At present a lack of suitable computer aided engineering tools to simulate multi-material vehicle structures involving magnesium components is an impediment in incorporating magnesium alloys into automotive structures.
This dissertation is devoted to the development of predictive capabilities for modeling the deformation of Mg alloys for three dimensional loading. Using two modeling approaches it is demonstrated that only by accounting for the combined effects of anisotropy and tension-compression asymmetry both at single crystal and polycrystal level, it is possible to explain and accurately predict the peculiarities of the behavior of magnesium and its alloys.
Two modeling frameworks, namely a self-consistent polycrystal model that accounts for tension-compression asymmetry introduced by twinning, and a macroscopic anisotropic plasticity model based on an orthotropic yield criterion that accounts for tension-compression asymmetry in plastic flow at polycrystal level were used. It was shown that unlike Hill's (1948) criterion, the latter macroscopic criterion quantitatively predicts the experimental results in torsion and axial crushing. Specifically, for the first time axial effects in torsion were predicted with accuracy using a polycrystalline framework. Moreover, it was shown that the observed experimental axial effects in torsion can be quantitatively predicted only if both slip and twinning are considered active , the level of accuracy being similar to that of the macroscopic model. However, if it is assumed that the plastic deformation is fully accommodated by crystallographic slip, the predicted axial strains are very close to that obtained with Hill (1948) criterion, which largely underestimates the measured axial strains in one orientation and cannot capture at all the development of axial strains in torsion along the normal direction.
For the first time, the unusual features of the buckling behavior of Mg AZ31 were explained. Furthermore, it was clearly demonstrated that the critical stress, the level of axial strain at buckling, and the deformed profiles can be predicted with accuracy. ( en )

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MULTI SCALE MODELING OF DEFORMATION OF A MAGNESIUM ALLOY BY NITIN CHANDOLA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2014

4 ACKNOWLEDGMENTS I would like to extend a humble thanks to my Ph.D. advisor Dr. Oana Cazacu for her invaluable guidance and support. Her contribution at every step, which goes beyond that of time and ideas , made my Ph.D. experience both productive and stimulating. Her passion and enthusiasm for research has been contagious and hi ghly motivational in the most difficult times . I also wish to express my gratitude to Dr. Benoit Revil Baudard, first and foremost for his friendship , which in itself has been a great gift . Every interaction with him has helped me gain new perspectives and improved my understanding of applied mechanics , especially plastic anisotropy, and numeric al methodologies for computational analysis . T his research work has been performed under a join t University Industry grant (GOALI) with General Motors supported by the National Science Foundation. My sincere gratitude goes to Dr. R aj K. Mishra and D r. Anil K. Sachdev at General Motors, Material's Research Laboratory (RML) for their feedback concerning this work. I am grateful to my fellow graduate students in our research group, Dr. Brad Martin, Geremy Kleiser and Philip Flater . I have enjoyed their camarader ie and great personalities. L ast but not the least , I wish to thank my mother, father , sister and grandparents for t heir unconditional love and support . Their encouragement has been the main reason for all my successful academic pursuits.

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5 TABLE OF CONTENTS Page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 ABSTRACT ................................ ................................ ................................ ................... 16 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 18 2 LITERATURE REVIEW ................................ ................................ .......................... 21 2.1 Polycrystalline Plasticity ................................ ................................ .................... 24 2.1.1 Visco plastic self consistent model for polycrystalline materials .............. 24 2.1.2 Crystal Plasticity Finite Element Model (CPFEM) ................................ .... 29 2.2 Macroscopic Models ................................ ................................ ......................... 35 2.2.1 Hill (1948) C riterion ................................ ................................ .................. 35 2.2.2 Hill (1979) C riterion ................................ ................................ .................. 36 2.2.3 Barlat et al. (1991) C riterion ................................ ................................ ..... 37 2.2.4 Hill (1993) C riterion ................................ ................................ .................. 37 2.2.5 Yld2000 2d C riterion (Barlat et al. 2003) ................................ ................. 39 2.2.6 Yld2004 18p C riterion (Barlat et al. (2005)) ................................ ............. 40 2.2.7 Cazacu and Barlat (2004) C riterion ................................ ......................... 41 2.2.8 Cazacu et al. (2006) C riterion ................................ ................................ .. 42 3 APPLICATION OF CAZACU ET AL.(2006) CRITERION TO MODEL PLASTIC DEFORMATION OF MG AZ31 ................................ ................................ ............... 44 3.1 Experimental Data on Mg AZ31 ................................ ................................ ........ 4 5 3.2 Elastic/ Plastic Constitutive Model ................................ ................................ ..... 52 3.2.1 Or thotropic Cazacu et al. (2006) Yield C riterion: Application to Mg AZ31 ................................ ................................ ................................ ................ 53 3.2.2 Identification P rocedure and New Evolution Laws for the Material Parameters for Mg AZ31 at Different Temperatures and Strain R ates ............ 57 3.3 Finite E lement Implementation of the Elastic/Plastic M odel .............................. 63 3.3.1 Rotation of the Anisotropy A xes ................................ .............................. 63 3.3.2 Return Mapping A lgorithm ................................ ................................ ....... 64 3.4 Results: Simulation of Uniaxial T ests ................................ ................................ 66

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6 4 MODELING EVO L U TION OF TEXTURE IN MG AZ31 ................................ ........... 68 4.1 New Procedure for Identification of VPSC Material P arameters for Mg AZ31 .. 72 4.2 Assessment of the Predictive Capabilities of the Polycrystal M odel ................. 86 4.3 Monotonic Simple Shear ................................ ................................ ................... 97 4.3 .1 Prediction of Slip and Twinning Activity in Simple S hear ......................... 98 4.3 .2 Polycrystalline Model S imulations ................................ ............................ 99 5 FREE END TORSION OF MG AZ31 TUBES ................................ ....................... 108 5.1 Preliminar ies: Swift E ffects in Isotropic M aterials ................................ ............ 110 5.2 Swift E ffect in Mg AZ31: Macroscopic A pproach ................................ ............ 114 5.3 Swift E ffect in Mg AZ31: Crystal plasticity F ramework ................................ .... 131 5.4 Observations ................................ ................................ ................................ ... 144 6 EFFECT OF TENSION COMPRESSION ASYMMETRY ON PLASTIC BUCKLING OF AXIALLY COMPRESSED CYLINDRICAL SHELLS .................... 146 6.1 Prediction of Plastic B uckling S tress in A xial C ompression of Cylindrical Shells of I sotropic M etals D isplaying T ension C ompression A symmetry ........ 148 6.2 Magnesium Alloy: Mg AZ31 ................................ ................................ ............ 171 6.2 .1 Axial C rushing of Mg AZ31 Cylindrical T ube: Experiments .................... 172 6.2 .2 Axial C rushing of Mg AZ31 Cylindrical T ube: FE Simulation ................. 175 6.2 .3 Effect of Tension Compression Asymmtery on B uc kling of Mg AZ31 .... 186 6.3 Observations ................................ ................................ ................................ ... 196 7 CONCLUSION AND FUTURE WORK ................................ ................................ .. 199 AP PENDIX : CAZACU ET AL. (2006) PARAMETERS FOR DIFFERENT STRAIN RATES AND TEMPERATURES ................................ ................................ ........... 202 LIST OF REFERENCES ................................ ................................ ............................. 204 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 211

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7 LIST OF TABLES Table page 3 1 Initial yield stresses in tension and compression for Mg AZ31 at room temperature (Khan et al. 2011) . ................................ ................................ .......... 47 3 2 Model parameters for M g AZ31 at room temperature and strain rate of 1/s. ...... 60 4 1 VPSC parameters determined by Walde and Reidel (2007) for Mg AZ31 .......... 70 4 2 VPSC parameters determined by Jain and Agnew (2007) for Mg AZ31 ............. 70 4 3 VPSC parameters determined by Wang et al. (2010) assuming affine linearization ................................ ................................ ................................ ........ 71 4 4 VPSC parameters determined by Wang et al. (2010) assuming n eff =10 ............. 71 4 5 Hardening parameters for the active deformation systems at room temperature ................................ ................................ ................................ ........ 77 4 6 VPSC parameters determined for simple shear tests. ................................ ........ 99 4 7 Latent hardening parameters in VPSC determined for simple shear tests. ...... 100 4 8 Observed and predicted twin volume fraction in simple shear at equivalent strain, / 3 =20%. ................................ ................................ .............................. 106 5 1 Anisotropy coefficients in Hill (1948) yield criterion (Equation (2 23)) determined for Mg AZ31 alloy. ................................ ................................ .......... 125 6 1 Material properties for AA 2024 T4 ................................ ................................ ... 158 6 2 Evolution of the SD parameter k with plastic strain for AA 2024 T4 ................. 159 6 3 Comparison of maximum stress at buckling in MPa ................................ ......... 167 6 4 Plastic dissipation per unit volume (in N/m 2 ) at buckling ................................ ... 169 6 5 Summary of the critical stress max , the axial strain at buckling av /L and the energy absorbed per unit volume W p predicted and experimentally reported for the axial crushing of Mg AZ31 tube ................................ ............................. 195 A 1 Values of the Cazacu et al. (2006) coefficient s for the Mg AZ31 specimens tested at 150 0 F and strain rate of 10 4 /s ................................ ............................ 202 A 2 Values of the Cazacu et al.(2006) coefficients for Mg AZ31 specimens tested at 300 0 F and strain rate of 10 4 /s. ................................ ................................ ..... 202

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8 A 3 Values of the Cazacu et al.(2006) coefficients for the Mg AZ31 specimens tested at 300 0 F and strain rate of 1/s. ................................ ............................... 203

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9 LIST OF FIGURES Figure page 2 1 Hexagonal (hcp) crystal structure. ................................ ................................ ...... 21 2 2 Kinematics of elastic and plastic deformation in crystal plasticity finite element model. ................................ ................................ ................................ ... 30 2 3 Decomposition of the deformation gradient tensor when deformation twinning is included as a plastic deformation mode. ................................ ......................... 34 3 1 Uniaxial test results (Khan et al. 2011) for strain rate 1/s in three in plane orientations showing the material's anisotropy ................................ ................... 47 3 2 Comparison between the mechanical response in uniaxial tension and compression at strain rate 1/s....................................................................... ...... 50 3 3 Uniaxial test results (Khan et al. 2011) at room temperature, 150 0 F and 300 0 F for s train rate of 10 4 /s. ................................ ................................ ............. 50 3 4 Uniaxial test results (Khan et al. 2011) at room temperature, 150 0 F and 300 0 F for s train rate of 1/s ................................ ................................ .................. 51 3 5 Theoretical yield surfaces at room temperature for several levels of accumulated plastic strain according to the Cazacu et al. (2006) yield criterion in c omparison with data. Strain rate of 1/s. ................................ ........... 57 3 6 Evolution of the anisotropy parameters with accumulated pla stic strain at room temperature for s train rate 1/s ................................ ................................ ... 59 3 7 Theoretical yield surfaces corresponding to different levels of accumulated plastic strain according to the anisotropic form of the Cazacu et al. (2006) cr iterion for test data measured at 300 0 F at a strain rate of 1/s. ......................... 60 3 8 Theoretical yield surfaces corresponding to different levels of accumulated plastic strain according to the anisotropic form of the Cazacu et al. (2006) c riterion for test data at 150 0 F and 300 0 F and strain rate of 10 4 /s. .................... 62 3 9 Comparison between theoretical predictions according t o the proposed model and experimental uniaxial stress strain response ................................ .... 67 4 1 Pole figures showing initial texture of Mg AZ31 sheet ................................ ........ 73 4 2 ODF generated for 5 0 increments of using EBSD scan and imposing orthotropic symmetry ................................ ................................ ......................... 74 4 3 Comparison between the measured texture and predicted texture. ................... 79

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10 4 4 D eformation response in RD tension. ................................ ................................ . 81 4 5 D eformation response in RD compression. ................................ ........................ 84 4 6 D eformation response in ND compression. ................................ ........................ 85 4 7 D eformation re sponse in T D tension ................................ ................................ .. 87 4 8 D eformation response in TD compression . ................................ ........................ 90 4 9 D eformation response in D D tension ................................ ................................ .. 92 4 10 D eformation response in D D compression . ................................ ........................ 94 4 11 D eformation response in ND tension ................................ ................................ .. 96 4 12 Direction of loading relative to c axis for simple shear tests. ............................ 101 4 13 D eformation response in R D shear. ................................ ................................ . 103 4 14 Pole figures in RD shear test at equivalent strain / 3 =20%. ........................... 104 4 15 D eformation response in T D shear . ................................ ................................ .. 105 4 16 Po le figures in T D shear test at equivalent strain / 3=20% ............................. 107 5 1 Projection in the plane of the von Mises yield surface along with the normal to the surface for shear loading . ................................ ........................... 114 5 2 Projection in the plane of the iso tropic form of Cazacu et al . (2006) yield surface along with the normal to the surface for shear loading. ....................... 114 5 3 Orientation of the material anisotropy axes relative to loading axes for free end torsion tests. ................................ ................................ .............................. 116 5 4 Theoretical yield surfaces , according to the orthotropic Cazacu et al. (2006) criterion in the ( 12 11 ) plane, corresponding to the different levels of accumulated p lastic strain. . ................................ ................................ .............. 118 5 5 T heoretical yield surfaces, according to the orthotropic Cazacu et al. (2006) criterion in the ( 13 11 plane), corresponding to different levels of accumul ated pl astic strain. . ................................ ................................ .............. 118 5 6 Theoretical yield surfaces, according to the orthotropic Cazacu et al. (2006) criterion in the 13 33 plane, corresponding to different levels of accumulated plastic strain.. ................................ ................................ ................................ .... 119

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11 5 7 Theoretical yield surfaces, according to the ortho tropic Cazacu et al. (2006) criterion in the 23 33 plane, corresponding to different levels of accumulated plastic strain. ................................ ................................ ................................ ..... 120 5 8 Sample geometry, dimensions (mm) and finite element mesh for free end torsion test. ................................ ................................ ................................ ....... 121 5 9 Comparison between experimental data (from Guo et al. (2013)) and the FE predictions obtained with Cazacu et al. (2006) yield criterion with evolving anisotropy coefficients ,for the long axis of the specimen along RD and ND . .... 123 5 10 Yield loci corresponding to fixed levels of accumulated plastic strain according to Hi ll (1948) orthotropic criterion against mechanical test data from Khan et al. (2011). ................................ ................................ .................... 125 5 11 Theoretical yield surfaces according to the Hill (1948) criterion in the 12 11 plane, corresponding to different levels of accumulated plastic strain. ............. 127 5 12 Theoretical yield surfaces according to the Hill (1948) criterion in the 13 11 plane, corresponding to different levels of accumulated plastic strain. ............. 127 5 13 Theoretical yield surfaces according to the Hill (1948) criterion in the 13 33 plane, corresponding to different levels of accumulated plastic strain .............. 128 5 14 Theoretical yield surfaces according to the Hill (1 948) criterion in the 23 33 plane, corresponding to different levels of accumulated plastic strain .............. 128 5 15 Comparison of the variation of the axial strain with shear strain observed in experiments during free end torsion along the rolling direction (RD) against the predictions according to the (i) orthotropic Cazacu et al. (2006) yield criter ion and isotropic hardening law (ii) Hill (1948) yield criterion and the same isotropic hardening law. ................................ ................................ .......... 130 5 16 Comparison of the variation of axial strain with shear strain observed in experiments during free end torsion along the normal direction (ND) against the prediction according to the (i) orthotropic Cazacu et al. (2006) yield criter ion and isotropic hardening law (ii) Hill (1948) yield criterion and the same isotropic hardening law. ................................ ................................ .......... 131 5 17 Comparison between variation of the axial strain vs. shear strain during free end torsion about RD and ND given by (i) experimental data and (ii) the numerical predictions by the VPSC model. ................................ ...................... 134 5 18 Predicted effective stress vs. effective strain response and evolution of the microstructure in RD shear using the VPSC mo del ................................ .......... 138 5 19 Final texture at equivalent strain / 3=20% in simple shear along the rolling direction ................................ ................................ ................................ ............ 139

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12 5 20 Comparison of the axial strain vs. shear strain observed experimentally and according to the VPSC m odel for all three cases in free end torsion ................ 141 5 21 Comparison of the (i) experimental variation of the axial strain with the shear strain during free end torsion (ii) numeric al prediction according to the orthotropic Cazacu et al. (2006) criterion and the VPSC model simulations considering that all sl ip modes and twinning are active ................................ .... 142 5 22 Comparison of the variation of the axial strain vs. shear strain during free end torsion against the predictions according to the (i) orthotro pic Hill (1948) yield criterion (ii) the VPSC model simulations, where twinning was neglected and all slip modes are active and (iii) experimental data from Guo et al. (2013). .... 143 6 1 Axisymmetric pla stic buckling of cylindrical shell of circular cross section (from Batterman, 1965). ................................ ................................ ................... 150 6 2 Evolution of the critical stress at buckling with the ratio R/h of tube geometry for three different isotropic materials : k= 0.5, k=0, k=0.5 . ................................ . 155 6 3 Comparison betwe en the experimental stress strain response in uniaxial tension and compression for A A 2024 T4 (data from Batterman, 1965). ........... 1 57 6 4 Bending test results for AA 2024 T4 (data from Batterman , 1965). ................... 157 6 5 Comparison between experimental stress strain response in un iaxial tension and compression and predicted response according to the isotropic form of the Ca zacu et al. (2006) model . Data from Batterman (1965). . ........................ 160 6 6 Comparison of yield loci according to isotropic Cazacu et al . (2006) and von Mises yield criteria at initial yield in the biaxial stress plane . . ........................... 160 6 7 Mechanical test specimen geometry. ................................ ............................... 161 6 8 Average stress strain curves from axial crushing tests on AA 2024 T4 cylindrical shells of different radius to thickness (R/h) ratios. ........................... 162 6 9 Comparison of critical buckling str ess observed in mechanical test s and given by theoretical prediction based on the analytical model developed according to the (i) von Mises yield criterion; (ii) Cazacu et al. (2006) yield criterion. ................................ ................................ ................................ ............ 164 6 10 Typical mesh used for the finite element simulation ................................ ......... 166 6 11 Comparison between FE results and analytical model predictions for different specimen geometries; in both FE and analytical models the von Mises yield criterion and Cazacu et al. (2006) criterion (new model) were used respect ively. ................................ ................................ ................................ ...... 168

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13 6 12 Axisymmetric cylindrical specimen tested under axial compression. ................ 173 6 13 Average stress axial strain data in axial crush of Mg AZ31 tube machined with the long axis along rolling direction. ................................ .......................... 174 6 14 Evolution of energy absorbed per unit volume in axial crush estimated from mechanical test data. ................................ ................................ ........................ 174 6 15 Mesh used for the finite element simulation. ................................ .................... 175 6 16 Comparison of average stress vs. average axial strain observed in experiments and predicted by Cazacu et al. (2006) criterion during axial crush of Mg AZ3 1 tube along rolling direction. ................................ ................. 176 6 17 Comparison of energy absorbed per unit volume vs. average axial strain observed in experime nts and predicted by Cazacu et al. (2006) criterion during axial crush of Mg AZ31 tube along rolling direction. .............................. 177 6 18 Deformed profile (outside surface) of Mg AZ31 in RD compression in the X Z plane , predicted by Cazacu et al. (2006) yield criterion . ................................ ... 178 6 19 Deformed profile (inside surface) of Mg AZ31 in RD compression in the X Z plane , predicted by Cazacu et al. (2006) yield criterion . ................................ ... 179 6 20 Deformed profile (outside surface) of Mg AZ31 in RD compression in the X Y plane , as predicted by Cazacu et al. (2006) yield criterion . .............................. 180 6 21 Deformed profile (inside surface) of Mg AZ31 in RD compression in the X Y plane , as predicted by Cazacu et al. (2006) yield criterion . . ............................. 180 6 22 Theoretical yield surfaces in TD ND plane according to the orthotropic Cazacu et al. (2006) criterion for Mg AZ31 alloy cor responding to different levels of equivalent plastic strain. ................................ ................................ ..... 182 6 23 Isocontours of equivalent plastic strain in the cross secti on at different values of average axial strain. ................................ ................................ ..................... 183 6 24 Deformed profile of the Mg AZ31 in RD compression in the X Y plane showing isocontours of third inva riant of Cauchy stress deviator as predicted by Cazacu et al. (2006) yield criterion. ................................ ............................. 185 6 25 Deformed profile of the Mg AZ31 in RD compression in the X Z plane showing isocontours of t hird inva riant of Cauchy stress deviator as predicted by Cazacu et al. (2006) yield criterion. ................................ ............................. 185 6 26 Center cross section (lower half) predicted at buckling fo r axial compression along RD . ................................ ................................ ................................ ......... 187

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14 6 27 Average stress vs. average axial stra in obtained from experiments , FE simula tion using Hill (1948) criterion and FE simulation using Cazacu et al. (2006) criterion during axial crush of Mg AZ31 tube along rolling direction. ..... 188 6 28 Energy absorbed per unit volume vs. average axial strain du ring axial crush along RD obtained from experiments , Cazacu et al. (2 006) yield criterion and Hill (19 48) yield criterion . ................................ ................................ ........... 189 6 29 Deformed profile of Mg AZ31 in RD compression in the X Z plane , as predicted by Hill (1948) yield criterion . ................................ .............................. 189 6 30 Deformed profile of Mg AZ31 in ND compression , as predicted by Cazacu et al. (2006) yield criterion . ................................ ................................ ................... 192 6 31 Deformed profile of Mg AZ31 in ND compression , as predicted by Hill (1948) yield criterion . ................................ ................................ ................................ ... 192 6 32 Cross section (upper half) predicted at buckling for axial compression along ND . .................. ................................ ................................ ................................ . 193 6 33 Average stress vs. average axial strain during axial crush of Mg AZ31 tube along ND predicted by Cazacu et al. (2006) yield criterion and predicted by Hill (1948) yield criterion. ................................ ................................ .................. 194 6 34 Energy absorbed per unit volume vs. average axial strain durin g axial crushing along ND predicted using Cazacu et al. (20 06) yield c riterion and predicted by using Hill (1948) yield criterion . ................................ .................... 194

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15 LIST OF ABBREVIATION S CPFEM Crystal Plasticity Finite Element Method DD Diagonal Direction EBSD Electron Backscatter Diffraction FE M Finite Element Method ND Normal Direction RD Rolling Direction RT R oom Temperature TD Transverse Direction TT Through thickness VPSC Visco plastic self consistent

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16 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MULTI SCALE MODELING OF DEFORMATION OF A MAGNESIUM ALLOY By Nitin Chandola August 2014 Chair: Oana Cazacu Major: Mechanical Engineering At present a lack of suitable computer aided engineering tools to simulate multi material vehicle structures involving magnesium components is an impediment in incorporating magnesium alloys into automotive structures. This dissertation is devoted to the development of predictive capabilities for modeling the def ormation of Mg alloys for three dimens ional loadings . Using two modeling approaches it is demonstrated that only by account ing for the combined effects of anisotropy and tension compression asymmetry both at single crystal and polycrystal level, it is possible to explain and accurately predict the peculiarities of the behavior of magnesium and its alloys. Two modeling frameworks, namely a self consistent polycrystal model that accounts for tension compression asymmetry introduced by twinning, and a macroscopic anisotropic plasticity model based on an orthotropic yield criterion that accounts for tension compression asymmetry in plastic flow at polycrystal level were used. It was shown that unlike Hill's (1948) criterion, the latter macroscopic criterion quantitatively predicts the experimental results in torsion and axial crushing.

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17 Specifically, for the first time axial effects in torsion were predicted with accuracy using a polycrystalline framework. Moreover, it was shown that the observed experimental axial effects in torsion can be quantitatively predicted only if both slip and twinning are c onsidered active , the level of accuracy being similar to that of the macroscopic model. However, if it is assumed that the plastic deformation is fully accommodated by crystallographic slip, the predicted axial strains are very close to that obtained with Hill (1948) criterion, which largely underestimates the measured axial strains in one orientation (along rolling direction) and cannot capture at all the development of axial strains in torsion along the normal direction. For the first time, th e unusual features of the buckling behavior of Mg AZ31 were explained. Furthermore, it was clearly demonstrated that the critical stress, the level of axial strain at buckling, and the deformed profiles can be predicted with accuracy.

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18 CHAPTER 1 INTRODUCTION In a quest to reduce fuel consumption and improve machine performance, the automotive and aerospace industries are seeking to expand the use of lightweight materials such as magnesium alloys. Wit h the use of such lightweight metals and their new alloys there arises a need to develop new macroscopic constitutive models capable of accounting for their unusual plastic flow characteristics. These models can then be further used to address the challeng es associated with every aspect of manufacturing products made of these materials. The key difficulty and stringent need is to incorporate the physics of deformation at lower length scales in the macroscopic level formulations to b e used for design of proc esses. Of all the new lightweight materials , the deformation and failure behavior of magnesium alloy AZ31 is least understood. The far reaching goal of this dissertation is to advance the state of the art in modeling and simulation of Mg alloys , in particular AZ31 . A multi disciplinary approach combining mechanics and computational material science is adopted. The outline of the dissertation is as follows. Chapter 2 is devoted to a survey of the major contributions in the description of plastic behavior of polycrystalline materials. Chapter 3 presents modeling of the mechanical response of AZ31 polycrystalline Mg alloy within the framework of the mathematical theory of plasticity. While t he yield criterion used is that of Cazacu et a l. (2006), which accounts for both anisotropy and strength differential effects , n ew hardening laws are developed to capture the distorsion

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19 of the yield surface associated with texture evolution in this material . The model consisting of the yield criterion and new hardening laws is implemented as a User Material Routine (UMAT) in the implicit FE code Abaqus. It is shown that the model captures all the key features of the mechanical response of the material for uniaxial loadings. In particular, for the first time, the unusual sigmoidal st ress strain response in in plane (RD TD plane ) compression is simulated with accuracy and it is predicted that the normal direction tension curve cannot have the concave down appearance which is typical f or metallic materials (e.g. Ti alloys, Nixon et al., 2010). Extensive v alidation of the model is provided by analyzing the mechanical response in torsion ( Chapter 5 ) . Chapter 4 discusses modeling Mg AZ31 within the framework of crystal plasticity. Using a combination of mechanical data and metallographic information, a new methodology for the determination of the single crystal plastic deformation mechanisms operational for different strain paths is developed. Furthermore, it is demonstrated that by using this methodology it is possible to model with great accuracy both the texture evolution and the stress strain response at the macroscopic level. Chapter 5 is devoted to the theoretical investigation of the response of the material in free end torsion. Fo r the first time, the unusual characteristics of the torsional response of Mg AZ31 are explained and the experimental data predicted with accuracy. Furthermore, correlation s between the response in uniaxial loading and torsional response are established. S pecifically, it is explained why axial strains develop when subjecting this material to torsion and why in one direction the material elongates while in the other it contracts . Furthermore, using the crystal plasticity model it is

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20 demonstrated that the obs erved experimental axial effects could be quantitatively predicted only if both slip and twinning are considered operational. Chapter 6 is devoted to the investigation of buckling of thin cylindrical tubes made of Mg AZ31 . For the first time , the effect o f tension compression asymmetry and anisotropy in plastic flow , on the critical buckling stress and the plastic energy that the material can accumulate ( prior to buckling ) is demonstrated . Chapter 7 presents the major conclusions of this research dissertation and future plans.

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21 CHAPTER 2 LITERATURE REVIEW As shown by Taylor (1938 ), in order to accommodate any irreversible shape change by slip a material should have at least five independent slip systems. In contrast to metals with cubic crystal structure which have 12 (FCC) and 24 (BCC) independent slip systems, respectively; Mg which has hexagonal close p acked (hcp) crystal structure with a ratio c/a less than 1.72 (Fig ure 2 1 ) , has less than five slip systems operational even for the simplest loading paths (e.g. uniaxial tension, early work of Reed Hill ( 1973 ), Kel ley and Hosford (1968) on single crystal and polycrystalline pure Mg and severa l Mg alloys, m ore recent studies on Mg alloys such as those by, Agnew et al., 2001 , Koike et al., 2003 , Koike, 2005 , SandlÃ¶bes, 2011 and Wu, 2008 ) . Deformation twinning may become a contributor to plastic deformation and even become a dominant mecha nism ( e.g,. in uniaxial compression Agnew et al., 2002 , Hong et al., 2010 , Jiang et al., 2007 , Martin, 2010 and Park et al., 2010 ) . Figure 2 1. Hexagonal (hcp) crystal structure.

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22 In contrast to crystallographic slip, twinning i s a directional shear mechanism, i.e. shear in one direction can cause twinning while shear in the opposite direction will not cause twinning . For example, in pure Mg and magnesium alloy sheets twinning is n ot active in uniaxial tensile loading along any direction in the plane of the sheet, bu t is easily activated in uniaxial compression. As a result the average initial in plane compressive yield stress is about half the average in plane tensile yield stress (e.g. Kelley and Hosford, 1968, Lou et al., 2007 , Walde et al. 2007, Khan et al. 2011 ). As a consequence, the yield surfaces are not symmetric with respect to the stress free condition. Since Mg and its alloy sheets exhibit strong basal textures ( Salem et al., 2003 , Agnew and Duygulu 2005, Lou et al. 2007, Proust et al. 2009, Nixon et al. 2010 , Khan et al. 2010, Tirry et al., 2011 and Knezevic et al., 2013 ) i.e. the c axis of majority of grains is oriented predominantly perpendic ular to the thickness direction , a pronounced anisotropy in yielding is observed. This difference in the type of deformation mechanisms active within the microstructure of the material for different loading path s significantly influences the overall mechanical properties of magnesium based materials. In general, the lack of understanding of the mechanical response and the causes of reduced formability of Mg materials at room temperature, limit the use of these m etals in structural applications ( Hilditch et al., 2009 ) . Recently, in an attempt to account for the effect of micro structural changes i.e. texture evolution during plastic deformation and to simultaneously estimate the ensuing macroscopic mechanical response of Mg AZ31, crystal plasticity based models have been used. The two main crystal plasticity approaches for modeling the mechanical response of m agnesium alloys have been (a) the visco plastic self consistent (VPSC)

25 compared to the plastic deformations and as such can be ignored. In this model, t he rate of deformation of the grain (single crystal) is described as: , (2 1 ) In Equation (2 1), s den otes a slip system which is characterized by a normal vector n s (normal to the slip or twinning plane) and a vector b s (Burgers vector or twinning shear direction) , is the local shear rate on the slip system s, is the local average of the deviatoric strain rate in the grain, is the local average of the stress in the grain, is a reference shear strain rate, is the Schmid tensor for the given system s, is the threshold shear stress for activation of the system s, and n is a parameter that accounts for the rate sensitivity of the plastic deformation. Note that the symbol " " denotes the d y adic product between any two vectors a and which is a second order of components: . The activation criterion for both slip and twinning is given by the expression in the parenthesis in Equation (2 1). Onset of plastic deformation corresponds to the resolved shear stress on the particular slip/twinning system exceeding a threshold value, for that slip/twinning system. T winning i s treated as a pseudo slip mode i.e. it has a critical resolved shear stress of activation in the twinning plane . However, t winning differs from slip in its directionality, and is modeled by allowing activation only if the resolved shear stress is positive (along the Burgers vector of the twin). Another aspect of twinning which is incorporated in the visco plastic self consistent model is the change of orientation of the twinned regions as opposed to regions which have not undergone

27 , (2 3 ) where , is the compliance matrix . Hardening on each deformation system (slip or twin) is given by a Voce type law , which describes the evolution of the critical resolved shear stress in the grain as a function of accumulated shear strain : (2 4 ) In Equation (2 4 ), is the initial critical resolved shear stress on system s, is the back extrapolated critical resolved shear stress on system s, is the initial slope of the hardening curve, is th e final slope of the hardening curve and is the accumulated shear strain for the system s. The accumulated shear strain for the slip system s is determined as , , (2 5 ) In the self consistent homogenization scheme, the interaction of each grain with its surroundings is based on the assumption that the grain is an inclusion embedded in the anisotropic homogenized (overall) , visco plastic aggregate . The boundary conditions such as strain or stress are then applied to this aggregate and the orientation of the grains is updated at each time increment along with the compliance matrix (relating stress and strain averaged over the entire volume , Equation 2 3 ) of the overall aggregate. The self consistent approach assumes lo cal stress equilibrium within the entire volume of the aggregate. The overall plastic compliance relating the overall (averaged over all grains) stress to the overall average strain rate is determined assuming either a secant model, tangent model, or an eff ective interaction model (combination of Secant and Tangent approaches) i.e.

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28 , (2 6 ) In Equation (2 6 ), is the polycrystal aggregate strain rate, is the polycrystal aggregate Cauchy stress. One can express : , (2 7 ) where, n eff is a parameter , its range being 0< n eff <1. Based on the averaged (polycrystal aggregate) interaction model and the single grain constitutive model ( Equation 2 1) the overall constitutive equation is given as: , (2 8 ) where, and denotes single crystal deformation rate and stress tensor , and and denotes aggregate deformation rate and stress tensor . Assum ing one of the given linearization assumptions ( Equation (2 6 )) the Eshelby tensor is calculated , , (2 9 ) In Equation (2 9 ), S E is the Eshelby tensor which is a function of the properties of the effective medium (M tg ) and the grain shape. The Eshelby tensor along with the secant compliance matrix and n eff is used to calculate the accommodation tensor relating stress deviations and strain rate deviations between each grain and the matrix. The self consistent equation, which allows adjustment of the macroscopic compliance by requiring a matching between the o verall averages of the local fields and the corresponding effective magnitudes, is given as , (2 10 )

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29 where Â· denotes average over the set of grains that represents the polycyrstal. T he localization tensor B g relates the local (grain) and effective stresses, , (2 11 ) The expression for the localization tensor is obtained as, , (2 12 ) The average tangent compliance matrix is determined such as to satisfy Eq uation ( 2 9 ) and Equation ( 2 11 ) . Hence, the VPSC model is able to account for the overall plastic deformation behavior of the polycrystalline material. 2.1 .2 Crystal Plasticity Finite Element Model (CPFEM) With the development of the finite element technique, it has become possible to attempt to build micro mechanical polycrystalline finite element models that can provide solutio ns of stress and strain fields in a given polycrystalline aggregate and also, in a weak numerical sense, satisfy both equilibrium and compatibility relations. It is claimed that this approach has an advantage over homogenized models such as VPSC because i t does not make use of a linear comparison material (e.g. Equation (2 2)) and as such it may better predict local phenomena occurring at the grain scale. However, it should be noted from the very beginning of this presentation that th e major drawback of existing CP FE models is that the kinematics of plastic deformation is accurately described only if deformation occurs by slip (the review by Kalidindi , 2004 ). The presentation that follows discusses the standard classic CPFE formulation in which plastic de formation is fully accommodated by slip. Neglecting elastic effects the deformation gradient F is expressed as,

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30 , (2 13 ) In E quation (2 13) , , represents the plastic deformation gradient, due solely to the cumulative effect of dislocation motion on active slip systems and represents the crystal lattice rotation (Figure 2 2) . Note that R is not obtained from the polar decomposition of the deformation gradient F . Figure 2 2. Kinematics of elastic and plastic deformation in crystal plasticity finite element model. Using Equation (2 13), the velocity gradient in the current configuration, , can be expressed as, , or, , (2 14 ) Since all plastic straining is due to slip on slip planes, L P is expressed as,

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31 , (2 15 ) where, is the shear rate of slip system s, and and are unit vectors in the reference configuration, directed along the slip direction and along the normal to the slip plane respectively. The summation is performed over the potentially active slip systems. Expressing the velocity gradient tensor as, , w ith is its antisymmetric part (skew Tensor) and its symmetric component , using Equatio n ( 2 14 ) and (2 15) one obtains , , (2 16 ) where, and are the symmetric and skew symmetric parts of the Schmid tensor. The lattice rotation evolution equation is obtained by solving Equation (5.2) for , (2 17 ) The slip system shear rate is expressed as a function of the traction acting on the slip plane in the slip direction. Using the transformation of coordinate system from the reference configuration to the current configuration, the slip direction and its normal are expressed in the current configuration as and , so is : (2 18 ) A power law approximation is generally used to relate the shear rate on system s to so we get,

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32 , (2 19 ) where is the flow stress on the slip system at the reference shear rate . In general, the power law expression with fixed exponent n is only accurate for a small range of strain rates. Substituting Equation (2 19) into Equation (2 16) , the deviatoric part o f the Cauchy stress, ' , can be solved from the nonlinear equation, (2 20 ) where, The critical flow stress for each slip system is updated through an evolution equation, for example in a power law type hardening i.e., , (2 21 ) where, is the initial hardening rate, is the initial yield stress and is the saturation stress. In the above we only present the special case of Taylor hardening, i.e., all slip systems contribute equally to hardening. This reduces the number of hardening state variables per crystal to one . Next, to calculate the response of the polycrystal l ine aggregate, a FE mesh is placed over the microstructure with each element representing a single grain, or part of a single grain. The response of the polycrystal, composed of grains for which the deformation behavior is described by constitutive relations in Eqs. (2 13) to (2 21), depends on the type of homogenization used . If , Taylor assumption (Taylor, 1938) is

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33 considered, the plastic strain of all grains within the polycrystal is the same, and hence equal to the macroscopic plastic strain. If the extended Taylor hypothesis is considered , the velocity gradient of each grain is equal to the macroscopic velocity gradient , or equivalently the deformation gradient in each grain is homogeneous and equal to the mac roscopic deformation gradient ( Taylor 1938 and Kok et al., 2002). The Taylor assumption ensures compatibility between all crystals, however local stress equilibrium is violated . Other assumptions which attempt to a ccount for both compatibility and equilibrium in the polycrystal l ine aggregate are also possible (Molinari et al., 1987; Raabe, 1995; Sarma and Dawson, 1996). With growing interest in hcp metals such as Mg and its alloys inclusion of deformation twinning i n the crystal plasticity finite element method framework has been given a lot of attention (Kalidindi (1992), Kalidindi (2004)). Figure 2 3, shows the interpretation of the multiplicative decomposition of the deformation gradient into elastic and plastic components when twinning is included as an additional mode of plastic deformation. For clarity, only one twin system is shown in the Figure 2 3 , it is however, possible for multiple twin systems to be activated in the same grain. Furthermore, the twinned r egion belonging to one twin system is idealized as a continous block, while in actuality, deformation twins occur as parallel elliptical regions. It should be noted that further slip is permitted in each of the twinned regions.

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34 Figure 2 3. D ecomposition of the deformation gradient tensor when deformation twinning is included as a plastic deformation mode . The quantities F , F* and F P in Figure 2 3 represent homogenized values at the grain scale. As such they do not represent the deformation gradients in either the matrix or the twinned regions alone. It is assumed that twinning causes a rotation of the lattice of part(s) of the grain relative to the untwinned regions. The relationship between the lattice orientation in the twinned region and the lattice orientation of the matrix is treated here as an additional solution dependent state variable, denoted by R t . For the twinned regions in the crysta l , the lattice orientation is now computed as, (2 22 ) Here, the special relationship of a 180 degree rotation about the twin habit plane normal for the matrix and the twinned orientations will only be satisfied just when the twin is produced. With further slip activity this relationship is lost. The most si ginificant limitation of the model, is that it is implied in the idealization that all of the twinned regions of the crystal belonging to a particular twin system would

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35 have a single lattice orientation in the current configuration. In actuality, since the twins are produced at different instances along the deformation path, they will not have the e xact same lattice orientation ( Kalidindi, 2004). More details of this deformation twinning mode l are given by Kalidindi (2004) . This model and its modifications h ave been used by Knezevic et al . 2010, Kaan and Mishra ( 2012 ) , Choi ( 2010, 2011 ), for modeling twinning in Mg and its alloy AZ31 using the crystal plasticity finite element analysis framework. 2.2 Macroscopic Models 2.2.1 Hill (1948) C riterion One of the first macroscopic models used for describing the plastic behavior of orthotropic metals was proposed by Hill (1948) . Hill (1948) yield criterion is written as: (2 23 ) In Equation (2 23), the coefficients , , , , and are material constants. The axes 1, 2 and 3 are the orthotropy axes (axes of symmetry of the material). For example, in case of a rolled sheet these axes (1, 2, and 3) are the rolling, transverse , and through thickness direc tion , respectively. The coefficients of this criterion can be determined from simple mechanical tests. If the tensile yield stresses for 1, 2, and 3 directions are denoted as X,Y and Z then according to Hill 's criterion ( Equation (2 2 4 )) : (2 24 ) Solving for F, G, and H we get: (2 25 )

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36 If the yield stresses in shear are denoted as R,S, and T corresponding to shear stresses in the 23 , 31 and 12 planes respectively, then: (2 26 ) This yield criterion will be used for modeling Mg AZ31 at room temperature in Section 3 and compared against Cazacu et al. (2006) m acroscopic model. Equation (2 23) reduces to the isotropic von Mises criterion for isotropy when F=G=H= 1/6 X 2 and L=M=N= 1/ X 2 , where X is the uniaxial tension yield stress along any direction in the material . 2.2.2 Hill (1979) C riterion Hill (1948) yield criterion although used successfully for the description of yielding of numerous metals , especially steels is unable to capture the behavior of some aluminum alloys which have an average value of Lankford coefficients less than 1 and the yield stres s in biaxial tension greater than the yield stress in uniaxial tension . In order to describe the plastic behavior of such alloys, Hill developed another yield criterion in 1979 , which is expressed in the following form: (2 27 ) In Equation (2 27) , Y is the yield stress in the rolling direction , and are the principal Cauchy stresses, m is a homogenization constant and the coefficients , , , , and are material constants . The major l imitation of this yield criterion is that the principal stress axes and the anisotropy axes must be superimposed for it to be applied. Any state involving shear stresses cannot be accounted for by this model.

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37 2.2.3 Barlat et al. (1991) C riterion Barlat et al. (1991) proposed a yield criterion which extends to orthotropy the following isotropic yield criterion : , (2 28 ) (proposed by Hershey in 1954 and Hosford in 1972) to orthotropy. To account for orthotropy , in the expression of the isotropic criterion the Cauchy stress components are given different weighting coefficients. This amounts to applying a fo urth order linear operator on the Cauchy stress tensor so t he yield criterion is given as: , (2 29 ) In Equation (2 29 ) , and are the principal values of the transformed Cauchy stress tensor. The transformation is carried out as where the transformation tensor L is given as: , (2 30 ) In the fourth order tensor L (Equation (2 30)) , c 1 , c 2, c 3, c 4, c 5, and c 6 are constants. It should be noted that the plastic anisotropy is represented by the same n umber of coefficients as Hill's (1948) criterion . 2.2.4 Hill (1993) C riterion In order to model yielding and plastic flow of textured sheets of orthotropic materials and to improve on the flexibility of the Hill (1948) yield criterion to capture

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38 orthotropic material behavior, Hill proposed another yield criterion (Hill, 1993) which is applicable only for 2 D plane stress problems and materials exhibiting planar anisotropy. This criterion is obtained by adding a particular pair of cubic terms to the 2 D form of the Hill (1948) criterion (E quation ( 2 23 ) in Section 2.2.1). The proposed yield criterion is of the form: , (2 31 ) In Equation (2 31 ), is the uniaxial tensile yield stress in the rolling direction, is the uniaxial tensile yield stress in the transverse direction (in plane direction normal to the rolling direction), is the yield stress under biaxial tension, and c , p and q are parameters . Using Equation (2 31 ), the physical significance of these parameters can be easily obtained. Specifically, (2 32 ) (2 33 ) (2 34 ) In Equation (2 31) to (2 33), is the r value (Lankford coefficient) for uniaxial tension in the rolling direction, is the r value for uniaxial tension in the in plane direction perpendicular to the rolling direction .

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39 2.2.5 Yld2000 2d C riterion (Barlat et al. 2003) A convex plane stress function that describes well the anisotropic behavior of sheet metals, in particular, aluminum sheets, called the Yld2000 2d was proposed by Barlat et al. ( 2003 ). This orthotropic yield function was obtained by application of two linear transformations to two isotropic yield functions. The two isotropic yield functions were obtained from the isotropic Hershey Hosford yield function given as: (2 35 ) In Equation (2 35 ), is the yield function, , and are the principal values of the stress deviator and a is the homogenization parameter. For 2 D loadings, this function reduces to: , with , (2 36 ) Next, the following two linear transformations are applied to the stress deviator: (2 37 ) (2 38 ) On substituting X in Equation (2 36) the anisotropic yield function is obtained: , (2 39 )

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40 It was proposed that this idea can be generalized to linear transformations, , and isotropic yie ld functions, . The resulting yield function is given by, (2 40 ) 2.2.6 Yld2004 18p C riterion ( Barlat et al. (2005) ) In 2005, Barlat et al. described the general aspects of applying the linear transformations operating on the Cauchy stress tensor to extend to anisotropy a pressure independent isotropic yield function. Th e principal values of the transformed tensor were then used to replace the principal values of the Cauchy stress deviator in the isotropic yield function to obtain the anisotropic yield function. The transformation is applied as: (2 41 ) In Equation (2 40), ' is the Cauchy stress deviator, is the transformed stress tensor, C is an anisotropic linear tensor, is the Cauchy stress tensor and T transforms the Cauchy stress tensor to the stress deviator. By applying up to two such transformations to the Cauchy stress deviator (using 18 independent coefficients) the Hershey and H osford yield criterion was extended to an orthotropic yield criterion Yld2004 18p. This is a three dimensional yield criterion as opposed to Yld2000. The expression of this criterion is given as: (2 42 )

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41 is obtained by a single transformation as and is obtained as . In order to determine the coefficients for the anisotropic materials, an error function was minimized. (2 43 ) In Equation (2 43 ), p represents the number of experimental flow stresses and q represents the number of experimental r values. Each term is weighted by w . This criterion was shown to represent the yield loci of aluminum alloys with accuracy. 2.2.7 Cazacu and Barlat (2004) C riterion So far all the macroscopic yield functions reviewed were developed only to account for the anisotropy present in metals and their alloys. However, no attention was given to modeling other phenomenon observed in low symmetry metals and their alloys which have fewer active slip systems. This phenomenon is the asymmetry in yielding which occurs due to the activity of twinning systems. Cazacu and Barlat (2004) proposed an isotropic yield criterion to acc ount for tension compression yield asymmetry which is of the form: (2 44 ) In Equation (2 44) J 2 and J 3 are the second and third invariant of the Cauchy stress deviator and c is a constant that accounts for the asymmetry in yielding based on the sense of loading and is given by: , (2 45 )

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42 where and are the uniaxial yield stresses in tension and compression, respectively. The yield criterion can be reduced to the von Mises criterion by reducing the constant c=0. To ensure convexity of the yield surface the condition imposed on the constant is: c . This isotropic yield function was later extended to include orthotropy using the generalized invariants approach of Cazacu and Barlat (2001, 2003) and applied to the description of Mg and its alloys. 2.2.8 Cazacu et al . (2006) C riterion To account for the tension compression asymmetry due to shear deformation mechanisms in metals, an isotropic yield criterion was proposed by Cazacu Plunkett Barlat (Cazacu et al., 2006) in the form: (2 46 ) In Equation (2 46), , i =1...3 denote the principal values of the Cauchy stress deviator, a is a homogenization paramet er and can range from 1 to . k is a material parameter which accounts for yield stress asymmetry based on the sense of loading and whose physical significance can be seen directly from uniaxial tension and compression tests. It should be noted that for a=2 and k=0 the yield criterion (Equation 2 46) reduces to the von Mises yield criterion. This yield criterion relates to a convex yield surface and is highly effective in modeling isotropic materials with strength differential effects. Next, in order to account for both asymmetry in yield stresses in tension and compression and anisotropy observed in low symmetry materi als the criterion (Equation

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43 2 46 ) was extended to include orthotropy. Following Hill (1948), the orthotropy was introduc ed by means of a linear transformation on the Cauchy stress deviator. The principal values of the Cauchy stress deviator in the isotropic criterion are substituted by the principal values of the transformed tensor as: ( 2 47 ) with i, j, k, l= 1...3, C is a fourth order tensor that accounts for anisotropy . This fourth order tensor C satisfies the major an d minor symmetry (C ijkl =C jikl =C klij =C lkij , for i, j, k, l =1...3) and the requirements of invariance with respect to th e orthotropy group. In Chapter 3 , this criterion (further abbreviated as CPB06 model ) is discussed in further detail and will be applied to model deformation behavior of Mg AZ31.

4 5 alloys even under monotonic lo ading conditions, Graff et al. (2010) and Steglich et al. (2012) hav e used the yield function proposed by Cazacu and Barlat ( 2004 ) in conjunction with a directionally dependent hardening model. Also, recently Mekonen et al. (2011) used the same criterion and evolution laws for each anisotropy parameter to describe the combined effects of texture evolution and temperature on the plastic flow of magnesium alloy AZ3 1 subjected to uniaxial tension and compression. To account for evolving anisotropy, a linear interpolation based approach was proposed by Plunkett et al. (200 6; 2008 ) in conjunction with the anisotropic yield function described by Cazacu et al. ( 2006 ) and applied to model the mechanical response of Zr under quasi static monotonic tension, compression and bending . The main focus of this research is to descri be the evolving anisotropy of Mg AZ31 and especially the unusual hardening characteristics of this material. The modeling work is based on the experimental data of Khan et al. (2011), which will be briefly summarized in the following. 3.1 Experimental Dat a on Mg AZ31 The material used in th is work is a commercial magnesium alloy, named Mg AZ31, (3wt%Al, 1wt%Zn, Mg bal.), which is initially orthotropic. Its initial texture was determined by electron backscatter diffraction (EBSD) and X ray diffraction (XRD), showing that the c axes of most of the grains were distributed along the normal to the plane of the sheet making an angle of 30 0 with the sheet normal. For more information on texture analysis , see Chapter 4 . Experimental data in uniaxial tension and uniaxial compression have b een reported by Khan et al. (201 1) for several strain rates ( 10 4 /s, 10 2 /s and 1/s ) and temperatures (room temperature, 150 0 F and 300 0 F ). Tension s pecimens in the sheet

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46 plane directions i.e. rolling, transverse and at 45 degree to the rolling directi on ( designated as RD, TD, and DD respectively ) were machined using ASTM E8 and sub sized dog bone shaped specimens with a gage length of 31.0 mm, gage wi dth of 6.4 mm, and 2.0 mm thickness. Quasi static compression specimens in the same directions were prepared by bonding two sheets with specimen dimensions of 12.7 mm length, 11.4 mm gage width, and 4.0 mm thickness. Compression specimens were also prepare d to perform experiments along the normal direction of the sheet. Figure 3 1 shows the stress strain curve s under a strain rate condition of 1/s at room temperature for uniaxial tension and uniaxial compression tests along the RD, TD and DD orientations, respectively . The plastic strains are obtained from the total strains by subtracting the elastic strains , computed using isotropic elasticity relations with the values for the Young modulus E = 45 GPa and Poisson ratio = 0.3 . Note that the Mg AZ31 alloy is orthotropic, and exhibits the highest strength in the transverse direction and the lowest strength in the rolling direction for both unixial tension and unixial compression loadings. The tensile stress ratio between these two dir ections is 1.245 at yielding, but this ratio decreases as the effective plastic strain increases. For uniaxial compression, the stress ratio is smaller at yielding = 1.0 62 , but increases with the effect ive plastic strain. The main particularity of the plastic behavior of the Mg AZ31 alloy is the S shape of the stress strain curve observed in uniaxial compression. Note that t he hardening rate suddenly increases at an effective plastic strain of 5%. Furthe rmore, comparing the stress strain response for uniaxial tension and uniaxial compression in each orientation (Figure 3 2), it is evident that Mg AZ31 alloy

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47 displays strong tension compression asymmetry (see also the yield values for each orientation and l oading condition reported in Table 3 1). Figure 3 1 . Uniaxial test results (Khan et al. 2011) for strain rate 1/s in three in plane orientations showing the material's anisotropy in A ) Tension and B ) Compression. Moreover, the tension compression asymmetry between uniaxial tensile and uniaxial compressive tests is evolving with accumulated plastic deformation, the strength differential effect being significant at low plastic strain ( and at yielding) , but saturates (i.e. almost zero) for effective plastic strain in the range of 9 10%. Table 3 1 . Initial y ield stresses in tension and compression for Mg AZ31 at room temperature (Khan et al. 2011) . Loading direction Tensile yield stress (MPa) Compressive yield stress (MPa) 0 167.85 98.08 45 196.05 101.36 90 208.90 104.38

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48 Mg AZ31 sheet specimens were also mechanically tested under different strain rates varying from 10 4 /s to 1/s at room temperature, 150 0 F and 300 0 F in uniaxial tension and compression tests along different directions. It should be noted that even at 300 0 F (or 148.9 0 C) the elastic properties such as the Y oung's modulus and P oisson ratio remain the same as at room temperature. Figure 3 2 . Comparison between the mechanical response in uniaxial tension and compression at strain rate 1/s along A ) Rolling direction (RD) . B ) 45 degree direction (DD) . C ) Transver se direction (TD) . Figure 3 3 and Figure 3 4 show the effect of temperature on the stress strain response for two different strain rates: 10 4 /s and 1/s, respecti vely. Note that , irrespective of temperature and strain rate , the material displays very strong anisotropy

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49 and tension compression asymmetry. As ob served in most metals, regardless of the strain rate, an increase of the temperature leads to lower strength in uniaxial tension and unixial compression. It is worth noting that hardening in compression is strongly depend ent on the temperature and the stra in rate. An increase of the temperature reduces the rate of hardening in uniaxial compression. Also only for certain combinations of strain rate and temperature (low strain rate and high temperature), the unusual hardening and S shape of the stress strain curve observed in uniaxial compression tends to disappear ( e.g. there is no sudden change in slope at 5% plastic strain for uniaxial compression in RD, DD and TD at 300 0 F and strain rate of 10 4 /s but a clear S shape and unusual hard ening at 5% plastic strain for uniaxial com pression at room temperature for the same strain rate ). In conclusion, the mechanical data indicate s that in order to correctly predict the plastic behavior of this Mg AZ31 alloy, the tension compression asymmetry and the orthotropy of this material has to be accounted for in the macroscopic modeling approach .

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50 Figure 3 3. Uniaxial test results (Khan et al. 2011) at room temperature , 150 0 F and 300 0 F for s train rate of 10 4 /s . A) rolling direction (RD). B) diagonal direction (45 0 ). C) transverse direction (TD) .

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51 Figure 3 4. Uniaxial test results (Khan et al. 2011) at room temperature , 150 0 F and 300 0 F for s train rate of 1 /s . A) rolling direction (RD). B) diagonal direction (45 0 ). C) transverse direction (TD) .

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52 Based on the mechanical data, it can be observed that the mechanical response depends on the relative orientation between the loading direction and the symmetry axes of the material. Hence, an anisotropic yield criterion must b e used to describe the material behavior of Mg AZ31. The description of the elastic plastic modeling framework ( S ection 3.2 ) and the specific expressions for the yie ld criterion and hardening laws (S ection 3.2.1 ) are given in Section 3.2 . The implemen tation of this model in the finite element framework is shown in S ection 3.3 . Additionally, it is imperat ive to account for the material's anisotropy and to correctly model the rotation of the orthotropy axes throu gh the deformation process. This issue is addressed in S ection 3.3.1 . 3.2 Elastic/ Plastic Constitutive Model To explain and model the plastic deformation response of Mg AZ31 along various strain paths , we consider an elastic plastic model with yielding based on the Cazacu et al . (2006) yield criterion . This criterion is used in conjunction with an isotropic hardening law. The governing equations are first presented. The usual convention used in metal plasticity i.e. , tensile strains and stresses are considered positive is adopted. The total r ate of deformation D (the symmetric part of , where F is the deformation gradient) is considered to be the sum of an elastic part and a plastic part . The elastic stress strain relationship is given by , , ( 3 1 ) with being the Green Naghdi rate which is an objective rate o f the Cauchy stress tensor ( Green and Naghdi (1965), ABAQUS, 2009) ,

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53 contracted product between the two tensors. In Equation ( 3 51 ), is the fourth order stiffness tensor, which is expressed with respect t o any coordinate system as , ( 3 2 ) ij being the Kronecker unit delta tensor while G and K are the shear and bulk modul i , respectively. The evolution of the plastic strain is given by an associated flow rule: , (3 3 ) where is the plastic multiplier. Hardening is considered isotropic and is governed by the accumulated plastic strain. Thus, the plastic potential in Equation ( 3 53 ) is of the form: , ( 3 4 ) where, is the effective stress associated with the anisotropic yield criterion while is the corresponding equivalent plastic strain calculated based on the work equivalence principle while is the hardening law. 3.2.1 Or thotropic Cazacu et al. (2006) Yield C riterion: Application to Mg AZ31 The anisotropic yield criterion used in this study is of the form : (3 5 ) where, are the principal values of C:s with s= tr ( )I ; I denoting th e second order identity tensor, tr being the trace operator tr ( )= kk and is the effective stress associated with this criterion . In Equation (3 5 ), k represents the strength differential (SD) parameter, a is the degree of homogeneity, while C is a fourth order

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54 orthotropic tensor (Cazacu et a l. , (2006)), while m is a constant defined such that the equivalent stress, , reduces to the tensile stress along x (or RD) direction. Thus, m is expressed in terms of the anisotropy coefficients C ij ,with i, j =1...3 and the material parameter k as follows: (3 6 ) where , , ; . The effects of micro structur al evolution during proportional loading will be incorporated into the yield function (Equation 3 5 ) by accounting for the evolution of the anisotropy coefficients and str ength differential (SD) paramete r with the effective plastic strain. For Mg AZ31, we consider t he hardening function Y( ) to be given by the experimental tensile stress strain curve along RD and use a Voce hardening law (E quation 3 4). , (3 7 ) where A 0 , B 0 , C 0 are constants . The anisotropy coefficients and SD parameter k evolve as a function of the effective plastic strain and are determined for several levels of effective plastic strain , , where corresponds to initial yi elding while is the highest level of effective plastic strain in the given mechanical test. For a 3 D stress state s and o rthotropic symmetry, the fourth order orthotropic and symmetric tensor has nine non zero components in the ( x , y , z ) coordinate system

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55 ( x , y and z represent the R D, TD and ND, respectively). Using Voigt notations, this anisotropic tensor is represented as: (3 8 ) The yield function is homogeneous of degree one in its arguments (E quation (3 5 )) . Thus , if we replace C ij by C ij , being any positive number, the respective expressions for the effective stress ( Equation ( 3 5 )) and Lankford coefficients remain unchanged. Hence , we can scal e the anisotropy coefficients by C 11 . The degree of homogeneity , a is fixed and set to 2 . The anisotropy coefficients C ij and strength differential parameter k, involved in the macroscopic yield criterion need to be determined at several strain level s from =0 ( initial yield ) to = 10%. This is achieved by minimizing the following error fun ction , (3 9 ) In Equation (3 9) , and denote the tensile and compressive yield stresses for uniaxial loading at an angle to the rolling direction, respectively; stands for weights given to the respective data , and E is the magnitude of the error vector . The data points used in the er ror minimization function (Equation 3 9 ) are taken from the experimental data (Khan et al. , (2011)) at initial yield and the following fixed levels of effective plastic strain = 3%, 5%, 6%, 8% and 10% for both tension and compression in the RD, DD

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56 (45 0 to the rolling direction), and TD respectively. For the ND (normal direction) only compression data were available and were used in the minimization. The Lankford coefficient in RD tension, (i.e. ) reported in Khan et al. 2011 varies from 0.28 3.5 with effective plastic strain. On the other hand , several authors have reported r values in the range 1 to 5 (Agnew et al. 2005 ; Lou et al., 2007, etc. ) in uniaxial tension tests along the in plane direction of the sheet . Recently, Kang et al. ( 2013 ) , showed using Digital Image Correlation (DIC ) measurements that the r value in the RD tension test are close to 2 at initial yield and increase up to 2.5 a t a strain of 20% . It is important to note that due to the absence of any reliable data on r value for Mg AZ31 sheets, r values are not directly considered in the determination of the parameters of the Cazacu et al. (2006) yield criterion. However, we ensure that the in pl ane (RD TD plane) r values always stay within reasonable bounds (1
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57 methodology used for updating the anisotropy coefficients and the strength differential parameter. 3.2. 2 Identification P rocedure and New Evolution L aws for the Material P arameters for Mg AZ31 at Different Temperatures and Strain R ates The identification of anisotropy parameters, i.e. the C ij coefficients and the SD parameter k for AZ31 is carried out using the objective function (Equation 3 9 ) and the interior point method algorithm in Matlab (2010b) . Figure 3 5. Theoretical yield surfaces at room temp erature for several levels of accumulated plastic strain according to the Cazacu et al.(2006) yield criterion (Equation 3 5 ) in comparison with data. Strain rate of 1/s. Figure 3 5 displays the biaxial plane projections ( ) of the yield loci for Mg AZ31 at room temperature for the different levels of accumulated plastic strain. Data plotted and used in the identification correspond to a strain rate of 1/s.

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58 It is worth noting that the shape of the yield surfaces evolve from a triangular shape toward s an elliptical shape to account for the evolving tension compression asymmetry of the Mg AZ31 alloy. T he evolution of the S D parameter (Figure 3 6), k , and of the anisotropy coefficients with the accumulated plastic strain can be approxim ated by a linear interpolation between coefficients as follows: (3 10 ) The interpolation parameter involved in Equation (3 10 ) is defined as , (3 11 ) Note that the evolution of the SD parameter also exhibits a S shape (F ig ure 3 6) . It follows that the anisotropic coefficients and the SD parameter k correctly reflect the specificities of the plastic behavior of the Mg AZ31 alloy. The numerical values of all the material parameters involved in the model are given in Table 3 2 for several effective plastic strain levels .

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59 Figure 3 6. Evolution of the anisotropy parameters with accumulated plastic strain at r oom temperature for s train rate 1/s.

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60 Table 3 2 . Model parameters for Mg AZ31 at room temperature and strain rate of 1/s. Strain C 22 C 33 C 12 C 13 C 23 C 44 k 0.03 1.1706 5.8933 1.2804 0.3914 0.029 3.1505 0.6039 0.05 1.1706 5.8933 1.2804 0.3914 0.029 3.1505 0.6039 0.06 1.0493 4.6933 1.6887 0.8782 0.6089 3.8512 0.4943 0.08 1.2324 2.288 2.3926 1.6062 1.2798 5.0012 0.1795 0.10 1.2639 0.872 2.3134 2.1241 1.7895 4.8756 0.125 Figure 3 7. Theoretical yield surfaces corresponding to different levels of accumulated plastic strain according to the anisotropic form of the Cazacu et al. (2006) criterion for tes t data measured at 300 0 F at a strain rate of 1/s . The same procedure is used to identify the yield loci at several temperatures . The yield surfaces corresponding to temperature of 300 0 F (same strain rate of 1/s) are

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61 plotted in the same biaxial plane (in Figure 3 7 ) as was done previously . As seen from the experimental stress strain curves shown in Figure 3 4, the tension compression asymmetry at 300 0 F is more pronounced in the large plastic strain range. It follows that the ev olution of the shape of the yield locus with the effective plastic strain is less drastic than at room temperature, and only a small deviation from triangular shape is observed. On the other hand the mechanical stress strain response at room temperature an d 150 0 F are almost the same (Figure 3 4 shows response at the same strain rate and 150 0 F ). Figure 3 8 show s the yield loci for a strain rate of 10 4 /s and temperature of 150 0 F and 300 0 F, respectively. It is worth noting that at room temperature, the plastic behavior of the Mg AZ31 alloy is not influenced by the strain rate (compare Figure 3 3 and Figure 3 4). By using the Cazacu et al. (2006) yield criterion to model the plastic behavio r of Mg alloys, the particularities of the plastic flow in tension and in compression are well captured. Additionally , the influence of the strain rate and the temperature can be accounted for by identifying the coefficients of this criterion (Appendix A) .

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62 A B Figure 3 8. Theoretical yield surfaces corresponding to different levels of accumulated plastic strain according to the anisotropic form of the Cazacu et al. (2006) criterion for test data at A ) 150 0 F and B ) 300 0 F and strain rate of 10 4 /s .

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63 3.3 F inite E lement I mplementation of the Elastic/Plastic M odel 3.3 .1 Rotation of the Anisotropy A xes The deformation of a solid is driven by the deformation gradient [ F ], which enables to determine the position and orientation of any vector in the deformed configuration from its original coordinates in the reference configuration (usually considered to be the initial configuration) . Let's consider the vector x in the current configuration which maps the vector X belonging to the initial configuratio n . The deformation gradient F is defined as (3 12 ) Any elastic plastic deformation c an be decomposed in a rigid body deformation and a stretch. Let's consider that the rigid body deformation could only be a rigid body rotation, defined by the orthogonal rotation matrix R . By the polar decomposition theorem, (3 13 ) where, U is called the right stretch tensor and V is called the left stretch tensor (Malvern, 1968 ). For orthotropic material s undergoing elastic plastic deformation, the orientation of the symmetry axes of the material must be defined for every deformation process. For orthotropic materials, it is usual to update the orientation of the orthotropy axes by rigid rotation. Thus, a rigid rotation will not influence the mechanical response. The anisotropy axes are updated at increment "n+1" by using the rotation tensor associated with the deformation gradient at increment "n". Let [R] n be the rigid body rotation

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64 associated with the deformation gradient at increment "n", then the anisotropy axes are updated as follows a 1 n+1 =[R] n a 1 n a 2 n+1 =[R] n a 2 n (3 14 ) a 3 n+1 = [R] n a 3 n where a 1 , a 2 , a 3 , denote the unit vectors associated with the orthotropy axes. It should be noted that the orthotropy properties of the material are thus conserved. 3.3.2 Return Mapping A lgorithm Knowing the state variables at time t n , in order to determine them at time t n +1 , first a trial state (elastic predictor) is computed. If , the stress state is elastic and then . If , there is plastic flow and the following non linear system must be solved for and : (3 15 ) Let m denote the local iteration counter, with m=0 corresponding to the elastic trial state ( and ) . T he plastic multiplier and the stress increment are updated as follows: , (3 16 ) where , denotes the variation of the variable between iterations and : (3 17 )

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65 The incremental variations of the plastic mu ltiplier and of the stress tensor are obtained through a Taylor expansion of the non linear system ( Equation 3 1 6 ): (3 18 ) The variation of the plastic multiplier between two iterations is expr essed by solving the system ( Equation 3 18 ): (3 19 ) w here denotes a fourth order tensor expressed as: (3 20 ) with being the fourth order identity tensor. Thus, it follows the variation of the stress increment is given as : (3 21 ) The stresses and the plastic strains are then updated until a specified tolerance of the yield function has been obtained (usually, is used as a

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66 convergence criterion). Once convergence is reached, the updated stresses and strains are accepted as the current state. The tangent matrix relates the current stress increment to the current total strain increment . It is used to predict the total strain increment for the next iteration: (3 22 ) 3.4 Results : Simulation of Uniaxial T ests The orthotropic yield criterion Cazacu et al . (2006) in conjunction with the proposed distortional hardening law (Equation 3 5 to 3 7 ) was implemented in the f inite element (FE) code ABAQUS using the fully implicit integration algorithm presented in Section 3.3 . Key novel aspects consisted of allowi ng the distortion of the respective yield surfaces ( (Equation 3 11)). Next, t ensile and compressi on tests were simulated by using a single continuum element with eight integration points (C3D 8) . The simulated normal stresses in the appropriate loading di rection versus the effective strain are compared to the experimental data (Figures 3 9 ( a c ) ). Note that CPB06 yield function accurately reproduces the data for each loading orientation. Hence, the anisotropic Cazcau et al. (2006) macroscopic yield criteri on along with the proposed evolution laws for its anisotropy coefficients and strength differential parameter and a hardening law in the rolling direction (RD) tension is able to capture the observed stress strain response in all the orientations of loadin g and in both tension

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67 and compression. Most importantly, for the first time the unusual hardening characteristics of the material were predicted with great accuracy. Figure 3 9. Comparison between theoretical predictions according to the proposed model (interrupted lines) and experimental uniaxial st ress strain response (symbols) . A ) Rolling direction . B ) Transverse direction . C ) Normal direction. Validation of the model and specifically the methodology used for the identification of the material parame ters along with the new evolution laws ( Equation (3 11)) will be provided in Chapters 5 and 6 for free end torsion and buckling behavior . In Chapter 4 , a polycrystalline approach is also used to model the material behavior. Particular attention is given to the challenging problem of modeling the texture evolution. It is worth noting that up to now the texture evolution of Mg AZ31 was not fully understood or modeled for different loading paths.