NUMERICAL SIMULATION OF THE ONE METER DROP TEST ON A ...

NUMERICAL SIMULATION OF THE ONE METER DROP TEST ON A BAR FOR THE CASTOR CASK Initial and sensitivity analysis

Nikola Jakˇ si´ c, Karl-Fredrik Nilsson

DG JRC Institute for Energy

2007

EUR 22470 EN

Mission of the Institute for Energy The Institute for Energy provides scientific and technical support for the conception, development, implementation and monitoring of community policies related to energy. Special emphasis is given to the security of energy supply and to sustainable and safe energy production. European Commission Directorate-General Joint Research Centre (DG JRC) http://www.jrc.ec.europa.eu/ Institute for Energy, Petten (the Netherlands) http://ie.jrc.ec.europa.eu/ Contact details: Nikola Jakˇsić +31 (0)224 565498 [email protected]

Karl-Fredrik Nilsson +31 (0)224 565420 [email protected]

Legal Notice

Neither the European Commission nor any person acting on behalf of the Commission is responsible for the use which might be made of this publication. The use of trademarks in this publication does not constitute an endorsement by the European Commission. The views expressed in this publication are the sole responsibility of the author(s) and do not necessarily reflect the views of the European Commission. A great deal of additional information on the European Union is available on the Internet. It can be accessed through the Europa server http://europa.eu/ EUR 22470 EN ISSN 1018-5593 Luxembourg: Office for Official Publications of the European Communities c European Communities, 2007

Reproduction is authorised provided the source is acknowledged. Printed in the Netherlands

Contents Abstract

4

1 Introduction

5

1.1

Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2.1

Transport regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2.2

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2.3

Simulation tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Overview of explicit dynamics in ABAQUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.3

2 Model description

11

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2

Model parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3

2.4

2.5

2.2.1

Cask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2

Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Model assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.1

Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.2

Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.3

Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.4

Contact definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Results of the simulation and comparison with the experimental data . . . . . . . . . . . . . . . . 25 2.4.1

Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.2

Quantitative assessment of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.3

Qualitative assessment of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1

2

CONTENTS

3 Model parameter sensitivity study

41

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2

Comparative analysis of Models a to j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3

3.2.1

Element size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.2

Bulk viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.3

Coefficient of friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.4

Element type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.5

Contact formulation variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.6

Hourglass control variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.7

Boundary conditions variation on the bar’s bottom surface . . . . . . . . . . . . . . . . . 70

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 Summary conclusions

77

A Model parameters and material description

79

A.1 Parameters and switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 A.2 Material data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 B Scripting in ABAQUS

93

B.1 Scripting inside the ABAQUS/CAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 B.2 Script files’ tree structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 B.3 Structure of the script file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 C Explicit dynamics in ABAQUS

101

C.1 Types of problems suited for ABAQUS/Explicit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 C.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 C.3 Explicit dynamics analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 C.3.1 Numerical implementation (Central difference method) . . . . . . . . . . . . . . . . . . . . 103 C.3.2 Nodal mass and inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 C.3.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 C.3.4 Dilatational wave speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 C.3.5 Time increment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

CONTENTS

3

C.3.6 Computational cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 C.3.7 Bulk viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 C.3.8 Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 C.3.9 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 C.3.10 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 C.3.11 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 C.3.12 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 C.3.13 Predefined fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 C.3.14 Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 C.3.15 Coulomb friction in contact formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 C.3.16 Methods for suppressing hourglass modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 D Material definition in ABAQUS

117

D.1 Material definition overview in ABAQUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 D.2 Metal plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 D.2.1 Yield surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 D.2.2 Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 D.2.3 Flow rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 D.2.4 Rate dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 D.2.5 Progressive damage and failure in ABAQUS/Explicit . . . . . . . . . . . . . . . . . . . . . 122 D.2.6 Shear and tensile dynamic failure in ABAQUS/Explicit . . . . . . . . . . . . . . . . . . . 122 D.2.7 Heat generation by plastic work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 D.3 Rate-dependent yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 D.3.1 Work hardening dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 D.3.2 Evaluation of strain-rate-dependent data in ABAQUS/Explicit . . . . . . . . . . . . . . . 125 D.3.3 Johnson-Cook plasticity material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 D.4 Material damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 D.4.1 Rayleigh damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 D.4.2 Artificial damping in explicit dynamic analysis . . . . . . . . . . . . . . . . . . . . . . . . 128 Bibliography

129

4

Abstract

Abstract The report presents the numerical analysis of two one meter drop tests of a single ductile cast iron cask on a steel bar. The cask comes from the CASTOR family with machined cooling fins in a region where impact occurs. In the first test, the impact is on the cask’s cooling fins whereas in the second test the impact is in an area where the ribs have been locally machined away. The numerical analysis is based on explicit dynamic analysis using the commercial finite element code ABAQUS extended with Python scripts to allow a parametric description of the problem. The analysis consists of two parts: a “blind-analysis” with assumed model parameters and a sensitivity analysis. The “blind-analysis” (basic model without any knowledge of test results) was performed for the two drop tests. The overall behavior of the model is qualitatively very similar to what was observed during the experiments. A longer impact duration, (between 3 ms to 5 ms) when the cask is dropped on the fins in comparison to the flat target, is observed in both the analysis and the test. The reaction force at the bar’s bottom surface from the model is qualitatively similar to that from the test. The peak force value is however overestimated by about 35%. On the contrary the measured strains inside the cask above the impact area are underestimated for the similar amount. Latter is mainly attributed to the fact that the lid wasn’t included in this version of the model. The maximum strain is about 20% higher for the flat impact area in both simulation and experiment. The sensitivity analysis was performed to study the influence of parameters which either cannot be or were not defined directly from the experimental data, such as the friction coefficient, or which are linked to the FE numerical procedures, like the bulk viscosity. Based on results, recommendations for an optimised set of parameters values are given.

Chapter 1

Introduction 1.1

Scope

The integrity of waste packages is crucial for the safe storage and transport of spent nuclear fuel and radioactive waste. The manufactures are regulated to demonstrate that the waste packages can withstand loads that could occur under operation and accident conditions. These tests are defined by IAEA. The manufacturer needs to prove the integrity by full-scale testing or simulation. Simulations will never completely replace full-scale tests, but the trend is to do as much as possible by numerical analysis. Simulation has of course the advantage that the cost for a “numerical test” is significantly lower than for an actual drop test, but the numerical simulation gives also a better understanding of the underlying physics and allows the user to check the influence of specific parameters. The analysis of a drop test is a highly nonlinear and dynamic event and it is difficult to prove the accuracy and completeness of the results. This report is result of the cooperation between JRC-IE (Joint Research Center - Institute for Energy) and GNS (Gesellschaft f¨ ur Nuklear-Service mbH) and presents the modelling of the 1m drop test on a steel bar of a Castor AVR cask for the nuclear waste transport and storage. This particular test was proposed by GNS to evaluate different modelling aspects. The problem has a direct practical implication since the effect of the cask’s cooling fins on the impact behaviour is not completely understood. The finite element analysis is based on the explicit version of the commercial finite element code ABAQUS which has been extended with our own routines. The report describes two sets of analyses. First a “blind-analysis” of the impact tests was performed using the basic parameters and data provided by GNS, but where details of the test outcome was not known. The computed results are compared with the actual test data. The second part of the analysis is concerned with the sensitivity of the results to different modelling parameters. The recommendations are given for optimizing further analyses with respect to accuracy as well as computational efficiency.

5

6

1.2

1. Introduction

Introduction

GNS is the manufacturer of CASTOR (CAsk for Storage and Transport Of Radioactive material) and CONSTOR cask for transport and storage of spent fuel and radioactive waste. The casks are used in the EU, Russia and the USA. The CASTOR is the family of the transport and storage cask for spent fuel and radioactive waste. Around 900 casks of the CASTOR family have been produced since 1979. The IAEA drop test system is made up of a series of well defined structural and destructive dynamic tests. It includes a 9 meter drop test on an unyielding surface as well as a 1 meter drop test on a bar. The tests are designed to simulate worst case scenarios during the cask transportation and handling. The manufacturer of casks need to verify that the casks’ integrity and functioning are maintained after drop tests. This can either be done by experimental drop tests or by computer simulations which are difficult to perform. The drop tests are extremely expensive to perform. The drop test is very complex to analyse and computationally extremely costly since it involves large deformations, dynamics and and material failure. The tools for impact analysis have developed significantly and numerical simulations are now performed in parallel with experiments. The numerical simulations are also very useful in design since they allow the influence of modifications and different parameters to be assessed. A further aspect is that the understanding of the physical phenomena often require a combination of dedicated tests and associated analyses.

1.2.1

Transport regulations

According to IAEA §726, the 1 m drop onto a steel bar and the 9 meter drop test should be followed by a thermal test. The sequence and boundary conditions (drop orientations) of the two mechanical tests should be selected in such a way that they cause the maximum damage. For example, the functionability of a penetration protection is to be maintained after the 9 m drop. The boundary conditions for the test with the 1 m drop onto a steel bar are defined in §727(b). According to IAEA §637 and §664, a temperature range of -40◦ C to 70◦ C is to be considered.

1.2.2

Experiment

Based on the requirements of transport regulations a drop of the transport/storage cask from a height of 1 m onto a steel bar must be assessed. High strains occur in the cask body due to the locally induced load force from the bar. The location of the impact may affect the global deformation. Transport casks usually have cooling fins on parts of its outer surface. A question is then to what extent the overall response to an impact is affected by whether the impact occurred at where fins are located or not. It is difficult to model this due to the very complex geometries involved. GNS therefore performed two 1 meter drop tests to quantify the influence of the circumferential fins on energy absorption and cask integrity. The experiments were performed on an old CASTOR AVR cask, Fig. 1.1, that scales almost perfectly to the targeted member of the CASTOR family (CASTOR V/19) by a geometrical ratio of 1:2. The fins manufactured were machined away at two distinct places, Fig. 1.2. The cask was dropped from 1 m height on the bar twice, Fig. 1.1. First with the fins facing the bar (finned target) and after that, with the flatten area facing the bar (flat target). The force under the bar was measured in the vertical direction only.

1. Introduction

7

The cask accelerations were measured in two directions at two different positions. Strains were measured at the inside surface of the cask directly above the impact area at 9 different locations by means of multi-axial strain gages.

Figure 1.1: The CASTOR AVR cask and the set-up of the experiment (courtesy of GNS).

Figure 1.2: The flat and finned target on the cask (courtesy of GNS).

The main conclusions of the experiment were: • impact duration is approximately 5 ms longer for the finned target drop,

8

1. Introduction • force in the force transducer below the bar is approximately 6% smaller for the finned target drop, • both drops produce only elastic strains at the measurement points, with 21% lower strain amplitude for the finned target compared to the flat target.

1.2.3

Simulation tools

The commercial code ABAQUS is the main tool used for the FEM analysis in the SAFEWASTE action. It is currently running on a Linux Cluster. We have both the implicit and explicit versions installed and licensed, together with ABAQUS/CAE and VIEWER for pre- and post- processing the analyses. The drop test is modelled with the parameter driven Python code inside the Abaqus/CAE software. ABAQUS/CAE is used as a pre-processor as well as a post-processor. ABAQUS/Explicit is used as solver for a given problem. ABAQUS [1] is a suite of powerful engineering simulation programs, based on the finite element method, that can solve problems ranging from relatively simple linear analyses to the most challenging nonlinear simulations. ABAQUS contains an extensive library of elements that can model virtually any geometry. It has an equally extensive list of material models that can simulate the behavior of most typical engineering materials including metals, rubber, polymers, composites, reinforced concrete, crushable and resilient foams, and geotechnical materials such as soils and rock. Designed as a general-purpose simulation tool, ABAQUS can be used to study more than just structural (stress/displacement) problems. It can simulate problems in such diverse areas as heat transfer, mass diffusion, thermal management of electrical components (coupled thermal-electrical analyses), acoustics, soil mechanics (coupled pore fluid-stress analyses), and piezoelectric analysis. ABAQUS/CAE [2] is a complete ABAQUS environment that provides a simple, consistent interface for creating, submitting, monitoring, and evaluating results from ABAQUS/Standard and ABAQUS/Explicit simulations. Python used by Abaqus/CAE is not the “standard Python” supported by the www.ptyhon.org, though the differences are few. ABAQUS Python adds more than 500 new data types to the “standard Python”. Python, see www.ptyhon.org, is an easy to learn, powerful programming language. It has efficient high-level data structure and a simple but effective approach to object-oriented programming. Python’s elegant syntax and dynamic typing, together with its interpreted nature, make it an ideal language for scripting and rapid application development in many areas on most platforms. In order to understand how the Python scripts are processed inside ABAQUS/CAE, the architecture of ABAQUS/CAE should be closely examined, figure 1.3. The Python script is created/entered into ABAQUS/CAE either through the Graphic User Interface (GUI) itself by browsing the menus and dialogs or by typing commands in the command line editor, which is a part of GUI. The third way to enter the script is by means of reading pre-prepared script files. The input is interpreted by built-in Python interpreter. At this stage the replay files (log files for commands used) are created. ABAQUS/CAE kernel appears to work very closely together with the Python interpreter, so it is not really clear whether all of the Python commands/data types are handled solely by the interpreter. Finally, the input file (*.inp) is created by ABAQUS/CAE before the job is submitted. The *.inp file is the text file containing the mesh, boundary conditions, contact definitions, etc., in short: everything that is needed for the solver to perform the analysis. The solver creates the output database, which is a binary file

1. Introduction

9

Figure 1.3: ABAQUS scripting flow-chart.

with the requested results. They can be read by ABAQUS/CAE in Visualisation mode (post-processing mode). The *.inp file can be easily created by hand and the ABAQUS Python scrip structure resembles the *.inp file structure to a large extent. There are two major reasons for using ABAQUS Python scripts: • to utilize the automatic meshing capabilities of ABAQUS/CAE and • to use parametric approach for the geometry and mesh formation. ABAQUS/Explicit is a solver for explicit dynamic problems and thus suited for the drop test simulation. The Explicit dynamics is discussed in the following section.

1.3

Overview of explicit dynamics in ABAQUS

This section is mainly a recapitulation of 6.3 section of [3]. A large class of stress analysis problems can be solved with ABAQUS. A fundamental division of such problems is in their predominantly static or dynamic behavior. Dynamic problems are those in which inertia effects are significant. The same analysis can contain several static and dynamic phases. Thus, a static pre-load might be applied, and then the linear or nonlinear dynamic response computed.

10

1. Introduction

ABAQUS/Explicit solves dynamic response problems using an explicit direct-integration procedure. It also provides heat transfer and acoustic analysis capabilities. The explicit dynamics procedure can be an effective tool for solving a wide variety of nonlinear solid and structural mechanics problems. The direct-integration dynamic procedure provided in ABAQUS/Standard uses the implicit Hilber-HughesTaylor operator for integration of the equations of motion, while ABAQUS/Explicit uses the central-difference operator. In an implicit dynamic analysis the integration operator matrix must be inverted and a set of nonlinear equilibrium equations must be solved at each time increment. Because displacements and velocities in an explicit dynamic analysis are calculated in terms of quantities that are known at the beginning of an increment, the global mass and stiffness matrices need not be formed and inverted, which means that each increment is relatively inexpensive compared to the increments in an implicit integration scheme. The size of the time increment in an explicit dynamic analysis is limited, however, because the central-difference operator is only conditionally stable, whereas the Hilber-Hughes-Taylor operator is unconditionally stable and, thus, there is no such limit on the size of the time increment that can be used for most analyses in ABAQUS/Standard (accuracy governs the time increment in ABAQUS/Standard). From a user standpoint, the distinguishing characteristics of the explicit and implicit methods concerning either dynamic or static problems are as follows. • Explicit methods require a small time increment size that depends solely on the highest natural frequencies of the model and is independent of the type and duration of loading. Simulations generally take on the order of 104 to 106 and even more increments, but the computational cost per increment is relatively small. • Implicit methods do not place an inherent limitation on the time increment size; increment size is generally determined from accuracy and convergence considerations. Implicit simulations typically take orders of magnitude fewer increments than explicit simulations. However, since a global set of equations must be solved in each increment, the cost per increment of an implicit method is far greater than that of an explicit method. For the explicit dynamic analysis can be said that it: • is computationally efficient for the analysis of large models with relatively short dynamic response times and for the analysis of extremely discontinuous events or processes; • allows for the definition of very general contact conditions; • uses a consistent, large-deformation theory; models can undergo large rotations and large deformation; • can use a geometrically linear deformation theory; strains and rotations are assumed to be small; • can be used to perform an adiabatic stress analysis if inelastic dissipation is expected to generate heat in the material; • can be used to perform quasi-static analyses with complicated contact conditions; • and allows for either automatic or fixed time incrementation to be used-by default, ABAQUS/Explicit uses automatic time incrementation with the global time estimator. For more details, see also [4] and the Appendixes to this report.

Chapter 2

Model description 2.1

Introduction

The numerical modeling presented in this report represents a “blind-simulation” of the experiment where only the geometry and material data were provided. Term “blind” annotates that the experimental results were not known at the time of the model development. The model has been developed using only provided data. No model fine tuning was performed. The second part of the analysis concerns how model parameters affect the results. These model parameters either cannot be or were not defined directly from the auxiliary experimental data, for instance the friction coefficient, or are linked to the numerical procedures within the FEM, such as the bulk viscosity. The explicit dynamic analysis using the commercial finite element code ABAQUS was extended with pythons scripts for a parametric description of the problem. Parametric model support allows the user to define geometry parameters, material parameters as well as analysis parameters such as mesh density in a way which can be easily modified for sensitivity studies. The model consists of the cask’s body (without the lid) and the bar fixed at its bottom face by boundary conditions. One symmetry plane is considered. Nevertheless one should be aware that in dynamics symmetry in model does not assure symmetry in model’s response. This is definitely true for the study of the structural vibrations. The phenomena can however be, in most cases, neglected in drop or crash tests i.e. for severe impact loads where the material plasticity and material failure plays an important role due to low influence of the elastic waves on the overall response of the model. The model description roughly follows the modelling philosophy of ABAQUS. The geometry description of the parts is given accompanied with the meshing description. The parts of the model are subsequently assembled and positioned. The analysis is then defined by initial and boundary conditions, loads and contact definition. The description of the model parameters, material data and structure of the script files can be found in the Appendices.

11

12

2.2

2. Model description

Model parts

The real system consist of 3 parts: the lid, the cask and the bar. The lid has not yet been modeled due to time constraints. The model consists of 7 parts. 6 parts make up the cask and remaining one the bar. Multiple parts are needed for the cask in order to mesh volumes more in accordance with their physical behaviour. Hence, different volumes are meshed differently. Volumes near the impact point are meshed with finer mesh than volumes farther away.

2.2.1

Cask

The geometries and meshes of the parts are shown in figures 2.1 – 2.6. The bottom half of the cask is modeled by the part 1 and the top half by the part 2. Parts 3 – 6 represent the finned sector of the cask between parts 1 and 2. Part 6 is the smallest and represents the target area, either finned or flat one. The finest mesh of the whole cask is found here. Hexagonal elements are used for meshing in the case of finned target and tetrahedral ones when target area is flat. At the top of the part 6 is part 5 which defines strain measurement points. The geometry of the part 5 is defined in a way to ensure that some nodes of the mesh are in the positions of the strain measurement points. Hence, part 5 is always meshed with tetrahedral elements. The rest of the finned ring around the cask circumference is represented by part 4. Part 3 (underneath part 4) represents the rest of the ring around the circumference of the cask. Both parts 3 and 4 are normally meshed with the hexagonal elements. The element size for each cask part is controlled by the meshing parameters, see table A.2. If the parameter anaElemType=1 then all cask parts are meshed with the tetragonal elements.

2.2.2

Bar

The bar is modelled as a single part normally meshed with the hexagonal elements. Tetragonal elements are used only in the case of anaElemType=1. The element size is controlled by the meshing parameter pin ElementSize, see table A.2.

2.3

Model assembly

The model is assembled as shown in Fig. 2.8. Parts of the cask are connected by the tie command, tieing adjacent surfaces of the parts. Tie constraints tie two separate surfaces together so that there is no relative motion between them. The nodes on the slave surface are constrained to have the same value of displacement, temperature, pore pressure, or electrical potential as the point on the master surface to which they are tied. ABAQUS/CAE generally selects the surface with the finer mesh to be the slave surface. The computation for the depth of the slave node adjustment zone for the tie constraint is based on the bounding dimensions of the interfacing regions. All 12 connections are shown in figure 2.10 where master surface is indicated with red mesh and slave surface with pink one.


13

2

2 3 1

3

1

2

2 3 1

3

1

Figure 2.1: The geometrical representation and the mesh of the part 1 of the cask.

14


2

2 3 1

3

1

2

2 3 1

3

1


2

2

3

3

1

1



2

15

2

3 3

1

1

2

2

3 3

1

1


16


2

2 3

3

1 1

2

2 3 1

3

1



17

2

2 3

3

1

1

2

2 3 1

3 1

Figure 2.6: The geometrical representation and the mesh of the part 6 of the cask. Finned target area on the left hand side and flat one on the right.

18


2 2

3 1

1

3

2 2

3 1

1 3

Figure 2.7: The geometrical representation and the mesh of the bar-part.


19

Y Y X

Z Z

2

1

2

3

3

1

Y Z Z X Y

X

3

2 3

2

1 1

Figure 2.8: Assembly of the model - the geometrical representation.

X

20


2 3 1

2 3 1

Figure 2.9: Assembly of the model - the expanded geometrical representation and mesh.


21

2

2

3

2

3

1

3

1

1

Parts: 1 and 3

Parts: 2 and 3

2

Parts: 3 and 4

2

3

2

3

1

3

1

1

Parts: 1 and 4

Parts: 2 and 4

2

Parts: 3 and 5

2

3

2

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1

3

1

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Parts: 2 and 5

2

Parts: 4 and 6

2

3

2

3

1

3

1

Parts: 1 and 6

1

Parts: 2 and 6

Parts: 5 and 6

Figure 2.10: Assembly of the model - connecting the cask parts.

22


2.3.1

Initial conditions

The drop (free fall) is modeled by prescribing the velocity of the cask as a rigid body just before the impact. It is computed by the conservation of the mechanical energy law. The computed velocity, equation (2.1), is introduced as the initial velocity of the cask in the initial condition specifications, Fig. 2.11. Hence, the cask is above the bar. The distance between them is defined with the eps parameter. Initial over-closures of the cask’s and bar’s meshes are thus avoided. The distance eps is taken into account when computing value of the initial velocity: vo =

p 2g(H − eps),

(2.1)

where vo is the initial velocity of the cask, g is the gravity constant, H is the drop height and eps is the initial small distance between the cask and bar.

2 3 1

Figure 2.11: Initial conditions.

2.3.2

Boundary conditions

There are two different kinds of the boundary conditions used: • ones for defining model’s symmetry plane and • others for defining model’s support. Symmetry boundary conditions The symmetry vertical plane, perpendicular to the axis 3, is adopted to reduce the computational time. The size of the problem (number of the degrees of freedom) is thus almost halved. The same is approximately true for the computational time due to lumped mass formulation in the explicit problem. The symmetry plane is modeled by the symmetry boundary conditions shown in figure 2.12. Support boundary conditions The bar is screwed on the force transducer in the experimental set-up, Fig. 1.1. The elasticity of the force transducer is neglected and the contact between the transducer and bar is modeled by two different boundary


23

2 3 1

Figure 2.12: Symmetry boundary conditions.

conditions: fixing all degrees of freedom on the bar’s bottom surface where the bolt is and fixing only the vertical displacement (axis 2) everywhere else over the bar’s bottom surface, Fig. 2.13. A schematic representation of the support boundary conditions is presented in the same figure on the right.

3

2

1

Figure 2.13: Support boundary conditions on the bar’s bottom surface; fixed degree of freedom in all directions (encastre) on the hatched surface and fixed ones only in the vertical (axis 2) direction on the rest of the surface. Schematic representation of the support boundary conditions on the right.

24

2.3.3


Loads

The only load defined in the model is volume force (acceleration) due to gravity, Fig. 2.14. There is no precalculated equilibrium state of the model due to the gravity load of the model because of negligible influence of the gravity acceleration in comparison to the accelerations during impact. The eps distance allows some time for the gravity load to influence the bar prior impact.

2 3 1

Figure 2.14: Loads of the model: gravity.

2.3.4

Contact definition

The general contact algorithm, which is very powerful and highly automated, is used for contact formulation during the impact. It searches for possible contact between faces and between edges as well between combination of edges and faces. It also searches for possible contact of the face or edge to itself. It is a more robust algorithm and even less computationally expensive then classical definitions of the contact pairs. The general contact algorithm was developed to facilitate multi-body contact in car crash test. All contact algorithms are based on the penalty method. The user should only define the behaviour in normal/tangential direction.


2.4

25

Results of the simulation and comparison with the experimental data

The switch and parameter values from Tables A.1 and A.2 respectively, as well as the material data in tables A.5 and A.4 are used when performing the numerical analysis/simulation with the model.

2.4.1

Mass

The total mass of the cask model is 8102 kg. The lid’s half mass is estimated to be 560 kg. Hence, the total model mass is estimated at 8662 kg, which is in good agreement with the real cask’s half mass of 8655 kg. The estimated difference is less than 0.1%. The mass of the cask model itself is some 6.3% smaller than the real cask’s half mass due to the fact that the lid is not modeled. Underestimated mass means underestimated inertia forces and, thus underestimated resulting stresses and forces. This should be kept in mind when doing quantitative comparison between the simulation and experimental results. A qualitative comparison is be possible due to relatively small mass difference between the model and real cask.

2.4.2

Quantitative assessment of the results

The numerical results are presented in numerical as well as graphical form. Graphs present time histories of different variables. The time histories are presented by thick and thin lines corresponding two different numerical simulations. The thick line is always associated with the results of the flat impact target and the thin line with the results on the finned impact target. Detailed description is given in the legend of each graph and the graph title.

2.4.2.1

Energies

Four different energies of the whole model are considered: kinetic, strain, total and artificial. The kinetic energy is linked to the motion of the model and total strain energy to the deformation of the model, total energy is measure of the energy balance and artificial energy is linked to the hourglass control. The latter is product of the numerical procedures and has nothing to do with any physical phenomena. It is an attempt to correct possible error introduced by the element shapes in FEM. Its true value should be zero. The formulation for reduced-integration elements considers only the linearly varying part of the incremental displacement field in the element for the calculation of the increment of physical strain. The remaining part of the nodal incremental displacement field is the hourglass field and can be expressed in terms of hourglass modes. Excitation of these modes may lead to severe mesh distortion, with no stresses resisting the deformation. Hourglass control attempts to minimize this problem without introducing excessive constraints on the element’s physical response. More details are provided later in the next chapter. As is shown in Fig. 2.15, the kinetic energy decreases from its initial value of 80 kJ until the cask rebounds, when it slightly increases. It does not reach zero value due to the elasticity of the cask enabling the propagation

26


Kinetic & Strain Energies 100

90

80

70

Energy [kJ]

60 Kinetic/flat Kinetic/fins Strain/flat Strain/fins

50

40

30

20

10

0

0

5

10

15

20

25

30

35

Time [ms]

Figure 2.15: Kinetic and total strain energies. Artificial & Total Energies 100

90

80

70

Energy [kJ]

60 Artificial/flat Artificial/fins Total/flat Total/fins

50

40

30

20

10

0

0

5

10

15

20

25

Time [ms]

Figure 2.16: Artificial and total energies.

30

35


27

of elastic waves. The elastic waves spread through the continuum due to the severe impact loading, which is considered to be spectrally wide and natural vibrations are triggered. It can also be seen in Fig. 2.15 that the minimum of the kinetic energy when simulating finned target occurs 2.9 ms later than for one of the flat target. The impact lasts longer for the finned target, which is in agreement with the experimental results in which it increased by approximately 5 ms. It is obvious from Fig. 2.15 that the total strain energy increases due the deformations during the impact. After reaching a local maximum, the restitution phase starts. The maximum of the strain energy for drop on the finned target lags 3 ms behind the drop on the flat surface, which is again in agreement with the experimental results. In the case of the flat target the maximum value of the artificial energy reaches 5.276 kJ which is 6.7% of the total initial energy. But in the case of the finned target the maximum value of the artificial energy is 9.238 kJ, which is 75.1% larger than for the flat target and 11.6% of its total initial energy. The contact problem in flat target case is numerically much easier to tackle than for the finned target, where a complex geometry is added to the already difficult contact problem. Hence, more effort is needed for hourglass control and this gives a rise to the larger artificial energy. Total energy slightly rises above the total initial energy towards the end of the impact, which is physically meaningless, but it can be attributed to roundoff errors and to procedures like hourglass control and the penalty method in the contact problem formulation.

2.4.2.2

Reaction forces

The reaction forces presented in figure 2.17 are computed as the sum of the nodal reaction forces in the vertical direction (axis 2) over the entire bottom surface of the bar (see boundary conditions definitions). That counts for the half of the true model value. The maximum half value of the reaction force in the vertical direction is 3.092 MN, 19.6 ms after the start of impact for the flat target, and 3.048 MN at 22.2 ms for the finned target. The slightly lower reaction force for the finned case is due to the longer impact duration and to the fact that more energy is dissipated through plastic deformation and contact frictional forces in directions other than vertical. The small jump in the vertical reaction force in the case of the finned target at the time history maximum is unexpected and only lasts for 0.1 ms. There is no obvious physical reason why this occurred. The maximum vertical reaction force measured during experiment was 4.75 MN at 19 ms after start of impact for flat target and 4.45 MN at 24 ms for the finned one, which is 6.3% less than at the flat target. There is very good agreement when comparing the time of the maximum vertical reaction force position between experiment and simulation. The difference in magnitude is significant; the maximum force for the flat target is 30.1% overestimated and for the finned target is also overestimated by 37.0%. This discrepancy can not be easily explained, even more so because of the missing mass in the model. However, the reaction force ratios between the finned and flat scenarios for the simulation and the experiment (3.092/3.048 and 4.75/4.45) are quite similar, and the time at which the maximum forces occur are also well captured: 19.6 ms and 22.2 ms compared to 19 ms and 24 ms.

28


Vertical reaction force 4 Force/flat Force/fins 3.5

3

Force [MN]

2.5

2

1.5

1

0.5

0

0

5

10

15

20

25

30

35

Time [ms]

Figure 2.17: Reaction forces in the vertical direction. Max principal strain @ CSK/D1 300 Strain/flat Strain/fins 250

Strain [µm/m]

200

150

100

50

0

0

5

10

15

20

25

30

Time [ms]

Figure 2.18: Maximal principal strains at the D1 strain measurement point.

35

2. Model description 2.4.2.3

29

Maximum principal strains

The maximal principal strains are computed at the strain measurement point D1 (Fig. A.4). The strains, computed at the inside of the cask at strain measurement points D1 to D9, are purely elastic strains. No plastic strains means no mechanisms of local energy dissipation (apart from the bulk viscosity). The structure in the vicinity of the strain measurement points exhibits linear elastic behavior: elastic waves occur, which implies natural vibrations. The effect is more visible in strains than in displacement due to the fact that strains are the first derivative of the displacements. The maximum principal strain of 221 µm/m is reached after 16.5 ms for the flat target whereas for the finned target the maximum of 184 µm/m is reached 5.3 ms later (21.8 ms). This estimation is however unreliable due to the time histories oscillations. The strain in the case of the finned target is 16.7% smaller than for the flat target. This difference is substantially larger than that for the reaction forces, which is in agrement with the experimental results, where the equivalent difference was 21.1%.

2.4.2.4

Displacements

The vertical displacements for two different positions on the cask are presented in figures 2.19 and 2.20. The point CSK1 is at the top of the cask and CSK3 is the contact point on the cask, see figure A.1. When dealing with the point CSK1 the maximal absolute displacement (64.4 mm) is larger for the finned target compared to the flat one (54.1 mm), due to the fact that fins are drastically deformed at the point of contact. In this case the finned target maximal absolute displacement lags 3.2 ms behind the flat target’s one. At the point CSK3, the picture is just opposite; the maximal absolute displacement (53.4 mm) is larger for the flat target than for the finned one (48.7 mm). It can be seen in figure 2.20 that in the case of finned target the fin in contact deforms first followed by the bar deformation similar to one at the flat target. In this case the finned target maximal absolute displacement again lags 3.1 ms behind that of the flat target.

2.4.2.5

Velocities

The vertical velocities for the two different positions on the cask are presented in figures 2.21 and 2.22. Location of the points CSK1 and CSK3 can be seen in figure A.1. One can notice that there is more “noise” (oscillations) present in time histories for the velocities than for the displacements. Numerical integration is acting as low-pass filter when integrating time history. Hence, only predominant dynamics are captured by the displacements. It can also be seen in figures 2.21 and 2.22 that the time history of the finned target velocity at the CSK3 is, by far, the noisiest one. This is mainly due to the complex contact geometry and penalty method used in the contact problem formulation and also, but a lesser extent, due to the stick-slip effect, associated with the Coulomb dry friction model in frictional contact. The oscillations seen in graphs of the velocities and maximum principal strains are physically plausible and are not a result of numerical errors.

30


Vertical displacement @ CSK1 0 Displacement/flat Displacement/fins −10

Displacement [mm]

−20

−30

−40

−50

−60

−70

0

5

10

15

20

25

30

35

Time [ms]

Figure 2.19: Vertical displacements at CSK1 measurement point. Vertical displacement @ CSK3 0 Displacement/flat Displacement/fins −10

Displacement [mm]

−20

−30

−40

−50

−60

−70

0

5

10

15

20

25

30

Time [ms]

Figure 2.20: Vertical displacements at CSK3 measurement point.

35


31 Vertical velocity @ CSK1 1 Velocity/flat Velocity/fins 0

Velocity [m/s]

−1

−2

−3

−4

−5

0

5

10

15

20

25

30

35

Time [ms]

Figure 2.21: Vertical velocities at CSK1 measurement point. Vertical velocity @ CSK3 1 Velocity/flat Velocity/fins 0

Velocity [m/s]

−1

−2

−3

−4

−5

0

5

10

15

20

25

30

Time [ms]

Figure 2.22: Vertical velocities at CSK1 measurement point.

35

32


2.4.3

Qualitative assessment of the results

2.4.3.1

Impact on the flat target

The shapes of the bar and cask after the drop between simulation and experiment are also compared in figures 2.23 and 2.24 respectively. Figures 2.25, 2.26 and 2.27 show the deformed shape of the detail between cask part 6 and the bar, the cask part 6 only, and the bar, respectively in a sequence of frames at 20 ms steps.

Step: dropStep Frame: 350 S, Mises (Avg: 75%) +4.437e+08 +4.253e+08 +4.068e+08 +3.883e+08 +3.698e+08 +3.513e+08 +3.328e+08 +3.143e+08 +2.958e+08 +2.773e+08 +2.589e+08 +2.404e+08 +2.219e+08 +2.034e+08 +1.849e+08 +1.664e+08 +1.479e+08 +1.294e+08 +1.109e+08 +9.245e+07 +7.396e+07 +5.547e+07 +3.698e+07 +1.849e+07 +0.000e+00 2

3 1

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ABAQUS/EXPLICIT Version 6.6−1

Fri Jan 26 16:13:23 CET 2007

Step: dropStep, Drop test step Increment 548993: Step Time = 3.5000E−02 Primary Var: S, Mises Deformed Var: U Deformation Scale Factor: no deformation

Figure 2.23: The shape of the bar after the experiment (left) and at the simulation’s end (right).

Step: dropStep Frame: 350 S, Mises (Avg: 75%) +3.111e+08 +2.981e+08 +2.852e+08 +2.722e+08 +2.592e+08 +2.463e+08 +2.333e+08 +2.204e+08 +2.074e+08 +1.944e+08 +1.815e+08 +1.685e+08 +1.555e+08 +1.426e+08 +1.296e+08 +1.167e+08 +1.037e+08 +9.074e+07 +7.777e+07 +6.481e+07 +5.185e+07 +3.889e+07 +2.592e+07 +1.296e+07 +0.000e+00 2

1



Fri Jan 26 16:13:23 CET 2007

Step: dropStep, Drop test step 3 Increment 548993: Step Time = 3.5000E−02 Primary Var: S, Mises Deformed Var: U Deformation Scale Factor: no deformation

Figure 2.24: The shape of the cask’s flat target after the experiment (left) and at the simulation’s end (right).

The deformation of the cask flat surface, figure 2.24, is difficult to notice from the figure for the experiment or the simulation. The shape of the bar at the end of simulation is similar to the shape of the bar after the experiment. There is an exception at the bottom of the bar, figure 2.24. The bottom surface is too “spread out” in the simulation case. This implies that the boundary conditions should be revised. At the moment no friction between the bar and the force transducer is taken into account, although it occurs at the experiment.


33

Step: dropStep Frame: 0

t = 0 ms

S, Mises (Avg: 75%)

1

Step: dropStep Frame: 0 S, Mises (Avg: 75%)

+4.437e+08 +4.253e+08 +4.068e+08 +3.883e+08 +3.698e+08 +3.513e+08 +3.328e+08 +3.143e+08 +2.958e+08 +2.773e+08 +2.589e+08 +2.404e+08 +2.219e+08 +2.034e+08 +1.849e+08 +1.664e+08 +1.479e+08 +1.294e+08 +1.109e+08 +9.245e+07 +7.396e+07 +5.547e+07 +3.698e+07 +1.849e+07 +0.000e+00 2 Drop test ODB: dt−afd.odb ABAQUS/EXPLICIT Version 6.6−1 Fri Jan 26 16:13:23 CET 2007 3 Step: dropStep, Drop test step Increment 0: Step Time = 0.0 Primary Var: S, Mises Deformed Var: U Deformation Scale Factor: no deformation

2


+4.437e+08 +4.253e+08 +4.068e+08 +3.883e+08 +3.698e+08 +3.513e+08 +3.328e+08 +3.143e+08 +2.958e+08 +2.773e+08 +2.589e+08 +2.404e+08 +2.219e+08 +2.034e+08 +1.849e+08 +1.664e+08 +1.479e+08 +1.294e+08 +1.109e+08 +9.245e+07 +7.396e+07 +5.547e+07 +3.698e+07 +1.849e+07 +0.000e+00 ABAQUS/EXPLICIT Version 6.6−1

t = 20 ms t = 40 ms

1

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Fri Jan 26 16:13:23 CET 2007

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2


Step: dropStep Frame: 60 S, Mises (Avg: 75%) +4.437e+08 +4.253e+08 +4.068e+08 +3.883e+08 +3.698e+08 +3.513e+08 +3.328e+08 +3.143e+08 +2.958e+08 +2.773e+08 +2.589e+08 +2.404e+08 +2.219e+08 +2.034e+08 +1.849e+08 +1.664e+08 +1.479e+08 +1.294e+08 +1.109e+08 +9.245e+07 +7.396e+07 +5.547e+07 +3.698e+07 +1.849e+07 +0.000e+00


Fri Jan 26 16:13:23 CET 2007


3

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+3.111e+08 +2.981e+08 +2.852e+08 +2.722e+08 +2.592e+08 +2.463e+08 +2.333e+08 +2.204e+08 +2.074e+08 +1.944e+08 +1.815e+08 +1.685e+08 +1.555e+08 +1.426e+08 +1.296e+08 +1.167e+08 +1.037e+08 +9.074e+07 +7.777e+07 +6.481e+07 +5.185e+07 +3.889e+07 +2.592e+07 +1.296e+07 +0.000e+00 2


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+3.111e+08 +2.981e+08 +2.852e+08 +2.722e+08 +2.592e+08 +2.463e+08 +2.333e+08 +2.204e+08 +2.074e+08 +1.944e+08 +1.815e+08 +1.685e+08 +1.555e+08 +1.426e+08 +1.296e+08 +1.167e+08 +1.037e+08 +9.074e+07 +7.777e+07 +6.481e+07 +5.185e+07 +3.889e+07 +2.592e+07 +1.296e+07 +0.000e+00

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+4.437e+08 +4.253e+08 +4.068e+08 +3.883e+08 +3.698e+08 +3.513e+08 +3.328e+08 +3.143e+08 +2.958e+08 +2.773e+08 +2.589e+08 +2.404e+08 +2.219e+08 +2.034e+08 +1.849e+08 +1.664e+08 +1.479e+08 +1.294e+08 +1.109e+08 +9.245e+07 +7.396e+07 +5.547e+07 +3.698e+07 +1.849e+07 +0.000e+00 2 Drop test ODB: dt−afd.odb ABAQUS/EXPLICIT Version 6.6−1 Fri Jan 26 16:13:23 CET 2007 3 Step: dropStep, Drop test step Increment 163004: Step Time = 1.0000E−02 Primary Var: S, Mises Deformed Var: U Deformation Scale Factor: no deformation

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Figure 2.25: Drop on the flat surface; 0 ms – 100 ms.



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34



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36


2.4.3.2

Impact on the finned target

The experimentally measured and predicted shapes of the bar and cask after the drop compared in figures 2.28 and 2.29 respectively. Figures 2.30, 2.31 and 2.32 show the deformed shape of the detail between the cask part 6 and the bar, the cask part 6 only, and the bar respectively in a sequence of frames at 20 ms steps.


2

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Drop test ODB: dt−ard.odb


Fri Jan 26 16:03:08 CET 2007


1

Figure 2.28: The shape of the bar after the experiment (left) and at the simulation’s end (right).


2

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Fri Jan 26 16:03:08 CET 2007

Step: dropStep, Drop test step 558399: Step Time = 3.5000E−02 3 Increment Primary Var: S, Mises Deformed Var: U Deformation Scale Factor: no deformation

Figure 2.29: The shape of the cask’s flat target after the experiment (left) and at the simulation’s end (right).

As observed for the flat target drop, the bottom surface of the bar is too “spread out” in the simulation case. The boundary conditions should be revised. At the moment no friction between the bar and the force transducer is taken into account, although it occurs at the experiment. The damage of the cask and the bar at their interface surfaces is extensive, as seen from the experiment. Although the bar model resembles the experimental results, it is obvious that the mesh shape is not optimal and should have been finer. In the case of the cask the extent of the fins’ failure couldn’t be modeled since no failure criteria are used. Moreover, the mesh of the cask part 6 is most probably too coarse (hence the fins are too rigid) to simulate the behavior observed during the experiment.


37



t = 0 ms


2

1

3



2

Fri Jan 26 16:03:08 CET 2007





Fri Jan 26 16:03:08 CET 2007

Step: dropStep, Drop test step Increment 0: Step Time = 0.0 3 Primary Var: S, Mises Deformed Var: U Deformation Scale Factor: no deformation

1


t = 20 ms

2

t = 40 ms

3



2

Fri Jan 26 16:03:08 CET 2007




Fri Jan 26 16:03:08 CET 2007


2



Fri Jan 26 16:03:08 CET 2007


2



Fri Jan 26 16:03:08 CET 2007


2



Fri Jan 26 16:03:08 CET 2007

t = 60 ms t = 80 ms t = 100 ms

3


2



Fri Jan 26 16:03:08 CET 2007





2

Fri Jan 26 16:03:08 CET 2007


Fri Jan 26 16:03:08 CET 2007


1

S, Mises (Avg: 75%)






3


S, Mises (Avg: 75%)


1


2

2


Fri Jan 26 16:03:08 CET 2007

S, Mises (Avg: 75%)

1

3



Fri Jan 26 16:03:08 CET 2007


1

S, Mises (Avg: 75%)






3


S, Mises (Avg: 75%)

Step: dropStep, Drop test step Increment 99837: Step Time = 6.0001E−03 3 Primary Var: S, Mises Deformed Var: U Deformation Scale Factor: no deformation

1


2

2


Fri Jan 26 16:03:08 CET 2007

S, Mises (Avg: 75%)

1

3



Fri Jan 26 16:03:08 CET 2007


1

S, Mises (Avg: 75%)






3


S, Mises (Avg: 75%)


1


2

2


Fri Jan 26 16:03:08 CET 2007

S, Mises (Avg: 75%)

1

3



Fri Jan 26 16:03:08 CET 2007


1

S, Mises (Avg: 75%)






3


S, Mises (Avg: 75%)


1


2

2

Step: dropStep Frame: 20 +4.572e+08 +4.382e+08 +4.191e+08 +4.001e+08 +3.810e+08 +3.620e+08 +3.429e+08 +3.239e+08 +3.048e+08 +2.858e+08 +2.667e+08 +2.477e+08 +2.286e+08 +2.096e+08 +1.905e+08 +1.715e+08 +1.524e+08 +1.334e+08 +1.143e+08 +9.526e+07 +7.621e+07 +5.715e+07 +3.810e+07 +1.905e+07 +0.000e+00

S, Mises (Avg: 75%)

1

3

1

S, Mises (Avg: 75%)


1



1




Fri Jan 26 16:03:08 CET 2007


3

1

2



Fri Jan 26 16:03:08 CET 2007


Figure 2.30: Drop on the finned surface; 0 ms – 100 ms.

38




t = 120 ms


2

1

3



2

Fri Jan 26 16:03:08 CET 2007





Fri Jan 26 16:03:08 CET 2007


1


t = 140 ms

2

3



2

Fri Jan 26 16:03:08 CET 2007




Fri Jan 26 16:03:08 CET 2007


2



Fri Jan 26 16:03:08 CET 2007


2



Fri Jan 26 16:03:08 CET 2007

t = 160 ms t = 180 ms t = 200 ms

3


2



Fri Jan 26 16:03:08 CET 2007





2

Fri Jan 26 16:03:08 CET 2007


Fri Jan 26 16:03:08 CET 2007


1

S, Mises (Avg: 75%)






3


S, Mises (Avg: 75%)


1


2

2


Fri Jan 26 16:03:08 CET 2007

S, Mises (Avg: 75%)

1

3



Fri Jan 26 16:03:08 CET 2007


1

S, Mises (Avg: 75%)






3


S, Mises (Avg: 75%)


1


2

2


Fri Jan 26 16:03:08 CET 2007

S, Mises (Avg: 75%)

1

3



Fri Jan 26 16:03:08 CET 2007


1

S, Mises (Avg: 75%)






3


S, Mises (Avg: 75%)


1


2

2


S, Mises (Avg: 75%)

1

3

1

S, Mises (Avg: 75%)


1



1




Fri Jan 26 16:03:08 CET 2007


3

1

2



Fri Jan 26 16:03:08 CET 2007




39



t = 220 ms


2

1

3



2

Fri Jan 26 16:03:08 CET 2007





Fri Jan 26 16:03:08 CET 2007


1


t = 240 ms

2

3



2

Fri Jan 26 16:03:08 CET 2007




Fri Jan 26 16:03:08 CET 2007


2



Fri Jan 26 16:03:08 CET 2007


2



Fri Jan 26 16:03:08 CET 2007

t = 260 ms t = 280 ms t = 300 ms

3


2



Fri Jan 26 16:03:08 CET 2007





2

Fri Jan 26 16:03:08 CET 2007


Fri Jan 26 16:03:08 CET 2007


1

S, Mises (Avg: 75%)






3


S, Mises (Avg: 75%)


1


2

2


Fri Jan 26 16:03:08 CET 2007

S, Mises (Avg: 75%)

1

3



Fri Jan 26 16:03:08 CET 2007


1

S, Mises (Avg: 75%)






3


S, Mises (Avg: 75%)


1


2

2


Fri Jan 26 16:03:08 CET 2007

S, Mises (Avg: 75%)

1

3



Fri Jan 26 16:03:08 CET 2007


1

S, Mises (Avg: 75%)






3


S, Mises (Avg: 75%)


1


2

2


S, Mises (Avg: 75%)

1

3

1

S, Mises (Avg: 75%)


1



1




Fri Jan 26 16:03:08 CET 2007


3

1

2



Fri Jan 26 16:03:08 CET 2007



40


2.5

Conclusions

The overall behaviour predicted by the model is qualitatively very similar to what was observed during the experiments. The longer impact duration, approximately between 3 ms to 5 ms, when the cask is dropped on the fins compared to the flat target drop, is observed in both analysis and test. The reaction force at the bar’s bottom surface from the computational model is qualitatively similar to the test. The peak force is overestimated by about 35%. On the contrary the measured strains inside the cask above the impact area are underestimated for the similar amount. The underestimation is partly attributed to the fact that the lid wasn’t modeled and therefore the total mass underestimated. The maximum strain is about 20% higher for the flat impact area in both simulation and experiment. On the basis of the simulation and the experimental results presented in previous section, the following conclusion can be drawn: • The predicted behaviour is in good agreement with the experiment. There is also a room for the model improvement in some details which are gathered in the following list. • Lid must be added to the model to ensure better inertia representation. This influences values of all variables under consideration. • The mesh of the finned target should be refined. • The mesh of the bar is also too coarse due to the same reasons as fins. • The mesh pattern of the bar is not really optimal for the shape of the other body in contact in the case of the finned target area. • The boundary conditions at the bar’s bottom surface assume no friction in the contact between bar and the force transducer. It was shown that the simulated shape of the bar at its bottom surface is qualitatively different to the one observed during the experiment. The upper transducer’s surface will have to be modelled and thus introducing another contact pair of surfaces. • The artificial energy exceeds 10% of the total initial energy for the finned target drop. The influence of the different hourglass procedures should be examined. • In order to be able to simulate the failure of the fins the model has to take into account the failure criteria. • Only the tensile test material data are considered. The compression test data, lacking at the present, should also be incorporated in the material model. • There is necessity of upgrading the model with the failure criterion and/or material damage model and/or with the adaptive meshing.

Chapter 3

Model parameter sensitivity study 3.1

Introduction

A parametric analysis of the model is presented in this section. The influence of selected parameters on the model response is studied. The analyses are set to determine the influence of the following parameters’ variations on the model response: • mesh element size; • bulk viscosity; • dry friction coefficient in contact formulation; • contact formulation; • hourglass control; and • boundary conditions at the bar’s bottom surface. These parameters either cannot be, or were not, defined directly from the experimental data, like the friction coefficient, or are linked to the numerical procedures within the FEM, like the bulk viscosity. To accommodate these variations, 11 additional models were developed. The basic model configuration, named a, was presented in the previous chapter. The set of the basic values of the analysis parameters for Model a and their variations for models from b to j are as follows: a - basic model data set: dropDuration = 35 ms dropHeight = 1.006-eps m dropFreq = 10 dropDataFreq = 1 anaMassScaling = 0 (= no) 41

42

3. Model parameter sensitivity study anaTimeStep = 10− 7 s anaLinBulkVis = 0.06 anaQuadBulkVis = 1.2 anaFricCoef = 0.3 anaElemType = 0 (= hex elements where possible) anaContactMethod = 0 (= penalty) anaHourglassControl = 0 (= enhanced) cskElementSize cskElementSize cskElementSize cskElementSize cskElementSize cskElementSize pinElementSize

b - finer mesh: cskElementSize cskElementSize cskElementSize cskElementSize cskElementSize cskElementSize pinElementSize

1 = 50 mm 2 = 50 mm 3 = 50 mm 4 = 50 mm 5 = 20 mm 6 = 15 mm = 10 mm

1= 2= 3= 4= 5= 6= =5

35 mm 35 mm 35 mm 35 mm 20 mm 5 mm mm

c: - half bulk viscosity value: anaLinBulkVis = 0.03 anaQuadBulkVis = 0.6 d: - zero bulk viscosity value: anaLinBulkVis = 0.0 anaQuadBulkVis = 0.0 e: - half friction coefficient value: anaFricCoef = 0.15 f: - zero friction coefficient value: anaFricCoef = 0.0 g: - elemnt type: anaElemType = 1 (= tet elements everywhere) h: - contact formulation: anaContactMethod = 1 (= frictionless penalty contact) i1: - hourglass control type: anaHourglassControl = 1 (= relaxed stiffness)

3. Model parameter sensitivity study

43

i2: - hourglass control type: anaHourglassControl = 2 (= stiffness) i3: - hourglass control type: anaHourglassControl = 3 (= viscous) j: - boundary conditions at the bar bottom surface: the BC of all nodes at the bar’s bottom surface is encastered. The common features such as time increment, duration of analysis and number of increments, are compiled in table 3.1. There are no significant deviations regarding the computational cost (duration), except for Model g and b. The Model b is very demanding due to the denser mesh. Table 3.1: Common analyses features.

Computed drop duration [ms]

Minimal time increment [ns]

Maximal time increment [ns]

Number of increments

CPU time [DD / HH:MM:SS]

Model

Target

a

flat fins

35 35

58 58

66 64

548,999 558,399

14:22:12 13:23:12

b

flat fins

35 35

14 14

16 16

2,278,269 2,282,805

13 / 11:23:41 8 / 23:17:06

c

flat fins

35 35

60 60

68 69

536,778 535,326

13:39:50 13:28:44

d

flat fins

35 35

60 60

71 70

511,450 520,335

13:17:42 12:05:55

e

flat fins

35 35

60 58

84 67

475,052 556,895

12:07:54 13:19:30

f

flat fins

35 35

65 60

84 68

466,010 554,790

11:57:41 12:07:39

g

flat fins

35 35

8 18

31 37

2,534,656 1,460,182

10 / 08:12:47 6 / 02:21:27

h

flat fins

35 35

62 57

84 68

466,010 554,790

11:48:34 13:51:05

i1

flat fins

35 35

55 55

72 72

528,965 549,111

13:36:17 12:41:57

i2

flat fins

35 35

52 57

84 69

494,826 534,307

12:06:22 11:45:55

i3

flat fins

35 35

58 58

71 71

533,391 546,629

12:53:07 12:29:04

j

flat

35

58

64

557,707

14:22:24

44

3.2


Comparative analysis of Models a to j

When analyzing graphs in this section, the reader should note that the thicker lines always belong to the reference model, Model a, and thinner ones to the model with changed parameter values. The results of the drop on the flat target are always depicted with continuous line whereas the dashed line describes the finned target drop results. The Model j is an exception as only the flat target drop is studied.

3.2.1

Element size

A finer mesh is considered with the Model b. The element sizes for the basic Model a and Model b are presented in table 3.2. The mesh comparisons for the whole assembly as well as for the bar and cask part 6 can be seen in figures 3.1, 3.2 and 3.3, respectively. Table 3.2: Values of the element size controlling parameters.

Model: cskElementSize 1 cskElementSize 2 cskElementSize 3 cskElementSize 4 cskElementSize 5 cskElementSize 6 pinElementSize

a 50 50 50 50 20 15 10

mm mm mm mm mm mm mm

b 35 mm 35 mm 35 mm 35 mm 20 mm 5 mm 5 mm

Four different energies of the complete model are considered and compared for models a and b: artificial, total, kinetic and strain are presented in figures 3.4 and 3.5 respectively. A comparison of the reaction forces in the vertical directions is shown in figure 3.6, together by the maximal principal strains at the D1 measurement point. Vertical displacements at the points CSK1 and CSK3 are compared in figure 3.7. Vertical velocities at the points CSK1 and CSK3 are compared in figure 3.8. It is evident from table 3.1 that Model b is computationally very costly due to the finer mesh. Apart from this Model b yields physically sounder results than Model a. Preservation of total energy, figure 3.4, of Model b is much stronger than of Model a and the artificial energy (in the same figure) is smaller. There are no significant differences in the kinetic and strain energies (figure 3.5). The vertical reaction force, figure 3.6, shows relatively low sensitivity to the change of the mesh density. The force time series are smoother for the Model b. On the other hand, the maximal principal strains shows higher sensitivity to the mesh density. The values of the maximal principal strains are higher for Model b than for Model a and also closer to the experimental results. The differences in the cask displacements at points CSK1 and CSK3 are more than 10%. Less oscillations can be observed in the Model b velocity time series at the contact point CSK3 (figure 3.8). The probable cause lies in the shorter time increment used in Model b, since this gives more response points for averaging of final response output.


45

2

2 3

3

1

1

Figure 3.1: The assembly’s basic and finer mesh.

2

2 3

3

1

1

Figure 3.2: The bar’s basic and finer mesh.

2

2 3

3

1

1

Figure 3.3: The cask’s part 6 basic and finer mesh.

46


Artificial Energy

Total Energy

15

90 flat/a fins/a flat/b fins/b

flat/a fins/a flat/b fins/b

88

86

84 10

Energy [kJ]

Energy [kJ]

82

80

78 5 76

74

72

0

0

5

10

15

20

25

30

70

35

0

5

10

15

Time [ms]

20

25

30

35

30

35

Time [ms]

Figure 3.4: The artificial energy of models a and b on the left and the total energy on the right.

Kinetic Energy

Strain Energy

90

100


80


90

80

70

70

60

Energy [kJ]

Energy [kJ]

60

50

40

50

40

30 30

20

20

10

0

10

0

5

10

15

20 Time [ms]

25

30

35

0

0

5

10

15

20

25

Time [ms]

Figure 3.5: The kinetic energy of models a and b on the left and the total strain energy on the right.


47

Vertical reaction force

Max principal strain @ CSK/D1

4

300


3.5

flat/a fins/a flat/b fins/b 250

3 200 Strain [µm/m]

Force [MN]

2.5

2

150

1.5 100

1 50

0.5

0

0

5

10

15

20

25

30

0

35

0

5

10

15

Time [ms]

20

25

30

35

Time [ms]

Figure 3.6: The vertical reaction forces of models a and b on the left and the maximal principal strains at the D1 measurement point on the right. Vertical displacement @ CSK1

Vertical displacement @ CSK3

0


−10

−10

−20

Displacement [mm]

Displacement [mm]

−20

−30

−40

−30

−40

−50

−50

−60

−60

−70


0

5

10

15

20

25

30

−70

35

0

5

10

15

Time [ms]

20

25

30

35

Time [ms]

Figure 3.7: The vertical displacements at the CSK1 measurement point of models a and b on the left and the vertical displacements at the CSK3 measurement point on the right. Vertical displacement @ CSK1


0


−10

−20

Displacement [mm]

Displacement [mm]

−20

−30

−40

−30

−40

−50

−50

−60

−60

−70


−10

0

5

10

15

20

25

30

35

−70

0

5

10

Time [ms]

15

20

25

30

Time [ms]

Figure 3.8: The vertical velocities at the CSK1 measurement point of models a and b on the left and the vertical velocities at the CSK3 measurement point on the right.

35

48

3.2.2


Bulk viscosity

The different bulk viscosity parameter values of models a, c and d are shown in table 3.3. The bulk viscosity is discussed in detail in the Appendix section C.3.7. Table 3.3: Values of the bulk viscosities parameters.

Model:

a

c

d

anaLinBulkVis anaQuadBulkVis

0.06 1.2

0.03 0.6

0.0 0.0

Four different energies are considered and compared for the models a, c and d: artificial, total, kinetic and strain energy presented in figures 3.9 to 3.12 respectively. The comparison of the reaction forces in the vertical directions is shown in figure 3.13, followed by the maximal principal strains at the D1 measurement point in figure 3.14. Displacements at the point CSK1 and CSK3 are compared in figure 3.15 and figure 3.16 respectively. Velocities at the point CSK1 and CSK3 are compared in figure 3.17 and figure 3.18 respectively. There is no significant difference in the plotted results. Yet it appears from figure 3.9 that Model c, exhibits smallest maximum (2.6% smaller than basic model a) of artificial energy for the finned target drop. The lower values of the bulk viscosity parameters, models c and d, also made the unexplained small jump at maximum value of the vertical reaction force for the finned target drop disappear. Model c exhibits 7.6% smaller amplitude of the vertical reaction force for the finned target drop in comparison to Model a. Hence, the values of the bulk parameters of Model c are preferred.


49

Artificial Energy

Artificial Energy

15

15 flat/a fins/a flat/c fins/c

flat/a fins/a flat/d fins/d

Energy [kJ]

10

Energy [kJ]

10

5

0

5

0

5

10

15

20

25

30

0

35

0

5

10

15

Time [ms]

20

25

30

35

30

35

Time [ms]

Figure 3.9: The artificial energy of models a and c on the left and of a and d on the right.

Total Energy

Total Energy

90


88

86

86

84

84

82

82 Energy [kJ]

Energy [kJ]

88

80

80

78

78

76

76

74

74

72

72

70

0

5


10

15

20 Time [ms]

25

30

35

70

0

5

10

15

20

25

Time [ms]

Figure 3.10: The total energy of models a and c on the left and of a and d on the right.

50


Kinetic Energy

Kinetic Energy

90


70

70

60

60

50

40

50

40

30

30

20

20

10

10

0

0

5

10

15

20

25

30


80

Energy [kJ]

Energy [kJ]

80

0

35

0

5

10

15

Time [ms]

20

25

30

35

30

35

Time [ms]

Figure 3.11: The kinetic energy of models a and c on the left and of a and d on the right.

Strain Energy

Strain Energy

100


90

80

80

70

70

60

60 Energy [kJ]

Energy [kJ]

90

50

50

40

40

30

30

20

20

10

10

0

0

5


10

15

20 Time [ms]

25

30

35

0

0

5

10

15

20

25

Time [ms]

Figure 3.12: The total strain energy of models a and c on the left and of a and d on the right.


51



4


3.5

3

3

2.5

2.5 Force [MN]

Force [MN]

3.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

5


10

15

20

25

30

0

35

0

5

10

15

Time [ms]

20

25

30

35

Time [ms]

Figure 3.13: The vertical reaction forces of models a and c on the left and of a and d on the right.



300


250

250

200 Strain [µm/m]

Strain [µm/m]

200

150

150

100

100

50

50

0


0

5

10

15

20

25

30

35

Time [ms]

0

0

5

10

15

20

25

30

35

Time [ms]

Figure 3.14: The maximal principal strains at the D1 measurement point of models a and c on the left and of a and d on the right.

52




0


−10

−10

−20

Displacement [mm]

Displacement [mm]

−20

−30

−40

−30

−40

−50

−50

−60

−60

−70


0

5

10

15

20

25

30

−70

35

0

5

10

15

Time [ms]

20

25

30

35

Time [ms]

Figure 3.15: The vertical displacements at the CSK1 measurement point of models a and c left and of a and d right.



0


−10

−10

−20

Displacement [mm]

Displacement [mm]

−20

−30

−40

−30

−40

−50

−50

−60

−60

−70


0

5

10

15

20 Time [ms]

25

30

35

−70

0

5

10

15

20

25

30

35

Time [ms]

Figure 3.16: The vertical displacements at the CSK3 measurement point of models a and c left and of a and d right.


53

Vertical velocity @ CSK1


1


0

0

−1 Velocity [m/s]

Velocity [m/s]

−1

−2

−2

−3

−3

−4

−4

−5


0

5

10

15

20

25

30

−5

35

0

5

10

15

Time [ms]

20

25

30

35

Time [ms]

Figure 3.17: The vertical velocities at the CSK1 measurement point of models a and c left and of a and d right.



1


0

0

−1 Velocity [m/s]

Velocity [m/s]

−1

−2

−2

−3

−3

−4

−4

−5


0

5

10

15

20 Time [ms]

25

30

35

−5

0

5

10

15

20

25

30

Time [ms]

Figure 3.18: The vertical velocities at the CSK3 measurement point of models a and c left and of a and d right.

35

54

3.2.3


Coefficient of friction

The different coefficients of friction of models a, e and f are shown in table 3.4 (see also Appendix section C.3.15). Table 3.4: Values of the parameters describing the coefficient of friction.

Model:

a

e

f

anaFricCoef

0.3

0.15

0.0

Four different energies are considered again and compared for models a, e and f : artificial, total, kinetic and strain energy presented in figures from 3.19 to 3.22 respectively. The comparison of the reaction forces in the vertical directions are shown in figure 3.23, followed by the maximal principal strains at the D1 measurement point in figure 3.24. Displacements at the point CSK1 and CSK3 are compared in figure 3.25 and figure 3.26, respectively. Velocities at the point CSK1 and CSK3 are compared in figure 3.27 and figure 3.28, respectively. Again, there is no significant difference in the plotted results. Figure 3.19 needs more explanation. The decreasing coefficient of friction significantly decreases the amount of the artificial energy (around 50% for Model e and more than 75% for Model f ) for the flat target but increases the amount of the artificial energy in the case of the finned target (up tom the 15.5% for Model f ). It appears that for the flat target the tangential contact behaviour is responsible for excitation of hourglass modes. The reaction force, figure 3.23, shows relatively high sensitivity to the change of the coefficient of friction in particular in the flat target drop. The maximal vertical reaction force of Model f in the case of the flat target drop reaches 88.2% of the maximal value of Model a. This may be a way, apart from the auxiliary experimentation, to establish the appropriate value of the coefficient of friction through inverse problem formulation.


55

Artificial Energy

Artificial Energy

15

15 flat/a fins/a flat/e fins/e

flat/a fins/a flat/f fins/f

Energy [kJ]

10

Energy [kJ]

10

5

0

5

0

5

10

15

20

25

30

0

35

0

5

10

15

Time [ms]

20

25

30

35

30

35

Time [ms]

Figure 3.19: The artificial energy of models a and e on the left and of a and f on the right.

Total Energy

Total Energy

90


88

86

86

84

84

82

82 Energy [kJ]

Energy [kJ]

88

80

80

78

78

76

76

74

74

72

72

70

0

5


10

15

20 Time [ms]

25

30

35

70

0

5

10

15

20

25

Time [ms]

Figure 3.20: The total energy of models a and e on the left and of a and f on the right.

56


Kinetic Energy

Kinetic Energy

90


70

70

60

60

50

40

50

40

30

30

20

20

10

10

0

0

5

10

15

20

25

30


80

Energy [kJ]

Energy [kJ]

80

0

35

0

5

10

15

Time [ms]

20

25

30

35

30

35

Time [ms]

Figure 3.21: The kinetic energy of models a and e on the left and of a and f on the right.

Strain Energy

Strain Energy

100


90

80

80

70

70

60

60 Energy [kJ]

Energy [kJ]

90

50

50

40

40

30

30

20

20

10

10

0

0

5


10

15

20 Time [ms]

25

30

35

0

0

5

10

15

20

25

Time [ms]

Figure 3.22: The total strain energy of models a and e on the left and of a and f on the right.


57



4


3.5

3

3

2.5

2.5 Force [MN]

Force [MN]

3.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

5


10

15

20

25

30

0

35

0

5

10

15

Time [ms]

20

25

30

35

Time [ms]

Figure 3.23: The vertical reaction forces of models a and e on the left and of a and f on the right.



300


250

250

200 Strain [µm/m]

Strain [µm/m]

200

150

150

100

100

50

50

0


0

5

10

15

20

25

30

35

Time [ms]

0

0

5

10

15

20

25

30

35

Time [ms]

Figure 3.24: The maximal principal strains at the D1 measurement point of models a and e on the left and of a and f on the right.

58




0


−10

−10

−20

Displacement [mm]

Displacement [mm]

−20

−30

−40

−30

−40

−50

−50

−60

−60

−70


0

5

10

15

20

25

30

−70

35

0

5

10

15

Time [ms]

20

25

30

35

Time [ms]

Figure 3.25: The vertical displacements at the CSK1 measurement point of models a and e left and of a and f right.



0


−10

−10

−20

Displacement [mm]

Displacement [mm]

−20

−30

−40

−30

−40

−50

−50

−60

−60

−70


0

5

10

15

20 Time [ms]

25

30

35

−70

0

5

10

15

20

25

30

35

Time [ms]

Figure 3.26: The vertical displacements at the CSK3 measurement point of models a and e left and of a and f right.


59



1


0

0

−1 Velocity [m/s]

Velocity [m/s]

−1

−2

−2

−3

−3

−4

−4

−5


0

5

10

15

20

25

30

−5

35

0

5

10

15

Time [ms]

20

25

30

35

Time [ms]

Figure 3.27: The vertical velocities at the CSK1 measurement point of models a and e left and of a and f right.



1


0

0

−1 Velocity [m/s]

Velocity [m/s]

−1

−2

−2

−3

−3

−4

−4

−5


0

5

10

15

20 Time [ms]

25

30

35

−5

0

5

10

15

20

25

30

Time [ms]

Figure 3.28: The vertical velocities at the CSK3 measurement point of models a and e left and of a and f right.

35

60


3.2.4

Element type

A mesh with the tetragonal elements is considered with Model g. The element sizes for the basic Model a and Model g are the same. The mesh comparisons of the entire assembly as well as of the bar and cask part 6 can be seen in figures 3.29, 3.30 and 3.31 respectively. Quantitative comparisons of some variables were made as follows: Fig. 3.32 showing the artificial and the total energy, Fig. 3.33 showing the kinetic and the total strain energy, Fig. 3.34 showing the vertical reaction forces and the maximal principal strains, Fig. 3.35 showing the vertical displacement and Fig. 3.36 showing the vertical velocities. The long computational times, table 3.1, together with unrealistic physical behavior predicted for some variables tell us to avoid using tetrahedral elements in spite of the their zero artificial energy (no hourglass modes exists for this shape of the element). In the case of the flat target drop the whole cask part 6 is meshed with the tetrahedral elements due to its shape. The mesh pattern should be changed in a way that at least the impact area is meshed with the hexagonal elements.

2

2 3

3

1

1

Figure 3.29: The assembly’s basic and finer mesh.

2

2 3 1

3 1

Figure 3.30: The bar’s basic and finer mesh.


61

2

2 3

3 1

1

Figure 3.31: The cask’s part 6 basic and finer mesh. Artificial Energy

Total Energy

15

90 flat/a fins/a flat/g fins/g

flat/a fins/a flat/g fins/g

88

86

84 10

Energy [kJ]

Energy [kJ]

82

80

78 5 76

74

72

0

0

5

10

15

20

25

30

70

35

0

5

10

15

Time [ms]

20

25

30

35

30

35

Time [ms]

Figure 3.32: The artificial energy of models a and g on the left and the total energy on the right. Kinetic Energy

Strain Energy

90

100


80


90

80

70

70

60

Energy [kJ]

Energy [kJ]

60

50

40

50

40

30 30

20

20

10

0

10

0

5

10

15

20 Time [ms]

25

30

35

0

0

5

10

15

20

25

Time [ms]

Figure 3.33: The kinetic energy of models a and g on the left and the total strain energy on the right.

62




4

300


3.5

flat/a fins/a flat/g fins/g 250


Force [MN]

2.5

2

150

1.5 100

1 50

0.5

0

0

5

10

15

20

25

30

0

35

0

5

10

15

Time [ms]

20

25

30

35

Time [ms]

Figure 3.34: The vertical reaction forces of models a and g on the left and the maximal principal strains at the D1 measurement point on the right.



0


−10

−10

−20

Displacement [mm]

Displacement [mm]

−20

−30

−40

−30

−40

−50

−50

−60

−60

−70


0

5

10

15

20

25

30

35

−70

0

5

10

15

Time [ms]

20

25

30

35

Time [ms]

Figure 3.35: The vertical displacements at the CSK1 measurement point of models a and g on the left and the vertical displacements at the CSK3 measurement point on the right.


63



0


−10

−10

−20

Displacement [mm]

Displacement [mm]

−20

−30

−40

−30

−40

−50

−50

−60

−60

−70


0

5

10

15

20

25

30

35

−70

0

5

10

15

Time [ms]

20

25

30

35

Time [ms]

Figure 3.36: The vertical velocities at the CSK1 measurement point of models a and g on the left and the vertical velocities at the CSK3 measurement point on the right.

3.2.5

Contact formulation variation

Frictionless contact formulation is used with Model h. The frictionless contact formulation is identical to the one with friction, if using zero coefficient of friction Model f. Hence, models h and f give the same responses.

3.2.6

Hourglass control variation

There are three different hourglass controls for the hexagonal elements, see Appendix section C.3.16 on the page 114. Compared to Model a, three different i models are defined as seen from table 3.5. Table 3.5: Values of the hourglass control parameter.

Model

anaHourglassControl

hourglass method

a i1 i2 i3

0 1 2 3

enhanced integral viscoelastic Kelvin - pure stiffness Kelvin - pure viscous

The artificial and the total energy are presented in figure 3.37. It is obvious that the integral viscoelastic, Model i1, and Kelvin pure stiffness formulation, Model i2, of the hourglass controls give rise to additional artificial energy for the flat target drop in comparison to the enhanced hourglass control, Model a. On the contrary, the Kelvin pure viscous hourglass control formulation, Model i3, decreases the artificial energy for the flat target drop in comparison to the enhanced hourglass control, Model a. For the flat target drop cask part 6 is meshed by the tetrahedral elements. Hence, the hourglass control applies only to the bar. In the case of the finned target drop the overall behaviour is much different. Model i1 causes the artificial energy to decrease in comparison to Model a. On the contrary Model i3 increases the artificial energy in

64


comparison to Model a. The Kelvin pure viscous hourglass control formulation, model i3, gives a qualitatively different artificial energy response in this case. It is evident that the Kelvin pure stiffness formulation, Model i2, gives a (significant) rise in artificial energy in any case and should be avoided. There is no major differences when comparing total energies, except for Model i3 for the finned target drop. In this case the total energy rises well above the initial energy level. The Kelvin pure viscous hourglass control formulation, Model i3, should not be used for the finned target drops. There are some differences when comparing the kinetic energies for the different hourglass control formulations, figure 3.38, but not as drastic as when comparing the artificial and total energy. The same is true for the total strain energies, figure 3.38, with the exception of the Kelvin pure viscous hourglass control formulation, model i3, for the finned target drop. It is desirable to use same hourglass control for fined and flat drop simulations. The influence of the hourglass control formulations on the values of the vertical reaction force are significant, figure 3.39, yet the hourglass control mechanism should not be determined on this basis. It is set to suppress physically artificial behaviour of hexagonal elements. The control is closely tied to the artificial energy. Hence, proper behaviour of the formulation in respect to the energies is decisive for choosing it. There are some minor differences concerning the maximal principal strains, figure 3.39, when comparing results of the different hourglass control formulations. The most pronounced differences could be found for the flat target drop with Model i1, and Model i2 in comparison to Model a. When comparing displacement responses, figure 3.40, it is evident that the quantitative differences between the results for the selected hourglass control formulations are significant. The displacements could be used as a benchmark for the analysis due to sensitivity and relatively easy measurement of the bar’s shape, at least in the case of the flat target drop. The velocities, figure 3.41, are difficult to analyze. The responses are qualitatively similar but due to the “scatter” or “noise” (higher frequency components in the time histories), difficult to compare quantitatively.


65

Artificial Energy

Total Energy

15

90 flat/a fins/a flat/i1 fins/i1

flat/a fins/a flat/i1 fins/i1

88

86

84

Energy [kJ]

82 Energy [kJ]

a & i1

10

80

78 5 76

74

72

0

0

5

10

15

20

25

30

70

35

0

5

10

15

Time [ms]

20

25

30

35

25

30

35

25

30

35

Time [ms]

Artificial Energy

Total Energy

15



88

86

84

Energy [kJ]

82 Energy [kJ]

a & i2

10

80

78 5 76

74

72

0

0

5

10

15

20

25

30

70

35

0

5

10

15

Time [ms]

20 Time [ms]

Artificial Energy

Total Energy

15



88

86

84

Energy [kJ]

82 Energy [kJ]

a & i3

10

80

78 5 76

74

72

0

0

5

10

15

20 Time [ms]

25

30

35

70

0

5

10

15

20 Time [ms]

Figure 3.37: The artificial energy of models a, i1, i2 and i3 on the left and the total energy on the right.

66


Kinetic Energy

Strain Energy

90

100


80

90

80

70

70

60

60

50

Energy [kJ]

Energy [kJ]

a & i1


40

50

40

30 30

20

20

10

0

10

0

5

10

15

20

25

30

0

35

0

5

10

15

Time [ms] Kinetic Energy

25

30

35

25

30

35

25

30

35

Strain Energy

90

100


80


90

80

70

70

60

60

50

Energy [kJ]

Energy [kJ]

a & i2

20 Time [ms]

40

50

40

30 30

20

20

10

0

10

0

5

10

15

20

25

30

0

35

0

5

10

15

Time [ms] Kinetic Energy

Strain Energy

90

100


80


90

80

70

70

60

60

50

Energy [kJ]

Energy [kJ]

a & i3

20 Time [ms]

40

50

40

30 30

20

20

10

0

10

0

5

10

15

20 Time [ms]

25

30

35

0

0

5

10

15

20 Time [ms]

Figure 3.38: The kinetic energy of models a, i1, i2 and i3 on the left and the total strain energy on the right.


67



4

300


3.5

flat/a fins/a flat/i1 fins/i1 250


Force [MN]

a & i1

2.5

2

150

1.5 100

1 50

0.5

0

0

5

10

15

20

25

30

0

35

0

5

10

15

Time [ms]

25

30

35


4

300


3.5

20 Time [ms]



250


Force [MN]

a & i2

2.5

2

150

1.5 100

1 50

0.5

0

0

5

10

15

20

25

30

0

35

0

5

10

15

Time [ms]

25

30

35


4

300


3.5

20 Time [ms]




Force [MN]

a & i3

2.5

2

150

1.5 100

1 50

0.5

0

0

5

10

15

20

25

30

35

0

0

5

Time [ms]

10

15

20

25

30

35

Time [ms]

Figure 3.39: The vertical reaction forces of models a, i1, i2 and i3 on the left and the maximal principal strains at the D1 measurement point on the right.

68




0


−10

−10

−20

Displacement [mm]

Displacement [mm]

a & i1

−20

−30

−40

−30

−40

−50

−50

−60

−60

−70


0

5

10

15

20

25

30

−70

35

0

5

10

15

Time [ms] Vertical displacement @ CSK1

−20

Displacement [mm]

Displacement [mm]

a & i2

−30

−40

−30

−40

−50

−50

−60

−60

0

5

10

15

20

25

30

−70

35

0

5

10

15

Time [ms]

20

25

30

35

Time [ms]



0


−10


−10

−20

Displacement [mm]

−20

Displacement [mm]

35


−10

−20

a & i3

30


−10

−30

−40

−30

−40

−50

−50

−60

−60

−70

25


0

−70

20 Time [ms]

0

5

10

15

20 Time [ms]

25

30

35

−70

0

5

10

15

20

25

30

35

Time [ms]

Figure 3.40: The vertical displacements at the CSK1 measurement point of models a, i1, i2 and i3 on the left and the vertical displacements at the CSK3 measurement point on the right.


69



1


0

0

−1 Velocity [m/s]

Velocity [m/s]

a & i1

−1

−2

−3

−4

−4

0

5

10

15

20

25

30

−5

35

20

25

30

35

25

30

35

25

30

35

1 flat/a fins/a flat/i2 fins/i2 0

−1

−2

−2

−3

−3

−4

−4

0

5

10

15

20

25

30

−5

35

0

5

10

15

20

Time [ms]

Time [ms]



1


0


−1 Velocity [m/s]

−1 Velocity [m/s]

15 Time [ms]

Velocity [m/s]

Velocity [m/s]

a & i2

10


−1

−2

−2

−3

−3

−4

−4

−5

5

Vertical velocity @ CSK1 flat/a fins/a flat/i2 fins/i2 0

−5

0

Time [ms] 1

a & i3

−2

−3

−5


0

5

10

15

20 Time [ms]

25

30

35

−5

0

5

10

15

20 Time [ms]

Figure 3.41: The vertical velocities at the CSK1 measurement point of models a, i1, i2 and i3 on the left and the vertical velocities at the CSK3 measurement point on the right.

70


3.2.7

Boundary conditions variation on the bar’s bottom surface

One of conclusions in Section 2.5 states that bar’s boundary conditions are not the most appropriate ones. An alternative is to have the bottom surface of the bar is totally fixed by applying the encastre boundary conditions as is derived in Model j. This is the opposite with regards to the boundary condition of Model a. The truth probably lies somewhere in between those two extremes. The analysis is made only for flat target drop. The simple qualitative comparison among the models a and j and the experimental shapes are presented in figures 3.42, 3.43 and 3.44 respectively. It is quite obvious that the shape of the bar of Model j, figures 3.42 and 3.43, resembles more that of the bar after the experiment (Fig. 3.44) than the bar of Model a. Step: dropStep Frame: 350


S, Mises (Avg: 75%)

S, Mises (Avg: 75%)

+4.437e+08 +4.253e+08 +4.068e+08 +3.883e+08 +3.698e+08 +3.513e+08 +3.328e+08 +3.143e+08 +2.958e+08 +2.773e+08 +2.589e+08 +2.404e+08 +2.219e+08 +2.034e+08 +1.849e+08 +1.664e+08 +1.479e+08 +1.294e+08 +1.109e+08 +9.245e+07 +7.396e+07 +5.547e+07 +3.698e+07 +1.849e+07 +0.000e+00 3 1

2



3

Fri Jan 26 16:13:23 CET 2007

1


2

Drop test ODB: dt−jfd.odb


Fri Feb 02 18:50:43 CET 2007


Figure 3.42: The shape of the bar at the end of the simulation, model a on the left and model j on the right.



S, Mises (Avg: 75%)

1

S, Mises (Avg: 75%)


1

+4.372e+08 +4.190e+08 +4.008e+08 +3.826e+08 +3.643e+08 +3.461e+08 +3.279e+08 +3.097e+08 +2.915e+08 +2.733e+08 +2.550e+08 +2.368e+08 +2.186e+08 +2.004e+08 +1.822e+08 +1.640e+08 +1.457e+08 +1.275e+08 +1.093e+08 +9.109e+07 +7.287e+07 +5.465e+07 +3.643e+07 +1.822e+07 +0.000e+00 2 Drop test ODB: dt−jfd.odb ABAQUS/EXPLICIT Version 6.6−1 Fri Feb 02 18:50:43 CET 2007 3 Step: dropStep, Drop test step Increment 557707: Step Time = 3.5000E−02 Primary Var: S, Mises Deformed Var: U Deformation Scale Factor: no deformation

Figure 3.43: The shape of the cask and the bar at the end of the simulation, model a on the left and model j on the right.

The total energies of models a and j are almost identical, figure 3.45, which is not true for the artificial energies. A 46.2% higher artificial energy is reported for Model j than for Model a. There are some differences, but not really significant ones, when comparing the kinetic and the total strain energies of the models a and j, figure 3.46. The different impulse duration should be noted though. When analyzing the kinetic energy, the minimum of Model j appears 1.0 ms earlier than for Model a. The difference of the 1.2 ms can be observed when analyzing the maximum value of total strain energy.


71

Figure 3.44: The shape of the bar after the experiment.

The maximal vertical reaction force (figure 3.47) of Model j is 9.5% smaller then for Model a and it occurs 1.4 ms earlier. The amplitudes maximal principal strains of Model j, figure 3.48, reach higher values than for Model a and occurs 1.4 ms earlier. No difference can be observed between models a and j, when comparing the vertical displacement at the CSK1 measure point, figure 3.49. Even the vertical velocities in this point are qualitatively the same, though some differences can be observed in figure 3.51. The same goes for the vertical velocities at the CSK3 measurement point, figure 3.52. The difference in the vertical displacement at the CSK3 measurement point (contact point) are depicted in figure 3.50. Model j experiences smaller displacement minimum, 4.3%, which is in agrement with its smaller vertical reaction force, when compared to the one of Model a. The minimum of Model j occurs 0.5 ms before the one of Model a.

72


Artificial & Total Energies 100

90

80

70

Energy [kJ]

60 Artificial/flat/vertical BC Artificial/flat/encastre BC Total/flat/vertical BC Total/flat/encastre BC

50

40

30

20

10

0

0

5

10

15

20

25

30

35

Time [ms]

Figure 3.45: Artificial and total energies. Kinetic & Strain Energies 100

90

80

70

Energy [kJ]

60 Kinetic/flat/encastre BC Kinetic/flat/encastre BC Strain/flat/vertical BC Strain/flat/encastre BC

50

40

30

20

10

0

0

5

10

15

20

25

Time [ms]

Figure 3.46: Kinetic and total strain energies.

30

35


73

Vertical reaction force 4 Force/flat/vertical BC Force/flat/encastre BC 3.5

3

Force [MN]

2.5

2

1.5

1

0.5

0

0

5

10

15

20

25

30

35

Time [ms]

Figure 3.47: Reaction forces in the vertical direction. Max principal strain @ CSK/D1 300 Strain/flat/vertical BC Strain/flat/encastre BC 250

Strain [µm/m]

200

150

100

50

0

0

5

10

15

20

25

30

Time [ms]

Figure 3.48: Maximal principal strains at the D1 strain measurement point.

35

74


Vertical displacement @ CSK1 0 Displacement/flat/vertical BC Displacement/flat/encastre BC −10

Displacement [mm]

−20

−30

−40

−50

−60

−70

0

5

10

15

20

25

30

35

Time [ms]

Figure 3.49: Vertical displacements at CSK1 measurement point. Vertical displacement @ CSK3 0 Displacement/flat/vertical BC Displacement/flat/encastre BC −10

Displacement [mm]

−20

−30

−40

−50

−60

−70

0

5

10

15

20

25

30

Time [ms]

Figure 3.50: Vertical displacements at CSK3 measurement point.

35


75

Vertical velocity @ CSK1 1 Velocity/flat/vertical BC Velocity/flat/encastre BC 0

Velocity [m/s]

−1

−2

−3

−4

−5

0

5

10

15

20

25

30

35

Time [ms]

Figure 3.51: Vertical velocities at CSK1 measurement point. Vertical velocity @ CSK3 1 Velocity/flat/vertical BC Velocity/flat/encastre BC 0

Velocity [m/s]

−1

−2

−3

−4

−5

0

5

10

15

20

25

30

Time [ms]

Figure 3.52: Vertical velocities at CSK1 measurement point.

35

76


3.3

Conclusions

In addition to the conclusions in section 2.5, the following conclusions are drawn from the parameter sensitivity tests. • The default values of the bulk parameters in ABAQUS/Explicit should be decreased in order to decrease the artificial energy. • The model with the finer mesh yields better results than the model with the coarser one, but needs a lot of time to calculate the response. • The friction coefficient value has significant influence, in particular on the vertical reaction force. No value is available, but it could potentially be computed with an inverse numerical procedure based on the experimental data of the vertical reaction force or it could be measured by dedicated test. • Tetrahedral elements should be avoided if possible. • The enhanced hourglass control seems to work relatively well for this analysis. The Kelvin - pure stiffness hourglass formulation should be avoided. The integral viscoelastic hourglass formulation gives the lowest artificial energy in the case of the finned target drop. The Kelvin - pure viscous hourglass formulation gives the lowest artificial energy in the case of the flat target drop. Due to relatively big discrepancies in displacement responses among models, verification with the experimental data must be done prior the final verdict. The analysis did not clearly indicate the optional hourglass control. However, the enhanced hourglass control is formulation of the choice at the moment. • The experimental behaviour of the bar’s bottom part lies in between the two extreme cases: free to slide with zero friction and encastered. A contact friction should be considered. • Mass scaling should be added to speed up lengthy analysis. The amount of mass scaling should be investigated first in order to avoid influencing the model’s response too much. A compromise between speed and reliability of the results should be found (if there is any at all).

Chapter 4

Summary conclusions The report presents the numerical analysis of two one meter drop tests of a single ductile cast iron cask on a steel bar. The cask comes from the CASTOR family with machined cooling fins in a region where impact occurs. In the first test, the impact is on the cask’s cooling fins whereas in the second test the impact is in an area where the ribs have been locally machined away. The numerical analysis is based on explicit dynamic analysis using the commercial finite element code ABAQUS extended with Python scripts to allow a parametric description of the problem. The overall behavior of the model is qualitatively very similar to what was observed during the experiments. A longer impact duration, when the cask is dropped on the fins in comparison to the flat target, is observed in both the analysis and the test. The reaction force at the bar’s bottom surface from the model is qualitatively similar to that from the test. The same is true for the measured strains inside the cask above the impact area. The sensitivity analysis was performed to study the influence of parameters which either cannot be or were not defined directly from the experimental data, such as the friction coefficient, or which are linked to the FE numerical procedures, like the bulk viscosity. Only the tensile test material data are considered. Summary of detailed conclusions are as follows: • The mesh of the finned target and the bar should be refined. The model with the finer mesh yields better results than the model with the coarser one, but needs a lot of time to calculate the response. The mesh pattern of the bar is not optimal for the shape of the other body in contact in the case of the finned target area. • The boundary conditions at the bar’s bottom surface assume no friction in the contact between bar and the force transducer. It was shown that the simulated shape of the bar at its bottom surface is qualitatively different to the one observed during the experiment. • The friction coefficient value has significant influence, in particular on the vertical reaction force. • Tetrahedral elements should be avoided if possible. • The enhanced hourglass control seems to work relatively well for this analysis. 77

78

4. Summary conclusions • The vertical reaction force is sensitive to change of several model parameters. It is not the most reliable physical quantity for bench-marking the model.

Based on results, recommendations for further model development and fine tuning are given. • Lid must be added to the model to ensure better inertia representation. • The upper transducer’s surface should be modelled and thus introducing friction contact pair of surfaces (the bar and the transducer). • The compression test data, lacking at the present, should also be incorporated in the material model. • The default values of the bulk parameters in ABAQUS/Explicit should be decreased in order to decrease the artificial energy. • No value of the the friction coefficient is available, but it could potentially be computed with an inverse numerical procedure based on the experimental data of the vertical reaction force or it could be measured by dedicated test. • Mass scaling should be added to speed up lengthy analysis. The amount of mass scaling should be investigated first in order to avoid influencing the model’s response too much. A compromise between speed and reliability of the results should be found (if there is any at all). • In order to be able to simulate the failure of the fins as seen in the experiment, the model has to be upgraded with the failure criterion and/or material damage model and/or with the adaptive meshing.

Appendix A

Model parameters and material description A.1

Parameters and switches

Parameters are used in the drop test, analysis, geometry, mesh, loads, etc. definitions. The switches are parameters used for decision making inside the script. For example, whether a part should be modeled (switch is 1) or not (switch is 0). The switches are presented in table A.1 and the parameters in table A.2. There is also reference to the file where switch/parameter can be found. The geometry parameters are also graphically described and the reference to the figure is given in the table. Table A.1: List of the switches. Switch name

Def. value

Description

File

sw csk

1

modelling the cask 0 = do not model the cask 1 = do model the cask

01-parameters.py

sw pin

1

modelling of the bar 0 = do not model the bar 1 = do model the bar

01-parameters.py

sw csk rc

1

modelling of the cask’s circumferential fins 0 = do not model the fins 1 = do model the fins

01-parameters.py

sw csk cc

0

modelling of the cask’s circumferential channel 0 = do not model the channel 1 = do model the channel

01-parameters.py

sw csk mm

0

method of modelling of the cask’s material 0 = tabular material data

01-parameters.py

79

80

A. Model parameters and material description Table A.1: (continued. . .)

Switch name

Def. value

Description

File

1 = Johnson-Cook model (NOT yet implemented) sw pin mm

0

method of modelling of the bar’s material 0 = tabular material data 1 = Johnson-Cook model (NOT yet implemented)

01-parameters.py

Table A.2: List of the parameters. Parameter name

Def. value

Unit

Description

File

Figure

01A-basic.py 01A-basic.py 01A-basic.py 01A-basic.py 01A-basic.py 01A-basic.py 01A-basic.py

/ / / / / / /

the job type definition 0 = data check 1 = analysis submit the job? 0 = no 1 = yes duration of the drop job the drop height the cask’s drop target 0 = drop on the finned target 1 = drop on the flat target number of frames per millisecond number of history data per frame

01B-analysis.py

/

01B-analysis.py

/

01B-analysis.py 01B-analysis.py 01B-analysis.py

/ / /

01B-analysis.py

/

01B-analysis.py

/

do the mass scaling? 0 = no 1 = yes minimal integration time step for mass scaling coefficient of the linear bulk viscosity

01B-analysis.py

/

01B-analysis.py

/

01B-analysis.py

/

Tolerances and basic numeric parameters 10−6 10−3 0.017 . . . 6.283 . . . 9.80665 10−4 10−2

m rad / / m/s2 m m

dropJobType

0

/

dropJobSubmit

0

/

35 1.006-eps 0

ms m /

dropFreq

10

/

dropDataFreq

1

/

0

/

anaTimeStep

10−7

s

anaLinBulkVis

0.06

/

dropTol dropTolAngle pidiv180 twopi gravity eps epsAng

tolerance of the length tolerance of the angle π/180 2π gravity constant small length small angle

Drop test parameters

dropDuration dropHeight dropFlat

Analysis parameters anaMassScaling

A. Model parameters and material description

81

Table A.2: (continued. . .) Switch name

Def. value

Unit

anaQuadBulkVis

1.2

/

anaFricCoef anaContactMethod

0.3 0

/ /

anaElemType

0

/

anaHourglassControl

0

/

180

◦

5 2645 1230 20

mm mm mm mm

csk Rb

45

mm

csk Hsa

700

mm

csk Hsb

976

mm

csk Hsc

644

mm

csk Dsa

658

mm

csk Dsb

632

mm

csk Dsc

632

mm

csk Hla

32

mm

csk Hlb

165

mm

Description

File

coefficient of the quadratic bulk viscosity coefficient of dry friction contact method formulation 0 = penalty 1 = frictionless element type 0 = HEX where possible 1 = TET everywhere type of the hourglass control 0 = enhanced 1 = relaxed stiffness 2 = stiffness 3 = viscous

01B-analysis.py

/

01B-analysis.py 01B-analysis.py

/ /

01B-analysis.py

/

01B-analysis.py

/

angle of the model’s cross-section rotation

01-parameters.py

/ /

01C0-cask.py 01C0-cask.py 01C0-cask.py 01C0-cask.py

/ A.1 A.1 A.1

01C0-cask.py

A.1

01C0-cask.py

A.2

01C0-cask.py

A.2

01C0-cask.py

A.2

01C0-cask.py

A.1

01C0-cask.py

A.1

01C0-cask.py

A.1

01C0-cask.py

A.2

01C0-cask.py

A.2

Parameters of the MODEL rotationAngle Parameters of the CASK csk csk csk csk

Rmin Lo Do Rt

minimal radius for fillets cask length cask diameter radius of the cask’s fillet at the top radius of the cask’s fillet at the battom height of the cask’s storage space A height of the cask’s storage space B height of the cask’s storage space C inner diameter of the cask’s storage space A inner diameter of the cask’s storage space B inner diameter of the cask’s storage space C height of the cask’s lid seat A height of the cask’s

82

A. Model parameters and material description Table A.2: (continued. . .)

Switch name

Def. value

Unit

csk Hlc

305

mm

csk Dla

990

mm

csk Dlb

940

mm

csk Dlc

665

mm

csk Rla

2.5

mm

csk Rlb

2.5

mm

csk Rlc

6

mm

csk Hv

10

mm

csk Dv

1040

mm

csk Hcc

16

mm

csk Dcc

652

mm

csk Pcc

16

mm

csk Rcc

2.5

mm

csk Lr

1319

mm

csk Nr

5

/

csk Sr csk Or

22.5 5

mm mm

csk Pr

2.5

mm

csk Rr

6.25

mm

csk Dr csk Lf

30 582

mm mm

Description lid seat B height of the cask’s lid seat C diameter of the cask’s lid seat A diameter of the cask’s lid seat B diameter of the cask’s lid seat C fillet radius of the cask’s lid seat A fillet radius of the cask’s lid seat B fillet radius of the cask’s lid seat C height of the cask’s bottom void diameter of the cask’s bottom void height of the cask’s circumferential channel diameter of the cask’s circumferential channel position of the cask’s circumferential channel fillet radius of the cask’s circumferential channel position of the center of the cask’s fins half number of the cask’s fins step of the cask’s fins half thickness of the cask’s fins half thickness of the cask’s fins at the top fillet radius of the cask’s fins at the bottom depth of the cask’s fins distance from the cask’s flat

File 01C0-cask.py

A.2

01C0-cask.py

A.1

01C0-cask.py

A.1

01C0-cask.py

A.1

01C0-cask.py

A.1

01C0-cask.py

A.1

01C0-cask.py

A.1

01C0-cask.py

A.2

01C0-cask.py

A.2

01C0-cask.py

A.2

01C0-cask.py

A.1

01C0-cask.py

A.2

01C0-cask.py

A.1

01C0-cask.py

A.2

01C0-cask.py

A.3

01C0-cask.py 01C0-cask.py

A.3 A.3

01C0-cask.py

A.3

01C0-cask.py

A.3

01C0-cask.py 01C0-cask.py

A.3 A.2


83

Table A.2: (continued. . .) Switch name

Def. value

Unit

Description target to the cask’s centerline distance of the accelerometer from the cask’s bottom inner diameter of the strain gauges measurement positions outer diameter of the strain gauges measurement positions

csk La2

1200

mm

csk Dsi

100

mm

csk Dso

200

mm

1 3 100 75 20

mm mm mm mm mm

minimal radius for fillets fillet radius bar height bar diameter bolt diameter

csk ElementSize 1

50

mm

csk ElementSize 2

50

mm

csk ElementSize 3

50

mm

csk ElementSize 4

50

mm

csk ElementSize 5

20

mm

csk ElementSize 6

15

mm

pin ElementSize

10

mm

approx. size of the elements of the cask’s part 1 approx. size of the elements of the cask’s part 2 approx. size of the elements of the cask’s part 3 approx. size of the elements of the cask’s part 4 approx. size of the elements of the cask’s part 5 approx. size of the elements of the cask’s part 6 approx. size of the elements of the bar’s part

File 01C0-cask.py

A.2

01C0-cask.py

A.4

01C0-cask.py

A.4

01C3-pin.py 01C3-pin.py 01C3-pin.py 01C3-pin.py 01C3-pin.py

/ A.5 A.5 A.5 A.5

01C-meshing.py

/

01C-meshing.py

/

01C-meshing.py

/

01C-meshing.py

/

01C-meshing.py

/

01C-meshing.py

/

01C-meshing.py

/

Parameters of the BAR pin pin pin pin pin

Rmin R H D Db

Meshing parameters

84


Figure A.1: The geometry parameters of the cask (1); with measurement points.


Figure A.2: The geometry parameters of the cask (2); with geometry definition auxiliary points.

85

86


Figure A.3: The geometry parameters of the cask (3); with geometry definition auxiliary points.


Figure A.4: The geometry parameters of the cask (4); with strain measurement points.

87

88


Figure A.5: The geometry parameters of the bar.


A.2

89

Material data

Most of the material data were provided by the GNS. The experiment – the drop test – was done at room temperature (RT). Hence, only the material data of the tensile test for the RT is considered. Material data are presented in tables A.3, A.5 and A.4. The graphical presentation of the Castor material data found in A.5 is given in figure A.6 and the graphical presentation of the bar material data found in A.4 is given in figure A.9. The detailed Castor tensile test data are presented in figure A.7. The data points, currently used in the ABAQUS script, are presented in figure A.8 by the thicker lines. There are 4 points (2nd ,3rd ,4th and 5th at each strain rate) omitted in material data ABAQUS input due to user-supplied material data regularization inside the ABAQUS/Explicit, see D.1. The problem is noted and will be resolved in the next model version. Table A.3: Basic material data.

Density Young’s modulus Poisson’s ratio

ρ [kg/m3 ] E [GPa] ν [/]

Castor

Bar

Data source

7065 164 0.3

7850 212 0.3

/ [7, 8] /

Table A.4: Bar tensile test material data at the RT; source: [8].

Test type

Static ˙ = 0.004 [1/s]

technical plastic strain p [%]

true strain total t [%]

1 2 3 4 5

0.0 0.1 0.2 0.5 1.0

6 7 8 9 10 11 12 13

true stress

Dynamic ˙ = 1 [1/s]

σ [MPa]


0.0844 0.2221 0.3208 0.6184 1.1142

179.2 259.9 257.9 256.6 258.3

2.0 3.0 4.0 5.0 7.5

2.0936 3.0735 4.0480 5.0122 7.3759

10.0 12.5 15.0

9.6789 11.9271 14.1247

true stress


σ [MPa]


0.0942 0.2440 0.3461 0.6441 1.1392

200.0 306.6 312.0 311.9 312.4

250.5 265.0 289.3 311.8 353.2

2.1210 3.0908 4.0597 5.0231 7.3887

380.3 400.1 417.3

9.6929 11.9412 14.1387

true stress

Dynamic ˙ = 100 [1/s]

σ [MPa]


true stress σ [MPa]

0.1081 0.2549 0.3548 0.6512 1.1459

229.5 329.9 330.8 327.1 326.9

0.1239 0.2795 0.3758 0.6714 1.1703

263.2 382.5 375.5 370.7 380.1

311.0 304.1 316.3 337.5 384.6

2.1334 3.1107 4.0788 5.0388 7.4000

338.6 349.0 360.2 374.3 412.5

2.1523 3.1287 4.0980 5.0579 7.4150

380.4 389.6 404.4 419.2 449.3

416.1 438.2 456.6

9.7026 11.9501 14.1470

441.3 462.3 480.1

9.7145 11.9606 14.1566

472.0 490.5 507.3

0.0000 0.0005 0.0010 0.0020 0.0050

0.0100 0.0200 0.0500 0.1000 0.2000

0.5000 1.0000 2.0000 5.0000 6.0000

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

technical plastic strain p [%]

Test type

0.6704 1.1743 2.1708 5.0901 6.0414

0.1420 0.1577 0.1968 0.2546 0.3623

0.1086 0.1179 0.1207 0.1245 0.1324


285.1 300.7 326.1 382.8 396.5

217.0 226.4 241.6 254.7 268.2

178.4 192.8 196.6 201.3 209.3

σ [MPa]

true stress

Static ˙ = 0.004 [1/s]

0.6830 1.1862 2.1825 5.1001 6.0508

0.1599 0.1766 0.2169 0.2746 0.3801

0.1205 0.1322 0.1355 0.1401 0.1491


306.1 320.7 346.1 401.0 413.9

246.4 257.6 274.7 287.8 297.6

197.9 216.4 221.1 226.9 236.9

σ [MPa]

true stress

Dynamic ˙ = 0.1 [1/s]

0.6944 1.1933 2.1864 5.1034 6.0545

0.1654 0.1822 0.2229 0.2820 0.3906

0.1275 0.1385 0.1417 0.1461 0.1548


324.9 332.7 352.8 407.2 420.9

255.5 266.7 284.5 299.9 315.0

209.5 226.8 231.2 236.8 246.3

σ [MPa]

true stress


0.7114 1.2126 2.2070 5.1214 6.0715

0.1808 0.1982 0.2400 0.2997 0.4081

0.1392 0.1516 0.1551 0.1600 0.1695


353.4 365.1 388.3 439.9 452.5

280.8 293.2 312.8 329.2 344.0

228.7 248.4 253.4 259.7 270.5

σ [MPa]

true stress


Table A.5: Castor material tensile test data at the RT; source: [7].

0.7359 1.2285 2.2190 5.1326 6.0825

0.2120 0.2315 0.2752 0.3343 0.4379

0.1591 0.1761 0.1808 0.1870 0.1988


394.3 391.9 408.9 460.3 472.9

332.3 348.1 371.0 386.5 393.4

261.5 288.7 295.6 304.3 318.7

σ [MPa]

true stress

Dynamic ˙ = 100 [1/s]

90 A. Model parameters and material description


91

Castor material properties @ RT 500

450

True stress [MPa]

400

350

300

250 d/dt εpl = 0.004 1/s d/dt εpl = 0.1 1/s 200

d/dt εpl = 1 1/s d/dt εpl = 10 1/s d/dt εpl = 100 1/s

150

0

1

2

3 True plastic strain [%]

4

5

6

Figure A.6: Castor tensile test material data. Castor material properties @ RT 400

True stress [MPa]

350

300

250

d/dt εpl = 0.004 1/s

200

d/dt εpl = 0.1 1/s d/dt εpl = 1 1/s d/dt εpl = 10 1/s d/dt εpl = 100 1/s

150

0

0.05

0.1

0.15

0.2 0.25 0.3 True plastic strain [%]

0.35

0.4

Figure A.7: Castor tensile test material data - detail.

0.45

0.5

92


Castor material properties @ RT 360

340

320

True stress [MPa]

300

280

260 d/dt εpl = 0.004 1/s d/dt εpl = 0.1 1/s

240

d/dt εpl = 1 1/s d/dt εpl = 10 1/s

220

d/dt εpl = 100 1/s used, d/dt εpl = 0.004 1/s

200

used, d/dt εpl = 0.1 1/s used, d/dt εpl = 1 1/s

180

used, d/dt εpl = 10 1/s used, d/dt εpl = 100 1/s

160

0

0.005

0.01

0.015 True plastic strain [%]

0.02

0.025

0.03

Figure A.8: Castor tensile test material data; experimental (thinner lines) and data as used inside the model (thicker lines). Bar material properties @ RT 550

500

True stress [MPa]

450

400

350

300

250 d/dt εpl = 0.004 1/s d/dt εpl = 1 1/s

200

d/dt εpl = 10 1/s d/dt εpl = 100 1/s 150

0

2

4

6 8 True plastic strain [%]

10

Figure A.9: Bar material data from the tensile test.

12

14

Appendix B

Scripting in ABAQUS B.1

Scripting inside the ABAQUS/CAE

The Abaqus/CAE is used as a pre-processor as well as a post-processor. It produces the input file (*.inp) for the ABAQUS solvers, either ABAQUS/Standard or ABAQUS/Explicit. Scripting inside the ABACUS/CAE is Python based. Approximately 500 objects are added to the standard Python by the ABAQUS. There is one major one advantage of the Python scripting over the direct creating of the *.inp file. The automatic mesher is available through the Python scripts. The internal version of the model is 08.

B.2

Script files’ tree structure

The structure of the script files follows closely the ABAQUS modeling philosophy and is presented in fig. B.1. The files’ tree description is as follows: 00-run.py The script is run by this file, which loads almost all files as seen in fig. B.1.

00-run.py

01-parameters.py The file defines some parameters and switches and loads the rest of the parameter defining files as seen in fig. B.1. 01-parameters.py 01A-basic.py The file defines basic numerical and tolerance parameters. 01B-analysis.py The file defines drop test and analysis parameters.

01A-basic.py

01B-analysis.py

01C-geometry.py The file loads the files defining the cask’s (01C0-cask.py) and bar’s (01C3-pin.py) geometry parameters. 01C-geometry.py 01C0-cask.py The file defines cask’s geometry parameters. 01C3-pin.py The file defines bar’s geometry parameters. 93

01C0-cask.py

01C3-pin.py

94

B. Scripting in ABAQUS

00-run.py | |--- 01-parameters.py | | | |--- 01A-basic.py | | | |--- 01B-analysis.py | | | |--- 01C-geometry.py | | | | | |--- 01C0-cask.py | | | | | |--- 01C3-pin.py | | | |--- 01D-mesh.py | | | |--- 01E-report.py | |--- 02-material.py | |--- 03-geometry.py | | | |--- 03A-cask.py | | | | | |--- 03A1-cask(1).py | | | | | |--- 03A2-cask(2).py | | | | | |--- 03A3-cask(3).py | | | | | |--- 03A4-cask(4).py | | | | | |--- 03A5-cask(5).py | | | | | |--- 03A6-cask(6).py | | | |--- 03D-pin.py | |--- 04-mesh.py | | | |--- 04A-cask.py | | | |--- 04D-pin.py | |--- 05-assembly.py | |--- 06-step.py | |--- 07-initial-conditions.py | |--- 08-boundary-conditions.py | |--- 09-loads.py | |--- 10-contact.py | |--- 11-connections.py | |--- 12-job.py

Figure B.1: The structure tree of the script files.


95

01D-mesh.py The file defines parameter for meshing all of the parts.

01D-mesh.py

01E-report.py The file reports the values of all parameters and switches.

01E-report.py

02-material.py The file defines material properties.

02-material.py

03-geometry.py The file loads the rest of the geometry defining files as seen in fig. B.1.

03-geometry.py

03A-cask.py The file defines some auxiliary parameters and measurement points sets. It also loads all files for cask geometry definition as seen in fig. B.1. 03A-cask.py 03A1-cask(1).py The file defines the cask part 1 geometry and sets.

03A1-cask(1).py

03A2-cask(2).py The file defines the cask part 2 geometry and sets.

03A2-cask(2).py


03A3-cask(3).py


03A4-cask(4).py


03A5-cask(5).py


03A6-cask(6).py

03D-pin.py The file defines some auxiliary parameters, geometry and measurement points sets. 03D-pin.py 04-mesh.py The file loads meshing definitions as seen in fig. B.1. 04A-cask.py The file meshes all cask parts. 04D-pin.py The file meshes bar part.

04-mesh.py 04A-cask.py

04D-pin.py

05-assembly.py The file creates assembly and instances of the parts. Instances are arranged or positioned into correct pre-impact possition. 05-assembly.py 06-step.py The analysis step is defined as well as the eventual mass scaling. The field and time histories requests are also defined here. 06-step.py 07-initial-conditions.py The initial conditions are defined. 08-boundary-conditions.py The boundary conditions are defined. 09-loads.py The loads are defined. 10-contact.py The contact definition is configured. 11-connections.py The connections between the cask’s instances are defined. 12-job.py The job is defined and submitted.

07-initial-conditions.py 08-boundary-conditions.py 09-loads.py 10-contact.py

11-connections.py 12-job.py

The file names at the right margin of this document contain whole code of the particular file, which can be downloaded by the double click on the name.

96

B.3


Structure of the script file

A script file is divided into header followed by the part, where user input is required, and at the end the processing part, where user intervention is neither required nor wanted. The file structure is sketched in fig. B.2. As an example the file 01B-analysis.py is presented here. The header contains very basic data about the file itself, like: purpose, version, author, creation date and the place inside the files’ tree structure presented in fig. B.1. Latter defines main file, the owner file and owned file(s), if any at all, see fig. B.3. The first part of the file, where user input is required, does not appear in all files in the tree structure. The file 01B-analysis.py introduces some drop test and analysis parameters that have to be specified by the user. The code is heavily commented as could be seen in fig. B.4, which presents only the drop test parameters. The part of the file where user intervention is required always ends with the note: “# END of the user input part” as seen in fig. B.5 The second part of the file, the processing one, where user input is not required, appears in all files in the tree structure. The file 01B-analysis.py converts some parameters and computes additional ones as could be seen in fig. B.6, which presents only the computation of the additional parameters.


97

File’s header User interface part END of the user interface part Processing part Figure B.2: A sketch of the script file structure.

""" #========================================================== # # SAFECASK Action # CASTOR AVR drop test # Version: 08 # #========================================================== # # 01B-analysis.py -> Input parameters script file # -> drop test parameters # #---------------------------------------------------------# # Author(s): Nikola Jaksic # # Started at: 15/11/2006 # Last change: 02/02/2007 # #========================================================== # # Comments: # # Main file: run.py # Owner file: 01-parameters.py # #========================================================== """

Figure B.3: An example of the header of the script file.

98


#------------------------------# Drop test parameters #------------------------------print ’ reading drop test parameters...’ # the job type? # 0=DATACHECK, 1=ANALYSIS dropJobType=0 # submitting the job? # 0=NO, 1=YES dropJobSubmit=0 # duration of the DROP step # unit: [ms] dropDuration=35 # drop height # unit: [m] dropHeight=1.006-eps # drop on flat surface # unit: [/] dropFlat=0 # 0 = drop on ribs # 1 = drop on the flat surface # drop fequency # no of frames per 1ms # unit: [/] dropFreq=10 # drop data fequency # no od history data per 1 frame # unit: [/] dropDataFreq=1

Figure B.4: An example of the user input part of the script file.

#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # END of the user input part #~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Figure B.5: Marking an end of the user interface area of the script file.


99

#------------------------------# Computed parameters #------------------------------# drop velocity # unit: [m/s] dropVelocity = sqrt(2*gravity*dropHeight) # number of data saved at the DROP step # dropFreq frames per 1ms # unit: [/] dropNoDataSaved=int(dropFreq*dropDuration*1000) # frequency of the history data saved at the DROP step # dropDataFreq of the history data per one frame # unit: [/] dropNoHistoryDataSaved=int(dropDataFreq*float(dropNoDataSaved))

Figure B.6: An example of the processing part of the script file.

100


Appendix C

Explicit dynamics in ABAQUS C.1

Types of problems suited for ABAQUS/Explicit

Before discussing the theoretical background and the way how the explicit dynamics procedure works, it is helpful to understand what classes of problems are well-suited to ABAQUS/Explicit. Each of the following types of analyses can include temperature and heat transfer effects.

High-speed dynamic events: The explicit dynamic methods were originally developed to analyze high-speed dynamic events that can be extremely costly to analyze using implicit programs, such as ABAQUS/Standard. As an example of such a simulation is the effect of a short-duration blast load on a steel plate. Since the load is applied rapidly and is very severe, the response of the structure changes rapidly. Accurate tracking of stress waves through the plate is important for capturing the dynamic response. Since stress waves are associated with the highest frequencies of the system, obtaining an accurate solution requires many small time increments.

Complex contact problems: Contact conditions are formulated more easily using an explicit dynamics method than using an implicit method. The result is that ABAQUS/Explicit can readily analyze problems involving complex contact interaction between many independent bodies. ABAQUS/Explicit is particularly well-suited for analyzing the transient dynamic response of structures that are subject to impact loads and subsequently undergo complex contact interaction within the structure.

Complex post-buckling problems: Unstable post-buckling problems are solved readily in ABAQUS/Explicit. In such problems the stiffness of the structure changes drastically as the loads are applied. Post-buckling response often includes the effects of contact interactions. An illustration of both situations is a tube crushing. As the tube is crushed, it buckles severely and folds over onto itself, causing self-contact on both its inside and outside surfaces.

101

102

C. Explicit dynamics in ABAQUS

Highly nonlinear quasi-static problems: For a variety of reasons ABAQUS/Explicit is often very efficient in solving certain classes of problems that are essentially static. Quasi-static process simulation problems involving complex contact such as forging, rolling, and sheet-forming generally fall within these classes. Materials with degradation and failure: Material degradation and failure often lead to severe convergence difficulties in implicit analysis programs, but ABAQUS/Explicit models such materials well. An example of material degradation is the concrete cracking model, in which tensile cracking causes the material stiffness to become negative. An example of material failure is the ductile failure model for metals, in which material stiffness can degrade until it reduces to zero, at which time the elements are removed from the model entirely.

C.2

Theoretical background

Dynamics of the solid continuum is governed by the hyperbolic partial differential equation(s) of the second order. The physical interpretation of the equation is that local change takes time to propagate through continuum. The space discretization of the continua is done by FEM (Finite Element Method). FEM is a mesh discretization method, which generally speaking, interpolates values of the nodes’ variables over the whole mesh. Hence, the partial differential equation(s) is transformed into the set of ordinary differential equations. Time discretization is done by the FDM (Finite Difference Method), which transforms the set of the ordinary differential equations into the set of the algebraic (difference) equations. The lumped formulation is used, meaning that the element mass matrix is diagonal. Physical interpretation is that all mass of the continuum is condensed in nodes and not spread over elements as usual. The discretization, as already described, is named explicit numerical integration due to the fact that there is no need to solve the set of equations inside each time step. The latter is true for implicit methods. At each time step the dynamical equilibrium is reached and this is the state from which new dynamical equilibrium is computed in the next time step. The statical equilibrium is never reached due to presence of the inertial and energy dissipation forces. Some other phenomena should also be considered when building the model. Contact formulation, nonlinear material properties and large deformations find their way into the integration scheme or into the constitutive equations of the problem.

C.3

Explicit dynamics analysis

The explicit dynamics numerical procedure performs a large number of small time increments efficiently. A FDM submethod, the central-difference time integration rule, is used; each increment is relatively inexpensive to compute, in comparison to the direct-integration dynamic analysis procedure available in ABAQUS/Standard. The explicit central-difference operator satisfies the dynamic equilibrium equations at the beginning of the increment, t; the accelerations calculated at time t are used to advance the velocity solution to time t + ∆t/2 and the displacement solution to time t + ∆t, where ∆t is the time step.


C.3.1

103

Numerical implementation (Central difference method)

The explicit dynamics analysis procedure is based upon the implementation of an explicit integration rule together with the use of diagonal (“lumped”) element mass matrices. The equations of motion for the body are integrated using the explicit central-difference integration rule u˙ N ˙N (i+ 1 ) = u (i− 1 ) + 2

2

∆t(i+1) + ∆ti N u ¨(i) , 2

(C.1)

N ˙N uN (i+1) = u(i) + ∆t(i+1) u (i+ 1 ) ,

(C.2)

2

where uN is a degree of freedom (a displacement or rotation component) and the subscript i refers to the increment number in an explicit dynamics step. The central-difference integration operator is explicit in the sense that the kinematic state is advanced using known values of u˙ N and u ¨N (i) from the previous increment. (i− 1 ) 2

The explicit integration rule is quite simple but by itself does not provide the computational efficiency associated with the explicit dynamics procedure. The key to the computational efficiency of the explicit procedure is the use of diagonal element mass matrices because the accelerations at the beginning of the increment are computed by N J −1 J J u ¨N ) (P(i) − I(i) ), (i) = (M

(C.3)

J J where M N J is the mass matrix, P(i) is the applied load vector, and I(i) is the internal force vector. A lumped mass matrix is used because its inverse is simple to compute and because the vector multiplication of the mass inverse by the inertial force requires only n operations, where n is the number of degrees of freedom in the model. The explicit procedure requires no iterations and no tangent stiffness matrix. The internal force vector, I J , is assembled from contributions from the individual elements such that a global stiffness matrix need not be formed.

C.3.2

Nodal mass and inertia

The explicit integration scheme in ABAQUS/Explicit requires nodal mass or inertia to exist at all activated degrees of freedom unless constraints are applied using boundary conditions. More precisely, a nonzero nodal mass must exist unless all activated translational degrees of freedom are constrained and nonzero rotary inertia must exist unless all activated rotational degrees of freedom are constrained. Nodes that are part of a rigid body do not require mass, but the entire rigid body must possess mass and inertia unless constraints are used. When degrees of freedom at a node are activated by elements with a nonzero mass density (e.g., solid, shell, beam) or mass and inertia elements, a nonzero nodal mass or inertia occurs naturally from the assemblage of lumped mass contributions. When degrees of freedom at a node are activated by elements with no mass (e.g., spring, dashpot, or connector elements), care must be taken either to constrain the node or to add mass and inertia as appropriate.

104


C.3.3

Stability

The explicit procedure integrates through time by using many small time increments. The central-difference operator is conditionally stable, and the stability limit for the operator (with no damping) is given in terms of the highest frequency of the system as 2 ∆t ≤ . (C.4) ωmax With damping, the stable time increment is given by 2 p 2 1 + δmax − δmax . (C.5) ∆t ≤ ωmax where δ is the damping ratio of the mode with the highest frequency. Contrary to our usual engineering intuition, introducing damping to the solution reduces the stable time increment. In ABAQUS/Explicit a small amount of damping is introduced in the form of bulk viscosity to control high frequency oscillations. Physical forms of damping, such as dashpots or material damping, can also be introduced. Bulk viscosity and material damping are discussed below. C.3.3.1

Estimating the stable time increment size

An approximation to the stability limit is often written as the smallest transit time of a dilatational wave across any of the elements in the mesh Lmin , (C.6) cd where Lmin is the smallest element dimension in the mesh and cd is the dilatational wave speed in terms of λ0 and µ0 , defined below. ∆t ≈

In general, for beams, conventional shells, and membranes the element thickness or cross-sectional dimensions are not considered in determining the smallest element dimension; the stability limit is based upon the midplane or membrane dimensions only. When the transverse shear stiffness is defined for shell elements, the stable time increment will also be based on the transverse shear behavior. This estimate for ∆t is only approximate and in most cases is not a conservative (safe) estimate. In general, the actual stable time increment chosen by ABAQUS/Explicit will be less than this estimate by a factor √ √ between 1/ 2 and 1 in a two-dimensional model and between 1/ 3 and 1 in a three-dimensional model. The time increment chosen by ABAQUS/Explicit also accounts for any stiffness behavior in a model associated with penalty contact. C.3.3.2

Stable time increment report

ABAQUS/Explicit writes a report to the status (.sta) file during the data check phase of the analysis that contains an estimate of the minimum stable time increment and a listing of the elements with the smallest stable time increments and their values. The initial stable time increments listed do not include damping (bulk viscosity), mass scaling, or penalty contact effects. This listing is provided because often a few elements have much smaller stability limits than the rest of the elements in the mesh. The stable time increment can be increased by modifying the mesh to increase the size of the controlling element or by using appropriate mass scaling.


C.3.4

105

Dilatational wave speed

The current dilatational wave speed, cp , is determined in ABAQUS/Explicit by calculating the effective hypoeˆ and G ˆ = 2ˆ lastic material moduli from the material’s constitutive response. Effective Lamé’s constants, λ µ, are determined in the following manner. Define ∆p as the increment in the mean stress, ∆S as the increment in the deviatoric stress, ∆vol as the increment of volumetric strain, and as the deviatoric strain increment. We assume a hypoelastic stress-strain rule of the form ˆ + 2ˆ ∆p = 3λ µ ∆vol , (C.7)

∆S = 2ˆ µ∆e.

(C.8)

The effective moduli can then be computed as ˆ + 2ˆ ˆ = 3λ 3K µ=

2ˆ µ=

∆p , ∆vol

∆S : ∆e , ∆e : ∆e

1 ˆ ˆ + 2ˆ λ µ= 3K + 4ˆ µ . 3

(C.9)

(C.10)

(C.11)

For shell elements defined by a shell cross-section that requires numerical integration, the effective moduli for the section are computed by integrating the effective moduli at the section points through the thickness. These effective moduli represent the element stiffness and determine the current dilatational wave speed in the element as s ˆ + 2ˆ λ µ , (C.12) cp = ρ where ρ is the density of the material. In an isotropic, elastic material the effective Lamé’s constants can be defined in terms of Young’s modulus, E, and Poisson’s ratio, ν, by ˆ = λ0 = λ

Eν (1 + ν)(1 − 2ν)

(C.13)

µ ˆ = µ0 =

E . 2(1 + ν)

(C.14)

and

C.3.5

Time increment

The time increment used in an analysis must be smaller than the stability limit of the central-difference operator. Failure to use a small enough time increment will result in an unstable solution. When the solution

106


becomes unstable, the time history response of solution variables such as displacements will usually oscillate with increasing amplitudes. The total energy balance will also change significantly. If the model contains only one material type, the initial time increment is directly proportional to the size of the smallest element in the mesh. If the mesh contains uniform size elements but contains multiple material descriptions, the element with the highest wave speed will determine the initial time increment. In nonlinear problems with large deformations and/or nonlinear material response the highest frequency of the model will continually change, which consequently changes the stability limit. ABAQUS/Explicit has two strategies for time incrementation control: fully automatic time incrementation (where the code accounts for changes in the stability limit) and fixed time incrementation.

C.3.5.1

Scaling the time increment

To reduce the chance of a solution going unstable, one can adjust the stable time increment computed by ABAQUS/Explicit by a constant scaling factor. This factor can be used to scale the default global time estimate, the element-by-element estimate, or the fixed time increment based on the initial element-by-element estimate; it cannot be used to scale a fixed time increment specified directly by an user.

C.3.5.2

Automatic time increment

The default time incrementation scheme in ABAQUS/Explicit is fully automatic and requires no user intervention. Two types of estimates are used to determine the stability limit: element by element and global. An analysis always starts by using the element-by-element estimation method and may switch to the global estimation method under certain circumstances, as explained below. a) Element-by-element estimation In an analysis ABAQUS/Explicit initially uses a stability limit based on the highest element frequency in the whole model. This element-by-element estimate is determined using the current dilatational wave speed in each element. The element-by-element estimate is conservative; it will give a smaller stable time increment than the true stability limit that is based upon the maximum frequency of the entire model. In general, constraints such as boundary conditions and kinematic contact have the effect of compressing the eigenvalue spectrum, and the element-by-element estimates do not take this into account. The concept of the stable time increment as the time required to propagate a dilatational wave across the smallest element dimension is useful for interpreting how the explicit procedure chooses the time increment when element-by-element stability estimation controls the time increment. As the step proceeds, the global stability estimate, if used, will make the time increment less sensitive to element size. b) Global estimation The stability limit will be determined by the global estimator as the step proceeds unless the elementby-element estimation method is specified, fixed time incrementation is specified, or one of the conditions


107

explained below prevents the use of global estimation. The switch to the global estimation method occurs once the algorithm determines that the accuracy of the global estimation method is acceptable. The adaptive, global estimation algorithm determines the maximum frequency of the entire model using the current dilatational wave speed. This algorithm continuously updates the estimate for the maximum frequency. The global estimator will usually allow time increments that exceed the element-by-element values. ABAQUS/Explicit monitors the effectiveness of the global estimation algorithm. If the cost for computing the global time estimate is more than its benefit, the code will turn off the global estimation algorithm and simply use the element-by-element estimates to save computation time. The global estimation algorithm will not be used when any of the following capabilities are included in the model: • Fluid elements • Infinite elements • Dashpots • Thick shells (thickness to characteristic length ratio larger than 0.92) • Thick beams (thickness to length ratio larger than 1.0) • The JWL equation of state • Material damping • Nonisotropic elastic materials with temperature and field variable dependency • Distortion control • Adaptive meshing

C.3.5.3

Fixed time increment

A fixed time incrementation scheme is also available in ABAQUS/Explicit. The fixed time increment size is determined either by the initial element-by-element stability estimate for the step or by a user-specified time increment. Fixed time incrementation may be useful when a more accurate representation of the higher mode response of a problem is required. In this case a time increment size smaller than the element-by-element estimates may be used. When fixed time incrementation is used, ABAQUS/Explicit will not check that the computed response is stable during the step. User should ensure that a valid response has been obtained by carefully checking the energy history and other response variables.

C.3.6

Computational cost

The computer time involved in running a simulation using explicit time integration with a given mesh is proportional to the time period of the event. The time increment based on the element-by-element stability

108


estimate can be rewritten (ignoring damping) in the form r ρ ∆t ≤ min Le , ˆ + 2ˆ λ µ

(C.15)

where the minimum is taken over all elements in the mesh, Le is a characteristic length associated with an ˆ and µ element, ρ is the density of the material in the element, and λ ˆ are the effective Lamé’s constants for the material in the element. The time increment from the global stability estimate may be somewhat larger, but for this discussion we will assume that the above inequality always holds (when the inequality does not hold, the solution time will be somewhat faster). For linear, nonisotropic elastic materials this stability limit is further scaled down by the square root of the ratio of the effective material stiffness to the maximum material stiffness in one particular direction. Since this effectively means that the time increment can be no larger than the time required to propagate a stress wave across an element, the computer time involved in running a quasi-static analysis can be very large: the cost of the simulation is directly proportional to the number of time increments required. The number of increments, n, required is n = T /∆t if ∆t remains constant, where T is the time period of the event being simulated. (Even the element-by-element approximation of will not remain constant in general, since element distortion will change and nonlinear material response will change the effective Lamé constants. But the assumption is sufficiently accurate for the purposes of this discussion.) Thus,  s  ˆ 1 λ + 2ˆ µ n ≈ T max  . (C.16) Le ρ In a two-dimensional analysis refining the mesh by a factor of two in each direction will increase the run time in the explicit procedure by a factor of eight - four times as many elements and half the original time increment size. Similarly, in a three-dimensional analysis refining the mesh by a factor of two in each direction will increase the run time by a factor of sixteen. In a quasi-static analysis it is expedient to reduce the computational cost by either speeding up the simulation or by scaling the mass. In either case the kinetic energy should be monitored to ensure that the ratio of kinetic energy to internal energy does not get too large - typically less than 10

C.3.6.1

Reducing the computational cost by speeding up the simulation

To reduce the number of increments required, n, we can speed up the simulation compared to the time of the actual process - that is, we can artificially reduce the time period of the event, T . This will introduce two possible errors. If the simulation speed is increased too much, the increased inertia forces will change the predicted response (in an extreme case the problem will exhibit wave propagation response). The only way to avoid this error is to choose a speed-up that is not too large. The other error is that some aspects of the problem other than inertia forces - for example, material behavior - may also be rate dependent. In this case the actual time period of the event being modeled cannot be changed.

C. Explicit dynamics in ABAQUS C.3.6.2

109

Reducing the computational cost by using mass scaling

Artificially increasing the material density, ρ, by a factor f 2 reduces n to n/f , just like decreasing T to T /f . This concept, called “mass scaling”, reduces the ratio of the event time to the time for wave propagation across an element while leaving the event time fixed, which allows rate-dependent behavior to be included in the analysis. Mass scaling has exactly the same effect on inertia forces as speeding up the time of simulation. Mass scaling is attractive because it can be used in rate-dependent problems, but it must be used with care to ensure that the inertia forces do not dominate and change the solution. Either fixed or variable mass scaling can be invoked. Mass scaling can also be accomplished by altering the density; however, the fixed and variable mass scaling capabilities provide more versatile methods of scaling the mass of the entire model or specific element sets in the model.

C.3.7

Bulk viscosity

Bulk viscosity introduces damping associated with volumetric straining. Its purpose is to improve the modeling of high-speed dynamic events. ABAQUS/Explicit contains two forms of bulk viscosity: linear and quadratic. Linear bulk viscosity is included by default in an ABAQUS/Explicit analysis. The bulk viscosity parameters b1 and b2 defined below can be redefined and can be changed from step to step. If the default values are changed in a step, the new values will be used in subsequent steps until they are redefined. Bulk viscosities defined this way apply to the whole model. For an individual element set the linear and quadratic bulk viscosities can be scaled by a factor by defining section controls.

C.3.7.1

Linear bulk viscosity

Linear bulk viscosity is found in all elements and is introduced to damp vibrations with the highest element frequency. This damping is sometimes referred to as truncation frequency damping. It generates a bulk viscosity pressure that is linear in the volumetric strain rate pbv1 = b1 ρcd Le ˙vol ,

(C.17)

where b1 is a damping coefficient (default=0.06), ρ is the current material density, cp is the current dilatational wave speed, Le is an element characteristic length, and ˙vol is the volumetric strain rate.

C.3.7.2

Quadratic bulk viscosity

The second form of bulk viscosity pressure is found only in solid continuum elements (except the plane stress element CPS4R). This form is quadratic in the volumetric strain rate 2

pbv2 = ρ (b2 Le ˙vol ) ,

(C.18)

where b2 is a damping coefficient (default=1.2) and all other quantities are as defined for the linear bulk viscosity. Quadratic bulk viscosity is applied only if the volumetric strain rate is compressive.

110


The quadratic bulk viscosity pressure will smear a shock front across several elements and is introduced to prevent elements from collapsing under extremely high velocity gradients. Consider a simple one-element problem in which the nodes on one side of the element are fixed and the nodes on the other side have an initial velocity in the direction of the fixed nodes. If the initial velocity is equal to the dilatational wave speed of the material, without the quadratic bulk viscosity, the element would collapse to zero volume in one time increment (because the stable time increment size is precisely the transit time of a dilatational wave across the element). The quadratic bulk viscosity pressure will introduce a resisting pressure that will prevent the element from collapsing.

C.3.7.3

Fraction of critical damping (damping ratio) due to bulk viscosity

The bulk viscosity pressure is not included in the material point stresses because it is intended as a numerical effect onlyit is not considered part of the material’s constitutive response. The bulk viscosity pressures are based upon the dilatational mode of each element. The fraction of critical damping (damping ratio) in the dilatational mode of each element is given by δ = b1 − b22

C.3.8

Le min (0, ˙vol ) . cd

(C.19)

Energies

Following total energies output can be requested for a whole model: ALLAE: “Artificial” strain energy associated with constraints used to remove singular modes (such as hourglass control). ALLCD: Energy dissipated by viscoelasticity. (Not supported for hyperelastic and hyperfoam material models.) ALLFD: Total energy dissipated through frictional effects. (Available only for the whole model.) ALLIE: Total strain energy. (ALLIE=ALLSE + ALLPD + ALLCD + ALLAE + ALLDMD+ ALLDC+ ALLFC.) ALLKE: Kinetic energy. ALLPD: Energy dissipated by rate-independent and rate-dependent plastic deformation. ALLSE: Recoverable strain energy. ALLVD: Energy dissipated by viscous effects. ALLWK: External work. (Available only for the whole model.) ALLIHE: Internal heat energy. ALLHF: External heat energy through external fluxes. ALLDMD: Energy dissipated by damage.


111

ALLDC: Energy dissipated by distortion control. ALLFC: Fluid cavity energy, defined as the negative of the work done by all fluid cavities. (Available only for the whole model). ETOTAL: Energy balance defined as: ALLKE + ALLIE + ALLVD + ALLFD + ALLIHE ALLWK ALLHF. (Available only for the whole model.)

C.3.9

Material

Material definition is described in the section D.1

C.3.10

Initial conditions

Initial conditions are specified for particular nodes or elements, as appropriate. The data can be provided directly; in an external input file; or, in some cases, by a user subroutine or by the results or output database file from a previous ABAQUS analysis. If initial conditions are not specified, all initial conditions are zero except relative density in the porous metal plasticity model, which will have the value 1.0. Among many different initial conditions, the initial velocity is especially important for the explicit impact dynamics.

C.3.10.1

Defining initial angular and translational velocity

One can prescribe initial velocities in terms of an angular velocity and a translational velocity. This type of initial condition is typically used to define the initial velocity of a component of a rotating machine, such as a jet engine. The initial velocities are specified by giving the angular velocity, ω; the axis of rotation, defined from a point a at Xa to a point b at Xb ; and a translational velocity, vg . The initial velocity of node N at XN is then vN = vg + ω

C.3.11

Xb − Xa × XN − Xa b a |X − X |

(C.20)

Boundary conditions

Boundary conditions: • can be used to specify the values of all basic solution variables (displacements, rotations, warping amplitude, fluid pressures, pore pressures, temperatures, electrical potentials, normalized concentrations, or acoustic pressures) at nodes; • can be given as “model” input data (within the initial step in ABAQUS/CAE) to define zero-valued boundary conditions; and

112


• can be given as “history” input data (within an analysis step) to add, modify, or remove zero-valued or nonzero boundary conditions. Relative motions in connector elements can be prescribed similar to boundary conditions. Only zero-valued boundary conditions can be prescribed as model data in the ABAQUS/CAE step. One can specify the data using either “direct” or “type” format. As described below, the “type” format is a way of conveniently specifying common types of boundary conditions in stress/displacement analyses. “Direct” format must be used in all other analysis types.

C.3.11.1

Using the direct format

One can choose to enter the degrees of freedom to be constrained directly.

C.3.11.2

Using the “type” format in stress/displacement analyses

The type of boundary condition can be specified instead of degrees of freedom. The following boundary condition “types” are available in both ABAQUS/Standard and ABAQUS/Explicit: • XSYMM Symmetry about a plane X = constant (degrees of freedom 1, 5, 6 = 0 ). • YSYMM Symmetry about a plane Y = constant (degrees of freedom 2, 4, 6 = 0 ). • ZSYMM Symmetry about a plane Z = constant (degrees of freedom 3, 4, 5 = 0 ). • ENCASTRE Fully built-in (degrees of freedom 1, 2, 3, 4, 5, 6 = 0 ). • PINNED Pinned (degrees of freedom 1, 2, 3 = 0 ).

C.3.12

Loads

External loading can be applied in the following forms: • Concentrated or distributed tractions. • Concentrated or distributed fluxes. • Incident wave loads. There are two ways of specifying distributed loads in ABAQUS: element-based distributed loads and surfacebased distributed loads. Element-based distributed loads can be prescribed on element bodies, element surfaces, or element edges. Surface-based distributed loads can be prescribed on geometric surfaces or geometric edges. In ABAQUS/CAE all distributed surface and edge loads except pressure are surface-based, while distributed body loads are prescribed on geometric bodies or element bodies. Pressure loads can be element-based or surface-based. Many types of distributed loads are provided; they depend on the element type and are described in [3] in Part VI, “Elements”.


C.3.13

113

Predefined fields

The following predefined fields can be specified, as described in Predefined fields, Section 27.6.1: • Although temperature is not a degree of freedom in explicit dynamic analysis, nodal temperatures can be specified. Any difference between the applied and initial temperatures will cause thermal strain if a thermal expansion coefficient is given for the material. The specified temperature also affects temperaturedependent material properties, if any. • The values of user-defined field variables can be specified. These values affect only field-variable-dependent material properties, if any.

C.3.14

Elements

All of the elements available in ABAQUS/Explicit can be used in an explicit dynamic analysis. The elements are listed in [3] in Part VI, “Elements”. If coupled temperature-displacement elements are used in an explicit dynamic analysis, the temperature degrees of freedom will be ignored.

C.3.15

Coulomb friction in contact formulation

An extended version of the classical isotropic Coulomb friction model is provided in ABAQUS for use with all contact analysis capabilities. The extensions include an additional limit on the allowable shear stress, anisotropy, and the definition of a “secant” friction coefficient. The standard Coulomb friction model assumes that no relative motion occurs if the equivalent frictional stress q τeq = τ12 + τ22 (C.21) is less than the critical stress, τcrit , which is proportional to the contact pressure, p, in the form τcrit = µp,

(C.22)

where µ is the friction coefficient that can be defined as a function of the contact pressure, p; the slip rate, γ˙ eq ; the average surface temperature at the contact point; and the average field variables at the contact point. Rate-dependent friction cannot be used in a static Riks analysis since velocity is not defined. In ABAQUS it is possible to put a limit on the critical stress: τcrit = min (µp, τmax ) ,

(C.23)

where τmax is user-specified. If the equivalent stress is at the critical stress (τeq = τcrit ), slip can occur. If the friction is isotropic, the direction of the slip and the frictional stress coincide, which is expressed in the form τi γ˙ i = , τeq γ˙ eq where γ˙ i is the slip rate in direction i and γ˙ eg is the magnitude of the slip velocity, q γ˙ eq = γ˙ 12 + γ˙ 22 .

(C.24)

(C.25)

114


The same laws can be used for anisotropic friction after some simple transformations. In ABAQUS/Explicit the relative motion in the absence of slip is always equal to zero if the kinematic contact algorithm is used with hard tangential surface behavior; at the end of each increment the positions of the nodes on the contact surfaces are adjusted so that the relative motion is zero. With the penalty contact algorithm in ABAQUS/Explicit the relative motion in the absence of slip is equal to the friction force divided by the penalty stiffness.

C.3.16

Methods for suppressing hourglass modes

The formulation for reduced-integration elements considers only the linearly varying part of the incremental displacement field in the element for the calculation of the increment of physical strain. The remaining part of the nodal incremental displacement field is the hourglass field and can be expressed in terms of hourglass modes. Excitation of these modes may lead to severe mesh distortion, with no stresses resisting the deformation. Hourglass control attempts to minimize this problem without introducing excessive constraints on the element’s physical response. The following methods are available in ABAQUS/Explicit for suppressing the hourglass modes: • integral viscoelastic approach; • Kelvin viscoelastic approach; and • Enhanced hourglass control approach.

C.3.16.1

Integral viscoelastic approach

The integral viscoelastic approach available in ABAQUS/Explicit generates more resistance to hourglass forces early in the analysis step where sudden dynamic loading is more probable. Let q be an hourglass mode magnitude and Q be the force (or moment) conjugate to q. The integral viscoelastic approach is defined as Z t dq sK(t − t0 ) dt0 , Q= (C.26) dt 0 where K is the hourglass stiffness selected by ABAQUS/Explicit, and s is one of up to three scaling factors ss , sr and sw that one can define (by default, ss = sr = sw = 1). The scale factors are dimensionless and relate to specific displacement degrees of freedom. For solid and membrane elements ss scales all hourglass stiffnesses. For shell elements ss scales the hourglass stiffnesses related to the in-plane displacement degrees of freedom, and sr scales the hourglass stiffnesses related to the rotational degrees of freedom. In addition, sw scales the hourglass stiffness related to the transverse displacement for small-strain shell elements. The integral viscoelastic form of hourglass control is available for all reduced-integration elements and is the default form in ABAQUS/Explicit, except for elements modeled with hyperelastic and hyperfoam materials. It is the most computationally intensive hourglass control method.

C. Explicit dynamics in ABAQUS C.3.16.2

115

Kelvin viscoelastic approach

The Kelvin-type viscoelastic approach available in ABAQUS/Explicit is defined as dq Q = s (1 − α)Kq + αC , dt

(C.27)

where K is the linear stiffness and C is the linear viscous coefficient. This general form has pure stiffness and pure viscous hourglass control as limiting cases. When the combination is used, the stiffness term acts to maintain a nominal resistance to hourglassing throughout the simulation and the viscous term generates additional resistance to hourglassing under dynamic loading conditions. Three approaches are provided in ABAQUS/Explicit for specifying Kelvin viscoelastic hourglass control. Specifying the pure stiffness approach: The pure stiffness form of hourglass control is available for all reduced-integration elements and is recommended for both quasi-static and transient dynamic simulations. Specifying the pure viscous approach: The pure viscous form of hourglass control is available only for solid and membrane elements with reduced integration. It is the most computationally efficient form of hourglass control and has been shown to be effective for high-rate dynamic simulations. However, the pure viscous method is not recommended for low frequency dynamic or quasi-static problems since continuous (static) loading in hourglass modes will result in excessive hourglass deformation due to the lack of any nominal stiffness. Specifying a combination of stiffness and viscous hourglass control: A linear combination of stiffness and viscous hourglass control is available only for solid and membrane elements with reduced integration. One can specify the blending weight factor α (0 ≤ α ≤ 1) to scale the stiffness and viscous contributions. Specifying a weight factor equal to 0.0 or 1.0 results in the limiting cases of pure stiffness and pure viscous hourglass control, respectively. The default weight factor is 0.5.

C.3.16.3

Enhanced hourglass control approach

The enhanced hourglass control method is available for first-order solid, membrane, and finite-strain shell elements with reduced integration. The enhanced hourglass control approach available in both ABAQUS/Standard and ABAQUS/Explicit represents a refinement of the pure stiffness method in which the stiffness coefficients are based on the enhanced assumed strain method; no scale factor is required. It is the default hourglass control approach for hyperelastic and hyperfoam materials in both ABAQUS/Standard and ABAQUS/Explicit and for hysteresis materials in ABAQUS/Standard. This method gives more accurate displacement solutions for coarse meshes with linear elastic materials as compared to other hourglass control methods. It also provides increased resistance to hourglassing for nonlinear materials. Although generally beneficial, this may give overly stiff response in problems displaying plastic yielding under bending. In ABAQUS/Explicit the enhanced hourglass method will generally predict a much better return to the original configuration for hyperelastic or hyperfoam materials when loading is removed. The enhanced hourglass control approach is compatible between ABAQUS/Standard and ABAQUS/Explicit. It is recommended that enhanced hourglass control be used for both ABAQUS/Standard and ABAQUS/Explicit for all import analyses.

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The enhanced hourglass method cannot be used with elements modeled with hyperelastic or hyperfoam materials that are included in an adaptive mesh domain. Thus, if one decide to use hyperelastic or hyperfoam materials in an adaptive mesh domain, one must specify section controls to choose a different hourglass control approach. The use of adaptive meshing in domains modeled with finite-strain elastic materials is not recommended since better results are generally predicted using the enhanced hourglass method and, for solid elements, element distortion control (discussed below). Therefore, for these materials it is recommended that the analysis be run without adaptive meshing but with enhanced hourglass control.

Appendix D

Material definition in ABAQUS D.1

Material definition overview in ABAQUS

The material library in ABAQUS is intended to provide comprehensive coverage of both linear and nonlinear, isotropic and anisotropic material behaviors. The use of numerical integration in the elements, including numerical integration across the cross-sections of shells and beams, provides the flexibility to analyze the most complex composite structures. Material behaviors fall into the following general categories: • general properties (material damping, density, thermal expansion); • elastic mechanical properties; • inelastic mechanical properties; • thermal properties; • acoustic properties; • hydrostatic fluid properties; • equations of state; • mass diffusion properties; • electrical properties; and • pore fluid flow properties. Some of the mechanical behaviors offered are mutually exclusive: such behaviors cannot appear together in a single material definition. Some behaviors require the presence of other behaviors; for example, plasticity requires linear elasticity.

117

118

D. Material definition in ABAQUS

When giving material properties for finite-strain calculations, “stress” means “true” (Cauchy) stress (force per current area) and “strain” means logarithmic strain. For example, unless otherwise indicated, for uniaxial behavior Z dl l = = ln . (D.1) l l0 If nominal stress-strain data (σnom ↔ nom ) for a uniaxial test is available and the material is isotropic, a simple conversion to true stress and true strain (logarithmic plastic strain) is σtrue = σnom (1 + nom )

(D.2)

and true = pl = ln (1 + nom ) −

σnom . E

(D.3)

where E is the Young’s modulus. Material data are often specified as functions of independent variables such as temperature. Material properties are made temperature dependent by specifying them at several different temperatures. In some cases a material property can be defined as a function of variables calculated by ABAQUS; for example, to define a work-hardening curve, stress must be given as a function of equivalent plastic strain.

Interpolation of material data in ABAQUS In the simplest case of a constant property, only the constant value is entered. When the material data are functions of only one variable, the data must be given in order of increasing values of the independent variable. ABAQUS then interpolates linearly for values between those given. The property is assumed to be constant outside the range of independent variables given. Thus, one can give as many or as few input values as are necessary for the material model. If the material data depend on the independent variable in a strongly nonlinear manner, one must specify enough data points so that a linear interpolation captures the nonlinear behavior accurately. When material properties depend on several variables, the variation of the properties with respect to the first variable must be given at fixed values of the other variables, in ascending values of the second variable, then of the third variable, and so on. The data must always be ordered so that the independent variables are given increasing values. This process ensures that the value of the material property is completely and uniquely defined at any values of the independent variables upon which the property depends.

Regularizing user-defined data in ABAQUS/Explicit Interpolating material data as functions of independent variables requires table lookups of the material data values during the analysis. The table lookups occur frequently in ABAQUS/Explicit and are most economical if the interpolation is from regular intervals of the independent variables. This regularization normally requires the expansion of given data. If there are multiple independent variables, the concept of regular data also requires that the minimum and maximum values (the range) be constant for each independent variable while specifying the other independent variables.


119

It is not always desirable to regularize the input data so that they are reproduced exactly in a piecewise linear manner. ABAQUS/Explicit uses an error tolerance to regularize the input data. The number of intervals in the range of each independent variable is chosen such that the error between the piecewise linear regularized data and each of defined points is less than the tolerance times the range of the dependent variable. In some cases the number of intervals becomes excessive and ABAQUS/Explicit cannot regularize the data using a reasonable number of intervals. The number of intervals considered reasonable depends on the number of intervals one defines. If one defined 50 or less intervals, the maximum number of intervals used by ABAQUS/Explicit for regularization is equal to 100 times the number of user-defined intervals. If one defined more than 50 intervals, the maximum number of intervals used for regularization is equal to 5000 plus 10 times the number of userdefined intervals above 50. If the number of intervals becomes excessive, the program stops during the data checking phase and issues an error message. One can either redefine the material data or change the tolerance value. The default tolerance is 0.03. Since strain rate dependence of data is usually measured at logarithmic intervals, ABAQUS/Explicit regularizes strain rate data using logarithmic intervals rather than uniformly spaced intervals by default. This will generally provide a better match to typical strain-rate-dependent curves. One can specify linear strain rate regularization to use uniform intervals for regularization of strain rate data.

D.2

Metal plasticity

The classical metal plasticity models: • use Mises or Hill yield surfaces with associated plastic flow, which allow for isotropic and anisotropic yield, respectively; • use perfect plasticity or isotropic hardening behavior; • can be used when rate-dependent effects are important; • are intended for applications such as crash analyses, metal forming, and general collapse studies (Plasticity models that include kinematic hardening and are, therefore, more suitable for cases involving cyclic loading are also available in ABAQUS); • can be used in any procedure that uses elements with displacement degrees of freedom; • can be used in a fully coupled temperature-displacement analysis or an adiabatic thermal-stress analysis such that plastic dissipation results in the heating of a material; • can be used in conjunction with the models of progressive damage and failure in ABAQUS/Explicit to specify different damage initiation criteria and damage evolution laws that allow for the progressive degradation of the material stiffness and the removal of elements from the mesh; • can be used in conjunction with the shear failure model in ABAQUS/Explicit to provide a simple ductile dynamic failure criterion that allows for the removal of elements from the mesh, although the progressive damage and failure methods discussed above are generally recommended instead;

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• can be used in conjunction with the tensile failure model in ABAQUS/Explicit to provide a tensile spall criterion offering a number of failure choices and removal of elements from the mesh; and • must be used in conjunction with either the linear elastic material model or the equation of state material model.

D.2.1

Yield surfaces

The Mises and Hill yield surfaces assume that yielding of the metal is independent of the equivalent pressure stress: this observation is confirmed experimentally for most metals (except voided metals) under positive pressure stress but may be inaccurate for metals under conditions of high triaxial tension when voids may nucleate and grow in the material. Such conditions can arise in stress fields near crack tips and in some extreme thermal loading cases such as those that might occur during welding processes. A porous metal plasticity model is provided in ABAQUS for such situations.

Mises yield surface The Mises yield surface is used to define isotropic yielding. It is defined by giving the value of the uniaxial yield stress as a function of uniaxial equivalent plastic strain, temperature, and/or field variables.

Hill yield surface The Hill yield surface allows anisotropic yielding to be modeled. One must specify a reference yield stress, σ 0 , for the metal plasticity model and define a set of yield ratios, Rij , separately. These data define the yield stress corresponding to each stress component as Rij σ 0 . Hill’s potential function is discussed in detail in [3] in “Anisotropic yield/creep”. Yield ratios can be used to define three common forms of anisotropy associated with sheet metal forming: transverse anisotropy, planar anisotropy, and general anisotropy.

D.2.2

Hardening

In ABAQUS a perfectly plastic material (with no hardening) can be defined, or work hardening can be specified. Isotropic hardening is available in both ABAQUS/Standard and ABAQUS/Explicit; Johnson-Cook hardening is available only in ABAQUS/Explicit. In addition, ABAQUS provides kinematic hardening for materials subjected to cyclic loading.

Perfect plasticity Perfect plasticity means that the yield stress does not change with plastic strain. It can be defined in tabular form for a range of temperatures and/or field variables; a single yield stress value per temperature and/or field variable specifies the onset of yield.

Isotropic hardening Isotropic hardening means that the yield surface changes size uniformly in all directions such that the yield stress increases (or decreases) in all stress directions as plastic straining occurs. ABAQUS provides an isotropic


121

hardening model, which is useful for cases involving gross plastic straining or in cases where the straining at each point is essentially in the same direction in strain space throughout the analysis. Although the model is referred to as a “hardening” model, strain softening or hardening followed by softening can be defined. If isotropic hardening is defined, the yield stress, σ 0 , can be given as a tabular function of plastic strain and, if required, of temperature and/or other predefined field variables. The yield stress at a given state is simply interpolated from this table of data, and it remains constant for plastic strains exceeding the last value given as tabular data. ABAQUS/Explicit will regularize the data into tables that are defined in terms of even intervals of the independent variables. In some cases where the yield stress is defined at uneven intervals of the independent variable (plastic strain) and the range of the independent variable is large compared to the smallest interval, ABAQUS/Explicit may fail to obtain an accurate regularization of the data in a reasonable number of intervals. In this case the program will stop after all data are processed with an error message that one must redefine the material data.

Johnson-Cook isotropic hardening Johnson-Cook hardening is a particular type of isotropic hardening in ABAQUS/Explicit where the yield stress is given as an analytical function of equivalent plastic strain, strain rate, and temperature. This hardening law is suited for modeling high-rate deformation of many materials including most metals. Hill’s potential function cannot be used with Johnson-Cook hardening.

D.2.3

Flow rule

ABAQUS uses associated plastic flow. Therefore, as the material yields, the inelastic deformation rate is in the direction of the normal to the yield surface (the plastic deformation is volume invariant). This assumption is generally acceptable for most calculations with metals; the most obvious case where it is not appropriate is the detailed study of the localization of plastic flow in sheets of metal as the sheet develops texture and eventually tears apart. So long as the details of such effects are not of interest (or can be inferred from less detailed criteria, such as reaching a forming limit that is defined in terms of strain), the associated flow models in ABAQUS used with the smooth Mises or Hill yield surfaces generally predict the behavior accurately.

D.2.4

Rate dependence

As strain rates increase, many materials show an increase in their yield strength. This effect becomes important in many metals when the strain rates range between 0.1 and 1 per second; and it can be very important for strain rates ranging between 10 and 100 per second, which are characteristic of high-energy dynamic events or manufacturing processes. There are multiple ways to introduce a strain-rate-dependent yield stress.

Direct tabular data Test data can be provided as tables of yield stress values versus equivalent plastic strain at different equivalent

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plastic strain rates (¯˙pl ); one table per strain rate. Direct tabular data cannot be used with Johnson-Cook hardening. Yield stress ratios Alternatively, one can specify the strain rate dependence by means of a scaling function. In this case one enters only one hardening curve, the static hardening curve, and then express the rate-dependent hardening curves in terms of the static relation; that is, we assume that (D.4) σ ¯ ¯pl , ¯˙pl = σ 0 ¯pl R ¯˙pl where σ 0 is the static yield stress, ¯pl is the equivalent plastic strain, ¯˙pl is the equivalent plastic strain rate, and R is a ratio, defined as R = 1 at ¯˙pl = 0. This method is described further in “Rate-dependent yield”, section D.3.

D.2.5

Progressive damage and failure in ABAQUS/Explicit

In ABAQUS/Explicit the metal plasticity material models can be used in conjunction with the progressive damage and failure models. The capability allows for the specification of one or more damage initiation criteria, including ductile, shear, forming limit diagram (FLD), forming limit stress diagram (FLSD), M¨ uschenbornSonne forming limit diagram (MSFLD), and Marciniak-Kuczynski (M-K) criteria. After damage initiation, the material stiffness is degraded progressively according to the specified damage evolution response. The model offers two failure choices, including the removal of elements from the mesh as a result of tearing or ripping of the structure. The progressive damage models allow for a smooth degradation of the material stiffness, making them suitable for both quasi-static and dynamic situations. This is a great advantage over the dynamic failure models discussed next.

D.2.6

Shear and tensile dynamic failure in ABAQUS/Explicit

In ABAQUS/Explicit the metal plasticity material models can be used in conjunction with the shear and tensile failure models that are applicable in truly dynamic situations; however, the progressive damage and failure models discussed above are generally preferred. Shear failure The shear failure model provides a simple failure criterion that is suitable for high-strain-rate deformation of many materials including most metals. It offers two failure choices, including the removal of elements from the mesh as a result of tearing or ripping of the structure. The shear failure criterion is based on the value of the equivalent plastic strain and is applicable mainly to high-strain-rate, truly dynamic problems. Tensile failure The tensile failure model uses the hydrostatic pressure stress as a failure measure to model dynamic spall or a pressure cutoff. It offers a number of failure choices including element removal. Similarly to the shear failure model, the tensile failure model is suitable for high-strain-rate deformation of metals and is applicable to truly dynamic problems.


D.2.7

123

Heat generation by plastic work

ABAQUS optionally allows for plastic dissipation to result in the heating of a material. Heat generation is typically used in the simulation of bulk metal forming or high-speed manufacturing processes involving large amounts of inelastic strain where the heating of the material caused by its deformation is an important effect because of temperature dependence of the material properties. It is applicable only to adiabatic thermal-stress analysis or fully coupled temperature-displacement analysis. This effect is introduced by defining the fraction of the rate of inelastic dissipation that appears as a heat flux per volume.

D.3

Rate-dependent yield

Rate-dependent yield: • is needed to define a material’s yield behavior accurately when the yield strength depends on the rate of straining and the anticipated strain rates are significant; • is available only for the isotropic hardening metal plasticity models (Mises and Johnson-Cook), the isotropic component of the nonlinear isotropic/kinematic plasticity models, the extended Drucker-Prager plasticity model, and the crushable foam plasticity model; • can be conveniently defined on the basis of work hardening parameters and field variables by providing tabular data for the isotropic hardening metal plasticity models, the isotropic component of the nonlinear isotropic/kinematic plasticity models, and the extended Drucker-Prager plasticity model; • can be defined through specification of user-defined overstress power law parameters or yield stress ratios for the isotropic hardening metal plasticity models, the extended Drucker-Prager plasticity model, or the crushable foam plasticity model; • cannot be used with any of the ABAQUS/Standard creep models (metal creep, time-dependent volumetric swelling, Drucker-Prager creep, or cap creep) since creep behavior is already a rate-dependent mechanism; and • in dynamic analysis should be specified such that the yield stress increases with increasing strain rate.

D.3.1

Work hardening dependencies

¯ for the crushable foam model), is dependent on work hardening, Generally, a material’s yield stress, σ ¯ (or B which for isotropic hardening models is usually represented by a suitable measure of equivalent plastic strain, ¯pl ; the inelastic strain rate, ¯˙pl ; temperature, θ; and predefined field variables, fi : σ ¯=σ ¯ ¯pl , ¯˙pl , θ, fi (D.5) Many materials show an increase in their yield strength as strain rates increase; this effect becomes important in many metals and polymers when the strain rates range between 0.1 and 1 per second, and it can be very important for strain rates ranging between 10 and 100 per second, which are characteristic of high-energy dynamic events or manufacturing processes.

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Defining hardening dependencies for various material models Strain rate dependence can be defined by entering hardening curves at different strain rates directly or by defining yield stress ratios to specify the rate dependence independently.

Direct entry of test data Work hardening dependencies can be given quite generally as tabular data for the isotropic hardening Mises plasticity model, the isotropic component of the nonlinear isotropic/kinematic hardening model, and the extended Drucker-Prager plasticity model. The test data are entered as tables of yield stress values versus equivalent plastic strain at different equivalent plastic strain rates. The yield stress must be given as a function of the equivalent plastic strain and, if required, of temperature and of other predefined field variables. In defining this dependence at finite strains, “true” (Cauchy) stress and log strain values should be used. The hardening curve at each temperature must always start at zero plastic strain. For perfect plasticity only one yield stress, with zero plastic strain, should be defined at each temperature. It is possible to define the material to be strain softening as well as strain hardening. The work hardening data are repeated as often as needed to define stress-strain curves at different strain rates. The yield stress at a given strain and strain rate is interpolated directly from these tables.

Using yield stress ratios Alternatively, and as the only means of defining rate-dependent yield stress for the Johnson-Cook and the crushable foam plasticity models, the strain rate behavior can be assumed to be separable, so that the stress-strain dependence is similar at all strain rate levels: σ ¯ = σ 0 ¯pl , θ, fi R ¯˙pl , θ, fi

(D.6)

where σ 0 ¯pl , θ, fi (or B ¯pl , θ, fi in the foam model) is the static stress-strain behavior and R ¯˙pl , θ, fi is the ratio of the yield stress at nonzero strain rate to the static yield stress (so that R (0, θ, fi ) = 1). Two methods are offered to define R in ABAQUS: specifying an overstress power law or defining R directly as a tabular function. In addition, in ABAQUS/Explicit an analytical Johnson-Cook form can be specified to define R. • Overstress power law The Cowper-Symonds overstress power law has the form n ¯˙pl = D (R − 1)

for

σ ¯ ≥ σ0

¯ ≥ B in the crushable foam model), (or B

(D.7)

where D(θ, fi ) and n(θ, fi ) are material parameters that can be functions of temperature and, possibly, of other predefined field variables. • Tabular function Alternatively, R can be entered directly as a tabular function of the equivalent plastic strain rate (or the axial plastic strain rate in a uniaxial compression test for the crushable foam model), ¯˙pl ; temperature, θ; and field variables, fi . • Johnson-Cook rate dependence Johnson-Cook rate dependence has the form 1 ˙¯pl = ˙0 exp (R − 1) for σ ¯ ≥ σ0 , C

(D.8)


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where ˙0 and C are material constants that do not depend on temperature and are assumed not to depend on predefined field variables. Johnson-Cook rate dependence must be used with Johnson-Cook hardening (it cannot be used in conjunction with the isotropic hardening metal plasticity models, the extended Drucker-Prager plasticity model, or the crushable foam plasticity model).

D.3.2

Evaluation of strain-rate-dependent data in ABAQUS/Explicit

Rate-sensitive material constitutive behavior may introduce nonphysical high-frequency oscillations in an explicit dynamic analysis. To overcome this problem, ABAQUS/Explicit computes the equivalent plastic strain rate used for the evaluation of strain-rate-dependent data as ¯˙pl |t+∆t = ω

∆¯ pl + (1 − ω) ¯˙pl |t . ∆t

(D.9)

Here ∆¯ pl is the incremental change in equivalent plastic strain during the time increment ∆t, and ¯˙pl |t and pl ¯˙ |t+∆t are the strain rates at the beginning and end of the increment, respectively. The factor ω, (0 < ω < 1) facilitates filtering high-frequency oscillations associated with strain-rate-dependent material behavior. One can specify the value of the strain rate factor, ω, directly. The default value is 0.9. A value of ω = 1 does not provide the desired filtering effect and should be avoided.

D.3.3

Johnson-Cook plasticity material model

The Johnson-Cook plasticity model [5, 6]: • is a particular type of Mises plasticity model with analytical forms of the hardening law and rate dependence; • is suitable for high-strain-rate deformation of many materials, including most metals; • is typically used in adiabatic transient dynamic simulations; • can be used in conjunction with the Johnson-Cook dynamic failure model; • can be used in conjunction with the tensile failure model to model tensile spall or a pressure cutoff; • can be used in conjunction with the progressive damage and failure models in ABAQUS/Explicit to specify different damage initiation criteria and damage evolution laws that allow for the progressive degradation of the material stiffness and the removal of elements from the mesh; and • must be used in conjunction with either the linear elastic material model or the equation of state material model. A Mises yield surface with associated flow is used in the Johnson-Cook plasticity model.

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Johnson-Cook hardening Johnson-Cook hardening is a particular type of isotropic hardening where the static yield stress, σ 0 , is assumed to be of the form h n i 1 − θˆm , (D.10) σ 0 = A + B ¯pl where ¯pl is the equivalent plastic strain and A, B, n and m are material parameters measured at or below the transition temperature, θt . θˆ is the nondimensional temperature defined as   0 for θ < θt  θ−θt ˆ θ= (D.11) for θt ≤ θ ≤ θm , θ −θ   m t 1 for θ > θm where θ is the current temperature, θm is the melting temperature, and θt is the transition temperature defined as the one at or below which there is no temperature dependence on the expression of the yield stress. The material parameters must be measured at or below the transition temperature. When θ ≥ θm , the material will be melted and will behave like a fluid; there will be no shear resistance since σ = 0. The hardening memory will be removed by setting the equivalent plastic strain to zero. If backstresses are specified for the model, these will also be set to zero. 0

If the annealing behavior is included in the material definition and the annealing temperature is defined to be less than the melting temperature specified for the metal plasticity model, the hardening memory will be removed at the annealing temperature and the melting temperature will be used strictly to define the hardening function. Otherwise, the hardening memory will be removed automatically at the melting temperature. If the temperature of the material point falls below the annealing temperature at a subsequent point in time, the material point can work harden again. Johnson-Cook strain rate dependence Johnson-Cook strain rate dependence assumes that σ ¯ = σ 0 ¯pl , θ R ¯˙pl

(D.12)

and ¯˙pl = ˙0

1 (R − 1) C

for

σ ¯ ≥ σ0 ,

(D.13)

where σ ¯ ¯˙pl ˙0 and C σ 0 ¯pl , θ R ¯˙pl

is the yield stress at nonzero strain rate; is the equivalent plastic strain rate; are material parameters measured at or below the transition temperature, θt ; is the static yield stress; and is the ratio of the yield stress at nonzero strain rate to the static yield stress (so that R(˙0 = 1.0)).

The yield stress is, therefore, expressed as pl h n i ¯˙ σ ¯ = A + B ¯pl 1 + C ln 1 − θˆm . ˙0

(D.14)

The use of Johnson-Cook hardening does not necessarily require the use of Johnson-Cook strain rate dependence, but the use of Johnson-Cook strain rate dependence does require the use of Johnson-Cook hardening.


D.4

127

Material damping

Defining inelastic material behavior, dashpots, etc. will introduce energy dissipation into a model. In addition to these mechanisms, general (“Rayleigh”) material damping can be introduced. Adding damping to a model, especially stiffness proportional damping, βR , may significantly reduce the stable time increment. Material damping can be defined: • for direct-integration (nonlinear, implicit or explicit), direct-solution steady-state, and subspace-based steady-state dynamic analysis; or • for mode-based (linear) dynamic analysis in ABAQUS/Standard.

D.4.1

Rayleigh damping

In direct-integration dynamic analysis one very often define energy dissipation mechanisms - dashpots, inelastic material behavior, etc. - as part of the basic model. In such cases there is usually no need to introduce additional “structural” or general damping: it is often unimportant compared to these other dissipative effects. However, some models do not have such dissipation sources (an example is a linear system with chattering contact, such as a pipeline in a seismic event). In such cases it is often desirable to introduce some general damping. ABAQUS provides “Rayleigh” damping for this purpose. Rayleigh damping can also be used in direct-solution steady-state dynamic analyses and subspace-based steady-state dynamic analyses to get quantitatively accurate results, especially near natural frequencies. To define Rayleigh damping, one specifies two Rayleigh damping factors: αR for mass proportional damping and βR for stiffness proportional damping. In general, damping is a material property specified as part of the material definition. For the cases of rotary inertia, point mass elements, and substructures, where there is no reference to a material definition, the damping can be defined in conjunction with the property references. A nonstructural feature is assumed to have no damping. For a given mode i the fraction of critical damping (damping ratio), δi , can be expressed in terms of the damping factors and as: δi =

βR ωi αR + , 2ωi 2

(D.15)

where ωi is the natural frequency at this mode. This equation implies that, generally speaking, the mass proportional Rayleigh damping, αR , damps the lower frequencies and the stiffness proportional Rayleigh damping, βR , damps the higher frequencies.

Mass proportional damping The αR factor introduces damping forces caused by the absolute velocities of the model and so simulates the idea of the model moving through a viscous still fluid, so that any motion of any point in the model causes damping. This damping factor defines mass proportional damping, in the sense that it gives a damping contribution proportional to the mass matrix for an element. If the element contains more than one material in

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ABAQUS/Standard, the volume average value of αR is used to multiply the element’s mass matrix to define the damping contribution from this term. If the element contains more than one material in ABAQUS/Explicit, the mass average value of is used to multiply the element’s lumped mass matrix to define the damping contribution from this term. αR has unit of (1/time).

Stiffness proportional damping The βR factor introduces damping proportional to the strain rate, which can be thought of as damping associated with the material itself. βR defines damping proportional to the elastic material stiffness. Since the model may have quite general nonlinear response, the concept of “stiffness proportional damping” must be generalized, since it is possible for the tangent stiffness matrix to have negative eigenvalues (which would imply negative damping). To overcome this problem, βR is interpreted as defining viscous material damping in ABAQUS, which creates an additional “damping stress”, σd , proportional to the total strain rate: σd = βR Del , ˙

(D.16)

where ˙ is the strain rate. For hyperelastic and hyperfoam material’s Del is defined as the elastic stiffness in the strain-free state. For all other linear elastic materials in ABAQUS/Standard and all other materials in ABAQUS/Explicit, Del is the material’s current elastic stiffness. Del will be calculated based on the current temperature during the analysis. This damping stress is added to the stress caused by the constitutive response at the integration point when the dynamic equilibrium equations are formed, but it is not included in the stress output. As a result, damping can be introduced for any nonlinear case and provides standard Rayleigh damping for linear cases; for a linear case stiffness proportional damping is exactly the same as defining a damping matrix equal to βR times the (elastic) material stiffness matrix. Other contributions to the stiffness matrix (e.g., hourglass, transverse shear, and drill stiffnesses) are not included when computing stiffness proportional damping. βR has units of (time).

D.4.2

Artificial damping in explicit dynamic analysis

Rayleigh damping is meant to reflect physical damping in the actual material. In ABAQUS/Explicit a small amount of numerical damping is introduced by default in the form of bulk viscosity to control high frequency oscillations, as mentioned before.

Bibliography [1] ABAQUS, Getting Stated with ABAQUS, v6.6 [2] ABAQUS, ABAQUS/CAE User’s Manual, v6.6 [3] ABAQUS, Analysis User’s Manual, v6.6 [4] E. Paffumi, Report on dynamic structural integrity analysis of nuclear waste packages, IE, DG-JRC, EC, Petten, November 2005 (internal document: NSU/EP/200511401) [5] G.R. Johnson and W.H. Cook, (1983). A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. Proc. 7th Int. Symp. Ballistics, 541–547. [6] G.R. Johnson and W.H. Cook, (1985). Fracture characteristics of three metals subjected to various strains, strain rates, temepratures and pressures. Engineering Fracture Mechanics, 21(1), 31–48. [7] GNS, Technical report on Castor’s material priperties, Nr. E 2005/0085, 22.08.2005 [8] GNS, Technical report on bar’s material priperties, Nr. E 2006/0080, 30.08.2006

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European Commission EUR 22470 EN - DG JRC - Institute for Energy NUMERICAL SIMULATION OF THE ONE METER DROP TEST ON A BAR FOR THE CASTOR CASK Authors: Nikola Jakˇsić Karl-Fredrik Nilsson Luxembourg: Office for Official Publications of the European Communities 2007 - 129 pp. - 21 x 29.7 cm EUR - Scientific and Technical Research Series; ISSN 1018-5593

Abstract The report presents the numerical analysis of two one meter drop tests of a single ductile cast iron cask on a steel bar. The cask comes from the CASTOR family with machined cooling fins in a region where impact occurs. In the first test, the impact is on the cask’s cooling fins whereas in the second test the impact is in an area where the ribs have been locally machined away. The numerical analysis is based on explicit dynamic analysis using the commercial finite element code ABAQUS extended with Python scripts to allow a parametric description of the problem. The analysis consists of two parts: a “blind-analysis” with assumed model parameters and a sensitivity analysis. The “blind-analysis” (basic model without any knowledge of test results) was performed for the two drop tests. The overall behavior of the model is qualitatively very similar to what was observed during the experiments. A longer impact duration, (between 3 ms to 5 ms) when the cask is dropped on the fins in comparison to the flat target, is observed in both the analysis and the test. The reaction force at the bar’s bottom surface from the model is qualitatively similar to that from the test. The peak force value is however overestimated by about 35%. On the contrary the measured strains inside the cask above the impact area are underestimated for the similar amount. Latter is mainly attributed to the fact that the lid wasn’t included in this version of the model. The maximum strain is about 20% higher for the flat impact area in both simulation and experiment. The sensitivity analysis was performed to study the influence of parameters which either cannot be or were not defined directly from the experimental data, such as the friction coefficient, or which are linked to the FE numerical procedures, like the bulk viscosity. Based on results, recommendations for an optimised set of parameters values are given.

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