELLING AND SIMULATION

MODELLING, SIMULATION AND PERFORMANCE

Multiphysics Simulation of Thermoelectric Systems - Comsol

LNCS 4487 - Simulation of Multiphysics Multiscale Systems

multiphysics

MULTIPHYSICS

Intelligent Techniques for Simulation and Modelling

Modelling and Simulation of Mechanisms for ... - ScienceDirect.com

Supporting Modelling and Simulation for VRAPP Objectives

Modelling and Simulation of DSTATCOM for Power