... who always encouraged me in my life. ...... behavior bring in the application of magnesium alloys, there is a growing interest in usage of this material in the ...
A CONTINUUM DISLOCATION DYNAMICS FRAMEWORK FOR PLASTICITY OF POLYCRYSTALLINE MATERIALS
By HESAM ALDIN ASKARI
A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY
WASHINGTON STATE UNIVERSITY School of Mechanical and Materials Engineering May 2014
To the Faculty of Washington State University:
The members of the Committee appointed to examine the dissertation of HESAM ALDIN ASKARI find it satisfactory and recommend that it be accepted.
Hussein M. Zbib,, Ph.D., Chair
David Field Ph.D.
Sinisa Mesarovic, Ph.D.
Dongsheng Li, Ph.D.
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ACKNOWLEDGMENT
I would like to express my sincere thanks to my advisor, Prof. Hussein M. Zbib, for all his guidance and endless support. I am honored to have the opportunity to be his student and I am indebted to him for all he taught me from research skills to professional development. I would also like to thank Prof. David F. Field for his thoughtful support and encouragement. His valuable advices have benefited me throughout this research. I am also thankful to Prof. Sinisia Mesarovic for the valuable comments about my work and for all I have learnt in his classes. Many thanks to Dr. Dongsheng Li for his great support throughout this research. A very special thank you to my collaborators Prof. David P. Bahr and Michal Maughan at Purdue University and Prof. Antonios Kontsos and Dr. Kavan Hazeli at Drexel University for all their good work and great comments. I would also like to thank Dr. Ioannis Mastorakos who introduced me to my research and for his never ending support. I also like to thank Dr. Iman Salehinia for his helpful discussions and comments. Last, but by no means least, my thanks go to my wonderful wife, Niaz Abdolrahim, my amazing parents, Shahnaz Ejazi and Mohammad Askari, and all my lovely friends who always encouraged me in my life.
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A CONTINUUM DISLOCATION DYNAMICS FRAMEWORK FOR PLASTICITY OF POLYCRYSTALLINE MATERIALS
Abstract By Hesam Aldin Askari, Ph.D. Washington State University May 2014
Chair: Hussein M. Zbib
The objective of this research is to investigate the mechanical response of polycrystals in different settings to identify the mechanisms that give rise to specific response observed in the deformation process. Particularly the large deformation of magnesium alloys and yield properties of copper in small scales are investigated. We develop a continuum dislocation dynamics framework based on dislocation mechanisms and interaction laws and implement this formulation in a viscoplastic self-consistent scheme to obtain the mechanical response in a polycrystalline system. The versatility of this method allows various applications in the study of problems involving large deformation, study of microstructure and its evolution, superplasticity, study of size effect in polycrystals and stochastic plasticity. The findings from the numerical solution are compared to the experimental results to validate the simulation results. We apply this framework to study the deformation mechanisms in magnesium alloys at moderate to fast strain rates and room temperature to 450 °C. Experiments for the same range of
iv
strain rates and temperatures were carried out to obtain the mechanical and material properties, and to compare with the numerical results. The numerical approach for magnesium is divided into four main steps; 1) room temperature unidirectional loading 2) high temperature deformation without grain boundary sliding 3) high temperature with grain boundary sliding mechanism 4) room temperature cyclic loading. We demonstrate the capability of our modeling approach in prediction of mechanical properties and texture evolution and discuss the improvement obtained by using the continuum dislocation dynamics method. The framework was also applied to nano-sized copper polycrystals to study the yield properties at small scales and address the observed yield scatter. By combining our developed method with a Monte Carlo simulation approach, the stochastic plasticity at small length scales was studied and the sources of the uncertainty in the polycrystalline structure are discussed. Our results suggest that the stochastic response is mainly because of a) stochastic plasticity due to dislocation substructure inside crystals and b) the microstructure of the polycrystalline material. The extent of the uncertainty is correlated to the “effective cell length” in the sampling procedure whether using simulations and experimental approach.
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TABLE OF CONTENTS
ACKNOWLEDGMENT............................................................................................................... III
ABSTRACT .................................................................................................................................. IV
LIST OF TABLES ......................................................................................................................... X
LIST OF FIGURES ...................................................................................................................... XI
DEDICATION ........................................................................................................................... XVI
CHAPTER ONE: INTRODUCTION ............................................................................................. 1
CHAPTER TWO: A STUDY OF THE HOT AND COLD DEFORMATION OF TWIN ROLL CAST MAGNESIUM ALLOY AZ31 ................................................................................ 9 1. Abstract ....................................................................................................................................... 9 2. Introduction ............................................................................................................................... 10 3. Experimental Procedure ............................................................................................................ 13 4. Viscoplastic Self-Consistent Model .......................................................................................... 15 5. Continuum Dislocation Dynamics Coupled with Viscoplastic Self-Consistent Model ........... 19 6. Results and Discussion ............................................................................................................. 23 6.1 Experimental Results........................................................................................................... 23 vi
6.2 VPSC Simulation results ..................................................................................................... 33 6.3 CDD-VPSC Simulation results ........................................................................................... 42 7. Concluding Remarks ................................................................................................................. 46
CHAPTER THREE: MODELING AND SIMULATION OF FLOW STRESS PROPERTIES OF AZ31 MAGNESIUM ALLOY AT HIGH TEMPERATURE ....................... 48 1. Introduction ............................................................................................................................... 49 2. Modeling approach ................................................................................................................... 51 2.1 Modeling dislocation slip .................................................................................................... 51 2.2 Grain Boundary Sliding (GBS) model ................................................................................ 53 2.3 Viscoplastic self consistent scheme .................................................................................... 56 3. Experimental approach ............................................................................................................. 58 4. Results and discussion .............................................................................................................. 58
CHAPTER FOUR: MICROSTRUCTURE-SENSITIVE INVESTIGATION OF MAGNESIUM ALLOY FATIGUE ............................................................................................. 72 1. Abstract ..................................................................................................................................... 72 2. Introduction ............................................................................................................................... 73 3. Methodology ............................................................................................................................. 74 3.1 Experimental approach ........................................................................................................ 74 3.2 Modeling approach .............................................................................................................. 76 vii
4. Results and discussion .............................................................................................................. 82 4.1 Monotonic behavior ............................................................................................................ 82 4.2 Cyclic behavior ................................................................................................................... 83 4.3 Crystal plasticity modeling.................................................................................................. 85 5. Concluding remarks .................................................................................................................. 92
CHAPTER FIVE: STOCHASTIC CRYSTAL PLASTICITY FOR DEFORMATION IN MICRO-SCALE POLYCRYSTALLINE MATERIALS ............................................................. 93 1. ABSTRACT .............................................................................................................................. 93 2. Introduction ............................................................................................................................... 94 3. Stochastic crystal plasticity model ............................................................................................ 98 4. Experimental Approach .......................................................................................................... 102 5. Results ..................................................................................................................................... 106 5.1 Modeling Results............................................................................................................... 106 5.2 Experimental Results......................................................................................................... 109 6. Discussion ............................................................................................................................... 112 7. Conclusions ............................................................................................................................. 116
CHAPTER SIX: SUMMARY AND FUTURE WORK ............................................................. 119 Appendix A : Multiscale Modeling of Inclusions and Precipitation Hardening in Metal Matrix Composites: Application to Advanced High Strength Steels ......................................... 124 viii
A.1 ABSRTRACT ...................................................................................................................... 124 A.2 INTRODUCTION ............................................................................................................... 125 A.3 THEORY AND METHODS ............................................................................................... 129 A.3.1 Mulitscale Dislocation Dynamics Framework .................................................................. 129 A.3.2 Precipitate Effects ............................................................................................................. 133 A.3.3 Eigenstrain Inclusion Method (EIM) ................................................................................ 133 A.4 PROBLEM SET................................................................................................................... 135 A.5 RESULTS AND DISCUSSION .......................................................................................... 139 A.5.1 Comparison of the EIM results to FEM ............................................................................ 139 A.5.2 Atomistic Simulation of dislocation – precipitate interaction .......................................... 140 A.5.3 Dislocation Dynamics Simulation Results ........................................................................ 144 A.6 CONCLUSION ................................................................................................................... 150 Appendix B : Calculation of fourth-order interaction tensor
................................................ 152
REFERENCES ........................................................................................................................... 153
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LIST OF TABLES
Table 2.1 - CRSS and hardening responses of the four deformation mechanisms as a function of temperature. ..................................................................................................34 Table 2.2 - CDDVP parameters for AZ31 at room temperature. Note that α2 = α3 =1, α5 =0.002 and Rc=25b for all slip systems. Values of K are reported from Raeisinia et al. [63] ............................................................................................................44 Table 3.1 – List of parameters used in the simulations .....................................................62 Table 5.1 CDD-VPSC parameters for AZ31. Values of K are reported from Raeisinia et al.[63] .............................................................................................................87 Table 5.1 – Parameters from cumulative fraction data calculation of copper.................101
x
LIST OF FIGURES
Figure 2.1 - a) Pole figures showing initial texture of the as received material along the normal direction. b) Orientation map showing the initial microstructure .........24 Figure 2.2 - Effect of temperature on the material response in RD at constant strain rate of 10-2 (s-1). Breakage marked by x ...................................................................25 Figure 2.3 - True stress vs. true strain response at room temperature for rolling and transverse directions at different strain rates. Breakage shown by marker x. .............27 Figure 2.4 - True stress vs. True strain Curve in Rolling Direction at 410°C ..................29 Figure 2.5 - a) Yield stress(0.2% offset) vs. strain rate; b) calculated strain rate sensitivity parameter plotted in solid lines and percent elongation plotted in dashed line .........................................................................................................................31 Figure 2.6 - Calculation of activation energies of deformation in different temperatures at strain rate of 0.01(s-1) ...............................................................................32 Figure 2.7 - Critical Resolved Shear Stress of the deformation systems with respect to the temperature. Values for 20°C-200°C were used from Jain and Agnew[52]. ........................................................................................................................35 Figure 2.8 - a) Model results for different formulations compared to the experimental results shown by markers. b) Activity of deformation systems predicted from the model. ..................................................................................................37 Figure 2.9 - Model results at 370°C (a) and 450°C(c) for different strain rates compared with the experimental results shown by markers. Percent activity of deformation systems obtained from the model at 370°C (b)and 450°C (d) ......................39
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Figure 2.10 - Pole figures obtained from the model (left column) and experimental results (right column) at a) room temperature, b) 450⁰C with 10-2 (s1
) and c) 450⁰C with 10-3 (s-1) ............................................................................................40
Figure 2.11 - Stress –strain curve predicted by the CDD-VPSC model compared to the experimental curve. Calculated dislocation densities are shown with open markers. ..............................................................................................................................45 Figure 2.12 - Comparison of the predicted basal pole figures obtained by VPSC (a) and CDD-VPSC (b) with the experimental results (c). The max. intensities are 6.69, 8.44, 9.96 respectively ..............................................................................................46 Figure 3.1 – Simplified grain structure representing the microstructure showing major sliding planes ...........................................................................................................55 Figure 3.2 - a) Orientation map showing the initial microstructure using OIM analysis software. b) Pole figures showing initial texture of the as received material along the normal direction using MTEX [90]. ....................................................59 Figure 3.3 – Experimentally obtained flow stress at different strain rates and temperatures .......................................................................................................................60 Figure 3.4 – comparison of experimental results and modeling results (a) Excluding GBS mechanism. (b) Including GBS mechanism. ...........................................64 Figure 3.5 – Percentage activity of GBS mechanism predicted by numerical model compared to the experimental results of Panicker et al. [16] ..................................66 Figure 3.6 – Predicted textures from the model compared to the experimental texture from interrupted tests at strain rate of 10-3 (s-1). ....................................................68
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Figure 4.1 - Specimens configuration of tested compression, tension and fatigue coupons ..............................................................................................................................75 Figure 4.2 - Measured stress-strain curve for compression and tension specimen cutting from different direction ..........................................................................................83 Figure 4.3- Asymmetric hysteresis loop for varying applied strain amplitudes of 0.25, 0.33, 0.42, 0.5 and 0.58 under R 1 loading parallel to the transverse direction for (a)1st cycle, (b) 2nd cycle, (c) Half-life (d) last loop before final failure. ................................................................................................................................85 Figure 4.4 - Measured and predicted stress-strain curve under fatigue (Compression-tension): first cycle for (a) 0.25% (b) 0.5 % (c) 0.58% and (d) 1% applied cyclic strain amplitudes. ........................................................................................88 Figure 4.5 - Comparison of experimental and modeling results for the first and second cycle (Strain amplitude of 0.5%) ...........................................................................90 Figure 4.6 - Relative activity of various deformation mechanisms during cyclic compression-tension loading in (a) first cycle (b) first and second cycle combined. ........91 Figure 5.1 - Relevant geometric parameters of an indent. ..............................................104 Figure 5.2 – (a) EBSD inverse pole figure map of the Cu2 sample, and (b) TEM micrograph of the Ti sample. [(a) is a placeholder until better image can be obtained. ...........................................................................................................................105 Figure 5.3 – Yield strength vs. effective cell length for 100 MC calculations showing (a) yield stress scatter and size effect for different grain sizes and (b) grain size compensated yield stress and coefficient of variation (CV) versus effective cell length. .........................................................................................................108
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Figure 5.4 – Yield data from the simulations showing the effect of microstructure and collective behavior from the effect of stochastic plasticity and microstructure. .......109 Figure 5.5 – Indentation hardness, elastic modulus and coefficient of variation of the indentation hardness for the three samples tested. .....................................................110 Figure 5.6 - Yield strength versus effective cell length for the Cu1, Cu2 and Ti samples. ............................................................................................................................111 Figure 5.7 – Three simulation data compared to experimental nanoindentation measurements. ..................................................................................................................114 Figure 5.8 – Yield strength coefficient of variation versus the ratio of effective length to grain diameter (a) plotted on linear axes and (b) plotted with logarithmic axes. .................................................................................................................................115 Figure A. 1 - TEM image of the deformed structure showing dislocation pinning ........136 Figure A. 2 - Interaction of the dislocations with particles. (a) intermediate configuration before yielding. (b) An instance of final Configuration after yielding. (c) formation of superdislocations as also reported in [188] ............................138 Figure A.3 - Comparison of the EIM and FEM results for disturbance stress in a loaded matrix containing a 10nm spherical TiC particle. ................................................139 Figure A.4 - (a) Initial relaxed structure (left), dislocation pinned by particle (middle) and finally clearing the particle(right). (b) Front view of the dislocation after clearing the particle showing cross slip mechanism. (Coloring indicates common neighbor analysis parameter.) ...........................................................................142 Figure A.5 - Stress strain curve obtained by MM for different particle sizes.................143
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Figure A.6 - Stress vs. Strain curve for different volume fraction of precipitates: (a) D = 8nm, ....................................................................................................................145 Figure A.7 - Model results for constant volume fraction f=6% and low dislocation density ..............................................................................................................................147 Figure A.8 - Model results for different volume fractions and dislocation densities.( α=0.571, ρ=6.8nm, n=1.322) ..........................................................................148 Figure A.9 - (a) DD simulation box containing random distribution of cluster of precipitates with fixed particle size D but with different cluster size of radius R. (b) Comparison of stress and strain curve to the case of random distribution and various cluster sizes. ........................................................................................................149
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Dedication
To my parents, Mohammad and Parvaneh, who always put their children first
To my wife Niaz for being the cornerstone of my life
To my brothers Jafar and Alireza
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CHAPTER ONE: Introduction The plastic deformation in polycrystalline materials can generally be attributed to two distinct
categories
of
mechanisms;
intragranular
mechanisms
including
dislocation
glide/twinning and intergranular mechanisms such as grain boundary sliding. Though the deformation conditions and material properties can lead to complete suppression of one category, in certain conditions both of these mechanisms can be active. Common examples include deformation of materials with ultrafine grains or deformation at high temperatures relative to the material melting point. Though the share of each category will be determined by test conditions such as temperature or strain rate, both of these mechanisms can contribute to the plastic deformation. This phenomenon is particularly important in superplastic materials such as magnesium alloys where a change in the strain rate or deformation temperature can change the activity of each set of mechanisms mentioned above. Magnesium alloys demonstrate a transition between brittle rate-insensitive material at room temperature to a more ductile and rate-sensitive material at higher temperatures. Also magnesium crystals have a strong anisotropy due to their hexagonal close-packed (HCP) crystal structure and extensive twinning capability that intensifies the mechanical anisotropy of the polycrystalline material when texture is not random. Most of the manufacturing techniques for creating sheets or extruded products lead to a heterogeneous texture that intensifies anisotropy of mechanical properties. Despite the complexities that these behavior bring in the application of magnesium alloys, there is a growing interest in usage of this material in the applications that involve weight reduction and structural optimization. The higher strength to weight ratio in magnesium alloys can provide weight reduction of up to 50% [1]. Due to emergence of new forming processes called quick plastic forming, the manufacturing process
1
can be easier and more economical. Since the forming processes are often carried out at different temperatures and include various strain rates in order to form complex geometries in the final product, development of a temperature and strain-rate dependent material model that can precisely predict the flow properties of the material and can include the effects of texture is particularly important. When embedded in a finite element code, such model can predict the mechanical properties of each element and enable modeling of forming processes including material’s viscoplastic response and texture. The purpose of this study is to build such model that is capable of predicting the material behavior at a wide range of temperatures and strain rates and apply the numerical method to the deformation of magnesium alloys. Inspired by discreet dislocation dynamics (DDD) formulation [2] and observations of dislocation mechanics, we develop a continuum dislocation dynamics (CDD) based on the same approach. The main difference between the two methods is that we do not follow individual dislocations in the crystal during the course of deformation rather we obtain the average dislocation velocity in the slip planes given the loading conditions and dislocation densities in the crystal. Then we investigate the evolution of dislocation densities and threshold stresses based on the dislocation interaction laws and average dislocation velocities. This approach leads to very fast calculation time compared to DDD that enables high computational efficiency when used in a polycrystalline solution scheme. In the DDD method the dislocation are represented by piecewise continuous array of segmented lines whose dynamics is governed by a Newtonian type equation of motion with a driving force comprising of external and internal forces. The governing equation that describes the motion for each dislocation segment is given by
2
(1)
Here F is the force acting on the dislocation segment, m* is the effective mass per unit dislocation, M is the dislocation mobility which could depend on both temperature T and pressure p and some crystal structures and vi is dislocation velocity with dot over the symbol indicating derivative with respect to time. Therefore, the dislocation lines move in 3-D space and the rate of the swept area by the dislocation movement in each slip plane is the shear rate in that slip system. The total strain rate tensor ε ij is obtained by the summation of shear rates on each slip system as given in Eq. (2):
ε ij m sij γ s
(2)
m sij (n i b j n j b i )/2
where m sij is the Schmid orientation tensor for slip system (s) and γ s is the shear rate in slip system (s) defined by unit normal vector (n) and unit Burgers vector (b). The glide force in Eq.(1) can be due to multiple sources based on the type of problems. While forces such as Peirels forces, dislocation-dislocation forces and dislocation self forces are included in all the solutions, additional forces due to dislocation-obstacle interactions and forces from boundary surfaces on the dislocations can be included in the solution as well. For instance the application of this method in studying the hardening behavior in advanced high strength steels when dislocations are interacting with ceramic particles are presented in Appendix A. In the CDD approach the main difference is that we do not keep track of individual dislocation lines in the calculations and the dislocations are represented by their average dislocation densities in each slip system. The average dislocation velocity in each slip system is
3
obtained by a power law and the shear rate in the system follows an Orowan type relationship as in Eq.(3):
γ s ρ sm b s v s τs v v0 s τ th
1/
s
(3)
Sign( τ s )
s
s
where ρ sm , b and v represent mobile dislocation density, Burgers vector and velocity of mobile dislocations respectively. This velocity is defined in terms of resolved shear stress ( τ s ), threshold stress for dislocation glide ( τ sth ), strain rate sensitivity for dislocation glide ( ) and a scalar constant ( v 0 ). The strain rate in the crystal is then calculated according to Eq.(2). While evolution of dislocation densities and the resistances to dislocation movement is explicitly obtained in the DDD method, in the CDD method additional relationships for the evolution of these variables are needed. The reason is that the geometry and location of individual dislocations are not known and therefore these values cannot be calculated directly. We assume the threshold stresses in Eq.(3) is a function of an initial threshold stress τ 0 , total dislocation density in the slip planes ρ total , dislocation interaction constants defined as a matrix
Ω sr and grain size d. The rate of mobile and immobile dislocation densities is also represented as the sum of the dislocation reaction rates based on the type of dislocation mechanism or reactions. We represent this total rate in terms of dislocation generation rate, annihilation rate and cross slip (for mobile dislocations only). Therefore the evolution laws have the general forms presented in Eq.(4).
4
τ sth f ( τ s0 , ρ stotal , Ω sr , d) ρ sm ρ sm,Growth rate ρ sm,Anihilation rate ρ sNet dislocation density gained or lost due to cross slip
(4)
ρ si ρ si, Growth rate ρ si, Anihilation rate
The formulations discussed above are able to calculate the strain rate and stresses in a single crystal. But for the case of polycrystals the interactions between the grains and the variations in the strain rate and stresses among grains should be considered in the calculations. While simplified assumptions such as relaxed boundary assumption - Sachs model where stress is assumed uniform in all grains [3] - or fixed boundary assumption – Taylor model where strain is assumed uniform in all grains [4] – bring in the additional relationships required to solve the polycrystal system, they tend to form lower bound and upper bound to the actual solution of the polycrystal system. Other models that allow for some interaction condition between the grains have been proposed in various studies. In this study, we use a self-consistent viscoplastic formulation to calculate grain interactions and bring in the effects of texture in our numerical approach. In this sense, each grain is treated as an inclusion inside an otherwise homogeneous media and the interactions between each grain and the surrounding media is calculated according to the Eshelby’s equivalent inclusion method [5]. Details on the implementation of this solution method are outlined in Chapter 2. The application of this method to magnesium alloys at different deformation conditions are presented in Chapters 2-4. In Chapter 1 we apply the VPSC method to study the deformation of AZ31 at room temperatures and at high temperatures. Also the experimental studies performed at room temperature to 450 °C for strain rates ranging from 10-4 to 0.1 (s-1) are presented and the deformation mechanisms present in each range are discussed. Calculations of properties such as strain rate sensitivity parameter and activation energies are presented in this
5
chapter. The electron backscatter diffraction data (EBSD) obtained by our collaborator John Young at WSU is presented to discuss important features of the microstructure before and after deformation. The data from EBSD analysis are used in the modeling method to represent the initial microstructure. The mechanical response and texture outputs from the model are compared to the experimental data and a complete discussion on the strength and weakness of VPSC is presented. Also an attempt to apply this solution method to deformation at high temperature is presented in this chapter and the limitations of the model are discussed. The implementation of the CDD-VPSC model is presented in Chapters 2 and 3. In Chapter 2, we construct the details of the CDD approach and the differences and connections between the two methods are discussed in detail. Then we use the CDD-VPSC model to study the deformation of magnesium alloys at room temperature. The mechanical properties obtained from the model and texture predictions are compared to the experimentally obtained data. In Chapter 3 we use a combination of dislocation mechanisms and grain boundary sliding (GBS) mechanism to study the deformation properties of magnesium alloys at high temperature. In our numerical approach, we define pseudo slip systems to model the shear rates due to the GBS mechanism. This approach treats the GBS systems and the slip systems in a similar manner but evidently with different parameters. Therefore, an integrated framework for deformation at high temperature including GBS is obtained. We apply this numerical solution method to study deformation of AZ31 magnesium alloy at high temperature and we compare the findings from the model to the reported results in the literature and to our experimental approach. Detailed discussions on the improvements obtained by including GBS mechanism for prediction of mechanical properties and texture evolution are presented in this chapter.
6
The CDD-VPSC method is capable of solving various loading conditions. Chapters 2 and 3 contain simulations with unidirectional loading. In Chapter 4 we apply the method to study the cyclic loading of AZ31 magnesium alloy at room temperature. At these test conditions, twinning and detwinning are known to be a major contributor to the deformation in magnesium alloys. In this chapter we present a model for twinning and detwinning. The main distinction in our model is that we allow twinning recovery in a twin system if proper resolved shear stress is active on the twin system and twin volume fraction is positive in that system. We apply this method to the experimental cyclic data on AZ31 magnesium alloy provided by our collaborators Dr. Hazeli and Prof. Kontsos at Drexel University. We show in this chapter that our simplified model is capable of predicting the stress-strain response in the cyclic loading of magnesium alloy in a variety of strain amplitudes. Our model also captures the anomalous difference between the first and second cycle which is attributed to the twinning and detwinning mechanism. CDD-VPSC is a general solution framework that can be applied to study any crystal structure. While the focus in the first four chapters are on HCP crystal structures, the method is applicable to other crystal structures by defining proper base vectors and slip systems that define the crystal. In Chapter 5 we apply this solution method to the deformation of face centered cubic (FCC) materials. We add a stochastic crystal plasticity formulation based on Monte Carlo approach in our CDD-VPSC approach and study the stochastic response of polycrystalline FCC copper at small length scales. Using previously published data on copper using DDD method and also experimental data provided by our collaborators Michael Maughan and Prof. Bahr at Purdue University, we study the deformation of polycrystalline copper at small length scales by numerical and experimental approach. Our emphasis in this chapter is to utilize these data to
7
identify and explain the source of the observed stochastic response in the polycrystalline materials at small length scales. Ancillary results are presented in Appendix A where the strengthening effect of precipitates in BCC metals is investigated within a multiscale approach that utilizes models at various length scales, namely, Molecular Mechanics (MM), Discrete Dislocation Dynamics (DDD), and an Equivalent Inclusion Method (EIM). Through this multiscale approach, we model the precipitates by considering their interactions with the dislocations through their long range stress field as well as their short range resistance to dislocation glide. As an application to this method, the mechanical behavior of Advanced High Strength Steel (AHSS) is investigated and the results are then compared to the experimental data published in previous study. Our results show that the finely dispersive precipitates can strengthen the material by temporarily pinning the dislocations until a threshold shear stress is reached. The DDD results show that strengthening due to nano-sized particles is a function of the density and size of the precipitates. This size effect is then explained using a mechanistic model developed based on dislocationparticle interaction. We also study the effect of agglomerations on the strengthening extent of the precipitates using DDD calculations. The work presented in this dissertation covers computational methods applicable to major crystal structures for metallic materials from atomistic simulations to polycrystalline continuum level.
8
(Philosophical Magazine, 2014)
CHAPTER TWO: A study of the hot and cold deformation of twin roll cast magnesium alloy AZ31 Hesam Askari1, John Young1, David Field1, Ghassan Kridli2, Dongsheng Li3, Hussein Zbib1 1
School of Mechanical and Materials Engineering, Washington State University, Pullman, WA, 99164-2920 2
3
Department of Mechanical Engineering, Texas A&M University of Qatar, Doha, Qatar
Pacific Northwest National Laboratory, Computational Sciences and Mathematics Division, P.O. Box 999, Richland, WA 99352, USA
1. Abstract Recent advances in Twin Roll Casting technology of magnesium have demonstrated the feasibility of producing magnesium sheets in the range of widths needed for automotive applications. However, challenges in the areas of manufacturing, material processing and modeling need to be resolved in order to fully utilize magnesium alloys. Despite the limited formability of magnesium alloys at room temperature due to their hexagonal close-packed crystalline structure, studies have shown that the formability of magnesium alloys can be significantly improved by processing the material at elevated temperatures and by modifying their microstructure to increase ductility. Such improvements can potentially be achieved by processes such as superplastic forming along with manufacturing techniques such as TRC. In this work we investigate the superplastic behavior of twin-roll cast AZ31 through mechanical testing, microstructure characterization and computational modeling. 9
Validated by the
experimental results, a novel continuum dislocation dynamics (CDD) based constitutive model is developed and coupled with VPSC to simulate the deformation behavior. The model integrates the main microstructural features such as dislocation densities, grain shape and grain orientations within a self-consistent viscoplasticity theory with internal variables. Simulations of the deformation process at room temperature show large activity of the basal and prismatic systems at the early stages of deformation and increasing activity of pyramidal systems due to twinning at the later stages. The predicted texture at room temperature is consistent with the experimental results. Using appropriate model parameters at high temperatures, the stress-strain relationship can be described accurately over the range of low strain rates.
2. Introduction The need to produce fuel efficient vehicles, reducing emissions and lower production and assembly costs has led the automotive industry to pursue the use of the lightweight materials. One of the important candidates for this purpose are magnesium alloys due to their high specific strength, good castability and damping abilities and abundance of magnesium in nature [6]. Despite this growing interest, low ductility and formability of these alloys at low temperatures has been an obstacle. The ductility of these alloys is inadequate due to their limited number of slip systems in the hexagonal close packed (HCP) crystalline structure. Although twinning acts as an additional deformation mechanism, the total strain due to twinning is minor [7]. These facts lead to brittle failure and poor performance at low temperatures for magnesium alloys. For this reason usage of these materials has been limited in the past to the parts manufactured by diecasting methods; however the majority of parts in automotive applications are manufactured from sheet. The use of magnesium alloys through the hot superplastic forming (SPF) of sheets is
10
a promising method to achieve this goal as it will result in good formability at high temperatures and considerable weight saving in the final product. Magnesium alloy sheets used in SPF should have certain qualities. Most importantly small equiaxed grains, since high tensile ductility associated with superplasticity occurs when the grain size is typically less than 10 microns [8-11]. Conventional methods of thermomechanical processing of wrought material to obtain the required microstructure for SFP such as hot or cold rolling and asymmetric rolling are expensive. They typically lead to a pronounced basal texture [12] and produce cracks and non-uniform surfaces. Therefore they are costly in terms of processing and energy consumption. Conversely Twin Roll Casting (TRC) has become increasingly popular recently. Since this process is simply the solidification of molten material with high cooling rates and simultaneous rolling of the solidified material, beneficial features such as small grains, formation of non-equilibrium/metastable phases [13] and equiaxed microstructure can be obtained while keeping the cost and energy consumption low. With further optimization of the TRC method, magnesium alloy sheets can be produced for use with SPF. There is an extensive amount of ongoing research to achieve this goal. The aim of this study is to understand the thermomechanical properties of the as-cast AZ31 Magnesium alloy sheets produced by TRC through mechanical testing, microstructure characterization and computational modeling. We use unidirectional tensile tests to investigate the material response at both room and high temperatures. Microstructural characterization is performed to reveal the features of the as-received material and the fracture surface of the tensile test specimens. These data are then used in a polycrystalline material model to investigate the deformation mechanisms and the effect of microstructure on the response of the material.
11
This paper provides experimental and modeling data to represent the mechanical properties and microstructural characteristics of the deformation process over a wide range of strain rates and high temperature and can be used in different types of models. We also introduce a new continuum dislocation dynamics formulation for the polycrystalline self-consistent model. In the polycrystalline model we assume that the deformation mechanisms consist of both crystallographic slip and twinning and the actual solution to the boundary value problem depends on the shape and orientation of the crystallites which evolve during the deformation process. Most of the polycrystalline models invoke single crystal constitutive equations and relate the deformation behavior to the overall aggregate response. In principle, rigorous solutions to this type of problem can be pursued. However the most productive approach is through the models which identify the polycrystal response with appropriate averaging of the constituent crystallites [14]. These models are used to simulate the stress–strain behavior and texture evolution in different crystalline materials under different loads. The most established models are the Taylor model [4] which assumes homogenous strain for all grains, and the Sachs model [3] which assumes homogenous stress. These are upper and lower bound models respectively. Other models include intermediate models [15-18] and self consistent models [19-21]. The intermediate models utilize a new approach for a more realistic description of the deformation. Self-consistent polycrystal plasticity models consider the interaction between a given grain and a ‘‘homogenized matrix” representing the rest of the material. In this study we use the viscoplastic self consistent model (VPSC) developed by Tome and Lebensohn [19, 20]. This model has the advantage of considering the effect of grain shape in the calculations by assuming each grain is an ellipsoidal inclusion surrounded by a homogeneous matrix. The model then uses the response of the single crystal to solve for local quantities and homogenizes these local results to calculate
12
the response of the polycrystalline material. Another beneficial feature of this model is the prediction of the final texture after deformation and calculation of the contribution of different deformation mechanisms during the process.
However, existing VPSC models use
phenomenological laws that simply relate hardening to the accumulated plastic strain. Such models can be adjusted to fit experimental data but do not relate directly to the evolving microstructural features such as dislocations. In this paper we develop an internal state variable model based on the dislocation activities on each of the slip systems in HCP crystals along the lines discussed by Li et al. [22] for BCC crystals. Particularly we assume that each slip system in each grain has a net total dislocation density consisting of mobile and immobile dislocations. The evolution of the dislocation densities is a natural consequence of dislocation slip activities as well as dislocation reactions. We model the dislocation slip by calculating dislocation glide velocity at each time step within each grain and then we update the dislocation densities accordingly. Since strength is directly related to the dislocation substructures, the evolution of slip resistances can be captured by evolving the dislocation densities. There are a number of dislocation models developed for various applications, see for example [23, 24], [25-27] [25, 2830]. Here we develop a rigorous continuum model based on the dislocation mechanics that can be explicitly identified and quantified in discrete dislocation dynamics. The results obtained from the model are compared to the experimental results and the deformation mechanisms are discussed in detail.
3. Experimental Procedure Commercially available as-cast TRC AZ31 sheets with thickness of 4.3 mm were tested and characterized. The chemical composition of this material is 2.97% Al, 0.82%Zn, 0.28%Mn,
13
0.012% Si, 0.0036% Fe, 0.0004% Ni, 0.003% Cu, 0.0008% Be, and balance Mg (wt. percentages). Tensile test specimens according to the ASTM E2442 [31] were prepared. The specimens having gauge section dimensions of 25×6×4 (mm) were cut from sheets in directions parallel and perpendicular to the rolling direction by water-jet cutting process. This procedure is precise and does not cause an increase in the temperature of the sheet
and therefore no
microstructural changes are expected to occur. A displacement control load frame equipped with a three zone furnace was used to carry out tensile tests. Unidirectional tensile tests were carried out over constant true strain rates ranging from 10-4 to 10-1 (s-1) and temperatures ranging from room temperature to 450⁰C. The temperature of the specimen was measured separately by means of two thermocouples attached next to it. The furnace was preheated to the test temperature before mounting each specimen and 10 minutes reheating time with additional 10 minutes hold time was used to obtain a uniform temperature state in the test piece. No significant grain growth was observed due to this heating time. Microstructural characterization of the base metal and the post tensile test specimens was performed using electron backscatter diffraction (EBSD) with a field emission scanning electron microscope (FESEM). Base metal specimens were cut from the plate using a water jet and then cold mounted in epoxy resin for observation along the normal direction. For characterization of the post tensile test specimens, a sample was cut from the gauge of the failed tensile specimen. These specimens were mounted in epoxy for observation of the normal direction. EBSD scans of the failed tensile specimens were run within 1mm of the fracture surface. The specimens were wet sanded from 240-1200 grit and then polished using 1 μm and then 0.25 μm diamond paste and an alcohol based lubricant. Sequential grinding and polishing steps continued until no evidence of the previous step could be observed using an optical microscope. The final polishing
14
step involved 0.02 μm colloidal silicon on a MultiTex™ cloth. This final step continued until high quality Kikuchi patterns could be observed by EBSD. EBSD data were collected using an accelerating voltage of 20 kV and a probe current of ~10 mA. Scans were run at a magnification of 250 times and a step size of 2 μm. A filter removing suspicious data points with confidence index less than 0.1was the only cleanup procedure applied in this study.
4. Viscoplastic Self-Consistent Model In order to model the deformation process of the material under tension, we adopt the VPSC formulation developed by Tome and Lebehnson [19]. The VPSC model can simulate the plastic deformation of polycrystalline aggregates subjected to external strains and stresses and was originally developed for applications involving low-symmetry materials (hexagonal, trigonal, orthorhombic,…) while it also performs well on cubic materials. Deformation mechanisms in this model include slip and twinningwhich accounts for grain interaction effects. As a result, it predicts the evolution of hardening behavior and texture development associated with plastic deformation gradients. In VPSC, grains in the polycrystal are represented by their crystallographic orientations and their corresponding weight as obtained from the EBSD analysis. The weight of each orientation represents the volume fraction of grains aligned in that direction. Moreover the grains are treated as viscoplastic inclusions embedded inside a fully anisotropic viscoplastic matrix. The response of each grain is derived from a rate dependent crystal plasticity formulation including slip and twinning mechanisms and the activation of these systems is determined according to the Schmid’s law. Note that since twinning modes are polar mechanisms, they require a positive resolved shear stress in the twinning direction in order to be activated. Although the general form 15
of the formulation for twinning is the same as in slip systems, the activity of the twinning system is forced to be zero if negative resolved shear stress is obtained in that specific system. Also reorientation of grains due to twinning is incorporated in the formulations based on Predominant Twin Reorientation (PTR) scheme introduced by Tome et. al. [32]. The velocity gradient tensor L in the crystal axis in a grain in the intermediate configuration is represented by:
L γ s bs ns
(1)
s
where γ s , bs and ns are shear rate, slip/twin direction (unit vector in the direction of the Burgers vector) and unit vector normal to the slip/twin system s, respectively. The velocity gradient is further decomposed into symmetric and skew symmetric parts, leading to
D γ s m s s
W γ s q s s
1 m bs ns bs ns 2 1 q s bs ns bs ns 2 s
(2)
where D is the rate of deformation and W is the plastic spin rate. Transformation of the D and W to the current configuration gives the quantities in the off crystal axis. In the VPSC model, the plastic shearing rate
τs m s : σ
, where
τs γ γ 0 s τ th s
σ
γ s
is assumed to have a power law dependence on the resolved shear stress
is the stress tensor, i.e.
1/m
(3)
16
Here γ o is reference strain rate τ sth is the threshold stress and m is the strain rate sensitivity parameter. The deformation rate at any point ( x ) in the crystal can be defined as follows. m s : σ (x) D(x) γ 0 m τs s th N
1/ m
s
(4)
where N is the total number of the slip and twin systems. The strain hardening is characterized by the evolution of the threshold stress τ sth through the relationship: τ sth h sr γ r
(5)
r
where the components of the matrix hsr are the strain hardening moduli. The model also allows the incorporation of latent hardening by defining coupling coefficients that represent the hardening due to formation of obstacles created by dislocation activity of system r on the dislocation activity of system s. The sum ranges over all slip/twin systems. The diagonal terms (r = s) are the self hardening moduli and the off diagonal terms ( r s ) are the latent hardening terms. For the self hardening terms in slip and twin systems we adopt a Voce [33] type hardening law of the form: τ *s h ss γ τ *s τ s (τ s θ s γ)(1 exp( γ θ s τ s )) 0 1 1 0 1
(6)
t
where γ γ s dt is the accumulated shear in the grain and the parameters τ s0 , θ s0 , θ1s , (τs0 τ1s ) s
0
are the initial CRSS, the initial hardening rate, the asymptotic hardening rate and the backextrapolated CRSS, respectively. The latent hardening moduli are given by h rs q r h ss (γ) and qr
17
is the latent hardening parameter (here there is no summation over repeated indices and r s ). This is a phenomenological model and the parameters can be determined empirically. The Voce model is convenient to use for defining the single crystal properties since the hardening parameters can be determined separately for each slip/twin system to model different hardening or softening behavior of the system with increasing strain, but the drawback is that the parameters are not directly based on the physical quantities such as grain size and dislocation densities and their interactions. In order to solve a polycrystal composed of single crystals with the constitutive equation above, Equation (4) should be linearized. The deviatoric strain rate in grain “g” can be written in terms of a fourth order compliance tensor M (g) and a back extrapolated term Do(g) as follows: D(x) M (g) : σ l (x) Do(g)
(7)
Depending on the linearization assumption, M (g) and D o(g) can be defined using different linearization approaches defined by Lebensohn and Tome [19]. The same type of relationship can be assumed for the homogeneous media, from which one can then determine the average macroscopic quantities M and D o for the polycrystal. These moduli are unknown and need to be adjusted self consistently using the concept of Eshelby’s equivalent inclusion method (EIM) [5]. This formulation uses an iterative procedure to calculate the stress and strain rate tensors of the polycrystal starting from a Taylor guess for the initial calculations of viscoplastic compliance tensor and back extrapolated term. This in turn, enables calculation of the general macroscopic constitutive law, which includes deviations arising from grain shape by using Eshelby’s tensor. This new set of calculations yield the updated viscoplastic compliance tensor and back extrapolated term in an iterative scheme. Once convergence is reached, the next deformation step
18
is calculated. Further details on the linearization scheme and EIM can be found in the literature [19, 20, 34, 35].
5. Continuum Dislocation Dynamics Coupled with Viscoplastic Self-Consistent
Model It is well established that magnesium alloys deform due to crystalline slip/twinning and grain boundary sliding depending on the deformation conditions [36, 37]. While the GBS deformation requires specific microstructures and usually high temperatures for activation, the deformation due to slip and twinning are active at all deformation conditions either as main mechanism or as an accommodating mechanism. Therefore it is important to identify the contribution of slip and twinning mechanisms to the overall deformation at any temperature in order to distinguish between the fraction of deformation promoted by slip and the fraction due to GBS [9, 38]. In the current work we develop the model for deformation at room temperature where GBS is not active. The model can be extended to high temperature with appropriate modification to be addressed in the future. Moreover, within the current model we address slip and twinning mechanisms and their corresponding hardening response in a physically based approach than what presented in equation 3. In this section we integrate a continuum dislocation dynamics (CDD) model with VPSC. Instead of using phenomenological models such as the ones described in Eqs.(4-6), we develop an internal state variable model based on the dislocation activities on each of the slip systems in the HCP crystal, along the lines discussed by Li et al. [22] for BCC crystals. Particularly
we
assume that the dislocations on each slip system “s” in each grain can be divided into two families: mobile ρ sm and immobile ρ si , and develop corresponding evolution equations based on
19
the dislocation mechanics that can be explicitly identified and quantified in discrete dislocation dynamics. Then the plastic shearing γ s on each slip system can be determined from the Orowan equation, as follows. 1/ v s v 0 τ s /τ sth Sign( τ s ) γ ρ b v ; s v 0
s
s m
s
s
for τ s τ sth for τ τ s
(8)
s th
where we assume a power law dependency of glide velocity to the normalized resolved shear s
s
stress on each system. Here b is the magnitude of the Burgers vector, v is the dislocation glide velocity on slip system “s”, v0 is normalization factor which is in the order of 10-5 and the power factor η is typically in the order of 0.012 [22]. Next we assume that the threshold stress at each slip system has contributions from an initial critical resolved shear stress (CRSS), grain size hardening (Hall-Petch effect) and forest dislocation hardening, i.e. τ sth τ s0 τ sHP αμb Ω sr ρ r
(9)
where τ s0 is initial CRSS, τ sHP is Hall-Petch hardening and the third term is forest hardening (see for example Ohashi [39]). Using the experimentally measured average grain size D, the HallPetch strengthening term is calculated as τsHP K s D0.5 . In Equation (9), α is hardening coefficient typically in the order of unity, μ is shear modulus, ρ r is the total dislocation density in slip system “r” which is the sum of mobile and immobile dislocation densities( ρ r ρ rm ρ ir ) and the interaction matrix rs relates the hardening effect of slip system “r” on the slip system “s” based on the strength of possible interaction of the dislocations in these two systems. The initial value for τ s0 can be obtained by using single crystal experimental studies such as in Reed-Hill and Robertson [40], Wonsiewicz and Backofen [41], Yoshinaga and Horiuchi 20
[42] and Obara et al.[43] for low dislocation density content. The advantage of using Equation (9) is that the self and latent hardening contributions are directly considered in the calculations. Our experimental results in this study are based on the TRC AZ31 which has high dislocation density content. When the dislocation density is high, the contribution of forest dislocations to the CRSS and hardening response becomes considerably important as shown by Lavrentev and Pokhil [44]. These effects are calculated in the third term of Equation (9). Correct identification of the components of the matrix Ω for the range of experimental dislocation densities is the key challenge for modeling of strain hardening in the crystal. The components of this matrix can be calculated using discrete dislocation dynamics as suggest by Alankar et al. [45]. Here we use the values reported by Lavrentev [46]. The evolution equations for the mobile and immobile dislocation densities are developed based on the various mechanisms that can be explicitly captured in discrete dislocation dynamics (Alankar et al, 2012; Li et al 2013). Mobile dislocations evolve due to movement of resident dislocations as well as emission of dislocations from Frank-Read sources at the rate of 1ρ(ms ) vg(s ) / l where l is dislocation mean free path and v (s) is the average dislocation velocity. g
Note that we use the same constitutive model as stated in Equation (3) to describe twinning deformation. Here we assume twin boundaries are impenetrable walls and they reduce the volume of the crystal in which the dislocations glide by the amount of twin volume fraction ftw, and hence the dislocation mean free path decreases due to twinning. Dislocation mean free path can be derived similar to the approach presented by Kalidindi [47] as: l
1 f tw ρ total
(10)
21
The twin volume fraction rate in a grain is calculated by ftw ( γ tw )/S tw as the shear stain rate contributed by twin systems calculated by Equation (3) divided by the characteristic shear of the twin S tw [32]. Mobile dislocations of opposite signs can annihilate at the rate of 2 2 R c (ρ(ms ) ) 2 vg(s ) , where Rc is the critical radius for annihilation. Also annihilation of mobile and
immobile dislocations happens at the rate of 3R cρ(ms )ρi(s ) vg(s ) . Mobile dislocations can become immobilized due to reactions such as formation of dipoles and or junctions at the rate of 4 ρ(ms ) vg(s ) / l . Immobile dislocations can also break from their pinning point under high stress values and become mobilized at the rate of (s ) * 5 (| s | * ) r ρi(s ) vg(s ) / l where ρ i is the immobile dislocation density and τ is the threshold
stress. Cross slip is not considered here but can be included in the formulations as described by Li et al (2013). Upon combining properly the dislocation mechanisms discussed above we obtain the following evolution equations for the mobile and immobile dislocation densities, respectively. (s) (s) (s) 2 (s) (s) (s) (s) (s) (s) s * r (s) (s) ρ (s) m α1 ρ m v g / l 2α 2 R c (ρ m ) v g α 3 R c ρ m ρ i v g α 4 ρ m v g / l α 5 (| τ | /τ ) ρ i v g / l
(s) (s) (s) (s) s * r (s) (s) ρ i(s) α 4ρ(s) m vg / l α 3 R cρ m ρi vg α 5 (| τ | /τ ) ρi vg / l
(11)
The parameters α1 to α5 in Equation (11) can be determined using discrete dislocation dynamics, as outlined by Li et al (2013) or fitting to single crystal data for the specific material system.
22
6. Results and Discussion 6.1 Experimental Results
The experimental results are summarized in Figures 1-5. Figure 1a shows typical (0002) and (10 1 0) pole figures of the initial texture of the TRC plate. Generated from six scans taken at a magnification of 250 times, looking along the normal direction, it reveals a weak basal texture. The corresponding orientation imaging micrograph is presented in Figure 1b. This supports the weak basal texture seen in the pole figures and also shows the heterogeneous nature of the initial microstructure. While the majority of the grains are oriented with the basal plane normal parallel to the normal direction, there are pockets of disordered grains much as there are pockets of both large and small grains. This is expected as the TRC process has been found to produce a rather heterogeneous microstructure [48] when compared to wrought alloys [49] and rolled sheet [50]. True stress versus true strain curves for TRC AZ31 at various temperatures and a strain rate of 10-2 s-1 are shown in Figure 2. Comparing with the rolled sheet reported by Khan et al.[50], TRC specimens show smaller yield stress and brittle fracture at 150⁰C, attributed to larger grain size in TRC specimens. The figure indicates that the material response is strongly temperature dependent. This dependence is much more pronounced at temperatures below 300⁰C. There are two distinct responses at the two extremes of the curve. At room temperature, the material shows extensive strain hardening after yielding followed by sudden failure. There is limited strain rate dependence as shown in Figure 3. The considerable hardening is likely attributed to the limited number of slip systems in the HCP crystal structure which results in induced stresses in the grains due to deformation of neighboring grains [51]. Also the hardening due to twinning is a major contributor to the observed ultimate tensile strength (UTS) due to the strong interaction of dislocation with twin boundaries as formulated in Equation (10) . Increasing
23
a)
b)
Figure 2.1 - a) Pole figures showing initial texture of the as received material along the normal direction. b) Orientation map showing the initial microstructure
24
Figure 2.2 - Effect of temperature on the material response in RD at constant strain rate of 10-2 (s-1). Breakage marked by x
temperature reduces the twinning activity [52] and results in less interaction of the dislocations with twin boundaries. This is the reason the latent hardening coefficient for twins presented in Table.1 decreases with increasing temperature. There was no strain softening observed and the specimens broke due to quasi-brittle failure.
25
Qualitatively as can be seen from Figure 3, the transverse direction (TD) specimens show smaller yield stress but a more rapidly hardening region compared to the RD specimens, which is leading to the same ultimate tensile strength for the two kinds of specimens. The total elongation in the TD samples was generally less than that of the RD samples. Smaller yield stress in the TD specimens can be attributed to the presence of a dense region of (0002) poles at about 45⁰ to the TD region in the Figure 1, which results in a larger resolved shear stress for deformation in TD and earlier activity of the basal systems in this loading case.
26
Figure 2.3 - True stress vs. true strain response at room temperature for rolling and transverse directions at different strain rates. Breakage shown by marker x.
Figure 2 shows in contrast to the room temperature deformation, the failure at higher temperatures is completely ductile and a plateau region similar to perfectly plastic material response is observed which results in higher elongations. This region is followed by a strain
27
softening region and a fully ductile failure in a manner similar to conventional superplastic materials. Also with increasing temperature, a decrease in the slope of the elastic portion of the curve is observed which is consistent with the results reported by Frost and Ashby [53]. Increasing strain rate at high temperature is observed to increase flow stress and decrease maximum elongation as shown in Figure 4. The strain rate sensitivity can be calculated as the slope of the logarithmic plot of flow stress versus strain rate [54] as shown in Figure 5a.Temperature and strain rate can affect both the strain rate sensitivity and ductility. In the range of the presented experimental data, higher deformation temperature results in a higher strain rate sensitivity parameter and higher elongation at failure (Figure 5b). Also higher strain rate reduces strain rate sensitivity. Figure 5a shows that at moderate strain rates (10-3 to 5×10-2 s-1) the strain rate sensitivity has the highest values, suggesting that superplastic behavior may be taking place. These values are in accordance with those reported by Wu and Liu [55] for coarse grain material. Calculation of the activation energy of the deformation process can reveal the underlying mechanisms of deformation in this range of strain rates. Assuming an Arrhenius type relationship between strain rate and absolute temperature, the activation energy can be found from [56]:
ε exp(
Qa ) A σ1/m RT
(12)
28
Figure 2.4 - True stress vs. True strain Curve in Rolling Direction at 410°C
where R is the gas constant, σ is the flow stress and T is the temperature in Kelvin. The stress exponent 1/m , where m is the strain rate sensitivity and can be calculated using average values determined for m in Figure 5. Taking the natural logarithm on both sides of Equation (12) and
29
rewriting the equation for Q a , the apparent activation energy can be calculated from the following relationship: Qa
1 lnσ R m (1 T)
(13)
The plot of the ln σ versus 1 T diagram is shown in Figure 6. This figure reveals two linear parts with different slopes with a transition temperature around 340⁰C. The different slopes are indications of different deformation mechanisms. Since the strain rate sensitivity parameter varies with both temperature and strain rate, an average value for m has been used in Equation (12). Using the results from Figure 5a, the average m values were calculated as 0.18 for temperatures over 340⁰C and 0.10 for lower temperatures at a strain rate of 10-2 (s-1). For strain rate of 10-3(s-1), the average values of m were 0.22 and 0.12 respectively for the same two temperature ranges. Dividing each section by its respective average m value and multiplying by the gas constant R (8.314 J mol-1 K-1), the apparent activation energy for the temperatures above 340⁰C is found to be 145 kJ/mol which is close to the previously reported values for this alloy [55, 57, 58] and is also close to the lattice diffusion activation energy of Qav =135kJ/mol for magnesium [53], suggesting that the deformation process in this temperature and strain rate range is driven by diffusion accommodated grain boundary sliding. On the other hand, the calculated activation energy of about 95 kJ/mol for the temperature range of 200⁰C to 340⁰C suggests that the deformation mechanism is due to the grain boundary and pipe diffusion when compared to those values for magnesium (Qab=Qac=92 kJ/mol [53]).
30
0.2% Yield Stress (MPa)
a)
True Strain Rate b)
Figure 2.5 - a) Yield stress(0.2% offset) vs. strain rate; b) calculated strain rate sensitivity parameter plotted in solid lines and percent elongation plotted in dashed line
31
Figure 2.6 - Calculation of activation energies of deformation in different temperatures at strain rate of 0.01(s-1)
32
6.2 VPSC Simulation results
The experimentally measured initial texture of the material obtained by EBSD analysis from Figure 1 is used as an input to the VPSC code to define the initial texture along with single crystal parameters and boundary conditions. The single crystal parameters and the hardening parameters for the Voce model in Equation (6) are obtained from the literature [34, 52, 59] to model the material response at room temperature and those for high temperatures were fitted to the experimental results presented in the previous section as listed in Table 2.1. The hardening parameters were retrieved by fitting for strain rate of 10-2 (s-1) at each temperature. The same set parameters were applied to simulate the deformation behavior at other strain rates. The trend reported in the literature [52] up to 200⁰C was extrapolated to obtain approximate values for higher temperatures and the result is shown in Figure 7. Here we assumed that the basal system behaves athermally [60], and that the CRSS for the non-basal systems saturates after 300⁰C as suggested by Wonsiewicz and Backofen [41] and Partridge [61]. The twining mechanism was assumed to harden with temperature as observed in metals with twin dominated deformation. Compression studies of the AZ31 show a slight increase in the plateau stress with an increasing temperature which confirms this assumption [52]. This leads to suppression of the twinning mechanism at high temperatures as discussed in detail by Khan et al. [50].
33
Table 2.1 - CRSS and hardening responses of the four deformation mechanisms as a function of temperature. Temperature (⁰C) 20 [52]
Mode Basal Prismatic Pyramidal Twin
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