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Multi-Scale Modeling of Deformation of a Magnesium Alloy

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Title:
Multi-Scale Modeling of Deformation of a Magnesium Alloy
Creator:
Chandola, Nitin
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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english
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1 online resource (211 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
CAZACU,OANA
Committee Co-Chair:
SANKAR,BHAVANI V
Committee Members:
KUMAR,ASHOK V
BOGINSKIY,VLADIMIR L
Graduation Date:
8/9/2014

Subjects

Subjects / Keywords:
Alloys ( jstor )
Anisotropy ( jstor )
Asymmetry ( jstor )
Axial strain ( jstor )
Buckling ( jstor )
Compressive stress ( jstor )
Deformation ( jstor )
Modeling ( jstor )
Plasticity ( jstor )
Plastics ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
az31 -- buckling -- cpb06 -- deformation -- vpsc
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mechanical Engineering thesis, Ph.D.

Notes

Abstract:
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 )
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: CAZACU,OANA.
Local:
Co-adviser: SANKAR,BHAVANI V.
Statement of Responsibility:
by Nitin Chandola.

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UFRGP
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Copyright Chandola, Nitin. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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LD1780 2014 ( lcc )

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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

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© 2014 Nitin Chandola

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To my m other , f ather and s ister

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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 , Sandlöbes, 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)

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23 model (Agnew and Duygulu (2005), Jain et al. (2007), Walde and Reidel (2007), Wang et al.(2010 )) originally proposed by Molinari et al. (1987) and later extended by Lebensohn and Tom é (1993) and (b) a crystal plasticity based finite element method (CPFEM) (Choi et al. (2010, 2011) , Dawson (2004) , Prakash et al. (2009), Izadbakhsh et al. (2011), Kal idindi et al. (1992), Kok et al. (2002), Inal and Mishra (2012)). The visco palstic self consistent model (Lebensohn and Tomé (1993)) was specifically developed for low symmetry materials (hexagonal, trigonal, orthorhombic) but has also been successfully employed for cubic materials. In addition to providing the macroscopic stress strain response, it accounts for hardening, reorientation and shape change of individual grains while ensuring that the average response of all grains is consistent with the mac roscopically imposed conditions. Very recently, Steglich et al . (2012) used the VPSC approach to generate yield surfaces for Mg AZ31 at different values of plastic work. However, the simula ted surfaces have sharp corners. I n particular , at plastic work pe r unit volume, W P =20 MP a the surface displays a " fish tail " shape. In contrast to V PSC , CPFE based model ing has the potential to provide information on local effects (e.g. local microstructural features or stress concentration in the individual grains ). For crystal plasticity modeling of Mg alloys also refer to Lévesque et al. (2010) and Inal and Mishra (2012) . Both (VPSC and CPFEM) the crystal plastcity theory based models are further discussed in S ection 2.1. Anisotropic yield functions that captu re with increased accuracy both the anisotropy in yield ing and the anisotropic distributio n of the Lankford coefficients ( ratios of the thick ness to transverse strain rate in uniaxial tension) of metals with cubic crystal structure have been developed ( Cazacu and Barlat, 2003 , Cazacu and Barlat, 2004 ,

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24 Barlat et al., 2005 and Hu, 2005 , Cazacu et al. 2006). For example, elastic plastic deformation mod els have been developed for Alumnium and its alloys (Hill 1979, Hill 1990, Barlat et al. 1991, Barlat 1997, Lee et al. , 1997, Barlat et al. 2003, Cazacu and Barlat 2004, Bron and Besson 2004, Yoon et al. 2010, Soare and Barlat, 2011) in order to improve t heir performance in forming operations and to evaluate their structural capability. As a result, this metal and its alloys have been used to form a variety of components from food cans to automotive parts. However, these models developed for metals with cu bic crystal symmetry are unable to account for the unique features of the plastic deformation of hexagonal materials, namely the significant difference between tension and compression response . As a result, there was a need for development of new a nisotropic yield functions to account for the deformation behavior associated with hcp metals (Cazacu et al . (2006), Plunkett et al. (2008 ) for Mg alloys , Nixon et al., (2010) for titanium ). In the following, the main contributions in both poly crystal plas ticity framewo rk and that of the mathematical theory of plasticity are presented. 2.1 Polycrystalline P lasticity 2.1.1 Visco plastic self c onsistent m odel for polycrystalline materials Crystal plasticity aims at obtaining the plastic deformation in a poly crystalline material while taking into account explicitly the details of geometry and physics of deformation at the single crystal level. The self consistent homogenization based crystal plasticity model proposed by Molinari et al.(1987) and implemented by Lebensohn and Tomé (1993) is based on the assumption that the elastic deformations are small

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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

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26 twinning. Thes e twinned regions contribute to higher values of the latent hardening coefficients describing slip twin and twin twin interactions. With regards to texture evolution due to twinned fractions the PTR (Predominant Twin Reorientation) scheme proposed by Tomé (1991) is used. Under this scheme within each grain g the shear strain contributed by each twin system t, and the associated volume fraction in the grain are tracked. The sum over all the twin systems associated with a given twin mode and over all grains, represents the 'accumulated twin fraction' , V acc,mode , in the aggregate for the particular twin mode. Since it is not numerically feasible to consider ea ch twinned fraction as a new orientation a statistical approach is used to determine new orientations. The re orientation of the grain depends on the following condition: , (2 2 ) In E quation (2 2) , is the threshold volume fraction, is the 'effective twinned fraction' i.e. is the volume associated with the fully reoriented grains for the given mode, A th1 and A th2 are parameters which are to be determined. If the highest accumulated volume fraction, , for the given mode of twinning deformation is larger than then the grain is allowed to re orient and and are both updated. This process is repeated until either all grains are randomly checked or until the effective twin volume exceeds the accumulat ed twin volume. The grain constitutive Equation (2 1) is linearized as,

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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 · 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.

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44 CHAPTER 3 APPLICATION OF CAZACU ET AL. (2006) CRITERION TO MODEL PLASTIC DEFORMATION OF MG AZ31 With increased demand for extending the use of Mg alloys there arises a need to develop new macroscopic constitutive models capable of accounting for their unusual plastic flow characteristics . A comprehensive set of experimental data on the mechanical behavior of Mg AZ31 sheets subjected to monotonic quasi static uniaxial te nsile and compression tests has recently been published (Khan et al. , (2011)). It has been shown that, Mg AZ31 rolled sheets displays anisotropy i n the plane of the sheet ( rolling direction transverse direction plane) and a pronounced difference between the mechanical response in tension and compression tests at the macroscopic level , which are associated with the evolution of its microstructure ( Hosford 1993 , Tomé et al. 2001 , Kaiser et al. 2003 , Agnew and Duygulu 2005 , Cazacu et al. 2006, Plunkett et al. 2008 ). In the literature, in order to model the macroscopic behavior of Mg AZ31, yielding is described using the anisotropic yield function s developed at University of Florida by Cazacu and collaborators (Cazacu and Barlat, 2004; Cazacu et al. , (2006 ) , see Chapter 2 for the expressions of these criteria). The rationale for using such criteria is that they have been developed specifically for low symmetry materials ( Cazacu and Barlat (2004); Cazacu et al. (2006), Plunkett et al. (2008), Nixon et al., (2010)) such as magnesium and its alloys ( Cazacu and Barlat (2004 ), Cazacu et al. (2006)) and ha ve been shown to account for anisotropy as well as the diffe rence in response between tension and compression loading . Furthermore, to account for evolving microstructure which induces evolving anisotropy in magnesium

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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.

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68 CHAPTER 4 MODELING EVO L U TION OF TEXTURE IN MG AZ31 Numerous st udies have been conducted to identify the deformation mechanisms operational under quasi static deformation of Mg AZ31 alloy plates at room temperature. Early studies on Mg and its alloys reported type dislocations on the basal <11 20> planes and non basal planes, such as prismatic {10 10} (Ward Flynn et al. 1961) and pyramidal {10 11} (Reed Hill and Robertson, 1957 58) planes. The existence of non basal dislocations along and axes for Mg AZ31 at room temperature was only recently confirmed ( Agnew and Duygulu, 2005). These authors further attempted to capture the observed anisotropy, in particular the very large values of the Lankford coefficient corresponding to the rolling direction ( RD ) in Mg AZ31 at room temperature which are a direct result of the strong initial basal texture and of the obse rved non basal slip mechanisms, by using the visco plastic self consistent crystal plasticity model with the VPSC code developed by Lebensohn and Tom é (1993) . However, texture was measured and pred icted in only uniaxial tension tests along two orientations, RD and TD. From the results presented, all the deformation mechanisms active within Mg AZ31 when subject to these strain paths were not determined. In particular, the role played by mechanical t winning on the deformation of Mg AZ31 in uniaxial compression was established only in later studies (Jain and Agnew 2007, Knezevic et al. 2010, Khan et al. 2011). Although the existence of small fractions of contraction twins has also been reported in unia xial compression (Knezevic et al. 2010), the unusual sigmoidal shape (S shape) of the stress strain curve (Figure 3 2) was mainly attributed to the activation of {10 12} extension twinning and it was hypothesized that the contraction twins only serve to fu rther strain harden the material. It was

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69 established that twinning activity gives rise to a radical reorientation of the volume fraction of the grain that has twinned thus producing substantial modification of texture, which affects subsequent deformation by slip ( Lou et al., 2007). Four different approaches have been developed in order to model twinning in the framework of crystal plasticity: (a) the predominant twin reorientation (PTR) scheme ( Van Houtte, 1978); (b) the volume fraction transfer (VF T) method ( Tomé et al, 1991); (c) the so called Lagrangian approach (Wu et al., 2007), and the composite grain scheme (Proust et al, 2009). All these methods have been used in conjunction with either Taylor type (1938a, 1938b) or self consistent homogeniza tion methods. For further details on the VPSC modeling framework the reader is referred to the literature review presented in Section 2. In particular, the VPSC model of Lebensohn and Tom é (1993) has been used previously for modeling the deformation behavior of magnesium AZ31. For example, Agnew and collaborators (e.g. Agnew and Duygulu, 2005, Jain and Agnew, 2007) have calibrated the VPSC model using as input the initial texture and unia xial tension and compression stress strain curves and further used the model to demonstrate the importance of the slip (also refered to as cross slip) on texture evolution along uniaxial loading paths. Walde and Reidel (2007) used the VPSC model to gain understanding on how the starting texture affects earing formation during cup drawing at elevated temperatures. Choi et al. (2009) identified the VPSC model parameters using the uniaxial curves along the in plane axes of symmetry (i.e. RD and TD) and further simulated the microstructure evolution along the same uniaxial strain paths. It is to be noted that there is a large discrepancy/scatter concerning the values of the material parameters used in the VPSC model for Mg AZ31 by different authors. For

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70 example, the values of the initial critical resolved shear stress and of the hardening parameters involved in the single crystal hardening law of the Voce type ( Equation 2 4) for the various deformation mechanisms as reported by different authors is given in Table 4 1 to 4 4. It is also to be noted that there is a discrepancy between investigators as to which systems need to be active in the microstructure in order to accommodate the applied strain for each deformation process studied. For example, Walde an d Reidel (2007) considered only three deformation systems active in uniaxial loading, while Jain and Agnew (2007) and Wang et al. (2010) cite four deformation systems (three slip modes and tensile twinning) to be active. Table 4 1 . VPSC parameters dete rmined by Walde and Reidel (2007) for Mg AZ31 Deformation Mode Pa Pa Basal 40 40 250 0 Pyramidal II 100 100 500 0 Tensile Twinning 40 0 30 30 Table 4 2 . VPSC parameters det ermined by Jain and Agnew (2007) for Mg AZ31 Deformation Mode Pa Pa Basal 25 12.5 2000 75 Prismatic 8 0 50 50 0 0 Pyramidal II 125 150 12500 0 Tensile Twinning 37.5 0 0 0

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71 Table 4 3. VPSC parameters determined by Wang et al. (2010) assuming affine linearization Deformation Mode Pa Pa Basal 9 1 5000 25 Prismatic 79 40 590 50 Pyramidal II 100 100 5000 0 Tensile Twinning 47 0 0 0 Table 4 4. VPSC parameters determined by Wang et al. (2010) assuming n eff =10 Deformation Mode Pa Pa Basal 17 6 3800 100 Prismatic 77 33 650 50 Pyramidal II 148 35 9600 0 Tensile Twinning 33 0 0 0 It has not been demonstrated in any of the research efforts on polycrystalline modeling of Mg AZ31 that both the stress strain response and texture evolution can be captured using the VPSC modeling framework. Furthermore, the predictive capabilities of the model have also never been demonstrated. Specifically, in all the publications cited, no attempt was made to predict the mater ial response along strain paths that were not used for the identification of the model parameters. The objectives of the research conducted as part of this dissertation are to :

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72 Provide a new identification procedure for the viscoplastic self consistent polycrystalline model. Verify the robustness of the identification procedure through comparison of the VPSC predictions with mechanical data used for calibration (i.e stress strain curves) as well as relative activity of the deformation mechanisms, and tex ture evolution during a given loading path. Predict the mechanical response and texture evolution for uniaxial loading at orientations different from the ones used to calibrate the model in order to demonstrate the capability of the model to account for me chanical anisotropy. Predict the mechanical response and texture evolution in simple shear. 4.1 New Procedure for I dentification of VPSC Material P arameters for Mg AZ31 All the experimental results on the Mg AZ31 sheet that will be used for calibration, validation, and for comparison with the predictions of the polycrystal plasticity model were reported in Khan et al. (2011). Specifically, this data set consists of experimental stress strain curves in uniaxial ten sion and compression for three in plane orientations: rolling direction (RD), 45 o to the rolling direction (DD), 90 o (TD) to the rolling direction , in plane simple shear, and uniaxial compression along the normal to the sheet (ND) (see also the stress stra in curves and the discussion of experimental data provided in Chapter 3) . Since in a typical polycrystal plasticity simulation, the polycrystal is represented as a discrete set of grains, in order to correctly capture the initial texture of the material it is necessary to take as input the orientation of a sufficient ly large number of grains

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73 with correctly determined area fractions. To this end, the entire EBSD scan (784x808 m 2 area) was used to generate 2762 weighted orientations. The corresponding init ial texture to be used in the simulations (Figure 4 1B) is similar to that reported by Khan et al. ( Figure 4 1A ). Note that the material displays a strong basal texture with an almost equal fraction of grains having the c axis slightly tilted away from the sheet normal towards +RD and RD, respectively (around ±30 0 ). Figure 4 1. Pole figures showing initial texture of Mg AZ31 sheet . A ) Reported in Khan et al. 2011 . B ) Measured from a large EBSD scan and used as input in the polycrystal model. It is impo rtant to also note that the rolled sheet displays orthotropic symmetry. Figure 4 2 shows the orientation distribution function ( odf ) plot generated using the entire EBSD scan data . In these plots due to crystal symmetry, the angular range of 2 is limited to 60 o ( Engler and Randle, 2010). It is concluded that the Mg AZ31 rolled sheet has an orthotropic material symmetry.

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74 Figure 4 2. ODF generated for 5 0 increments of using EBSD scan an d imposing orthotropic symmetry One of the objectives of this research is to provide a new step by step procedure for the determination of the material parameters that characterize the plastic deformation at the grain level, namely the critical resolved shear stress (CRSS) and the hardening parameters associated with each deformation system that may be operational. To this end, we need to isolate as much as possible the deformation systems . We start by determining the parameters that characterize the basal and prismatic
slip systems using the stress strain curve in uniaxial tension along RD. From the uniaxial compres sion stress strain curve in ND, we identify the parameters associated with the pyramidal slip system. Finally, from the uniaxial

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75 compression test along RD, we identify the extension twinning sy stem and estimate the validity of the identification of the parameters characterizing the system that was done on the basis of the ND compression test. The rationale for such a procedure is explained in the following. RD tension . As already mentio ned , in uniaxial tension along RD, the potentially active systems are prismatic slip and the basal
slip, the latter system being the most easily activated (e.g. Jain and Agnew, 2007). Note that b asal slip leads to a rapid rotation of the grain such tha t the basal planes lie parallel to the she et surface and the tensile axis. Therefore, this exhausts the possibility to activate other twinning systems. This conclusion is confirmed by the analysis of the measured final texture (Fig ure 4 2 B) which indicates that there is no change in the c axis orientation of the majority of the grains. Thus, it can be considered that the dominant deformation mechanisms are the basal and prismatic slip systems . Therefore, simulation of RD tension is conducted keeping th e initial critical resolved shear stress on all other deformation systems sufficiently high . The values of the critical resolved shear stress and of the parameters associated with hardening of basal and prismatic slip systems are calibrated such as the simulation results describe well the experimental macroscopic stress strain curve. ND compression . Since the c axis of most grains is aligned with the normal direction or deviates by a very small angle from it, plastic deformation along the c axis can be accommodated by twinning or cross slip . However, experimentally only a very small percentage of compres sion twins have been observed ( Proust et al. 2009, Knezevic et al. 2010, Khan et al., 2011). Therefore, it is considered that twinning activity can be

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76 neglected, so in calibrating the VPSC model the only active deformation mode for this strain path is considered to be pyramidal slip. Hence, the parameters associated with pyramidal slip are obtained by simulating the uniaxial compression test along ND and comparing the numerical stress strain response with the macroscopic experimental one . During such calibration the parameters determined for basal and prismatic
slip systems ar e kept fixed to the values determined in uniaxial tension test along RD . RD compression . A ~86 o reorientation of the basal poles during uniaxial compression along RD has been observed in experimental tests and is attributed primarily to the activation of t he {1 0 12} tensile twinning system ( Duygulu and Agnew 2005, Lou et al. 2007, Khan et al 2011). Hence, the stress strain data in uniaxial compression tests along RD are used to calibrate the parameters for th is system. Once all four deformation systems have been individually calibrated as outlined above the three tests (RD tension, ND compression, and RD compression) are simulated once again in order to achieve the closest possible prediction of stress strain response for all loadings . F or these four def ormation systems, t he values of the initial threshold stress, , of the initial hardening rate, asymptotic hardening rate, and back extrapolated threshold stress , (i.e. the parameters which are involved in the Voce hardening law , Equation (2 4)) are given in Table 4 5 .

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77 Table 4 5. Hardening parameters for the active deformation systems at room temperature Deformation Mode Pa Pa Basal 17.5 5 3000 35 Prismatic
85 33 550 70 Pyramidal II 148 50 8500 0 Tensile Twinning 40 0 0 0 In addition, in the model we allow for self and latent hardening using the coupling coefficients , , which empirically account for the obstacles that new dislocations (or twins) associated with system s ' create for the propagati on of dislocations (or twins) on system s . Specifically, the increase in the threshold stress of the system s , is calculated as: (4 1 ) For Mg AZ31, the latent hardening coefficients associated with the interaction between basal and non basal slip and twinning has been set to a value of 2 . This reflects that the low or lack of slip activity is the cause of twin formation. Given that in uniaxial compression along RD it is observed that twinning saturates at high values of strain (approx. 8%) and the slip systems are active again, the tensile twinning system must self harden . Thus, the self hardening parameter is also set to 2. On the other hand, the latent hardening coefficient associated with interaction between pyramidal slip and tensile twinning is set to 1 in order to capture the very lit tle twinning activity observed i n ND compression .

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78 Figure 4 3 A and Figure 4 3B show s the comparison between the predicted textures and a ll the experimentally available textures following RD tension and RD compression, respectively . Note that the simulated textures reproduce very well the measured textures for both strain paths . Al though no experimental data was available for comparison, the prediction of the texture at 6% deformation in ND compression is also given in F ig ure 4 3 C to show that the model predicts a pronounced difference between the final texture obtained in ND compression and RD compression, which is consistent with the relative activity of the deformation systems given in Fig ure 4 4 B and Figure 4 5 B , resp ectively. It is to be noted that this is the first time such an excellent agreement between model and experiments is reported.

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79 Figure 4 3. Comparison between the measured texture and predicted texture . A) uniaxial tension al ong RD at 13% strain (~fail ure). B) uniaxial co mpression along RD at 8% strain. C) ND compression (no measured texture available). Although accurate predictions of the available final textures is an indication of the robustness of the calibration of the VPSC model, in the following we also verify that the mechanical response is well described and that the predicted microstructure evolution and the predicted relative activities of the s lip and twin modes contributing to plastic deformation corroborate and that the se predictions are in agreement with general trends reported in the literature on Mg AZ31 .

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80 For RD tension comparison between the polycrystalline model predictions of the stress strain response (solid line) and data (symbols) is shown in Figure 4 4 A. On the same figure the predicted texture evolution is shown while in Figure 4 4 B the relative system activity is presented. Note that the experimental stress strain response ( solid line) is well reproduced by the model (symbols).

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81 Figure 4 4. D eformation response in RD tension . A) Stress strain response and evolution of the microstructure according to calibrated VPSC model (line) in comparison wit h mechanical test data (symbol) . B ) R elative activities of each deformation mode. Analysis of the predicted relative activities shown in Figure 4 4 indicates that the basal and prismati c
slip is the dominant mode of deformation, which corroborates with the experimental evi dence as well as with the predicted microstructure evolution which indicates alignment of the basal plane along RD. Furthermore, the activity of basal and prismatic slip systems inhibits the activation of tensile twinning and pyramidal II slip. The value of the initial threshold stress, , of the basal slip mode is the lowest ( Table 4 5 ) as it should be, given that this is the most easily activated slip mode in Mg AZ31 . The predicted RD tension curve exhibits a decreasing work hardening rate, which also indicates that the plastic deformation is dominated by slip. The predicted mechanical response and microstructure evolution in RD compression is shown in Figure 4 5 A, Figure 4 5B and Figure 4 5C . Note that the p redicted (0002) pole figures shows a clear evolution of texture. As should be expected,

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82 basal slip and extension twinning are the dominant modes of deformation for this strain path and their relative activity gives the characteristic ~86 0 rotation of the b asal plane which has also been observed in the mechanical test sample at 4% strain in RD compression ( Figure 4 5 A and Figure 4 5B )). Also, it is important to note that the predicted twin volume fraction is 73.5%, which is very close to the experimental tw in volume which is ~80% (Khan et al. 2011). Furthermore, the predicted evolution of the twin volume fraction as a function of plastic strain shown in Figure 4 5 C corroborates with the t exture evolution shown in Figure 4 5 A. Deformation twins, while growing and accommodating plastic strain, reorient grains from softer to harder orientations ( rotation of the basal plane in Figure 4 5 A ) and induce texture hardening hence the need to activate the hard pyramidal slip to further ac comodate plastic deformation . Saturation of twinning is predicted at about 9% strain, which corroborate s with the final experimental texture ( Figure 4 3 B ). The interaction between twinning and slip systems also explains the peculiar changes in work hardeni ng (the "knee" between ~3% and 5% strain and the concave up appearance of the stress strain curve above 6% strain) both unusual features being well reproduced by the model (Figure 4 5 A ). Figure 4 6 shows the stress strain response obtained from mechanica l tests (symbols) and the predictions of the VPSC model (lines) in ND compression, along with, the predicted evolution of texture and slip/twin activity.

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83

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84 Figure 4 5. D eformation response in RD compression . A ) Stress strain response and e volution of the microstructure according to calibrated VPSC model (line ) in comparison wit h mechanical test data (symbol) . B ) R elative activities of each deformation mode . C ) Predicted t win volume frac tion evolution . From the predicted basal pole figures ( Figure 4 6 A ), it can be seen that the c axis of the majority of grains further aligns itself along the sheet normal as the plastic strain increases . Thus, the observed changes in microstructure can be attributed mainly to crystallographic slip activities . We note much less texture evolution during ND compression as compared to RD compression. This is due to substantial activity of the pyramidal slip in ND compression, a system which inhibits twinning activity ( Figure 4 6 B ). Note that although pyramidal has the highest initial CRSS ( Table 4 5), in ND compression, this is the only system that can produce deformation along the c axis.

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85 Figure 4 6. D eformation response in ND compression . A ) Evolution of micro structure according to the calibrated VPSC (line) model . B ) R elative activities of ea ch deformation mode .

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86 Although the predominant twinning reorientation model used represents a simplified scheme for treating twinning compared to e.g. the composite grain model (Proust et al., 2009) and that we use a rather simple hardening law (Voce type hardening law given by Equation (2 4) compared to some more sophisticate d hardening laws based on dislocation densities such as tho se proposed by Bey erlein and Tomé (2008), the key features of the material's response are captured very well. Furthermore, the agreement between model predictions and the experimental stress strain response , measured textures, and final twin volume fraction is excellent. N ote also that the predicted microstructure evolution corroborates very well with the relative activity of the deformation systems and leads to a correct description of final textures. 4.2 Assessment of the Predictive Capabilities of the Polycrystal M odel In the following section, for the first time the polycrystalline model is used to predict the material's mechanical response for strain paths that were not considered in the model calibration. In particular, we assess the model capabilities to predict the strong anisotropy of the material. Next, we apply the model to predict the material behavior for uniaxial tension and compression along the in plane directions at 45 0 (DD), and 90 0 ( TD ) from RD , and uniaxial tension along ND, respectively. In all the simulations we use the same values of the material parameters (i.e. the values of the parameters given in Table 4 5). In each simulation, initially all the systems are considered active. Figure 4 7 shows the stress strain response obtained from mechanical tests (symbols) and the predictions of the model (lines) in TD tension, along with, the predicted evolution of texture and relative activity of the deformation systems .

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87 Figure 4 7. D eformati on response in TD tension. A) Stress strain response and evolution of the microstructure according to the VPSC model (line) in comparison with mechanical test data (symbol) . B) Relative activities of each deformation mode.

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88 The predicted deformation modes show that in TD tension plastic deformation is completely accommodated by slip , which is consistent with the experimental observations. The predicted texture evolution suggests rapid alignment of the grain s such that the bas al planes lie parallel to the sheet surface along the loading direction (i.e. TD). Note also that the intensity contours in TD tension are rotated at 90 0 to those in RD tension ( compare Figure 4 4 A and Figure 4 7 A , respectively ) . Figure 4 8A shows the e xperimental stress strain response (symbols) in comparison with the predictions of the VPSC model (lines) in TD compression, along with the predicted evolution of texture . The agreement between model and data is excellent. T he predicted relative activity o f the deformation systems is give n in Figure 4 8 B .

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89

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90 Figure 4 8. D eformation response in TD compression. A) Stress strain response and evolution of the microstructure according to calibrated VPSC model (line) in comparison with mechanical test data (symbol) . B) Relative activities of each deformation mode. C) Predicted t win volume fraction evolution. Note that the model predicts that at higher strains ( i.e. above 4%), the c axis of the majority of grains is rotating towards the transverse direction and become s aligned with TD at ~ 9%. Basal slip and tensile twinning are the dominant deformation mechanisms as was the case in RD compression. It is to be noted that w hile there are many studies devoted to the investigation of Mg AZ31 using VPSC , the ca pabilities of this model to describe the anisotropy of the mechanical response of the material, in particular the stress strain for in plane orientations other than the axes of orthotropy were not assessed. Here , for the first time the material's respons e in uniaxial loading along the diagonal direction (DD) is simulated and compared with experimental data . For this purpose, we rotated the starting texture by 45 0 about the through thickness direction (i.e. ND) and impose strain along the corresponding direction, while enforcing zero average stress along the two lateral directions of the sample. The

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91 predicted stress strain curve in uniaxial tension and uniaxial compression shown in Figure 4 9 A and Figure 4 10 A , respe ctively are in excellent agreement with the data. Although no measured textures were available, t he prediction of the material's texture evolution shown in Figure 4 9 B and Figure 4 10 B , respectively are also provided in order to gain an understanding of th e microstructural changes that occur for these strain paths and the relative activities of the deformation systems.

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92 Figure 4 9. D eformation response in DD tension. A) Stress strain response and evolution of the microstructure according to the VPSC model (line) in comparison with mechanical test data (symbol). B) Relative activities of each deformation mode. Note that the RD TD axe s shown in the pole figures (Figure 4 9A) are in fact the original RD TD axes rotated by 45 0 about ND i.e. in Figure 4 9 B and Figure 4 10 B the rotated RD axis is the original DD direction. Note that in DD tension, the predicted relative activit y of the d eform ation modes shown in Figure 4 9 B indicates that deformation is mainly accommodated by the slip systems.

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93

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94 Figure 4 10. Deformation response in DD compression. A) Stress strain response and evolution of the microstructure according to the VPSC model (line) in comparison with mechanical test data (symbol). B) Relative activities of each deformation mode. C ) Predicted t win vo lume fraction evolution. On the other hand, i n DD compression there is a very strong texture evolution, with drastic changes occurring between 4% and 9% strains. It is worth noting that this corroborates very well with the predicted relative activity of the systems shown in Figure 4 10 B , which indicates that twinning initiates at about 4% strain and it is the dominant deformation mode for strains up to 9%. This correlates also with the observation of the reorientation of the basal pole by ~ 86 0 . It is imp ortant to note that the initial yield and subsequent hardening rate predicted by the model for DD tension and compression tests are higher than that predicted for the RD direction a nd lower than tho se predicted for the TD. This trend matches well the expe rimentally observed in plane anisotropy of the stress strain response (Figure 3 1 ). VPSC predictions in ND tension . Besides the stress strain response, no information concerning microstructure evolution along ND tension was reported in Khan et al. ( 2011 ) . However, to get a complete picture of the mechanical response and

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95 microstructural changes in Mg AZ31 , the response in uniaxial tension test along ND is also simulated. Figure 4 11 A shows the stress strain response predictions of the polycrystalline mode l (lines) in ND tension, along with, the predicted evolution of texture and relative activity of the deformation modes .

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96 Figure 4 11. Deformation response in ND tension. A ) Evolution of microstructure predicted by the VPSC model (line) . B ) Predicted relative activities of each deformation mode . C ) Predicted t win volume fraction evolution. It is predicted that the c axis of the majority of grains is undergoing extension, tensile twinning being a major contributor to the deformation of the m aterial. The effect of twinning on the macroscopic stress strain response is evident in Fig ure 4 11 A . Note that it is predicted that in tension along ND, the stress strain curve should have a sigmoidal S shape curve and a very unusual strain hardening b eha vior. The results

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97 need to be confirmed by direct experimental measurements. Partial verification/validation of the trends observed was very recently provided by Guo et al (2013) for a plate of Mg AZ31 with initial texture similar to that of the Mg AZ31 sheet investigated. Note that t he mode activity plot clearly demonstrates that initially basal slip and tensile twinning are the dominant deformation mechanisms. However, as deformation progresses and c axis orientation of the majority of grains changes , the slip mechanisms (basal, prismatic
and pyramidal II ) become more dominant. These conclusions corroborate with the predicted texture evolution and the unusual macroscopic strain hardening response (Figure 4 11). Indeed, the drastic change i n hardening rate observed between 4% and 9% strains is due to the drastic reorientation of the basal poles by ~86 0 which is due to tensile twinning. 4.3 Monotonic Simple Shear While the room temperature experimental response in simple shear along the RD (Lou et al., 2007) and both RD and TD ( Khan et al. 2011 ) has been studied at room temperature , the observed stress strain response was not explai ned by either crystal plasticity or macroscopic plasticity models. In both experimental studies i t has been shown that twinning is also operational during simple shear , although to a lesser extent than in compression but to a larger extent than in tension . The volume fraction of deformation twins during simple shear in the rolling direction is given as 0.506 at 20% equivalent strain (i.e. von Mises equivalent strain) and in the transverse direction as 0.408 at 20% equivalent strain .

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98 Concerning the polycr ystalline model simulation of the shear response of Mg and its alloys, most of the efforts have been focused on understanding the high temperature behavior ( ~250 0 C) . At high temperature , when the twinning activity is negligible, the plastic deformation is predicted to be accommodated by both pyramidal
and pyramidal slip (e.g. for simulation of pure Mg, Agnew et al. 2005; for pure Mg and AZ71 Mg, Beausir et al. 2009) in addition to basal and prismatic slip . However, currently no work has been done on modeling the mechanical response of Mg AZ31 in simple shear loading at room temperature using the VPSC model. In S ection 4.3 .1 , the different slip and twinning system activities during simple shear of Mg AZ31 at room temperature are discussed in further detail. 4.3 .1 Prediction of Slip and Twinning A ctivity in Simple S hear The step by step procedure used to determine CRSS values of the basal , prismatic , pyramidal II slip system and the tensile twinning mode for Mg AZ31 is given in Section 4.1 and 4.3 .1 . It is reasoned that , in simple shear, in the absence of pyramidal slip, twinning would be the dominant mechani sm active in the microstructure and the twin volume fraction would be higher than that observed experimentally , hence, it is necess ary to activate this slip system ( Backofen (1964), Reed Hill(1973), Khan et al. (2011)) in addition to those identified in T able 4 5. The experimental stress strain curves and measured texture used for comparison with simple shear test simulation results were reported by Khan et al . 2011. It is important to note that a small amount of compression stress is applied to the si mple shear specimen in the experimental set up.

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99 4.3 .2 Polycrystalline Model S imulations Figure 4 12 shows the orientation of the applied displacement with respect to the orthogonal material axes in the simple shear test . Using the stress strain response information from mechanical tests the visco plastic self consistent model is calibrated for simple shear in rolling direction and transverse direction by keeping four slip systems and tensile twinning system simultaneously active in the polycrystalline model. The twelve parameters associated with basal, prismatic and pyramidal II slip are exactly the same as determined in the uniaxial tension and comp ression test simulations. Table 4 6 gives the parameters used for s hear test simulation. Table 4 6. VPSC parameters determined for simple shear tests. Mode Basal 17.5 5 3000 35 Prismatic
85 33 550 70 Pyramidal 100 30 30 10 Pyramidal II 148 50 8500 0 Tensile Twinning 52 0 0 0 It must be noted that, the parameters associated with the PTR (Pre dominant twin reorientation) scheme i.e. Ath 1 =0.8 and Ath 2 =0.0 are the same as used in uniaxial test simulations . It is important to note that , for the pyramidal slip system the value of the CRSS must be higher than the more easily availabl e basal and prismatic glide, h owever, it should still be lower than the most difficult to deform direction along the c axis i.e. lower than the pyramidal II slip . The exact valu es of the hardening

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100 parameters for pyramidal
slip and tensile twinning system are determined by studying both the stress strain response and the predicted texture evolution in simple shear according to the polycrystalline model ( Table 4 6). Again as se en in case of uniaxial tests , the general lack of slip system activity is the cause of twin formation. Hence, twin formation must be accompanied by lower slip deformation. In order to capture this observation, slip systems must have a latent hardening para meter higher than unity relating slip and twinning activity (set to 2). However, in simple shear along RD, it is observed that twinning saturates at higher values of von Mises equivalent strain (approx. 10%) and the slip systems are active again. Hence, th e tensile twinning system must also self harden in order to saturate and allow for slip system activity at higher strains. Table 4 7 lists the latent hardening parameters used. Table 4 7. Latent hardening parameters in VPSC determined for simple shear test s. Mode h ' s1 h ' s2 h ' s3 h ' s4 h ' s5 (i) Basal 1 1 1 1 2 (ii) Prismatic 1 1 1 1 2 (iii) Pyramidal 2 1 1 1 2 (iv) Pyramidal II 1 1 1 1 2 (v) Tensile Twinning 1 1 1 1 2 The simple shear test in the rolling direction is equivalent to applying displacement boundary condition, (Figure 4 12 ) , as the only non zero boundary condition in the polycrystalline model simulation .

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101 Figure 4 12. Direction of loading relative to c axis for simple shear tests. The imposed velocity gradient matrix in the cartesian co ordinate system is given as: (4 2 ) Figure 4 13 shows the stress strain response obtained from mechanical tests p lotted with the predictions of the fully calibrated polycrystalline model in RD shear, along with, the predicted evolution of texture and slip/twin activity in the matrix against von Mises equivalent strain .

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102

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103 Figure 4 13. Deformation response in RD shear. A) Stress strain response and evoluti on of the microstructure according to the VPSC model (line) in comparison with mechanical test data (symbol). B) Relative activities of each deformation mode. C) Predicted twin volume fraction evolution and experimentally observed value (x). The predicte d stress strain response matches well with the response obtained from mechanical tests. The deformation mode activity plot shows that tensile twinning is initially activated and then saturates at higher strains. It is highest at approximately 3.5% von Mis es equivalent strain where a slight knee in the stress strain response can be observed indicating a change in slope. It saturates at around 10% equivalent strain and correspondingly the slope of the stress strain curve remains constant thereafter. The fina l texture observed in mechanical test s in RD shear is very close to that predicted by VPSC. Figure 4 1 4 shows this comparison. It is important to note that Lou et al. 2007 also observed a knee in the stress strain curve in simple shear at approximately 2.5 % equivalent strain which was attributed to tensile twinning.

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104 Figure 4 14. Pole figures in RD shear test at equivalent strain / 3 = 20% . A ) measured by Khan et al. 2011 . B ) Obtained from polycrystalline simulations . I n case of transverse direction simple shear loading , 12 is applied as a boundary condition and the velocity gradient matrix is given as: (4 3 )

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105 Figure 4 15. Deformation response in TD shear. A) Stress strain response and evolution of the microstructure according to the VPSC model (line) in comparison with mechanical test data (symbol). B) Relative activities of each deformation mode.

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106 Figure 4 1 5 shows the stress strain response obtained from mechanical tests plotted with the predictions of the fully calibrated polycrystalline model in TD shear, along with, the predicte d evolution of texture and relative slip and twinning mode activity. It is impo rtant to note the evolution of tex ture give n in Figure 4 15 B . Although for a rolled sheet generally an orthotropic crystal symmetry is expected, since the reported data (as reported by Khan et al. 2011) does not conform with this expected symmetry, the pre dicted texture is also plotted assuming triclinic symmetry (i.e. no effect of processing history on texture ). Table 4 8 shows the comparison of the observed (Khan et al. 2011) and predicted twin volume fraction at von Mises equivalent strain of 20% . Ther e is an excellent agreement between the predicted and observed values along both RD and TD shear. Table 4 8. Observed and predicted twin volume fraction in simple shear at equivalent strain , / 3 =20% . Direction of simple shear Observed twin volume fraction Predicted twin volume fraction RD 0.506 0.441 TD 0.408 0.437 Note the excellent agreement between the observed and predicted texture at equivalent strain , / 3 =20% (Figure 4 16).

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107 Figure 4 16. Comparison of final texture for TD shear at equivalent strain / 3 =20% . A ) as measured b y Khan et al. 2011 . B ) as predicted by polycrystalline simulation .

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108 CHAPTER 5 FREE END TORSION OF MG AZ31 TUBES Swift (1947) observed that in the plastic range of deformation , specimens made of various materials with cubic crystal structure (e.g. stainless steel, aluminum, copper) and with different geometries (solid rods or tubes) elongate in the direction of the axis about which the specimen is being twisted. Swift (1947) att ributed the occurrence of these axial effects to strain hardening, explanation that was later invalidated by the experiments of Billington (1977 a c ). Hill (1948) stated that if a material is isotropic the specimen should not change its length under torsion and that Swift effects are the result of texture induced anisotropy. However, Billington (1977 a c ) reported that there are metals with cubic crystal structure which display significant Swift effects but remain isotropic over the entire range of plastic de formation. Very recently, Cazacu et al. (2013) presented both analytical and numerical results that point to a new interpretation of Swift effects in isotropic materials. Specifically, it was shown that the occurrence of axial effects is related to a slight difference between the uniaxial tension and compression yield of these materials. For initially anisotropic materials, Swift effects were modeled mainly in the framework of crystal plasticity. A majority of the se studies were devoted to materials wi th cubic crystal structure. Thus, analysis was done assuming that the plastic deformation is fully accommodated by slip, the homogenized response of the polycrystal individua l grains are proportional to those which are imposed on the polycrystal), Taylor approximation ( deformation gradient within each grain has a uniform value throughout

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109 the aggregate) or a viscoplastic self consistent formulation (Molinari et al. 1987) and co de developed by Lebensohn and Tomé (1993) ( Chapter 2) . However, axial effects were only qualitatively predicted and only at very large shear strains (e.g. Toth et al.1990; Habraken and Duchene, 2004; Duchene 2007 ), the general conse nsus being that the sim ulation results largely underestimate the experimental data. As concerns polycrystalline materials with hexagonal crystal structure, as already mentioned, the past decade has witnessed a renewed interest in improving the fundamental understanding of their plastic behavior. Focus however has been on modeling the uniaxial ten sion and compression behavior ( review of the literature in Chapter 5). As concerns the torsional response, emphasis was on studying the effect of temperature on the plastic response of the material (e.g. Beausir et al., 2009; Biswas et al., 2013 for results on Mg AZ71 ). At room temperature , shortening of the specimens was observed in free end torsion tests along a given orientation (in plane orientation) . The occu rrence of axial strains in free end torsion was attributed to texture evolution due to plastic deformation by pyramidal and < a > slip. Very recently, Guo et al. (2013) rep orted experimental data on Mg AZ31 subjected to free end torsion at room temperature. Tests were done for two orientations. It was observed that if twisted along the rolling direction (RD) the specimen shortens, while if twisted along the normal direction (ND) the specimen elongates. The observed axial strains were attributed to tensile twinning, which in turn induces texture evolution in the material. In summary, initial anisotropy or induced anisotropy due to texture evolution is considered to be the only cause of Swift effects in polycrystalline metals. However, no explanation exists as to why in certain materials irreversible elongation is observed

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110 while in others permanent shortening occur s. Furthermore, on the basis of data gathered from other mechanical tests, presently it is not possible to predict whether Swift effects may occur and how important the se effects are . The purpose of this work is to demonstrate theoretically and validate b y comparison with data that there is a correlation between the Swift phenomenon in torsion and the stress strain behavior in uniaxial tension and compression. This correlation is established by describing the elastic p lastic response of the material by usi ng the macroscopic Cazacu et al. (2006) yield criterion in conjunction with isotropic hardening ( Chapter 3). The influence of the tension compression asymmetry on torsional response for metallic materials is also confirmed at a different modeling scale us ing the crystal plasticity framework. 5.1 P reliminaries: Swift E ffects in Isotropic M aterials In all existing models plastic anisotropy is considered to be the unique cause of Swift effects . Specifically, these models, whether phenomenological or physi cally based, can predict Swift effects only if the material is initially anisotropic or if hardening is anisotropic. However, in Cazacu et al . (2013), using the isotropic form of Cazacu et al. (2006) yield criterion and isotropic hardening it was shown that using an isotropic macroscopic model one can predict the occurrence of axial strains under monotonic free end torsion in isotropic ( untextured ) materials using FE simulations . Furthermore, a nalytical results can also be obtained, providing physical i nsights and explanation of the Swift phenomenon . In the following, the theoretical arguments are briefly presented followed by a detailed discussion and analysis of Swift effects in initially textured materials. For the first time, the peculiar Swift effec ts in Mg AZ31 are explained and

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111 furthermore it is demonstrated why up to now such effects were not reproduced within a polycrystalline plasticity framework. The correlation between strength differential effects and Swift effects in isotropic materials ca n be demonstrated analytically using the isotropic form of the Cazacu et al. (2006) yield criterion . According to this criterion, t he effective stress is: , ( 5 23 ) where k is an internal variable, its range of variation being ( 1, 1), a is a homogeneity constant while are the principal values of the Cauchy stress deviator , where ( I being the second order identity tensor) . In the expression of the effective stress according to the isotropic form of the CPB06 criterion ( Equation (5 1 )), B is a material constant defined such that reduces to the uniaxial tensi le flow stress T , i.e. ( 5 24 ) Note that for a = 2 and k = 0, B takes the value of and the isotropic form of the CPB06 criterion reduces to the v on Mises yield criterion . Consider a rod, loaded in torsion by twist ing it at one end. For simplicity, the cross section is assumed to be circular with initial radius denoted by R 0 . In a cylindrical coordinate system, the Cauchy stress tensor is: . Using the

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112 associ ative flow rule ( Chapter 3) in conjunction with the yield criterion ( Equation ( 5 1 )), it follows that the axial strain is (for more details, the reader is referred to Cazacu et al. 2013) : = . (5 25 ) Thus, the axial strain is non zero in general and, f urthermore, while , irrespective of the value of the homogeneity parameter a. This means that according to the model : (i) For k < 0: >0 , i.e. , extension along the z axis occurs (lengthening of the specimen) while the radial strain is negative; (ii) F or k > 0: < 0 , i.e. , contraction along the z axis (shortening of the specimen) and radial expansion. The same conclusions hold also for an elastic plastic material with isotropic hardening, since the axial strains are due solely to a non zero value of the internal variable k involved in the expression of the CPB06 yield criterion. Note that f or k = 0: = 0 (E quation ( 5 3 )), i.e. there are no length changes. This is to be expected, since for k = 0, the isotropic form of Cazacu et al. (2006) criterion reduces to the classical isotropic criterion, , which for a = 2, coincides with the v on Mises eff ective stress. For k different from zero, the isotropic CPB06 yield criterion has a strong sensitivity to the third invariant of the stress deviator.

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113 Figure 5 1 shows the projection of the yield surface for combined axial and torsion loading , according to the von Mises criterion ( Equation ( 5 1 ) , with k = 0 and a = 2 ) . The normal to the yield surface corresponding to shear loading ( 11 = 0), i.e. the strain rate vector , only has a shear component with the normal component equal to zero . This explains th at no length changes could be obtained with the von Mises yield criterion. On the other hand , Figure 5 2 shows the projection of the yield surfaces for combined axial loading torsion according to the Cazacu et al. (2006) yield criterion for a positive value of k (Figure 5 2 A with k=0.5 ( C =1.314) and a =2) and a negative val ue of the parameter k (Figure 5 2 B with k= 0.5 ( C =0.761) and a =2). Note that t he normal to the yield surface corresponding to shear loading ( ( 11 = 0), i.e. the strain rate vector , has an axial component. This explains the length changes (axial deformation) occurring under shear. Hence, for an isotropic material, for k>0 ( C >1 ) the specimen shortens whereas for k<0 ( C < 1 ) the material elongates axially in free end torsion.

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114 Figure 5 1. Projection in the plane of the von Mises yield surface along with the normal to the surface for shear loading . Figure 5 2. Projection in the plane of the isotropic form of Cazacu et al . (2006) yield surface along with the normal to the surface (plastic strain vector increment) for shear loading . A ) k=+0.5 (material with yield in tension T greater than the yield in compression C ) . B ) k= 0.5 (material with yield in tension T less than the yield in compression C ). 5.2 Swift E ffect in Mg AZ31: Macroscopic A pproach As seen in S ection 5.1 , the tension compression asymmetry , i.e. yield in tension T different than the yield stress in compression, C , results in the development of axial strains (Swift effect) during torsional loading of isotropic materials . In the following, it is demonstrated that this correlation between the Swift phenomenon in torsion and the stress strain behavior in uniaxial tension and compression also holds for t extured anisotropic materials. This is established by using the Cazacu et al. (2006) anisotropic yield criterion (( Equation (3 9)) for the description of the elastic plastic response of Mg AZ31 in conjunction with isotropic hardening law of Voce type: , (5 4)

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115 The parameters involved in the Voce type isotropic hardening law ( Equation ( 5 4 )) for Mg AZ31 were estimated from the axial stress vs. true strain curve in uniaxial tension in the rolling (RD or x ) direction as: A 0 = 315.4 MPa, A 1 = 140.6 MPa, A 2 =16.3. The values for the Young modulus and Poisson coefficient were taken as: E=45 GPa and , respectively. The material parameters for the yield criterion that will be used in the simulations have been identified alre ady on the basis of the mechanical test results in uniaxial tension and compression reported in Khan et al. (2011 ) ( Chapter 3). It is worth recalling that in order t o model the difference in hardening rates between tension and compression loadings observe d experimentally, all these material parameters were considered to evolve with the accumulated plastic deformation. The numerical values of these parameters corresponding to five individual leve ls of equivalent plastic strain (up to 0.1 strain) are those l isted in Table 3 2, the values corresponding to any given level of plastic strain being obtained by linear interpolation ( Equation 3 10 ). As shown in Figure 3 8 which presents theoretical yield surfaces (represented in the plane 11 22 , with 1 being the rolling direction (RD) and 2 the transverse direction (TD)) at room temperature for the orthotropic hcp Mg AZ31 alloy according to the orthotropic Cazacu et al. (2006) yield c riterion at different strain levels , (up to 10%) along with the experimental data (symbols) . T he model correctly predicts that at initial yielding and below 8% strain, the tension compression asymmetry is very pronounced (compare the tension tension and compression compression quadrants) and the surfaces have a triangular shape, while at 8% strain and beyond , the yield surfaces

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116 have an elliptical shape, and the difference in response between tension and compression becomes small. As demonstrated by comparison between model and stress st rain curves in uniaxial loading simulations (Figure (3 9)) the model describes remarkably well the particularities of the plastic behavior of Mg AZ31 alloy , specifically , the S shape of the experimental stress strain curve in uniaxial compression along RD and TD and in uniaxial tension for ND, respectively. As concerns the axial strain in torsi on, for an orthotropic material such as the Mg AZ31 alloy, it should depend on the direction about which the specimen is twisted. Before presenting the solution of the boundary value problem obtained using FE , the response in shear according to the elastic plastic model is first studied . For this purpose, c onsider a tube , loaded in t orsion by a given twist at its end. In a cylindrical coordinate system, the Cauchy stress tensor is: . Figure 5 3. Orientation of the material anisotropy axes relative to loading axes for free end torsion tests . A ) a specimen with long axis along the rolling direction . B ) a specimen with the long axis along the normal direction.

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117 For case (a) (Figure 5 3) th e orthotropy axes are given as 1 =RD , 2 =TD , 3 =ND , where e z is along the RD and the Cauchy shear stress is: . If the axis of the torsion test is the rolling direction (1 direction), the projection of the yield surface on the planes 12 11 and 13 11 will provide physical insights concerning the sign of the axial strain that develops during this torsional loading. Figure 5 4 shows the evolution of the theoretical yield surfaces in the shear stress normal stress i.e. 12 11 plane and Figure 5 5 shows the evolution of the theoretica l yield surfaces in the 13 11 plane. All the parameters of the anisotropic yield criterion which were identified based on uniaxial data, are given in Table 3 2. Note that for all the projections of the yield loci on these two specific planes, regardless of the level of the accumulated plastic strain, t he normal to the yield surface corresponding to shear loading ( 11 = 0), i.e. the strain rate vector d p has a negative axial component i.e. <0. Hence, an axial strain develops if the Mg AZ31 alloy tube is loaded in torsion about RD and this produces contraction of the sample (because <0).

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118 Figure 5 4. T heoretical yield surfaces, according to the orthotropic Cazacu et al. (2006) criterion in the ( 12 11 ) plane, corresponding to the different levels of accumulated plastic strain. Figure 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 accumulated plastic strain . If the z axis of the specimen coincides with the ND (normal direction), the projection of the yield surface on the planes 13 33 and 23 33 will provide physical insights concerning the sign of the axial strain that develop during torsi onal loading.

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119 Figure 5 6 shows the evolution of the theoretical yield surfaces in the shear stress normal stress 13 33 plane while Figure 5 7 shows the evolution of the theoretical yield surfaces in the 23 33 plane. For all the projections of the yield l ocus on these two specific planes, regardless of the level of the accumulated plastic strain, t he normal to the yield surface corresponding to shear loading ( 33 = zz = 0), has a tangential component which is positive i.e. the strain rate vector d p has a positive axial component ( >0). Figure 5 6. T heoretical yield surfaces, according to the orthotropic Cazacu et al. (2006) criterion in the 13 33 plane , corresponding to different levels of accumulated plastic strain . Hence, an axial strain that is positive will develop, so the specimen with long axis (z axis) along the ND will elongate under free end torsion.

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120 Figure 5 7. T heoretical yield surfaces, according to the orthotropic Cazacu et al. (2006) criterion in the 23 33 plane , corresponding to different levels of accumulated plastic strain . FE simulation of the torsional response of Mg AZ31 using Cazacu et al. (2006) yield criterion . As already mentioned, one of the objectives of this work is to investigate whether for anisotropic materials, in particular orthotropic Mg AZ31 , there is a correlation betwe en strength differential effect and the Swift phenomenon. The boundary value problem for free end torsion is solved numerically using the FE metho d . All the simulations were carried out using the commercial FE code ABAQUS and a user material subroutine (UMAT) that was developed ( Chapter 3 for details concerning the integration algorithm) for implementation of the anisotropic elastic plastic model with yielding described by Cazacu et al. (2006) criterion and the new evolution laws for the anisotropy coefficients . The geometry of the specimen and FE mesh used in all the calculations is shown in Figure 5 8 . The FE mesh consi sts of 1290 hexahedral

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121 elem ents (ABAQUS C3D8 ). The initial minimal section was meshed with 10 layers of elements and three elements were used along the wall thickness. The usual definitions of the axial and shear strains will be used in the present paper, namely: and (5 5 ) where , r is the current radius, L 0 is the initial length, u is the axial displacement, and is the angle of twist. In all the FE simulations, equal sized time increments t = 10 3 s were considered. Between five and six iterations per increment were necessary for convergence in the return mapping algorithm, the tolerance in satisfying the yield criterion being 10 7 (0.1 Pa). Figure 5 8. Sample geometry, dimensions (mm) and finite element mesh for free end torsion test . Comparisons between FE simulations and experiments for Mg AZ31 under free end torsion are presented in Figure 5 9 . The model predictions are in excellent agreement with the experimental observations. In part icular , the calculated initial slopes of the two predicted curves are almost in perfect agreement with the experimental slopes (Figure 5 9 ) . An even more remarkable result is that the model correctly

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122 reproduces the switch of the sign of the axial strain wi th the change in the loading direction of the torsion specimen. Furthermore, the sign of the axial strains that develop can be correlated to the tension compression asymmetry of the material in the direction about which the specim en is twisted. Indeed, i n RD, for which the flow stress is higher in uniaxial tension than in compression (Figure 3 2) , shortening of the specimen is observed. On the other hand, in ND, for which it is conjectured that the flow stress is higher in compression than in tension, len gthening of the specimen occurs. Thus, the sign of the axial strain that develops (elongation or contraction) under free end torsion depends on the ratio between uniaxial tension and compression in the given direction . In c onclusion, while for an isotropic material, the axial strain that develops during free end torsion testing is directly related to the strength diff erential which is the same in ever y direction, for an anisotropic material, the nature of the axial strains that develop (elongation or contra ction) is direction dependent. Whether a material contracts or elongates when twisted about a given direction depends on the ratio of the uniaxial tensile and compressive strengths corresponding to the twisting axis. In particular: o If the yield stress is higher in uniaxial tension than in uniaxial compression in a given direction, contraction of the specimen occurs under free end torsion along this direction. o If the yield stress is higher in uniaxial compression than in uniaxial tension in a given directio n, elongation of the specimen occurs under free end torsion along this direction.

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123 The most important finding is that the sign of axial strains that develop can be estimated only by analyzing stress strain curves obtained in a very few simple mechanical tests. Furthermore, the results presented in this paper are of significant practical importance in the design phase of material components, since based on limited mechanical test data it is possible to predict the response in t orsi on, namely whether Swift effects will occur and how important these effects will be in a structural member . Figure 5 9. Comparison between experimental data (from Guo et al. (2013)) and the FE predictions obtained with Cazacu et al. (2006) yield crit erion with evolving anisotropy coefficients , for the long axis of specimen alon g RD and ND.

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124 Importance of accounting for the tension compression asymmetry on the accuracy of prediction of the torsional response of Mg AZ31 . As seen previously , the numerical predictions of the torsional response for the Mg AZ31 alloy using the Cazacu et al. (2006) yield criterion with evolving anisotropy coefficients are in very good agreement with the experimental data. One characteristic of this yield criteri on is that it accounts for the tension compression asymmetry of the plastic flow. In this section, the importance of consideration of strength differential effects and material orthotropy in modeling will be investigated. For this purpose , we will also use Hill's (1948) orthotropic criterion (E quation ( 2 23 ) ( 2 26)) to describe the plastic flow of the same Mg AZ31 alloy . Note that Hill (1948) yield criterion accounts for the anisotropic behavior in tension, but predicts the same mechanical response in tensi on and in compression. The material parameters involved in Hill (1948) criterion are identified using the experimental flow stress data in uniaxial tension tests reported by Khan et al. (2011) ( Chapter 3). The numerical values of the anisotropy c oefficients involved i n Hill (1948) criterion (E quation ( 2 23 )) are given in Table 5 1 . The yield surfaces in the plane, associated with Hill (1948) yield criterion , corresponding to different leve ls of equivalent plastic strain are plotted in Figure 5 10 (stresses are in MPa) . Note that the shape of Hill's (1948) yield surface is always elliptical and does not capture the tension compression asymmetry of the material.

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125 Table 5 1. Anisotropy coefficients in Hill (1948) yield c riterion (Equation (2 23)) determined for Mg AZ31 alloy. Strain F G H N M L 0.03 0.195 0.276 0.724 2.089 11.186 12.429 0.05 0.206 0.288 0.712 2.167 11.813 15.245 0.06 0.217 0.297 0.703 2.184 12.184 15.873 0.08 0.228 0.298 0.702 2.165 12.210 16.136 0.10 0.246 0.312 0.688 2.167 11.816 15.725 Figure 5 10. Yield loci corresponding to fixed levels of accumulated plastic strain according to Hill (1948) orthotropic criterion (lines) against mechanical test data (symbols) from Khan et al. (2011). As seen previously, analyzing the projections of the yield surface described by the Hill (1948) yield criterion will provide physical insights concerning the predictive

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126 torsional response obtained with this criterion. Figure 5 11 and Figure 5 12 show the p rojections of the Hill (1948) yield surface for different accumulated plastic strains in the the 12 11 and 13 11 planes, respectively (i.e. the planes of im portance for a free end torsion specimen with the long axis along RD). The normal to the yield surface corresponding to shear loading ( 11 = 0 ) has a tangential component equal to zero.The same conclusion can be drawn by analyzing the projection of the yield surface in the 13 33 and 23 33 planes (i.e. the planes of importance for a free end torsi on test of a specimen with long axis along ND). The boundary value problem was also solved numerically using the FE method and the torsional response predicted by both yield criteria used in conjunction with the same Voce type isotropic hardening law ( Eq uation (3 9)) was compared. Note that FE simulations using Hill (1948) yield criterion show occurrence of very small axial strains at very large values of shear strain (e.g. Figure 5 15 for RD torsion , Figure 5 16 for ND torsion ). It is important to emp hasize that these axial strains are due solely due to the kinematics of the deformation process . Indeed, d uring the torsion test, the relative orientation between the orthotropy axes of the material and the axis of t wist will evolve.

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127 Figure 5 11. T heoretical yield surfaces according to the Hill (1948) criterion in the 12 11 plane , corresponding to different levels of accumulated plastic strain . Figure 5 12 . Theoretical yield surfaces according to the Hill (1948) criterion in the 13 11 plane , corresponding to different levels of accumulated plastic strain .

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128 Figure 5 13. T heoretical y ield surfaces according to the Hill (1948) criterion in the 13 33 plane , corresponding to different levels of accumulated plastic strain . Figure 5 14. T heoretical yield surfaces according to the Hill (1948 ) criterion in the 23 33 plane , corresponding to different levels of accumulated plastic strain . Indeed, examination of the simulation results of RD torsion indicate that although both models predict an axial contraction of the specimen, there is a large difference between the le vel of axial strain predicted ( Figure 5 15) . The axial length change

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129 predicted by the Cazacu et al. (2006) yield criterion which accounts for both the tension compression asymm etry and the anisotropy of the plastic behavior is in very good agreement with the experimental data, while the Hill (1948) yield criterion drastically underestimate s it. Examination of the simulation results of ND torsion indicate that although both model s predict an axial elongation of the specimen, only the Cazacu et al. (2006) yield criterion which accounts for both the tension compression asymmetry and the anisotropy of the plastic behavior is in very good agreement with the experimental data (Figure 5 16). The Hill (1948) yield criterion predicts negligeble swift effect in ND torsion. Thus, quantitative agreement between experimental data and numerical predictions for a free end torsion test can only be obtained if the yield criterion accounts for th e tension compression asymmetry. In this section , using the Cazacu et al. (2006) yield criterion in conjunction with isotropic hardening, it was demonstrated that there exists a correlation between Swift effects in torsion and the differential stress strai n behavior in uniaxial tension and compression of the polycrystalline material. Furthermore, an explanation as to why a material elongates or contracts under free end torsion is provided for the very first time .

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130 Figure 5 15. Comparison of the variation of the axial strain with shear strain (symbols) observed in experiments during free end torsion along the rolling direction (RD) again s t the predictions according to the (i) orthotropic Cazacu et al. (2006) yield criterion and i sotropic hardening law (Equation ( 3 7 ) , ( 5 4 ) ) (ii) Hill (1948) yield criterion and the same isotropic hardening law .

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131 Figure 5 16. Comparison of the variation of axial strain with shear strain observed in experiments (symbols) during free end torsion along the normal direction (N D) against the prediction according to the (i) orthotropic Cazacu et al. (2006) yield criterion and isotropic hardening law (E quation 3 7) (ii) Hill (1948) yield criterion and the same isotropic hardening law . 5.3 Swift E ffect in Mg AZ31: Crystal Plasticity F ramework In S ection 5.2 , the influence of the tension compression asymmetry on the torsional response of the Mg AZ31 alloy has been revealed. In S ection 5.3 , a crystal plasticity model will be used to solve the boundary va lue problem and confirm the importance of correctly modeling both the anisotropy and tension compression asymmetry of the material. Specifically, the VPSC polycrystal model that was calibrated f or this material will be used ( Chapter 4). Let us recall here that at room temperature, t hree slip systems (basal, prismatic, and pyramidal) and tensile twinning were shown to

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132 be active in the material when subjected to uniaxial tension and compression along different loading orientations. Next, the importance of con sideration of tensile twinning when simulating the response of the material in shear loading is assessed , namely how it affects the accuracy of the prediction of Swift effects and texture evolution. As already mentioned, at present few efforts have been devoted to understand Swift effects that develop at room temperature in Mg and its alloys. Existing studies have focused on the torsional/shear response of Mg and its alloys at high temperature, where the twinning activity becomes negligible. Thus, at hig h temperature, it was assumed that plastic deformation is fully accommodated by slip: basal, prismatic, pyramidal
and pyramidal slip (e.g. for simulation of pure Mg, Agnew et al. 2005; for pure Mg and AZ 71 Mg, Beausir et al. 2009)). To simulat e the Swift phenomenon (i.e. plastic behavior in free end torsion) we will use the VPSC parameters given in Table 4 6 and Table 4 7 , i.e. the same parameter values for the deformation systems that were identified based on uniaxial test data. It is worth r ecalling that with this set of parameters, for the first time it was possible to obtain with VPSC a very good agreement with experimental data in uniaxial tension, compression and simple shear along various orientations, both in terms of stress strain cur ves ( Chapter 4), and final twi n volume fraction ( T able 4 8 gives the simulation results of shear tests ). For modeling the Swift effect in this an isotropic hcp material, the boundary condition imposed in the VPSC code is of simple shear . It i s assumed tha t this boundary condition is equivalent to that observed in the free end torsion test . Thus, the velocity gradient imposed in the local c artesian coordinate reference system i s:

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133 ( 5 6 ) where , x corresponds to the tube axis. The calculations were terminated when the shear strain reached the value of =0.2. Comparisons between FE simulations and experiments for Mg AZ31 under free end torsion for a sp ecimen with the long axis along the RD and ND are presented in Figure 5 17 . The VPSC model predictions are in excellent agreement with the experimental observations. An even more remarkable result is that the model correctly reproduces the difference in th e sign of the axial strain (elongation or contraction) depending on the twist dir ection (Figure 5 17). While using the macroscopic yield criterion to describe the mechanical behavior of Mg AZ31 alloy the reasons for specimen elongation in one direction and contraction in the other can be clearly related to mechanical properties (strength differential effects) , such direct links are very difficult to draw using the crystal plasticity model . With the aim of establishing such correlations, the role of the physical deformation mechanisms active in the microstructure (slip or twinning) on the accuracy of the predictions of Swift effects must be further studied.

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134 Figure 5 17. Comparison between v ariation of the axial strain vs. shear strain during free end torsion a bout RD and ND given by (i) experimental data (points) and (ii) the numerical predictions (lines) by the VPSC model . Role of the deformation mechanisms on prediction of Swift effects in Mg AZ31 . As seen in Figure 5 17 , using VPSC with the set of p arameters given in Table 4 6 and Table 4 7 , the Swift effect are correctly described. However, it is difficult to see w hich deformation mechanism play a major role in the accurate predictions of the axial strains that develop. For this purpose, in this se ction, crystal plasticity simulation are performed assuming different scenarios in terms of activation of deformation mechanisms: Case (1): Tensile twinning and all the slip systems (basal, prismatic and pyramidal , and pyramidal
) are considered potentially active.

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135 Case (2): Twinning is neglected but all four slip systems (basal, prismatic and pyramidal , and pyramidal
) are considered potentially active. Case (3): Twinning is neglected and only three slip systems (basal, prismatic and pyr amidal ) are considered potentially active. Figure 5 18 shows a comparison between the predicted shear stress shear strain curve and the experimental data for all the three cases described above. It is worth noting that irrespective of the deformati on systems considered active the shear stress shear strain response of the Mg AZ31 alloy is captured . This is because the following three slip systems: basal, prismatic and pyramidal slip systems (Figure 5 18 C ) are the dominant slip deformation mechanisms that accommodate plastic deformation during simple shear. The role of the pyramidal slip becomes evident by comparing Fig ure 5 18 B and Figure 5 18 C . It appears that unless pyramidal slip is considered, the softe r response for strain greater than > 0.173 observed experimentally cannot be captured. However, the role of the deformation mechanism s becomes clear when analyzing the final texture obtained considering the differe nt case (Figure 5 19) . It is worth noting that whe n all the five systems (basal, prismatic, pyramidal and pyramidal slip, and tensile twinning) are considered active, the final observed and predicted textures match extremely well. However, if twinning is not operating (Case (2 , 3 )), only a rotat ion of the initial texture occurs when the material is subjected to simple shear. More specifically, the polycrystal model predicts that the basal pole intensity is elongated 45 0 away from the shear direction (which is the RD direction), while the experime ntally observed rotation of the axis is not captured at all. Furthermore, comparison between Figure 5 19C and Figure 5 19 (d), shows the

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136 specific role played by pyramidal
slip . The pyramidal slip system inhibits the rotation of the basal planes along the two mutually perpendicular directions at 45 0 to RD. The results presented in Figure 5 18 and Figure 5 19 demonstrate clearly the role of the deformation mechanism on the shear response for the Mg AZ31 alloy, showing that even if the stress stra in curve looks similar to the experimental curve, the texture evolution is totally depend e nt on the slip system s or twinning system being operational .

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137

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138 Figure 5 18. Predicted effective stress vs. effective strain response and evolution of the microstructure in RD shear using the VPSC model (solid line) . A ) twinning and all four slip systems active (Case (1 )). B ) twinning neglected and all slip modes active (Case (2 )) . C ) twinning neglected and only three slip systems operational: basal, prisma tic and pyramidal slip (Case (3 )).

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139 Figure 5 19. Final texture at equivalent strain / 3 = 20% in simple shear along the rolling direction (RD). A) Experiments. B ) VPSC predi ction assuming twinning and all 4 slip systems active . C ) VPSC prediction assuming that all slip modes are active. D ) VPSC pred iction assuming only basal, prismatic and pyramidal slip system are active .

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140 For all the different cases (case (1) to case (3)) studied, the predicted axial strain that develop during free end torsion of Mg AZ31 along the rolling direction is plotted in Figure 5 20 A and 5 20 B . The experimental data obtained by Guo et al. (2013) is also reported in Figure 5 20 A and Figure 5 20 B . It is important to note that irrespective of the plastic deformatio n systems that are considered to be active, shortening ( in RD) or lengthening (in ND) of the specimen along the direction of twist is qualitat ively predicted ( Figure 5 20 ) by the VPSC model. However, quan t it ative agreement between the crystal plasticity prediction and the experimental data can only be obtained by the activation of twinning. Otherwise, if on ly slip systems are activated (c ases ( 2 ) and ( 3 )), the Swift effect is largely underestimate d . It is very interesting to note that Hill's (1948) predic tions are close to the polycrystalline predictions in which twinning is neglected, and only slip deformation (all 4 slip systems active) is considered to be operational ( Figure 5 22). On the other hand, predictions of the Cazacu et al. (2006) model are ver y close to the polycrystalline simulations in which twinning is also considered to be active (Figure 5 21 A and Figure 5 21 B ). In conclusion, for the first time, the preponderant role of tensile twinning on predicting/explaining the Swift phenomenon is sh own . Moreover, the correlation between tension compression asymmetry and Swift effects that was established based on the predictions of Cazacu et al. (2006) macroscopic model. It is also demons trated that only by considering that all slip systems (basal, p rismatic and pyramidal , and pyramidal
) and tensile twinning are active , both the mechanical behavior and the microstructure evolution of the Mg AZ31 alloy during free end torsion could be correctly predict ed.

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141 A B Figure 5 20. Comparison of t he axial strain vs. shear strain observed experimentally ( symbols ) and according to the VPSC model (lines) for all three cases in free end torsion . A ) R olling direction and B ) N ormal direction . Note that only when tensile twinning and all slip modes are active, axial contraction of the specimen is accurately described.

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142 A B Figure 5 21. Comparison of the (i) experimental variation of the axial strain with the shear s train during free end torsion of Mg AZ31 ( symbols ) (ii) numerical prediction according to the orthotropic Cazacu et al. (2006) criterion and the polycrystal model simulations considering that all slip modes and twinning are active . A ) RD torsion . B ) ND torsion .

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143 A B Figure 5 22. Comparison of the variation of the axial strain vs. shear strain during free end torsion of specimens against the predictions according to the (i) orthotropic Hill (1948) yield criterion (solid line) , (ii) the VPSC model simulations, where twinning was neglected and all slip modes are active ( interrupted lines ), and (iii) e xperimental data from Guo et al. (2013) (symbols). A ) R olling direction torsion. B ) N ormal direction torsion.

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144 5.4 Observations In C hapter 5 , the understanding of Swift effects in M g AZ31 has been improved at different scales, using macroscopic modeling and the crystal plasticity framework. Independent of the modeling framework used , the correlation between the tension compression asymmetry of the material and the sign and level of t he axial strains that develop during torsion in Mg AZ31 all oy was demonstrated. C omparison between the prediction of axial strain vs . shear strain for torsion in RD , obtained using the VPSC model with twin n ing considered active, Cazacu et al. (2006) yield criterion in conjunction with appropriate evolution laws for the anisotropy coefficients and the SD parameter k, and the e xperimental data from Guo et al. (2013) show s that both models capture the unusual behavior of Mg AZ31 in torsion with great accuracy. Moreover, t he predictions obtained with the macroscopic model and VPSC where twinning was considered active are very close. This reinforces the fact that the Cazacu et al. (2006) yield criterion that accounts for the evolution of the tension compression asymmetry at the macroscopic scale correctly captures the evolution of the microstructure. On the other hand, for Hill's (1948) yield criterion that doesn't capture the tension compression asymmetry of the material and largely un derestimates the Swift phenomenon, the level of axial strains predicted are very close to that obtained using VPSC with twinning neglected. Most importantly , an explanation as to why a material elongates or contracts under free end torsion was provided. S pecifically, the following explanation of the Swift effect was given: For an anisotropic material, the tension compression asymmetry is direction dependent. Thus, the sign of the axial strains that develop (elongation or contraction) is dependent on the d irection of loading relative to the material axis

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145 of anisotropy . Whether a material contracts or elongates when twisted about a given direction depends on the ratio of the uniaxial tensile and compressive strengths in the direction about which the material is twist ed . Specifically : o If the yield stress is higher in uniaxial tension than in uniaxial compression in a given direction, contraction of the specimen occurs under free end torsion about this direction. o If the yield stress is higher in uniaxial compr ession than in uniaxial tension in a given direction, elongation of the specimen occurs under free end torsion about this direction. Most importantly, the sign of axial strains that develop can be estimated only by analyzing stress strain responses in a f ew very simple mechanical tests. The results presented in this paper are of significant practical importance in the design phase of material components, since based on limited mechanical tests it is possible to predict the axial response of the material in torsion, namely , whether Swift effects will occur and the extent to which the material will elongate or shorten .

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146 CHAPTER 6 EFFECT OF TENSION COMPRESSION ASYMMETRY ON PLASTIC BUCKLING OF AXIALLY COMPRESSED CYLINDRICAL SHELLS Failure of slender or thin structures is mainly d ue to buckling . Assuming linear elastic behavior, at the beginning of the eighteenth century Euler derived t he first critical load of a compressed beam by posing and solving an eigenvalue problem . Koiter (1945) was the first to provide an analysis of the post buckling behavior, namely he calculat ed the bifurcation modes and the post critical equilibrium branches. Moreover, he reveale d the role played by geometry , and performed an imperfection sensitivity analysis for an elastic rod. As concerns plastic buckling, the first significant results were presented by Shanley (1947), who analyzed the response of a rod supported by two elasto plastic springs at the bottom and subjected to an axial compressive force . Although he used a discrete model, his analysis provided insights concerning the buckling modes of a continuum structure undergoing plastic deformation. Hill (1958) extended Shanley's theory to a 3D continuum by using the conc ept of a "linear comparison elastic solid" and attempted to propose general theorems for uniqueness and stability. Buckling of cylindrical shells under axial compression and spherical shells under uniform pressure has been studie d intensively in the mid 1960's to the mid 1970's, most notable contribut ions including Hutchinson, 1965; Batterman, 1965, 1968; Budiansky and Hutchinson,1969 ; Hutchinson, 1970, 1974; Tvergaard, 1976, and others . In the past three decades , the focus has been on understanding and describing non eccentric buckling of axially compressed shells (e.g. Bushnell, 1982, Bushnell, 1985, Gellin, 1979, Tvergaard, 1983 (a,b) ) . It is worth noting that in all the paper s mentioned it

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147 is assumed that the plastic response is governed by either von Mises yield criterion or Gurson's (1977) yield criterion. While Kyriakides et al. (2005), Bardi and Kyriakides (2006) reported experimental data on plastic buckling of textured (anisotropic) aluminum alloys, interpretation of t he experimental result s was done using the isotropic v on Mises yield criterion that cannot capture the influence of the direction of axial loading with respect to the material's orthotropy axes on the critical load. Most recent post bifurcation analyses, e .g. Grognec and Le van (2008) also neglect anisotropy in plastic flow. In C hapter 6 , for the first time fundamental issues related to plastic buckling are addressed. The importance of the consideration of the specificities of the plastic flow on buckli ng predictions is demonstrated. Most importantly, the very unusual buckling behavior of Mg AZ31 tubes is explained for the first time and accurately simulated. The outline of C hapter 6 is as follows. First, analysis of plastic buckling is done assuming t hat the plastic flow is described by the isotropic form of Cazacu et al. (2006). To gain insights into the importance of consideration of strength differential effects on plastic buckling, an analytical s olution is provided (Section 6 1 ). The treatment is based on a plastic hinge model introduced by Batterman (1965). It is shown that in the case when SD effects are neglected, known results based on J 2 flow theory are recovered. Correlation between SD effects in plastic flow and the energy absorbed by the ma terial is clearly established. Moreover, FE simulations based on Cazacu et al. (2006) for buckling of a tube made of an aluminum alloy that displays SD effects (data after Batterman, 1965) are presented. It is shown that even if SD effects on plastic flow are very small, they do affect buckling behavior.

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148 In Section 6 2 buckling of cylindrical shells made of Mg AZ31 is analyzed. The u nusual buckling behavior is revealed and explained. A summary of the main findings are presented in Section 6 3 . 6.1 Prediction of Plastic Buckling Stress in Axial C ompression of C ylindrical Shells of Isotropic Metals Displaying Tension Compression A symmetry For thin cylindrical shells, compressed into the plastic range obeying von Mises yield criterion and isotropic h ardening, an analytical expression of the critical load was obtained by Batterman (1965) using a plastic hinge model. His analysis was complemented with a series of tests on shells having a wide range of radius to thickness ratios (R/h), in the range 10
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149 (i) the first critical bifurcation mode is axisymmetric, and the cylinder will buckle into a series of outward sine waves (F igure 6 1), (ii) Buckling occurs under equilibrium at constant load, i.e. = 0, where the dot stands for time derivative while is the membrane stress along the cylinder axis (F igure 6 1). Specifically, the assumed form of the outward velocity is: , (6 1) where, m (mode number) is an integer and the amplitude T 1 is a constant. The corresponding non zero strain rate measures are the rate of change of extensions in the axial and circumferential directions, respectively, , (6 2)

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150 Figure 6 1. Axisymmetric plastic buckling of cylindrical shell of circular cross section ( from Batterman, 1965). T here is a change in curvature only in the axial plane given as . The stress state in the shell is approximately plane, i.e. only the in plane components and are non zero. The corresponding membrane resultants (N x , N ) and bending moments (M x , M ) (Figure 6 1) being expressed as:

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151 . (6 3) Using the principle of virtual work, the requirement of equilibrium is specified in rate form. Based on assumptions (i) and (ii), the only equilibrium equation is: (6 4) As already mentioned, the constitutive equation that relates the strain increments to stress increments is elasto plastic, the plastic flow being considered to obey von Mises criterion. Using this constitutive relation in conjunction w ith Equation (6 1) and (6 2), further substituting in Equation (6 4) and imposing the condition = 0, leads to the following differential equation for the outward velocity of a material obeying von Mises criterion: , (6 5) where E is the elastic moduli and is the poisson's ratio and E T is the tangent plastic modulus defined as the slope of the stress vs. strain curve in the plastic domain. For the differential equation (Equation (6 5)) in conjunction with the boundary conditions: x = 0 and x= L: v n = 0 and = 0 ,

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152 to have a solution of the form given by Equation (6 1), it follows that: (6 6) Equation (6 6) gives the expression of the critical load for a von Mises material under the assumption of continuous plastic loading (i.e. no unloading takes place anywhere in the shell). Aluminum alloys such as AA2024 T4 (Batterman, 1965) or OFHC copper (Lenhart, 1955) are some example of isotropic metals displaying tension compression asymmetry in plastic flow. Hence, the re exists a need to assess the influence of such asymmetry on plastic buckling behavior. Fo r this purpose, we will carry out the analysis using the quadratic isotropic form of the Cazacu et al. (2006) yield criterion. Since the stress state in the shell is approximately plane, with 0< < (compr essive stresses are negative), the effective stress according to this criterion becomes , (6 7) with Further use of the decomposition of the total strain into an elastic and plastic part and application of the flow rule leads to: (6 8)

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153 with = E / E T and the Poisson coefficient. Further substitution in the expressions of the resultants and bending moments gives: ( 6 9 ) ( 6 10 ) ( 6 11 ) Imposing the buckling condition: = 0 using Equation (6 9) it follows that (6 12) Using Equation (6 1) in conjunction ( Equation (6 9 ) (6 11 )) and further substituting in the equilibrium equation in rate form ( Equation (6 4)) leads to the differential equation governing the velocity field v n for a isotropic material displaying a tension compression asymmetry. , (6 13) For E quation (6 13) to admit a solution of the form with m integer, it should be of the form , (6 14)

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154 H ence by comparing Equation (6 13) and (6 14) it follows that the critical load at which the tub e buckles should be, (6 15) It should be noted that if =1, the critical load N CPB reduces to the critical load for a von Mises material obtained by Batte rman (1965). This is expected since von Mises criterion corresponds to the SD parameter k=0, for which =1 in Equation (6 7). Note that the critical axial stress cr at buckling for an isotropic material with SD effects is, ( 6 16) and depends on the elastic properties of the material (elastic constants E and the specificities of the plastic flow, i.e. the tension compression asymmetry parameter k and the plastic tangent modulus ( ) as well as the geometry of the tube. Since the plastic tangent modulus E T depends on the equivalent plastic strain, to determine the critical stress at buckling a Newton Raphson scheme needs to be used to estimate the equivalent plastic strain from the consistency condition. To illustrate the effect of the tension compression asymmetry of the plastic flow on the critical stress at buckling, we determine the critical buckling stress for three isotropic materials having the same elastic pr operties and stress strain response in uniaxial tension, specifically (E = 73 GPa, = 0.3), and hardening law in uniaxial tension given by: (6 17)

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155 with A 0 = 298.28 , A 1 = 7370.53 and A 2 = 13.74 . F or k = 0, we recover a von Mises material. Critical buckling stress of three isotropic materials are shown in Figure 6 2 for different tube geometry R/h. T he first material displaying a tensile strength lar ger than the compressive streng t h (k=0.5), the second being a von Mises mat erial and the third one displaying a tensile strength smaller than the compressive strength (k= 0.5). Figure 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 . For a material characterized by tension strength greater than the compressive strength, the critical stress will be smaller when compare d to a von Mises material. This difference increas es as the ratio R/h increases. O n the other hand, if the mate rial displays a compressive strength greater than the tension strength, the critical stress will be larger than a von Mises material. Batterman (1964), in his seminal work, reported critical buckling stress for A A 2024 T4 aluminum alloy tubes of differ ent geometries i.e. for varying ratios between

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156 the inner radius R to thickness h . Furthermore, to characterize the plastic behavior of the AA 2024 T4 alloy, Batterman (1964) conducted an uniaxial tension and an uniaxial compression test, and found that this aluminum alloy displays a slight tension compression asymmetry ( Figure 6 4). To further confirm that this slight asymmetry in yielding is not due to experimental error, Batterman (1965) also conducted a bending test and reported the true stress total strain response as obtained from measurements on the top and bottom surface of the specimen (Figure 6 5) . The stress strain response at the center of the beam was considered to be the average of the response on the top and bottom of the specimen. It is w orth noting that the stress at the bottom of the specimen is higher than the average stress in the center. Thus, the test results confirmed that up to a strain of ~ 2%, the yield strength in tension is higher than the yield strength in compression. On the basis of all the test results (i.e. uniaxial tension, uniaxial compression and bending), it can be concluded that the observed strength differential is an intrinsic property of AA 2024 T4. As demonstrated in the previous section, this tension compression asymmetry influences the buckling stress.

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157 Figure 6 3 . Comparison between the experimental stress strain response in uniaxial tension and compression for AA 2024 T4 (data from Batterman, 1965) . Figure 6 4 . Bending test results for AA 2024 T4 (data from Batterman, 1965) . Table 6 1 gives the numerical values for the elastic constants as well as yield stresses in uniaxial tension and compression for this material .

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158 Table 6 1 . Material properties for AA 2024 T4 Young's Modulus (MPa) Poisson's Ratio Yield stress in uniaxial tension (MPa) Yield stress in uniaxial compression (MPa) 73084.446 0.3 3 303.12 282.68 To account for the tension compression asymmetry of the AA2024 T4 aluminum alloy, the isotropic form of the Cazacu et al. (2006) yield criterion will be used to model its plastic behavior. It is worth recalling that in its isotropic form, the only material parameter involved in the Cazacu et al. (2006) yield criterion could be directly related to the strength differential effect (E quation 3 8). For AA 2024 T4, at yielding, the ratio between the yield strength in tension and the yield strength in compression is T / C =1.075, which corresponds to k=0.085. Note also that, the uniaxial data reported in Fig ure 6 4 indicates an evolution of the strength differential effect from k=0.085 at yielding to k=0.013 at 2% plastic strain . L inear interpolation will be used to determine the value of the coefficient k for any accumulated plastic strain: ( 6 18 )

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159 Table 6 2. Evolution of the SD parameter k with plastic strain for AA 2024 T4 Plastic strain T / C k 0 1.075 0.085 0.0002 1.030 0.050 0.02 1.009 0.013 From the experimental true stress true strain curve (Figure 6 4) it follows that the hardening law can be expressed using Equation (6 17). T he expression for the tangent modulus E T is given as, (6 19 ) w here A 0 , A 1 , A 2 are constants, which are identified based only on one tensile uniaxial test . For AA 2024 T4 alloy the hardening parameters are given as: A 0 =298.28 , A 1 = 7370.53 and A 2 = 13.74 . Comparison between experimental and stress strain response predicted by the Cazacu et al. (2006) criterion with k given by the evolution law (6 18 ) and the isotropic hardening law (6 4 ) in uniaxial tension and uniaxial compression is shown in Figure 6 5 . It is worth noting t hat the model describes well the particularities of the plastic behavior of AA2024 T4 alloy, specially the tension compression asymmetry. Furthermore, The projection of the yield surface in the biaxial x plane ( x =0 ) according to the Cazacu et al. (2006) yield criterion and the von Mises yield criterion at initial yielding for A A2024 T4 are shown in Figure 6 6 . In cylindrical coordinates, x corresponds to the axial direction and corresponds to the circumferential direction.

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160 Figure 6 5 . Comparison between experimental stress strain response in uniaxial tension (x) and compression (o) and predicted response according to the isotropic form of the Cazacu et al . (2006) model (lines). Data from Batterman (1965). Figure 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 (x corresponds to the axial direction; circumferential direction).

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161 Data on Axial Compression of Thin Cylindrical Shells of AA 2024 T4 . Batterman (1965) has also reported results on low rate axial compression of cylindrical tubes of AA 2024 T4. Each geometry is defined by the length L of the specimen, the inner radius R and the thickness of th e tube h, as shown in Figure 6 7 . T he specimens were tested between flat ended smooth bearing blocks , in a 120,000 lb capacity Riehle testing machine. The change in length and change in diameter of the specimen at the center were measured throughout the test along with the displacement of the machine head . As an example, the average stress strain curves for several specimen ge ometries are given in Figure 6 8 . Buck ling of the tube specimen is reached when the slope of the stress strain curve changes sign, from positive to negative. Figure 6 7 . Mechanical test specimen geometry.

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162 Figure 6 8 . Average stress strain curves from axial crushing tests on AA 2024 T4 cyl indrical shells of different radius to thickness (R/h) ratios . From the data reported in the mechanical tests, the following conclusions can be drawn: Bucklin g strongly depends of the ratio R/h. As the radius to thickness ratio of the specimen increases t he critical buckling stress decreases. The length of the specimen does not have a significant effect on the critical buckling stress. This is observed from testing several specimens of the same radius to thickness ratio R/h=9.7, but of different initial le ngths (L=4 in, L=1.5 in and L=1 in) . Additionally, tests on specimens having the same radius to thickness ratio, R/h=19.7 and R/h=44.69 , but different lengths confirm this observation.

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163 Analytical modeling of the buckling of AA2024 T4 tubes . T he analytical model developed in the previous section will be used to describe the plastic buckl ing of AA2024 T4 alloy specimens for several geometries. Note that the boundary conditions considered for the analytical derivation, namely that all degrees of fre edom at the upper and lower surface boundary of the cylinder are constrained , are the same as in mechanical tests. Furthermore, in the analytical analysis, the tube cross section is assumed to be fully plastic when buckling occurs. Thus, the analytical mod el can be applied to geometries defined by R/h ratios in the 10 60 range. At higher ratios i.e. for very thin shells elastic buckling occurs . On the other hand, for low radius to thickness R/h ratios, the assumption of thin shells is not valid anymore and the analytical model will over predict the buckling load by a large amount. Note that the analytical expression for the buckling load being based on a hinge model, the predicted critical stress at buckling will be more conservative than the buckling load corresponding to a perfect cylinder . This is due to the fact that in the hinge model it is assumed that at critical load the cylinder has already deformed in an outward or inward sine wave form. Another hypothesis of the analytical model is of neutral equi librium at constant load i.e. the rate of change of axial force on the mid surface at buckling is zero. According to the analytical model ( Equation 6 16 ) ; th e buckling stress depends on the plastic deformation. T he yield criterion used to model the plastic behavior of a given material strongly influences the buckling stress predictions. In particular, we will investigate the importance of accounting for the stress differential in plastic flow.

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164 Figure 6 9 shows a comparison between the measured buckl ing stresses as a function of radius to thickness ratio and the predictions based on the analytical model using both the von Mises and Caz acu et al. (2006) yield criterion , respectively. Note that model ing the plastic behavior with a yield criterion that a ccounts for the tension compression asymmetry, the analytical predictions are closer to the experimental data. As seen previously, a material displaying a tensile stren gth greater than the compressive strength will buckle at lower loads than a material tha t does not display tension compression asymmetry . It is worth noting that even a very slight tension compression asymmetry has a visible impact on the buckling load. Figure 6 9 . Comparison of critical buckling stress observed in mechanical test s (symbol 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 .

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165 In the following section, the uniaxial compression of a tube boundary problem will b e solved using the finite element method. Finite element modeling of axial compression of AA 2024 T4 tubes . T he boundary value problem is solved numerically using the finite element (FE) method in conjunction with the von Mises yield criterion and the Cazacu et al . ( 2006) yield criterion, respectively. The FE predictions are compared with experimental data on isotropic AA 2024 T4. All the simulations are carried out using the commercial FE code ABAQUS and the user material routine (UMAT) that was develo ped ( Chapter 3) . As an example, the FE mesh used in a typical calculation for a specimen with a radius to thickness ra tio R/h=9.7 is shown in Figure 6 10 . Due to the symmetry of the problem, only one fourth of the specimen is mesh ed . The lower nodes ( z = 0) were pinned, i.e. , no displacement was allowed, while the upper nodes ( z = L 0 ) we re tied to a rigid tool. Axial displacement wa s imposed by the translation of this tool along the tube axis. The use of a rigid tool ensure d that all the upper nodes experience the same boundary conditions. The FE mesh used in the simulation consists of 2050 quadratic hexahedral elements with full integration (ABAQUS C3D20). The initial cross section was meshed with 5 0 layers of elements and t wo elements were used along the wall thickness. A mesh refinement study was carried out to ensure that the results are mesh independent. The usual definitions of the average axial strain is used , namely, ( 6 20 ) w here , L 0 is the initial length and u is the axial displacement. The reaction force and the applied displacement are measured at every step on the reference point of the rigid tool

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166 to give the load displacement history during the simulation. In the FE simulations, equal sized time increments = 10 3 s were considered. Between five and six iterations per increment were necessary for convergence in the return mapping algorithm, the tolerance in satisfying the respective yield criterion being 10 7 (0.1 Pa). Figure 6 10 . Typical mesh used for the finite element simulation . In order to study the influence of the tension compression asymmetry of the plastic behavior, both von Mises and Cazacu et al. (2006) criteria have been used to determine the buckling load. The buckling load is reached when the load displacement curve reac hes an inflection point ( i.e. the slo pe becomes zero). Ta ble 6 3 compares the prediction of the buckin g stresses obtained from experiments and numerically using the FE method for several geometries considering that the plastic behavior is characterized by the von Mises and the Cazacu et al. (2006) yield criteria, respectively. In addition, Fig ure 6 12 shows the critical buckling stress for several geometries c haracterized by different R/h ratios and a length L=1 in ch for both analytical and FE based numerical predictions according both von Mises and Cazacu et

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167 al. (2006) yield criteria. It is worth noting that both analytical and numerical models predict lower crit ical stresses when the yield criterion used captures the material's tension compression asymmetry. Table 6 3. Comparison of maximum stress at buckling in MPa Geometry Experimental (MPa) F E simulation (MPa) Von Mises CPB06 R/h=9.7, L=4 480.08 491.75 484.2879 R/h=14, L=2 442.02 454.66 449.4004 R/h=19.7, L=1 43 4.58 440 .93 433.5562 R/h=89.3, L=2 303.02 3 6 7 .95 3 6 0.8536 R/h=19.7, L=2 434 .89 449 .08 444.5879 R/h=44.2, L=1 4 0 9.82 4 2 8 .08 4 2 1 .2474 R/h=54.9, L=2 349.15 388 .18 381 .6172 The discrepancies between the analytical and numerical predictions are of the same order independently of the yield criter ion considered (Figure 6 11 ). This observation confirms that the differences between the analytical model and the FE simulation come from the assumption made in th e analytical model relative to the geometry of the specimen (shell hypothesis) and to the fact that the cross section is assumed to be fully plastic . For geometry that perfectly respects these assumptions (for ratio R/h less than 25), the analytical and n umerical critical stresses are in very good agreement.

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168 Figure 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 respectively. Another advantage of using the analytical and numerical approaches is that it not only helps in predicting the critical stress, but also in knowing the plastic energy absorbed by the material prior to pl ast ic buckling . T he automotive industry is looking for new materials such as to increase the energy absorbed during compression and thus improve the safety of the passengers in case of a collision. Hence , the influence of the tension compression asymmetry on the energy absorption will be studied. For each geometry , the plastic energy per unit volume at buckling was calculated. The plastic energy at buckling is the area under the average applied strain vs. overall stress curve. Table 6 4 compares the prediction s of the plast ic energy dissipated per unit v olume obtained numerically for several geometries considering that the plastic behavior is characterized by the von Mises and the Caz acu et al. (2006) yield criterion , respectively. For geometry characterized by ratios R/h less than 25, the experimental and numerical values of the plastic energy absorbed per unit volume are

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169 in very good agreement. Furthermore, it is worth noting that using von Mises criterion the predicted dissipation is more than that predicted using Cazacu et al. (2006) criterion. These results confirm the previous predictions given in Section 6.1.2 about the role of the tension compression asymmetry on the energy absorbed. If we consider two different materials having the same response in uniaxial tension, but different response in uniaxial compression, the material with the lower compressive strength will dissipate less energy per unit volume . Table 6 4. Plastic d issipation per unit volume ( in N/m 2 ) at buckling Geometry FE Analysis Experiment Von Mises CPB06 R/h=9.7, L=4 13146627.23 12238973.77 11206490.77 R/h=14, L=2 8789023.247 7935769.42 69366080.42 R/h=19.7, L=1 5423898.927 5049355.012 36504310.08 R/h=44.7, L=2 819690.1468 727052.1843 7 0 8062.1843 R/h=56.5, L=1 520071.5624 498835.7092 36 8835.7092 R/h=89.3, L=2 278093.1513 262311.0512 Accounting for the tension compression asymmetry of the plastic behavior during axial compression of a tubular structure thus allows a more accurate prediction of the buckling stress than with the usual von Mises yield criterion. Furthermore, even if the d ifference in critical stress is slight, the plastic energy dissipated per unit volume by the material prior to loss of its load carrying capacity is more closely predicted by

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170 accounting for t he strength differential effect . The new model hence gives a more accurate representation of the overall load displacement behavior. It is to be noted that using the isotropic form of Cazacu et al. (2006) criterion and isotropic hardening, it is demonstrated that during axial crushing of thin axisymmetric cylindrical sh ells the buckling stress as well as the energy input required to initiate buckling is correlated to the sign and the value of the strength differential parameter k (for more details and mathematical proofs see the beginning of this section). For isotropic materials with k > 0 (harder in tension than in compression) less energy is absorbed by the material prior to buckling failure whereas the reverse is true for materials with k < 0 (i.e. harder in compression than in tension) . In comparison, for k=0 i.e. fo r the von Mises yield criterion the predicted buckling stress and energy input required for buckling are always higher than that for a material with yield stress in tension higher than in compression (k>0) . With regards to the influence of initial imperfections, it must be acknowledged that the geometry of the specimen plays a very important role on the buckling behavior. Any surface defects or small imperfections would also influence the bu ckling characteristics discussed here. However, at this time the effect of imperfections is not studied for two reasons. The primary focus of this work has been to study the influence of material properties on plastic buckling behavior of perfect thin cyli ndrical specimens subjected to axial crushing. Moreover, for all the specimens studied, a regular axisymmetric mode of failure was observed indicating that surface defects were negligible .

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171 In S ection 6.3 , the analytical and FE model pre dictions are compared with experimental data on axial crushing of an anisotropic Mg AZ31 alloy which displays both anisotropy and a strong tension compression asymmetry irrespective of loading direction. 6.2 Magnesium Alloy: Mg AZ31 The main purpose of t his section is to assess the capabilities of the Mg AZ31 alloy to be used to design structural parts. As mentioned previously, the automotive industry is looking for new materials that can improve the fuel efficiency without losing any mechanical quality, specially the ability to absorb the maximum amount of energy during compression to enforce the safety of the passengers during a crash. In the previous section, the influence of the tension compression asymmetry on the mechanical behavior has been demonstr ated using an analytical development and F E simulations. As seen previously, the plastic behavior of the Mg alloys is characterized by a strong anisotropy and a large strength differential between uniaxial ten sion and uniaxial compression ( Chapter s 3 and 4 ). In this section, the mechanical behavior of the Mg AZ31 alloy will be modeled using the Cazacu et al. (2006) yield criterion and the Hill (1948) yield criterion in conjunction with an isotropic hardening law. For the identification of the material par ameters, the reader is referred to Chapter 3 and 5, specifically the material parameters of the Cazacu et al. (2006) criterion are defined in T able 3 2 , while the parameter for Hill (1948) criterion are given in Table 5 3.

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172 6.2 .1 Axial C rushing of Mg AZ31 C ylind rical T ube : Experiments The axial crush experiments presented here were performed at Material Research Laboratory (RML), General Motors, under a low strain rate (~0.001/s) loading and at room temperature. As seen in the Chapter 3, at room tempe rature the mechanical behavior of Mg AZ31 can be considered strain rate insensitive for strain rates upto 1/s . The initial specimen used was a tube of an initial length L=76.2 mm, an inner radius of R in =10.7 mm, and a thickness h = 2 mm, which lead s to th e ratio R/h= 5.35. The tube was machined in such a way that the axis of the specimen is oriented along the rolling direction. It is to be noted that no other experimental data are available, i.e., there is no data on crushing of a tube with its long axis a long the normal direction. The initial geometry and the specimen after failure are shown in Figure 6 12 . It is worth noting that by looking at the fractured specimen, no buckling seems apparent. But, the average stress axial strain curve reported for the t est clearly show s that the loss of load carrying capacity occurs at an average axial strain av /L= 0.096 for a critical stress of 252 MPa, and is immediately followed by the failur e of the specimen (Figure 6 13 ). Furthermore, the shape of the experimental overall stress vs. average applied strain data is very similar to the mechanical behavior of the Mg AZ31 alloy in compression.In particular, the same S shape that was reported in uniaxial compression is also re ported for the stress strain curve during buckling experiments. This is expected , since during the axial crushing of a tube, the stress state should activate the same deformation mechanisms in the microstructure as in uniaxial compression. As explained pre viously, to correctly capture the plastic behavior of a material during a crush test, the plastic dissipated energy should be well described . The evolution of the energy absorbed per

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173 unit volume is plotted in Fig ure 6 14 , the maximum energy absorbed by the tube per unit volume being ~14.75 N/mm 2 . In the following section, the experimental critical stress and the energy dissipated at buckling wil l be compared to numerical predictions obtained using the FE method . Figure 6 12 . A xisymmetric cylindrical speci men tested under axial compression. A ) Schematic Diagram . B ) Photographed at failure.

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174 Figure 6 13 . Average stress axial strain data in axial crush of Mg AZ31 tube machined with the long axis along rolling direction. Figure 6 14 . Evolution of energy absorbed per unit volume in axial crush estimated from mechanical test data.

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175 6.2 .2 Axial C rushing of Mg AZ31 Cylindrical T ube : FE Simulation The same elastic plastic model with a set of material parameters (Table 3 2) identified for Mg AZ31 to capture the uniaxial behavior is used to capture the mechanical behavior of this alloy under axial crushing. Only one fourth of the specimen was meshed with 2736 quadratic hexahedral elements with full integration (ABAQUS C3D20 ) , as shown in Figure 6 15 . The initial minimal section was meshed with 7 0 layers of elements and two elements were used along the wall thickness , h=2 mm. The boundary conditions are applied using rigid tools, as explai ned in the previous section. Figure 6 15 . Mesh used for the finite element simulation . Figure 6 16 shows the simulated average stress vs. axial strain curve in comparison to the one reported experimentally in axial crushing of a tube oriented along RD. The Cazacu et al. (2006) yield criterion based model prediction is in excellent agreement with the experimental observations . An even more remarkable result is th at the model correctly reproduces the characteristic S shape of the average stress vs. average axial strain . This S shape is also observed in experimental uniaxial

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176 compression tests along RD (or uniaxial tension along ND). According to the numerical predic tions, the buckling state is reached for a critical stress of 262 MPa and an axial strain of av /L= 0.121 , compared to max = 252 MPa and av /L= 0.096 for the experimental data. Figure 6 16 . Comparison of a verage stress vs. average axial strain observed in experiments (x) and predicted by Cazacu et al. ( 2006 ) criterion (line) during axial crush of Mg AZ31 tube along rolling direction.

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177 Figure 6 17 . Comparison of e nergy absorbed per unit volume vs. averag e axial strain observed in experiments (x) and predicted by Cazacu et al. ( 2006 ) criterion (line) during axial crush of Mg AZ31 tube along rolling direction. Figure 6 17 shows the evolution of the absorbed energy before buckling obtained from the FE simul ations using the Cazacu et al. (2006) yield criterion, compared to the experimental data. The dissipated plastic energy W p per unit volume is calculated using the incremental method: (6 2 1 ) where , are respectively the absorbed energy at the increment n and n+1, is the increment of the absorbed energy , and dt is the time step . At each time step, the increment of the absorbed energy is calculated u sing Gauss integration over the integration points: (6 22 ) where, N E is t he number of integration points and w i the weight associated to the integration point i. The model prediction is in very good agreement with experimental data , the absorbed energy per unit volume before buckling predicted is W P =18. 2 5 N/mm 2 , where as the experimental observation is W P =14.75 N/ m m 2 . The Cazacu et al.

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178 (2006) yield criterion correctly captures th e plastic behavior of a crushed tube of Mg AZ31 alloy. N umerical simulations will now be used to analyze the local behavior of the Mg alloy AZ31 at buckling. Figures 6 18, 6 19 show the isocontours of the equivalent plastic strain and the hydrostatic pressure p= 1/3 tr( ) in the plane (RD, ND) at buckling of a Mg AZ31 tube crushed along RD. Figure 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 . A ) isocontours of equivalent plastic strain. B) isocontours of p ressure . Note that X, Y and Z denote the rolling direction, transverse direction and the normal direction, respectively. By comparing the relative location of the zone of the maximal equivalent plastic strain and the zone of the minimal hydrostatic pressure, it can be concluded that at the location where the plastic strain is higher , the press ure is higher i.e. mean stress is lower.

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179 Figure 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 . A ) isocontours of equivalent plastic strain. B ) isocontours of pre ssure . The evolution of the hydrostatic pressure through the thickness o f the tube, shown in Figure 6 19 , reveals that outward buckles form when a pressure gradient exist s through the wall thickness, i.e. the tube will bulge if the press ure at the inner radius is higher than the pressure at the outer radius. At the buckling state, an outward buckle is observed in the plane (RD, ND). Similarly, the isocontours of the equivalent plastic strain and the hydrostatic pressure p= 1/3 tr( ) in the (RD, TD) plane at buckling of the Mg AZ31 tube crushed al ong RD is plotted in Figure 6 20; Figure 6 21 . The observed correlation between pressure and equivalent plastic strain are confirmed. I n the (RD, TD) plane, an inward buckle is observed, located where a pressure gr adient occurs.

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180 Figure 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 . A ) isocontours of e quivalent plastic strain . B ) isocontours of pressure . Figure 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 . A ) isocontours of equivalent plastic strain. B ) isocontours of pressure .

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181 The orthotropic plastic behavior of the Mg AZ31 alloy is clearly seen by comparing the shape of the cross section at the location of the buckle. The cross section, initially circular, becomes elliptic al , the radius along the ND increases, while the radius along t he TD decreases (Figure 6 27 ). To better understand how this change of shape of the cross section appears, the yield surface in the (ND, TD) plane is plotted in Fig ure 6 22 . Axial compression of the tube leads to a tensile stress state in the cross section. Thus, if the tube is crushed al ong the RD, it follows that tension is present along ND and TD. Comparing th e yield limit in tension in TD and ND shows that there is a clear difference in strength between these two directions. In ND tension, the material has least strength to resist pla stic deformation since ND has very low yield strength in tension up to a strain of about 6%. Furthermore, due to the triangular shape of the yield surface, the plastic strain in both ND and TD directions will be completely different (i.e. the normal to t he surface differs largely between ND tension to TD tension stress state ), leading to an elliptica l cross section (Figure 6 22 ).

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182 Figure 6 22 . Theoretical yield surfaces in TD ND plane according to the orthotropic Cazacu et al. (2006) criterion for Mg AZ31 alloy corresponding to different levels of equivalent plastic strain. Figure 6 23 shows the evolution of the isocontours of the equivalent plastic strain predicted by the Cazacu et al. (2006) yield criterion at different axial strain, av /L =0.017, 0 .025, 0.03, 0.05, 0.08, 0.12 in the plane (RD ND). Furthermore, in this figure, the formation of the outward buckle could be analyzed revealing physical insights on the buckling of a Mg AZ31 tube.

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183 Figure 6 23 . Isocontours of equivalent plastic strain i n the cross section at average axial strain A ) av /L =0.017, B ) av /L =0.025, C ) av /L =0.03, D ) av /L =0.05, E ) av /L =0.08, F ) av /L =0 .12; predicted by Cazacu et al. (2006) yield criterion. For low axial strain av /L <0.05, by looking at the deformed profile of the tube, it appears that initially an inward buckle is created (Figure 6 23), however this trend inverses and a outward buckle is created (Figure 6 23). It is worth noting that between

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184 av /L =0.08 and av /L =0.12, the through thickness plastic strain along ND varies from accumulated plastic strains of, P = 0.05 to 0.07 . This level of plastic strain corresponds where the hardening rate on the ND tension curve in creases rapidly (Figure 6 14 ). Similarly, the stress strain curve reported experimentally shows the same increase in terms of hardening. Both unixial compression along RD and uniaxial tension along ND exhibit an S shape stress strain curve. Thus, it is very difficult to correlate the buckling to one or the other direction, but it is evident that the typical plastic behavior of the Mg AZ31 alloy, namely the strong evolution of the tension compression asymmetry, strongly influences the mechanical response under axial crushing. This brut al change in the plastic behavior strongly impact the way the Mg alloy buckles. Since t he Cazacu et al. (2006) criterion is strongly dependant on the third invariant J 3 of the stress deviator s , (J 3 = det( s )), isocontours of J 3 in the plane (RD, TD) and (R D, ND) are plotted in Figures 6 2 4 ; 6 2 5 , respectively. At the buckle location, it should be noted that the sign of J 3 varies from positive to negative, meaning that some finite elements see compressive stress state (J 3 <0), while the others tensile stres s state (J 3 >0). For t he Cazacu et al . (2006) yield criterion which accounts for the tension compression asymmetry, the resulting mechanical behavior in these zones will be totally different, leading to the very specific stress strain curve reported experim entally for the crushing of an Mg AZ31 tube along RD.

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185 Figure 6 24 . Deformed profile of the Mg AZ31 in RD compression in the X Y plane showing isocontours of third invariant of Cauchy stress deviator; predicted by Cazacu et al. (2006) yield criterion. Figure 6 25 . Deformed profile of the Mg AZ31 in RD compression in the X Z plane showing isocontours of third invariant of Cauchy stress deviator; predicted by Cazacu et al. (2006) yield criterion.

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186 6.2 .3 Effect o f Tension Compression A symmtery on B ucklin g of Mg AZ31 As seen previously , the numerical prediction obtained for the Mg AZ31 alloy under axial compression using the Cazacu et al. (2006) yield criterion are in very good agreement with the experimental data. One characteristic of this yield criter ion is to account for the tension compression asymmetry of the plastic behavior. In this section, the relative influence of the str ength differential effect and the orthotropic behavior on the buckling of Mg AZ31 tubes will be investigated. For this purpos e, predictions obtained with the classical Hill (1948) yield criterion, presented in Chapter 5 (the parameters given in Table 5 3) will be compared to the one obtained with the Cazacu et al. (2006) yield criterion. The final cross section of the axial cr ushing of a tube oriented along RD obtained with both yield criteria are shown in Figure 6 26 . Note that for anisotropic Cazacu et al. (2006) yield criterion the effective plastic strain > 6% (also Figure 6 22 ).

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187 A B Figure 6 26 . Center cross section (lower half) at buckling for axial compression along RD. A ) predicted by anisotropic Cazac u et al. (2006) yield criterion. B ) predicted by Hill (1948) yield criterion. As seen previously, according to the Cazacu et al. (2006) yield criterion, the final cross section is strongly elliptical, while the predicted cross section obtained with the Hill (1948) yield criterion is almost circular . Furthermore, the average stress average strain curves predicted by both yield criteria are plotted in Figure 6 27 . While the curve predicted by the Cazacu et al. (2006) yield criterion is in very good agreement with the experimental data, the prediction of the Hill (1948) criterion does not capture the key

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188 feature of the curve, namely the S shape and the axial strain where buckling is reached. Both yield criteria predict a critical stress of the same order ( max =247 MPa for Hill (1948), max =262 MPa for Cazacu et al. (2006) and max =252 MPa experimentally), but the predicted axial strain at b uckling strongly differ ( av /L =0.055 for Hill (1948), av /L =0.12 for Cazacu et al. (2006) and av /L =0.1 experimentally). Furthermore, the evolution of the absorbed energy per unit volume predicted by the two yield criteria is presented in Fig ure 6 28 . While the prediction of the Cazacu et al. (2006) yield criterion are in very good agreement with the ex perimental data, the prediction of the Hill (1948) criterion strongly over predicts the energy absorption before buckling. With neglecting the tension co mpression asymmetry in the modeling of the plastic behavior of Mg AZ31 alloy, the Hill (1948) yield criterion is not able to correctly predict the mechanical response under axial crushing of a tube. Figure 6 27 . Average stress vs. average axial strain o btained from experiments (x), FE simulation using Hill (1948) criterion ( black line) and FE simulation using Cazacu et al. (2006) criterion (red line) during axial crush of Mg AZ31 tube along rolling direction.

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189 Figure 6 2 8 . Energy absorbed per unit volume vs. average axial strain during axial crush of Mg AZ31 tube along RD obtained from experiments (x), Cazacu et al. (2006) yield criterion (red line) and Hill (1948) yield criterion (black line). Figure 6 29 . Deformed profile of Mg AZ31 in RD com pression in the X Z plane , as predicted by Hill (1948) yield criterion. A ) isocontours of e quivalent plastic strain . B ) isocontours of p ressure .

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190 Figure 6 29 show s the isocontours of the equivalent plastic strain and the hydrostatic pressure p= 1/3 tr( ) in the plane (RD, ND) at the buckling state of a Mg AZ31 tube crushed along RD predicted by the Hill (1948) criterion. As seen previously, the buckle appears where a gradient of pressure is seen though the thickness of the tube. It is worth noting that C azacu et al. (2006) and Hill (1948) do not predict the same fin al cross section at buckling or the same location of the buckle. According to the Cazacu et al. (2006) yield criterion, the buckle occurs at the middle of the specimen, while the Hill (1948) yi eld criterion predicts that the buckle is seen clos e to the boundary of the tube. This comparison shows the strong influence of the tension compression asymmetry on the mechanical response under axial crushing in terms of final geometry, critical stress, b uckling axial strain and of absorbed energy per unit volume. In the following description , the axial crushing of an Mg AZ31 tube oriented along ND will be studied, even if no experimental data is available for this test . Figure 6 30 and Figure 6 31 show the final profile and the final cross section of the tube before buckling according to both models . It is worth noting that the prediction of both models is very close for the axial crushing of a tube along ND. Both predict that the buckles will be l ocated closed to the boundary of the specimen, and the final cross section remain almost circular. While with the Hill (1948) yield criterion , the orientation of the tube has no influence on the final geometry of t he specimen (compare Figure 6 29 with Fig u re 6 31 ), with the Cazacu et al. (2006) yield criterion, a strong influence of the direction of crushing is expected, the location and the final cross section of the buckle being completel y different (compare Figure 6 19 with Figure 6 30 ). A circular final cross section is expected for both yield criteria, since if the tube is crushed along the ND, it

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191 follows that tension is presen t along RD and TD. The yield limit in tension along both RD and TD are quite similar, and the yield loci according to both yield criteri a have the same shape in the tension tension quadrant (Figure 3 17 for Cazacu et al. (2006) and Figure 5 12 for Hill (1948) yield criterion ). Figure 6 3 3 shows the prediction of average stress applied axial strain curve for Mg AZ31 cylinder compr essed along ND according to both the Hill (1948) yield criterion and the Cazacu et al. (2006) yield criterion. Concerning the critical stress and the axial strain at buckling, similar predictions are obtained with both yield criteria ( max =358 MPa, av /L = 0.061 for Hill (1948), max =332 MPa, av /L = 0.065 for Cazacu et al. (2006)). It's worth noting that the predictions of the critical stress are much larger than the prediction of the critical stress for an axial crushing along RD.

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192 Figure 6 30 . Deformed profile of Mg AZ31 in ND compression , as predicted by Cazacu et al. (2006) yield criterion. A ) isocontours of Equivalent plastic strain . B ) isocontours of pressure. Figure 6 31 . Deformed profile of Mg AZ31 in ND compression , as predicted by Hill (1948) yield criterion . A ) isocontours of e quivalent plastic strain . B ) isocontours of pressure .

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193 A B Figure 6 32 . Cross section (upper half) at buckle for axial compression along ND . A ) predicted by anisotropic Cazac u et al. (2006) yield criterion. B ) predicted by Hill (1948) yield criterion.

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194 Figure 6 3 3 . Average stress vs. average axial strain during axial crush of Mg AZ31 tube along ND predicted by Cazacu et al. ( 2006 ) yield criterion (solid line) and predicted by Hill (1948) yield criterion (Dashed line). Figure 6 3 4 . Energy absorbed per unit volume vs. average axial strain during axial crushing of Mg AZ31 tube along ND predicted using Cazacu et al. ( 2006 ) yield criterion (solid line) and predicted by using Hill (1948) yield criterion (dah sed line). Figure 6 3 4 shows the predicted evolution of the energy absorbed per unit volume for axial compression along ND according to both the Hill ( 1948) yield criterion and Cazacu et al. (2006) yield criterion. Both yield criteria show similar trends. All the results (critical stress, axial strain at buckling and absorbed energy) discussed in this

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195 section are summarized in the Table 6 5. It should be noted that the analytical model based on the assumption of isotropic material are not reliable any more for a strongly orthotropic material such as Mg AZ31 alloy. Table 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 Axis of crushing Analytical (E q. 6 16 ) Exp. FE prediction von Mises CPB06 Hill (1948) CPB06 RD max (MPa) 301.2 308.90 252.24 247.53 262.38 av /L 0.12 0.13 0.096 0.055 0.121 W p (N/mm 2 ) 36.6 26.6 14.75 11.7 18.25 ND max (MPa) 356.2 377 358.79 332.9 av /L 0.116 0.092 0.061 0.065 W p (N/mm 2 ) 21.1 29.8 15.6 17.8 From Table 6 5 the following observations can be made: For Mg AZ31, the assumption of isotropy leads to overprediction of the buckling stress and energy absorbed per unit volume. The anisotropic Hill ( 1948) yield criterion underpredicts the axial strain at buckling and the plastic work dissipation per unit volume at buckling in RD tests. Only the Cazacu et al. (2006) yield criterion is correctly able to c apture both the critical stress and axial strain at failure as well as able to predict the plastic work

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196 dissipation per unit volume. According to this criterion, i rrespective of the direction of loading the specimen fails after plastic dissipation of ~18 N/ mm 2 . 6.3 Observations C hapter 6 is devoted to understanding the effect of tension compression asymmetry on the plastic buckling of axially compressed cylindrical shells. For this purpose, the mechanical behavior has been modeled using Cazacu et al. (200 6) yield criterion. For isotropic materials, two approaches have been used: (a) an analytical approach assuming a simplified plastic hinge model, and (b) a finite element method based numerical approach. The analytical a pproach extends the results obtaine d by Batterman (1965) for isotropic materials obeying von Mises criterion. On the basis of the analytical study, it can be concluded that SD effects influence the critical stress. Depending on the ratio of the yield strength in uniaxial tension and uniaxia l compression the buckling stress is either higher (this is the case for the ratio c / T >1) or lower (for c / T <1) than that predicted for a von Mises material ( c / T =1) by Batterman (1965). This holds true for any tube geometry (i.e. any ratio R/h, Figure 6 9 ). It was shown that for an isotropic material such as AA 2024 T4 which displays strength differential effect the predictions based on the new model are closer to the experimentally observed buckling stress ( Table 6 3 and Figure 6 9 ). Most importantly, it was shown that the plastic dissipation per unit volume at buckling is strongly influence d by the tension compression asymmetry of a given material. Even a very small difference between the yield in tension and compression results in a large difference in the energy absorbed by the material prior to buckling (Table 6 4 ).

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197 Axial crushing of cylindrical shells of AA2024 T4 was also simulated using FE and the UMAT for Cazacu et al. (2006) yield criterion in conjunction with isotropic harde ning . It was confirmed that there does indeed exist a strong correlation between the critical stress at buckling and the asymmetry in behavior between uni axial tension and compression (Figure 6 11 ) . Moreover, the trends predicted by the new analytical mode l are the same as those predicted by the 3 D FE simulations. For an anisotropic material, the tension compression asymmetry is direction dependent. Hence , for the anisotropic Mg AZ31 alloy it is shown that the critical stress at buckling depends on the di rection of axial loading relative to the material axes of orthotropy. It was shown that Hill's (1948) yield criterion, which does not account for tension compression asymmtery is unable to predict the load vs. deformation behavior and the critical stress at buckling, irrespective of the direction of loading. While with the Hill (1948) yield criterion, the orientation of the tube has no influence on the final geometry of t he specimen (compare Figure 6 29 with Figure 6 31 ), with the Cazacu et al. (2006) yield criterion, a strong influence of the direction of crushing was predicted, the location and the final cross section of the buckle being completely different ( comp are Figure 6 19 with Figure 6 30 ). Most importantly, the very unusual behavior of a Mg A Z31 specimen when it is subject to axial crushing along the rolling direction (RD) was explained. The evolution of the hydrostatic pressure through the thickn ess of the tube (Figure 6 20 ) reveals that an outward buckles forms when a pressure gradient exist s through the wall thickness, i.e. the tube will bulge if the pressure at the inner radius is higher than the pressure at the

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198 outer radius. At the buckling state, an outward buckle is observed in the plane (RD ND), but an inward buckle occurs in the (RD TD ) plane. Furthermore, the unusual buckling behavior was correlated with the unusual uniaxial compression behavior in the rolling direction (unusual S shape curve instead of a typical concave down curve). For the specimen with long axis along normal direc tion, no data were available. A completely different buckling behavior than for the RD specimen is predicted using Cazacu et al. (2006) model. For the ND specimen, the buckles are located closed to the lower and upper boundaries of the specimen, and the fi nal cross section remain almost circular. In conclusion, only by accounting for the tension compression asymmetry in the material both the load vs. deformation behavior and the critical stress at buckling of Mg AZ31 can be captured. Most importantly, the energy absorbed by the material prior to buckling can be correctly described only if the strength differential effects and anisotropy of the material are taken into consideration.

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199 CHAPTER 7 CONCLUSION AND FUTURE WORK This work is devoted to further the state of the art in modeling deformation of Mg AZ31 alloy. The approaches that were used to simulate the deformation behavior of this magnesium alloy were: A crystal plasticity theory based models and (b) an ela stic plastic macroscopic model based on Cazacu et al. (2006) yield criterion . First , the parameters of Cazacu et al. yield criterion were determined for the given Mg AZ31 alloy. New hardening laws we re developed to capture the distort ion of the yield su rface associated with the texture evolution in the material . The elasto plastic model based on Cazacu et al. (2006) criterion and the new hardening laws was implemented as a user material subroutine in the FE code Abaqus (2009). Simulations of the materia l response for uniaxial loading in tension and compression for different orientations were compared to the experimental data. It can be concluded that the model describes very well both the anisotropy and strength differential effects observed in the mater ial. In particular, for the first time, the unusual sigmoidal stress strain response for in plane compression (RD TD plane) was simulated with greater precision than ever achieved before. Moreover, it was predicted that the normal direction tension curve d oes not have a concave down appearance; instead it is predicted to have a distinct S shape. The plastic deformation response of Mg AZ31 was also modeled within the framework of crystal plasticity. Using a combination of mechanical test data and metallographic information, a new methodology was developed for the determination of the single crystal plastic defo rmation mechanisms operational along different st rain

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200 paths. Furthermore, it wa s demonstrated that by using this methodology it is possible to model with great accuracy within the viscoplastic self consistent framework (VPSC) both the texture evolution and the stress strain response at the macroscopic level. For the first time, the texture evolution of the material in shear was descr ibed with accuracy. Further, both modeling approaches were used to describe the response of the material in torsion. T he unusual characteristics of the torsional response of Mg AZ31 we re explained and the experimental data predicted with accuracy. Specifi cally, the nature and magnitude of the axial strains develop ed in the material when subjected to torsion is explained . U sing the crystal plasticity model it was demonstrated that the observed experimental axial effects could be quantitatively predicted onl y if both slip and twinning are considered operational. For the first time, plastic buckling of materials with strength differential effects was investigated. An analytical derivation was conducted to understand the role of tension compression asymmetry o n the buckling stress of axially compressed cylindrical tubes. Further analysis of buckling of thin cylindrical tubes made of Mg AZ31 using the FE method to simulate axial crush highlighted the role of both anisotropy and the tension compression asymmetry present in the material . It was shown that both the critical buckling stress and the plastic energy that accumulate prior to buckling depend heavily on the strength differential effect observed during uniaxial tests along different loading paths . While wi th the Hill (1948) yield criterion, the orientation of the tube has no influence on the final geometry of the specimen (compare Figure 6 29 with Figure 6 31 ), with the Cazacu et al. (2006) yield criterion, a strong influence of the direction of

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201 crushing wa s predicted, the location and the final cross section of the buckle being completely different. As a general conclusion, it can be stated that the main features of deformation of this Mg alloy for simple and complex loading paths can only be understood a nd simulated if the unique anisotropy and strength differential characteristics of the plastic flow are accounted for. Future work will be directed towards localization and bifurcation analysis using the model based on Cazacu et al. (2006) yield criterion. Failure analysis using a dilatational extension of this model will also be pursued.

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202 APPENDIX CAZACU ET AL. (2006) PARAMETERS FOR DIFFERENT STRAIN RATES AND TEMPERATURES Table A 1. Values of the Cazacu et al . ( 2006) coefficients for the Mg AZ31 specimens tested at 150 0 F and strain rate of 10 4 /s Strain C 22 C 33 C 12 C 13 C 23 C 44 k 0.03 0.9848 4.2235 0.4129 0.2026 0.2909 1.5977 0.6389 0.05 0.93 3.196 0.3305 0.263 0.2377 1.8137 0.5558 0.06 0.9734 2.9662 0.4854 0.1629 0.2365 1.872 0.485 0.08 1.1066 2.1729 1.5155 0.5883 0.3771 3.1089 0.1539 0.10 1.2497 0.967 1.3828 1.2423 0.8685 3.1234 0.1711 Table A 2. V alues of the Cazacu et al.(2006) coefficients for Mg AZ31 specimens tested at 300 0 F and strain rate of 10 4 /s . Strain C 22 C 33 C 12 C 13 C 23 C 44 k Initial 0.9919 2.7964 0.7187 0.2503 0.1136 1.772 0.4536 0.0 3 0.9919 2.7964 0.7187 0.2503 0.1136 1.772 0.4536 0.0 6 0.9919 2.796 4 0.7187 0.250 5 0.1136 1.77 4 0.4537

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203 Table A 3. V alues of the Cazacu et al.(2006) coefficients for the Mg AZ31 specimens tested at 300 0 F and strain rate of 1/s . Strain C 22 C 33 C 12 C 13 C 23 C 44 k 0.03 1.0611 6.5038 3.2958 2.2147 2.2065 2.4482 0.4617 0.05 1.0611 6.5038 3.2958 2.2147 2.2065 2.4482 0.4617 0.06 1.0726 6.4937 3.3355 2.1841 2.2093 2.4481 0.449 0.08 0.4033 6.8055 3.9125 2.2551 1.7776 3.4363 0.2986 0.10 0.2184 6.0004 3.8979 2.0455 1.2346 3.6872 0.2709

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211 BIOGRAPHICAL SKETCH Nitin Chandola was born in New Delhi, India, in 1985 . He graduated with a Bachelor of Engineering in Mechanical Engineerin g from the University of Pune in August, 2007 . In August 2009, Nitin joined the group of Professor Cazacu as a graduate research assistant in the Department of M echanical and Aerospace Engineering at the Research Engineering and Education Facility (REEF).