Development of a Generalized Isotropic Yield Surface for Pressure

Home , Yield surface

Development of A Generalized Isotropic Yield Surface for Pressure Insensitive Metal Plasticity Considering Yield Strength Differential Effect in Tension, Compression and Shear Stress States

by Kivanc Sengoz

B.Sc. in Aeronautical Engineering, June 2001, Istanbul Technical University M.Sc. in Mechanical Engineering, May 2006, Istanbul Technical University

A Dissertation submitted to The Faculty of The School of Engineering and Applied Science of The George Washington University in partial fulfillment of the requirements for the degree of Doctor of Philosophy

January 31, 2017

Dissertation directed by

Azim Eskandarian Professor of Engineering and Applied Science

Cing-Dao (Steve) Kan Professor of Engineering and Applied Science The School of Engineering and Applied Science of The George Washington University certifies that Kivanc Sengoz has passed the Final Examination for the degree of Doctor of Philosophy as of June 29, 2015. This is the final and approved form of the dissertation.

Development of A Generalized Isotropic Yield Surface for Pressure Insensitive Metal Plasticity Considering Yield Strength Differential Effect in Tension, Compression and Shear Stress States

Kivanc Sengoz

Dissertation Research Committee:

Azim Eskandarian, Professor of Engineering and Applied Science, Dissertation Co-Director

Cing-Dao (Steve) Kan, Professor of Computational Solid Mechanics, Dissertation Co-Director

Majid T. Manzari, Professor of Civil and Environmental Engineering, Committee Member

Pedro F. Silva, Professor of Civil and Environmental Engineering, Committee Member

Tarek Omar, Chief Scientist and Technical Advisor for Federal Railroad Administration, U.S. Department of Transportation, Committee Member

ii Dedication

To my mother

iii Acknowledgments

I would like to thank Prof. Azim Eskandarian for his academic supervision, personal support, and constant inspiration throughout my studies in graduate school. I also would like to acknowledge the financial, academic, and technical support that he provided to me as the

Director of the National Crash Analysis Center (NCAC).

I would like to express my sincere gratitude to Prof. Cing-Dao (Steve) Kan, who has been my research advisor and supervisor during my doctoral research. I also would like to acknowledge the financial, academic, and technical support that he provided to me during his service at GWU as the Director of the National Crash Analysis Center (NCAC).

I would like to acknowledge Prof. Majid T. Manzari and Kim Roddis, as chairs of the

Civil & Environmental Engineering Department of the George Washington University for their invaluable support during my doctoral studies.

I also would like to take this opportunity to extend my gratitude to Dr. Gilbert (Chip)

Queitzsch for bringing the invaluable industrial and governmental insight to my research. His guidance has been very influential in helping me to understand the scientific impact and potential applications that can come out of doctoral research.

It has been a privilege having the opportunity to work with Mr. Paul A. Du Bois, who is an internationally known expert in computational mechanics, during the course of my research. His scientific approach to a problem has shown me admirable ways of being a true scholar.

iv This research was funded by the Federal Aviation Administration. I would like to acknowledge Mr. William Emmerling and Mr. Donald Altobelli of the Federal Aviation

Administration’s Aircraft Catastrophic Failure Prevention Research Program, and the

William J. Hughes Technical Center, Atlantic City, N.J., for their technical and financial support throughout this research.

I would like to acknowledge Dr. John O. Hallquist of Livermore Software Technology

Corp. (LSTC) for providing me the required tools during the implementation of the material model into LS-DYNA.

I would like to acknowledge Prof. Amos Gilat and Dr. Jeremy Seidt of Ohio State

University (OSU) for their cooperation within regarding the uncontained aircraft engine failure research program and for providing material test data.

v Abstract

Development of A Generalized Isotropic Yield Surface for Pressure Insensitive Metal Plasticity Considering Yield Strength Differential Effect in Tension, Compression and Shear Stress States

Due to effects of twinning and texture evolution, the yield surface for hexagonal close packed (HCP) metals displays an asymmetry between the yield in tension and compression, and significantly changes its shape with accumulated plastic deformation. Traditional von

Mises von Mises ( ) plasticity yield criteria and hardening assumptions can not accurately model these phenomena due to lack of stress state dependent yield stress and hardening approach. In von Mises plasticity, yield stress in uni-axial tension is always equal to yield stress in uni-axial compression, and yield stress in pure-shear is always 0.577 times the yield stress in uni-axial tension. The same assumptions are also valid for hardening behavior. In this dissertation, a generalized isotropic yield surface model that can describe both the yielding asymmetry and distortional hardening (stress state dependent hardening) on the pressure independent plastic response of metals is proposed. Yielding is described by a newly developed macroscopic phenomenological yield function that accounts for asymmetry between yielding in uni-axial tension and uni-axial compression states. This yield criteria also provides flexibility for the yield stress in pure-shear compared to von Mises ( ) plasticity; therefore it could be used by face-centered cubic (FCC) and body-centered cubic

(BCC) materials as well as HCP materials. The coefficients involved in this proposed yield criterion are considered to be functions of the accumulated plastic strain. The proposed model was implemented into the explicit finite element code LS-DYNA and used to simulate the uni-axial tension, uni-axial compression, and torsion tests of Al2024-T351 and Ti-6Al-

4V materials in the rolling direction using smooth dogbone, solid cylinder, and thin walled

vi tube specimens respectively. Tabulated inputs of characterized material test results are directly used for the constitutive treatment of the new material model. Comparisons between predicted and measured force-displacement curves and macroscopic strain fields show that the proposed model describes very well the yield strength differential effect and distortional hardening (stress state dependent hardening) based on the uni-axial tension, uni-axial compression and pure-shear stress states. On the other hand, the anisotropic nature of Ti-6Al-

4V causes directionally dependent plastic flow in some cases, and this phenomenon cannot be treated by the current material model due to the isotropic plasticity assumption.

The proposed model is then extended to include the effects of strain rate and temperature softening related with the temperate increase within the material due to adiabatic heating using tabulated input curves for yield-temperature and yield-strain rate dependency. Then a new, viscoplastic material model with generalized yield surface is implemented into a nonlinear explicit dynamics finite element code, LS-DYNA. In this material model, a higher order (higher than quadratic order) generalized yield function with isotropic and isochoric plasticity is utilized, and stress state dependent isotropic hardening, strain-rate hardening and temperature softening is considered. The model takes adiabatic heating and resulting softening into account due to the plastic work. The constitutive relation is specifically developed to cover pressure insensitive plastic behavior of metals as a function of state-of- stress, strain-rate, and temperature.

The proposed strain rate and temperature dependent model was implemented into LS-

DYNA and used to simulate yield-temperature dependency of Al2023-T351 and yield-strain rate dependency of Ti-6Al-4V. Component based specimen tests are used to validate and show the robustness, accuracy and efficiency of the new material model. The post

vii experiment data and simulation results are shown to be in very good agreement with each other.

It is shown that the new material model is capable of predicting yielding asymmetry between uni-axial tension and uni-axial compression and pure-shear stress states accurately within the limits of the convexity region. Stress state, strain rate, and temperature dependent plastic flow prediction capability of the model suggests that the new material model can be used as a promising tool for diverse applications of dynamic plastic deformation and subsequent ductile failure phenomenon. Therefore, the material model can be used in the numerical solution of the impact problems in industry such as fan blade-out containment events in turbofan engines, vehicle crashworthiness, and metal forming.

viii Table of Contents

Pages

Dedication ...... iii

Acknowledgments ...... iv

Abstract ...... vi

Table of Contents ...... ix

List of Figures ...... xii

List of Tables ...... xxi

List of Symbols ...... xxii

List of Acronyms ...... xxv

CHAPTER 1 - Introduction ...... 1

1.1 Introduction ...... 1

1.2 Background and Motivation ...... 3

1.3 Research Objectives and scope ...... 5

1.4 Original Contributions ...... 6

1.5 Outline of the Dissertation ...... 7

CHAPTER 2 - Literature Review ...... 9

2.1 Introduction to Yielding ...... 9

2.1.1 Isotropic yield criteria for metals ...... 9

2.1.2 Deviations from plasticity and higher order yield surfaces ...... 14

ix 2.1.3 Role of Third Deviatoric Stress Invariant ( ) in Modeling Asymmetry between Tensile and Compressive Yield ...... 19

CHAPTER 3 - Methodology and Theoretical Framework ...... 25

3.1 Fundamentals of continuum plasticity ...... 25

3.1.1 Perspectives on Plastic Constitutive Equations: Principle of Maximum

Plastic Resistance and Drucker’s Postulate ...... 30

3.1.2 Consequences of Drucker’s hypothesis ...... 34

3.2 Development of proposed generalized yield surface (GYS) plasticity model involving stress invariants and ...... 35 3.3 Associated flow rule, hardening rule, and definition of effective plastic strain for GYS plasticity model...... 43

3.4 Perspectives on dependent yield surfaces: GYS, Hosford, Drucker, Cazacu-Barlat ...... 52

3.5 Convexity of the GYS model and comparison between the GYS, Drucker and Cazacu-Barlat convexity regions ...... 67

3.6 Development of convexity algorithm for the GYS model ...... 91

CHAPTER 4 - Numerical Implementation of GYS Yield Criterion ...... 98

4.1 Constitutive relation of the proposed GYS model ...... 98

4.2 Integration algorithms for elasto-plastic constitutive equations ...... 104

4.2.1 Time integration and numerical implementation of GYS into LS-DYNA ...... 108

x 4.3 Verification of numerical results using single element test cases at different stress states...... 122

4.3.1 Single element analysis and verification of GYS material model ...... 122

4.3.1.1 Benchmark-problem definition ...... 122

CHAPTER 5 - Application and validations of GYS plasticity model ...... 136

5.1 Materials, specimen properties and finite element procedures for the numerical simulation of uni-axial tension, compression and torsion (pure- shear) test cases ...... 136

5.2 Experimental results versus numerical simulations for uni-axial tension, compression and torsion (pure-shear) tests...... 145

5.3 Influence of the GYS plasticity model on the failure response compared to the von Mises plasticity model ...... 159

CHAPTER 6 - Concluding remarks and Future Work ...... 166

References ...... 169

Appendix - A: Second condition of convexity for the GYS model ...... 173

Appendix - B: Derivative of GYS yield function with respect to the increment of the plastic strain multiplier ...... 176

Appendix - C: Single element verification cases of convexity projection ...... 177

Appendix - D: Comparison of MAT224 and MAT224_GYS models under the von Mises plasticity assumption considering proportional (uni-axial tension and plain strain tension) and non-proportional loading (plain strain tension- uni-axial tension) case ...... 185

xi List of Figures

Pages Figure 1-1: Applications of dynamic failure for different structures; (a) offset frontal crash of a passenger car (http://www.iihs.org), (b) uncontained aircraft engine failure (c) fan disk failure and (d) different failure modes after ballistic impact of projectiles with different nose shapes, adapted from [1]...... 2 Figure 1-2: Failure modes in impacted plates, after [8]...... 5 Figure 2-1: Tresca and von Mises yield criteria in the π-plane...... 12 Figure 2-2: Difference of Tresca and von Mises yield criteria for pure-shear state in the π-plane...... 13 Figure 2-3: Comparison of yield surfaces for different materials under combined loading; (a) after [17] and (b) after [21]...... 13 Figure 2-4: Plane stress yield loci for Drucker, Tresca and von Mises (normalized by uniaxial tension) after Drucker [24]...... 15 Figure 2-5: Plane stress yield loci for Hosford, Tresca, von Mises (a) n=1 Tresca, n=2 von Mises and n=10 Hosford (b) plane stress yield loci predictions for different n values between 1 and 2.767 as shown by Hosford in 1972 [23]...... 16 Figure 2-6:. Different isotropic plane stress yield surfaces between the von Mises yield surface and the lower bound (Tresca) after Karafillis and Boyce [26]...... 18 Figure 2-7: Different isotropic plane stress yield surfaces between the von Mises yield surface and the upper bound after Karafillis and Boyce [26]...... 18 Figure 2-8: The four reference points to construct the Vegter yield function in plane stress after [31]...... 19 Figure 2-9: Elastic deformation corresponding to pure stretching of the crystal lattice (top) and plastic deformation trough dislocation slip (bottom) [36] [37]...... 21 Figure 2-10: Crystal lattice reorientation due to mechanical twinning [36][37] ...... 21

Figure 2-11: Plane stress yield loci corresponding to / =2/3, 3/3 ( von Mises), 4/3 [40] ...... ...... 22 Figure 2-12: A comparison of force displacement curves between experimental results and simulation results for flat grooved specimen (Bai & Wierzbicki [41]) ...... 23

xii Figure 3-1: (a) Isotropic hardening and (b) kinematic hardening in plane stress yield loci [47] ...... 28 Figure 3-2: Hardening response of pure titanium according to theoretical yield surface experimental yield data corresponding to fixed values of equivalent plastic strain after [48] ...... 29 Figure 3-3: Stable and unstable material behavior on stress-strain curve according to Drucker’s stable material postulate [45] ...... 32 Figure 3-4: Properties of a yield surface with associated flow rule (a) normality (b) corner condition (c) convexity [47][50][51] ...... 33 Figure 3-5: Representation of Lode parameter and UT (uniaxial tension), UC (uniaxial compression) and PS (pure-shear) stress states (b) Comparison of yield surface models in deviatoric plane ( ᴨ–plane) (c) effect of tension ( σT)-compression ( σC) yielding asymmetry on the yield surface in deviatoric plane…………………………………………………..37 Figure 3-6: Uni-axial tension, uni-axial compression and pure-shear (torsion) tests required by GYS to determine model coefficients , , and ...... …41 , Figure 3-7: (a) Four internal curves computed by GYS for initial yield and subsequent hardening (b) four equations for the determination of model coefficients , , and ...... 42 , Figure 3-8: Cazacu-Barlat yield surface in the plane stress space corresponding to uni-axial tension/uni-axial compression yield ratios, 3/2, 3/3, 3/4...... 57 Figure 3-9: Comparison between the Cazacu-Barlat and GYS yield surface models in the plane stress space corresponding to uni-axial tension/uni-axial compression yield ratios, 3/2, 3/3, 3/4...... 57 Figure 3-10: related scale factor on von Mises stress for Hosford yield function ...... 60 Figure 3-11: Comparison of related scale factor on the von Mises stress for the GYS and Hosford yield functions ...... 61 Figure 3-12: Comparison of scale factor on von Mises stress for GYS with and without tension- compression asymmetry corresponding to order (m) 6 ...... 62 Figure 3-13: Hosford non-unique flow ...... 62 Figure 3-14: GYS non-convex flow ...... 63

xiii Figure 3-15: Example of the plane strain condition for the Hosford yield function and non-unique flow characteristic shown by the yield function as exponent m becomes higher ...... 64 Figure 3-16: a = value for plane strain condition of the Hosford yield function ...... / ...... 65 Figure 3-17: Example of plane strain condition for the GYS function and unique flow characteristic shown by the yield function as order m gets higher...... 66 Figure 3-18: a = value for plane strain condition of the GYS model ...... 66 Figure 3-19: Convexity / region of the GYS model in terms of yield stress ratio of uni-axial compression to uni-axial tension and pure-shear to uni-axial tension and Lode parameter ...... 77 Figure 3-20: Convexity region of the GYS model in terms of yield stress ratio of uni-axial compression to uni-axial tension and pure-shear to uni-axial tension and Lode parameter ...... 78 Figure 3-21: (a) Convexity region of GYS model with yield ratio uni-axial compression/uni-axial tension=1 (b) Lower and upper boundaries of the convexity region correspond to pure-shear yield to uni-axial tension yield ratio 0.545 and 0.611 with the yield ratio uni-axial compression/uni-axial tension=1 ...... 79 Figure 3-22: Convexity region of GYS model with yield ratio uniaxial compression/tension=0.95...... 81 Figure 3-23: Convexity region of GYS model with yield ratio uniaxial compression/tension=0.75 ...... 81 Figure 3-24: Convexity region of GYS model with yield ratio uniaxial compression/tension=1.1 ...... 83 Figure 3-25: Convexity region of GYS model with yield ratio uniaxial compression/tension=1.25 ...... 83 Figure 3-26: (a) Convexity region of GYS model with yield ratio compression/tension=2.00 (b) Convexity region of GYS model with yield ratio uniaxial compression/tension=0.50...... 85 Figure 3-27: Convexity curves of the GYS model corresponding to the Lode angle parameter (a) 0, (b) 1, (c) -1, respectively ...... 86

xiv Figure 3-28: (a) Intersection of the Lode angle parameter ( ) curves corresponding to 0, 1, -1 for the GYS model (b) Comparison of the convexity regions of the GYS model and von Mises, Drucker, and Cazacu-Barlat models...... 87 Figure 3-29: Min and Max points on the convexity region of GYS model in terms of yield stress ratios ...... 88 Figure 3-30: Min and Max tip points on the convexity region of GYS model ...... 92 Figure 3-31: Projection operation of convexity algorithm of GYS model ...... 93 Figure 4-1: Schematics of the return mapping algorithm in stress space ...... 105 Figure 4-2: Schematics of the return mapping algorithm in stress space (a) Fully implicit (b) semi-implicit tangent cutting plane ...... 106 Figure 4-3: The role of elasto-plastic return mapping algorithms ...... 110 Figure 4-4: Schematic illustration of the return mapping algorithm for the GYS ...... 110 Figure 4-5: Stress update scheme using trial stress and a plastic correction ...... 113 Figure 4-6: (a) Geometric interpretation of the radial-return in principle stress space (b) GYS return mapping where mapping direction is the yield surface normal at the trial stress point B ...... 115 Figure 4-7: Geometric interpretation of the return on the invariant plane ...... 116 Figure 4-8: Schematics of the solution interval and successive secant iterations to find the increment in plastic strain assuming that root of the function ( f=0) is found at 4 th iteration step [71] ...... 117 Figure 4-9: Successive secant iterations and dependence of function f on the increment of equivalent plastic strain. Note that curvature of the function f could be both ways (a) and (b) without effecting monotonic decrease of the function [70][71]...... 118 Figure 4-10: (a) Rate dependent tensile hardening curve from a dynamic test ...... 119 Figure 4-11: Coupling between GYS and MAT224 ...... 120 Figure 4-12: Single element model at different state of stresses ...... 122 Figure 4-13: Axial stress (Z stress) versus Effective stress (von Mises) ...... 123 Figure 4-14: GYS-Pure-shear case. Shear stress (YZ stress) versus Effective stress (von Mises) ...... 123

xv Figure 4-15: GYS-Uniaxial compression case. Axial stress (Z stress) versus Effective stress (von Mises) ...... 124 Figure 4-16: GYS-Plain strain tension case. Axial stress (Z stress) versus Effective stress (von Mises) ...... 124 Figure 4-17: GYS-MAT24 comparison under the von Mises plasticity assumption .... 125 Figure 4-18: Temperature dependent yield curves for the temperature softening case .. 125 Figure 4-19: Temperature softening—uniaxial tension single element case ...... 126 Figure 4-20: Temperature softening—uniaxial compression single element case ...... 127 Figure 4-21: Temperature softening—temperature increase on uniaxial compression single element case ...... 127 Figure 4-22: Temperature softening—temperature increase on uniaxial tension single element case ...... 128 Figure 4-23: Strain rate dependent yield curves for single element strain rate dependency case ...... 128 Figure 4-24: Strain rate single element test in compression [strain rate=1/s] ...... 129 Figure 4-25: Strain rate single element test in compression [strain rate=100/s] ...... 129 Figure 4-26: Stress increment and return mapping using a plastic direction estimated from the beginning of time step for (a) small strain increment and a strain increment more than an order of maginute larger than yield strain [60] ...... 130 Figure 4-27: Effect of time step on the GYS response-uniaxial tension single element case ...... 132 Figure 4-28: Effect of time step on the GYS response-uniaxial coompression single element case ...... 132 Figure 4-29: Effect of time step on the GYS response-simple shear single element case ...... 133 Figure 4-30: Effect of time step on the GYS response-nonproportional loading case-uniaxial tension-biaxial tension (axial stress) ...... 133 Figure 4-31: Effect of time step on the GYS response-nonproportional loading case-uniaxial tension-biaxial tension (lateral stress) ...... 134 Figure 4-32: Effect of time step on the GYS response-nonproportional loading case plain strain tension-uniaxial tension (axial stress) ...... 134

xvi Figure 4-33: Effect of time step on the GYS response-nonproportional loading case plain strain tension-uniaxial tension (lateral stress) ...... 135 Figure 5-1: Al2024-T351 and Ti-6Al-4V specimen manufacturing orientations for the 12.7 mm stock ...... 138 Figure 5-2: Reported composition of a 12.7 mm thick Al2024-T351 plate ...... 138 Figure 5-3: Reported composition of a 12.7 mm thick Ti-6Al-4V plate ...... 138 Figure 5-4: (a) Instron 1321 servo-hydraulic biaxial load frame located in OSU DMML (b) Elevated temperature test set up ...... 139 Figure 5-5: Tension and Torsion Split-Hopkinson bar located in OSU DMML ...... 139 Figure 5-6: Compression Split-Hopkinson bar located in OSU DMML ...... 139 Figure 5-7: Dimensions of a thin flat dog-bone spceimen for uni-axial tension test ...... 140 Figure 5-8: Dimensions of a thin walled tube specimen for torsion test ...... 140 Figure 5-9: (a) Tension and (b) Compression temperature test series data for Al2024-T351 from OSU DMML [16] ...... 142 Figure 5-10: (a) Tension and (b) Compression strain rate test series data for Al2024-T351 from OSU DMML [16] ...... 143 Figure 5-11: (a) Tension and (b) Compression temperature test series data for Ti- 6Al-4V from OSU DMML [76] ...... 144 Figure 5-12: (a) Tension and (b) Compression strainrate test series data for Ti- 6Al-4V from OSU DMML [76] ...... 145 Figure 5-13: Finite element simulation model and set up: (a) Tension simulation and (b) Compression simulation ...... 147 Figure 5-14: Finite element simulation model and set up: Torsion simulation ...... 147 Figure 5-15: Al2024-T351 uni-axial tension test and simulation at room temperature ...... 149 Figure 5-16: Maximum principal strain contours of Al2024-T351 tension test specimen at room temperature (a) simulation using GYS (b) Test (DIC) ...... 149 Figure 5-17: Al2024-T351 uni-axial compression test and simulation at room temperature ...... 150 Figure 5-18: Al2024-T351 uni-axial tension test and simulation at 150 oC ...... 150 Figure 5-19: Al2024-T351 uni-axial compression test and simulation at 150 oC ...... 151

xvii Figure 5-20: Al2024-T351 uni-axial tension test and simulation at 300 oC ...... 151 Figure 5-21: Al2024-T351 uni-axial compression test and simulation at 300 oC ...... 152 Figure 5-22: Al2024-T351 torsion test and simulation at room temperature ...... 152 Figure 5-23: Al2024-T351 torsion test at room temperature (a) simulation using GYS (b) Test (DIC) ...... 153 Figure 5-24: Ti-6Al-4V uni-axial tension test and simulation at strain rate of 1/s at room temperature ...... 153 Figure 5-25: Ti-6Al-4V uni-axial tension test and simulation at strain rate of 1/s at room temperature (a) simulation using GYS (b) Test (DIC data) ...... 154 Figure 5-26: Ti-6Al-4V uni-axial tension test and simulation at strain rate of 1/s at room temperature (a) simulation using GYS (b) Test (DIC data) ...... 154 Figure 5-27: Ti-6Al-4V uni-axial compression test and simulation at strain rate of 1/s at room temperature ...... 154 Figure 5-28: Axial strain distribution of Ti-6Al-4V uni-axial compression test at strain rate of 1/s at room temperature (a) simulation using GYS (b) Test (DIC data) .. 155 Figure 5-29: Ti-6Al-4V torsion test and simulation at strain rate of 1/s at room temperature ...... 156 Figure 5-30: Shear strain distribution on Ti-6Al-4V torsion test specimen at strain rate of 1/s at room temperature (a) simulation using GYS (b) Test (DIC data) ...... 156 Figure 5-31: Maximum principal strain distribution on Ti-6Al-4V torsion test specimen at strain rate of 1/s at room temperature (a) simulation using GYS (b) Test (DIC data) ...... 156 Figure 5-32: Ti-6Al-4V uni-axial tension test and simulation at strain rate of 1500/s at room temperature ...... 157 Figure 5-33: Ti-6Al-4V uni-axial compression test and simulation at strain rate of 1500/s at room temperature ...... 157 Figure 5-34: Axial strain distribution on Ti-6Al-4V uni-axial tension test specimen at strain rate of 1500/s at room temperature ...... 158 Figure 5-35: Axial strain distribution on Ti-6Al-4V uni-axial tension test specimen at strain rate of 1500/s at (a) simulation using GYS (b) Test (DIC data) ...... 158

xviii Figure 5-36: Axial stress for yield in uni-axial Compression/Tension=0.9 (uni-axial tension case) ...... 159 Figure 5-37: Axial stress for yield in uni-axial Compression/Tension=0.9 (uni-axial compression case) ...... 160 Figure 5-38: Equivalent plastic strain for yield in uni-axial Compression/Tension=0.9 (uni-axial compression case) ...... 160 Figure 5-39: Axial stress for yield in uni-axial Compression/Tension=1.1 (uni-axial compression case) ...... 161 Figure 5-40: Equivalent plastic strain for yield uni-axial Compression/Tension=1.1 (uni-axial compression case) ...... 161 Figure 5-41: Axial stress for a yield stress ratio of uni-axial compression to tension varies from 0.9 to 1.1 (uni-axial compression case) ...... 162 Figure 5-42: Effect of the Uni-axial Compression/Tension yield ratio on failure with the GYS uni-axial compression case ...... 162 Figure 5-43: Effect of the Pure-Shear/Uni-axial Tension yield stress ratio on failure using GYS (pure-shear case) ...... 163 Figure 5-44: Effect of the Pure-Shear/Uni-axial Tension yield stress ratio on failure using GYS (pure-shear case) ...... 163 Figure 5-45: Effect of the Pure-Shear/Uni-axial Tension yield stress ratio on failure using GYS (Non-proportional loading case pure shear-uniaxial tension) ...... 164 Figure 5-46: Effect of the Pure-Shear/Uni-axial Tension yield stress ratio on failure using GYS (Non-proportional loading case pure-shear-uniaxial tension) ...... 164 Figure 5-47: Effect of loading sequence on failure using GYS-maximum principal stress ...... 165 Figure 5-48: Effect of loading sequence on failure using GYS-equivalent plastic strain ...... 165 Figure C-1: The GYS convexity region and data point projection...... 177 Figure C-2: Verification cases of projection algorithm for the data set out of convexity region...... 177 Figure C-3: Verification case-2 ( =1.8) and projection point on the convexity region...... ...... / ...... 178

xix Figure C-4: Verification case-2 ( =1.8) (a) tension and compression effective stresses without scale factor (b) scale/ factor of 1.693 on the tension curve...... 179 Figure C-5: Verification case-3 ( =1.8) and projection point on the convexity region...... ...... / ...... 180 Figure C-6: Verification case-3 ( =0.5). (a) Uni-axial tension and uni-axial compression effective stresses without / scale factor. (b) Scale factor of 0.59 on the uni-axial tension curve ...... 181

Figure C-7: Verification case-4 =1.3 and projection point on the convexity region...... √3 /...... 181

Figure C-8: Verification case-4 =1.3 (a) Uni-axial tension and pure-shear effective stresses without scale factor.√3 / (b) Scale factor of 1.0588 on the uni-axial tension curve...... 182 Figure C-9: Verification case-8 =0.3-2.29 for data point projection for non- convex cases. (a) Uni-axial tension / and uni-axial compression effective stresses without scale factor. (b) Scale factor of 1.0588 on the uni-axial tension curve. (c) Uni-axial tension and pure-shear effective stresses without scale factor. (d) Scale factor of 1.059 on the uni-axial tension curve...... 184 Figure D-1: Loading cases and evolution of the stress-state values...... 185 Figure D-2: Uni-axial tension case – Plastic strain to failure, damage, triaxiality, effective stress...... 186 Figure D-3: Plain strain tension case – Plastic strain to failure, damage, triaxiality, effective stress...... 186 Figure D-4: Non-proportional loading case (Uni-axial tension-plain strain tension) case – Plastic strain to failure, damage, triaxiality, effective stress...... 187

xx List of Tables

Pages Table 3-1: Yield stress prediction of GYS at different state of stresses...... 54 Table 3-2: Comparison of properties of yield surfaces...... 55 Table 3-3: Comparison of convexity ranges of yield surfaces ...... 75 Table 4-1: MAT224_GYS input card in LS-DYNA...... 121

xxi List of Symbols

Roman Symbols

Material constants for the plasticity model of Gao [44] , , Fourth order positive definite isotropic stiffness tensor

GYS hardening parameters , , Elastic tensor in the component form

Specific heat

Deviatoric strain rates

Plastic deviatoric strain rate Modulus of elasticity

Yield function Shear modulus Plastic potential H Hessian matrix Second-order unit tensor

Fourth-order, symmetric unit tensor Stress invariants , , Deviatoric stress invariants

, , Bulk modulus k Order of the Karafillis yield function Current length Initial length, gauge length m Order of the Hosford yield function Moment

M Principal minors of Hessian matrix , , Prandtl-Reuss normal outward flow vector

NNNpq Curvature tensor n Yield surface normal n Order of the Hosford yield function Pressure

Pressure rate Updated pressure Trial pressure Load Deviatoric stress rates sss Deviatoric principle stresses , , Deviatoric stress tensor

Deviatoric trial stress rate

xxii Time increments , ⁄, Temperature

Torque Melting temperature

Homologous temperature ∗ Current temperature Room temperature

Trace of a tensor or matrix tr Plastic work Greek symbols

Taylor–Quinney coefficient

Engineering shear strain Kronecker delta

τ Yield stress in pure shear Plastic strain increment tensor Δ Strain tensor Strain rate (rate-of-deformation) tensor Effective plastic strain ̅ Effective plastic strain at the beginning of time step ̅ Elastic components of the rate of the strain tensor Plastic components of the rate of the strain tensor Volumetric strain rate Plastic volumetric strain rate True plastic strain in uni-axial tension, uni-axial compression, , , and pure-shear Lode angle

Lode-angle-parameter Plastic multiplier Rate of plastic multiplier

One of the Lamé constants Poisson’s ratio = The number Pi Density Eigenvalue Cauchy stress tensor Effective stress of GYS model

xxiii Equivalent stress inVegter model Principle stresses , , , Yield stress in uni-axial tension, uni-axial compression and shear , , Second-order stress tensor

Hydrostatic stress

Trace of the stress tensor

Mean stress

von Mises stress Yield, flow stress

Initial von Mises stress Updated von Mises stress Initial Yield stress Updated Yield stress Trial yield stress , Shear stress Twist angle and yield function in Karafillis-Boyce model Vegter yield function Subscript/superscript

Indices , , , , , Cartesian coordinate indices

, , Indices

1,2,3 … , Time increments

, + 1 Hydrostatic

H Plastic

von Mises trial Trial prediction Yield

xxiv List of Acronyms

AHSS Advanced High Strength Steels BCC Body-Centered-Cubic CDM Continuum Damage Mechanics CSHB Compression split Hopkinson bar DIC Digital Image Correlation DSP Digital Speckle Photography EOS Equation of State FAA The Federal Aviation Administration FAR Federal Aviation Regulation FBO Fan Blade-Out FCC Face-Centered-Cubic FE Finite Element FEA Finite Element Analysis GYS Generalized Yield Surface HCP Hexagonal Close Packed J-C Johnson-Cook Material Model NCAC National Crash Analysis Center PS Pure-Shear RT Room temperature SD Strength Differential SHLF Servo-Hydraulic Loading Frame SHPB Split Hopkinson Pressure Bar SSHB Shear split Hopkinson bar TSHB Tension split Hopkinson bar Tri Triaxiality UC Uni-axial compression UMAT User Material UT Uni-axial tension

xxv CHAPTER 1 - Introduction

1.1 Introduction Inelastic behavior of materials is a subject that has fascinated engineers and researchers over many decades. Sustained interest in this topic is due to its direct relevance in numerous engineering applications. The real challenge is to accurately predict inelastic deformation when the material is subjected to complex mechanical loading, where the material is subjected to a multi-axial state of stress, high strain rates and variable temperature conditions. The accurate representation of stress, strain, and temperature fields within the structural components depends strongly on the mathematical representations or constitutive equations of the inelastic behavior of these materials. It is vital to understand the dynamic plastic deformation characteristics of structures to design better engineered products. Designing lighter and cost effective structures under impact loading without any compromise from structural performance, safety, and integrity has been one of the most important areas of research in mechanics for several decades. Dynamic plastic deformation and failure applications can be extended to ‘aircraft hardening’, ‘spacecraft/satellite shielding’, ‘high speed machining’, and ‘armor penetration.’ Some of the potential applications of ductile dynamic deformation and failure are illustrated in Figure 1-1 for automotive-crashworthiness, aerospace-uncontained engine debris protection, fan blade containment during the fan blade-out event, metal forming-pressurized pipes, and defense- armor penetration. The circumstances that are illustrated in Figure 1-1 prove the importance of successfully predicting the dynamic plastic deformation and subsequent failure for engineering practice, where an unsuccessful prediction may put the engineering structure in jeopardy. For computational modeling of impact loading on metallic materials, ductile failure occurs after a significant amount of plastic deformation followed by material instability that leads to necking or shear localization. Therefore, the correct representation of yielding and subsequent plastic flow is crucially important on the numerical simulation of impact problems. For some metals, yield stress and subsequent flow stress might be different in uni- axial compression than in tension. This phenomenon is known as strength differential (SD)

1 effect in yielding and mostly seen with hexagonal close packed (HCP) metals such as magnesium and titanium alloys which offer great potential to reduce weight and thus replace the most commonly used materials.

Figure 1-1: Applications of dynamic ductile deformation and subsequent failure for different structures; (a) offset frontal crash of a passenger car (http://www.iihs.org), (b) uncontained aircraft engine failure (http://www.ntsb.gov), (c) fan disk failure and (d) different failure modes after ballistic impact of projectiles with different nose shapes, adapted from [1].

Currently, the use of HCP metals is restricted because of a lack of fundamental understanding of their complex flow behavior under a multi-axial stress state and a lack of a representative rate and temperature dependent phenomenological generalized yield surface model for computational analysis. Experimental programs underscore another need for a generalized yield function. Pure- Shear yield ratio of some metals relative to the uni-axial tension yield might be different than 0.577, which is a fixed ratio at classical (von Mises) plasticity. This situation can be observed in FCC, BCC and HCP metals. Yielding in shear stress is one of the most important states experienced by the structure at the points of shear localization before the ductile failure occurs. The SD (strength differential) effect in yielding and deviations from a fixed ratio of pure- shear yield relative to uni-axial tension yield for metals establish the need for material

2 models with a generalized yield function to be able to capture these effects in computational analysis. Considering that nonlinear explicit finite element analysis is the most popular tool used for the computational analysis of impact problems, commonly used material models in these codes such as Johnson-Cook or Piecewise Plasticity Mat24 in LS-DYNA use classical ( von Mises) plasticity [2]. It is clear that deviations from flow in metal plasticity cannot be captured within the capability of these material models. To address this requirement, in this dissertation, a new viscoplastic material model with a generalized yield surface is developed that incorporates state-of-stress, temperature, and strain rate effects to predict plastic response during ductile deformation. The new model is supported by an experimental program to characterize the material properties for the model input. Results of a parallel experimental program are used to validate the accuracy and robustness of the model.

1.2 Background and Motivation Uncontained turbine engine debris presents risks to an aircraft where high energy fragments impacting the aircraft need to be considered. Damage from an engine rotor burst can be catastrophic because the events may compromise structural integrity, initiate a fire, endanger critical systems, or put lives in danger [3][4]. The Federal Aviation Airworthiness Regulations (FAR) require aircraft be designed to minimize and mitigate these risks. FAR 25.903(d) (1) states: ‘Design precautions must be taken to minimize the hazards to the airplane in the event of an engine rotor failure.’ Therefore, there is significant interest in methods that can be applied to better understand events and improve aircraft designs. Following a catastrophic uncontained fan disk failure in commercial passenger service in 1989 [4], The Federal Aviation Administration (FAA) initiated a research program called Uncontained Engine Debris Mitigation Program, part of the Aircraft Catastrophic Failure Prevention Program, to investigate methods for analyzing and designing for uncontained engine debris mitigation. Different methods were suggested to minimize the hazards, although it was understood that absolute containment is unlikely and consequently, the minimization of fragment energies and developing mitigation strategies in the aircraft were the reasonable methods of compliance [5].

3 Civil aviation authorities also require that turbine engines in civil service be designed to contain the release of any single fan, compressor or turbine blade. To demonstrate compliance, FAR 33.94 requires every new civil engine design be tested to prove that the highest energy blade will be contained [6]. After successful demonstration of the highest energy blade, the other stages in the engine are allowed to be shown to comply by analysis. Considering the cost, difficulty and time demands of performing such tests, the engine manufacturers want to be sure that a new engine will pass the first time it is tested. Therefore predictive numerical models and analysis of the containment structures is used during the design process to reduce the risk of having to repeat the blade containment test. Computational mechanics has evolved to a level that performance of structures under impact loading can efficiently and effectively be predicted by employing numerical techniques with high accuracy if accurate material models are used [7][8][9]. Therefore, there is a significant amount of literature about the assessment of dynamic plastic deformation and failure on the impact performance evaluation of major aerospace materials, along with their capability to mitigate hazards from uncontained engine debris.The Johnson-Cook plasticity and fracture model is a phenomenological material model commonly used to simulate impact and penetration [10][11]. This constitutive model defines the effective material flow stress as a function of strain rate, temperature and effective plastic strain. But there is no difference between yielding in uni-axial tension and yielding in uni-axial compression in this model, and the ratio of pure-shear yield to uni-axial tension yield is fixed at a value of of 1/ √3 (0.577). This means that the material model uses classical ( von Mises) plasticity and cannot model yielding asymmetry or strength differential effect in uni-axial tension, uni- axial compression, and pure-shear stress states. Thus a stress state dependent generalized yield surface is needed to be able to capture the deviations from classical plasticity to make sure that stress state dependency of initial yielding and subsequent plastic flow are covered adequately treated by the material model. All of these models, whether phenomenological or physics-based, require experimental data for calibration of parameters. Since results from numerical simulations are dependent on the material models, they are also dependent on the experimental data used to calibrate these models. To be considered an improvement over the Johnson-Cook plasticity and fracture model, the proposed material model should have three basic input curves for most common tests of plasticity uni-axial

4 tension, uni-axial compression, and torsion (pure-shear) to be able to cover strength differential in yielding. Surely this ability will reflect positively to the parameter calibration process and will be helpful to overcome the shortcomings of a calibration procedure based on classical plasticity model which allow one to use only one test as an input for the parameter calibration process. Common failure mechanisms under impact loading were summarized by Backman [12] and Backman and Goldsmith [13], and an illustration of these failure modes are categorized by Zukas as shown in Figure 1-2 [8]. Accurate prediction of plastic deformation and subsequent failure modes by numerical simulations is achievable only if a material model is used that utilizes explicitly defined stress state dependent plastic deformation and failure characteristics, which should also incorporate strain-rate and temperature softening as addressed by Perzyna [14].

Figure 1-2: Failure modes in impacted plates, after [8].

1.3 Research Objectives and scope As discussed in the motivation and background, a yield surface based solely on the second invariant of deviatoric stress tensor ( ) is not able to correctly represent a material with a yield strength differential. Thus, the goal of the current work is to develop a

5 generalized yield surface based on second invariant of deviatoric stress tensor ( ) and third invariant of deviatoric stress tensor ( ) to account for the yield strength differential effect for uni-axial tension, uni-axial compression, and pure-shear stress states and to develop a tabulated viscoplastic material model with a generalized yield surface (GYS) that can incorporate high strain rate and temperature softening effects. Simulations of the deformation behavior of structures having length scales much larger than those of crystallographic grains cannot be performed effectively with crystal plasticity models but require phenomenological constitutive equations for yielding and hardening under multi-axial stress states. The new material model described in this work has been developed using a phenomenological model approach at the macro level. Nano-scale or micro-scale concerns and phase transformation effects are not considered in this work. A constitutive law of plastic deformation has been developed using small strain increment, rate and time independent plasticity concepts. Transformation of the constitutive rate equation into an incremental equation has been done using a suitable semi-implicit integration procedure. The necessary rotations of stress and strain for large strain calculations are taken care of by LS-DYNA. The new material model is implemented into an explicit dynamics finite element (FE) code, LS-DYNA. This model is expected to be used as a generic tool to simulate plastic flow of any ductile metal under impact loading. In order to meet the objectives, the following assumptions have been made. Isotropic yielding and hardening is assumed. Anisotropic or orthotropic effects are not the subject of this work. Associated plastic flow is preferred. Isochoric plasticity is assumed, so there is no volume change during plastic deformation. Dependence on stress triaxiality (Pressure/ von Mises) is excluded, so plastic flow is pressure insensitive. A multiplicative composition is assumed for the new proposed yield function in terms of and effects.

1.4 Original Contributions The original contributions accomplished within the course of this Dissertation can be summarized as: Development and implementation of a thermo-elastic/viscoplastic material model with a generalized yield surface in terms of the second and third stress deviator invariants for isotropic materials exhibiting yield strength differential effect and pressure insensitivity. A

6 new viscoplastic material model with higher order dependence on and is developed, which employs tabulated inputs of uni-axial tension, uni-axial compression and pure-shear stress-strain curves to construct the flow surface as a function of state of stress, strain-rate and temperature. The convexity requirement is considered and the constraints imposed on the material model are discussed. This level of detail in the plasticity model is important for simulations involving ductile deformation where the failure is preceded by intense localization of plastic strain. In industrial applications, a new material model will provide independent yielding in uni-axial tension, uni-axial compression, and pure-shear while remaining isotropic for the nonlinear finite element analysis of pressure insensitive metal plasticity. This provides greater flexibility to match experimental data in uni-axial tension, uni-axial compression, and pure-shear states at specimen level. Thus material characterization based on matching force-deflection curves at the specimen level will provide better characterized data for the agreement between test and finite element simulations at the component level regarding ductile deformation and failure. The model has generic modular input for rate and temperature dependent stress-strain curves, so the flow surface is explicitly constructed through uni-axial tension, uni-axial compression, and pure-shear hardening curves at different strain rates and temperature. The model can therefore be easily used for a wide range of applications including high velocity impact analysis, vehicle crashworthiness, airworthiness, vulnerability, and survivability predictions, etc. It is successfully shown by the test programs that the state-of-stress has a significant role in the dynamic deformation and subsequent failure behavior of ductile materials under impact loading [15][16][17]. Since the new material model shows successful prediction capability in stress state dependent plasticity, it has several advantages over previous models, particularly in predicting state dependent ductile failure and thus the new material model can also be utilized as a promising tool to evaluate the dynamic failure prediction of structures.

1.5 Outline of the Dissertation The dissertation consists of 5 interrelated chapters. Chapter 1 provides background information and motivation, defines objectives and scope, and presents an overview of the thesis.

7 Chapter 2 presents an overview of computational material models in structural metals. This chapter presents a review of the classical plasticity approaches currently used to model material plastic deformation. The general procedures used to implement these approaches are briefly described, and an assessment of the shortcomings associated with each approach is discussed. Then deviations from classical J 2 plasticity are addressed and the role of J 3 in material modeling of metal plasticity is discussed. Chapter 3 provides a step-by-step explanation of the theoretical framework involved in developing a and dependent material model with generalized yield surface. Convexity of proposed model has been discussed and details of convexity region have been evaluated from computational point of view. Chapter 4 describes the numerical implementation of the new material model and discuss validation cases and single element verification tests. Chapter 5 presents the validation procedures for the material model, where component based test specimens are simulated and compared with the test results of Al2024-T351 aluminum alloy and Ti-6Al-4V titanium alloy in the rolling direction. Force-deflection curves are used to compare test and numerical analysis considering both existing classical (von Mises) plasticity model and proposed GYS plasticity model. Also, the proposed material model is compared and benchmarked with an existing classical dependent material model using single element tests. Numerical stability, accuracy and robustness of the new model is benchmarked with an existing model, where various failure modes are compared by using proportional and non-proportional loading at different states-of-stress to show influence of the proposed GYS plasticity model on failure prediction. Finally, chapter 6 presents a summary of the main results. A list of contributions to the philosophy of engineering science and recommendations for future research and potential applications are also provided.

8 CHAPTER 2 - Literature Review

2.1 Introduction to Yielding The analysis of plastic deformation is important in many engineering applications including crashworthiness, impact analysis, and manufacturing problems, among many others. When materials undergo plastic deformations, permanent strains are developed when the load is removed. Many materials exhibit elastic-plastic behaviors, that is, the materials exhibit elastic behavior up to a certain stress limit, called the yield strength, after which plastic deformation occurs. For sufficiently small values of stress and strain, a metal reassume their original shape upon unloading. When loaded beyond this reversible (elastic) range, metals do not reassume their original shape upon unloading, but exhibits a permanent (plastic) deformation. In the plastic range, the flow stress for metals typically increases monotonically with accumulated plastic strain. After a metal has been subjected to a stress exceeding the yield limit that separates the elastic and plastic ranges, the current stress becomes the new yield limit when the material is reloaded. For a multi-axial state of stress, a material’s yield limit is mathematically described by a yield criterion. Considering the general case of a three dimensional stress state, for all combinations of three principal stresses, the locus of all stresses for which yielding occurs in principal stress space is called the yield surface. The yield surface is the three dimensional analog of the two-dimensional yield curve and of the one-dimensional yield point. A state of stress inside the yield surface is elastic, a state of stress on the surface is plastic, and states of stress outside the surface do not have a unique solution.

2.1.1 Isotropic yield criteria for metals

At the continuum level, the initial material behavior is taken to be linearly elastic up to the yield point. The yield surface delineates the current elastic region in stress space. For an elastically isotropic metal, yielding depends on stress, temperature, strain rate and internal state (described by hardening or internal state variables). Under isothermal conditions, initial yielding depends only on the stress state, which is often described by three stress invariants

9 (2.1) =

(2.2) 1 = 2

(2.3) 1 = 2 where is the first stress invariant, and are the second and third deviatoric stress invariants, respectively, and and denote the Cauchy and deviatoric stress components, respectively. The yield function can be expressed as

(2.4) = , , =

Physically, represents the hydrostatic stress, and represents the distortional energy in the material. Though no definite physical quantity is attributed to , it can be treated as a weighting parameter that induces asymmetry in yield and flow behavior between tension and compression. In this formulation, k is a constant. Historically, the oldest isotropic yield criterion was proposed by Tresca in 1864 [18]. He experimented with plastic flow of metals under high pressure (extrusion of a metal through die). He concluded that yielding depends on maximum shear stresses so hydrostatic pressure is insignificant. According to Tresca’s criterion, the material transitions to a plastic state when the maximum shear stress reaches a critical value. The Tresca criterion is given by

= (2.5) {| − |, | − |, | − | } where and are the principal stresses, and is the yield stress in uni-axial tension. , , Possibly the most widely used isotropic yield criterion is the one proposed independently by Huber [19] in 1904 and von Mises [20] in 1913. This criterion is usually referred to as the von Mises criterion. The von Mises criterion is based on the observation that a

10 hydrostatic pressure cannot cause yielding of the material. The plastic state corresponds to a critical value of the elastic energy of distortion, i.e.

(2.6) = where k is a constant and is second invariant of the Cauchy stress deviator given by

(2.7) 1 = − + − + − 6 or, alternatively

(2.8) 1 = + + 2 where and are the principal values of the Cauchy stress deviator, defined as , ,

(2.9) 1 = − 3 The von Mises yield criterion is also known as the maximum distortional energy criterion, and yielding is presumed to occur when the distortional strain energy density at some point in a body with a multiaxial stress state is equal to the to the distortional strain energy density at yielding under uniaxial tension or compression.

= (2.10) = √ − + − + − = 3 where is the uniaxial tensile yield strength and is the equivalent von Mises stress. In this equation, stresses are deviatoric stresses since only the distortional part of the energy is considered and due to its direct association with , the von Mises yield criterion is also known as the criterion.

11 In principal stress space, we can select a plane on which the hydrostatic stress on the plane is zero, namely the plane perpendicular to the axis through the origin. This plane is called π-plane or stress deviator plane and= it is= perpendic ular to hydrostatic stress axis. The projection of Tresca’s yield surface on the π-plane (the plane which passes through the origin of principal stress axis and is perpendicular to the hydrostatic axis) is a hexagon centered on the origin whose size depends on the magnitude of yield stress in uniaxial tension. The projection of the von Mises yield locus on the π-plane is a circle that circumscribes the Tresca hexagon as shown in Figure 2-1. Thus the implication of the Tresca criterion is that yielding takes place at or below the loads at which yielding takes place according to Mises yield criterion. These two yield criteria are easier to compare on the π- plane three-dimensional principal stress space. The two criteria coincide under uni-axial tension and uni-axial compression. However under pure shear the largest difference between two criteria occurs as shown in Figure 2-2. That difference is after considering of geometry of a regular hexagon, an amount √ . Thus von Mises criterion predicts yielding in - √ pure shear at a stress nearly 16% higher than the Tresca criterion.

Figure 2-1: Tresca and von Mises yield criteria in the π-plane.

Figure 2-2: Difference of Tresca and von Mises yield criteria for pure-shear state in the π-plane.

In separate studies, Lode [17] and Taylor and Quinney [21] have tested samples under combined loading and have checked the yield surfaces of the materials of interest. Lode tested steel, copper and nickel, and Taylor and Quinney tested copper, aluminum and mild steel. The results of their findings are illustrated in Figure 2-3(a) for Lode and Figure 2-3(b) for Taylor and Quinney, where it is shown that the materials of interest behave closer to the von Mises yield surface than to that from Tresca theory.

(a) (b) Figure 2-3: Comparison of yield surfaces for different materials under combined loading; (a) after [17] and (b) after [21].

13 2.1.2 Deviations from J 2 plasticity and higher order yield surfaces

A yield function is called quadratic if it is a function of the squares of the component of the stress tensor. The von Mises criterion ( plasticity) is the most general quadratic yield function for isotropic materials and requires only one constant, and yield stress from either a uni-axial tension or compression test. The multiaxial response of the material is fully determined by the results of a uni-axial test. Thus yield strength in the pure shear state relative to the tensile yield strength has a fixed ratio of 0.577 in (von Mises) plasticity. Also the yield stress in uni-axial tension is always identical to the yield stress in uni-axial compression. Mendelson [22] has shown the existence of bounds in an isotropic yield surface of a material with a fixed yield stress in uni-axial tension. These bounds are derived after symmetry and convexity considerations. The lower bound coincides with the limiting maximum shear stress yield surface as described by Tresca [18] in 1864, whereas the upper bound corresponds to a limiting value of the sum of the two greater diameters of Mohr’s circles as mentioned by Hosford [23] in 1972. Higher order (higher than quadratic order) generalized yield surfaces are needed to be able to capture the family of yield surfaces swapping stress space between lower and upper bound rather than the quadratic yield surfaces (von Mises). As a result, (von Mises) plasticity cannot describe certain phenomena in isotropic metals such as:

• Yield stress in pure shear different than 0.577 times the tensile yield stress. • Characterization of the family of yield surfaces between lower bound (Tresca) and von Mises or between von Mises and upper bound. • Tensile-Compressive asymmetry in yielding.

From the phenomenological viewpoint, for metallic materials, higher order nonquadratic yield functions were proposed to characterize the yield behavior under multiaxial loading conditions, for example, see Drucker [24], Hershey [25], Hosford [23], Karafillis and Boyce [26], and Cazacu and Barlat [27][28]. Higher order terms of the stresses in these yield functions are needed to characterize the rounded vertices of the yield surface in the stress space.

14 Drucker [24] in 1949 extended plasticity theory to account for deviations from classical plasticity. Drucker compared experimental data on aluminum alloy tubes to classical Tresca and von Mises yield criterion. While neither the Tresca criterion, nor the Mises criterion, agreed well with the experimental data, a function of the form:

(2.11) , = − was proposed and a yield surface corresponding to a value of 2.25 of the parameter c, gave an excellent correlation with the experimental data on 24S-T aluminum alloy [24]. This was one of the first yield functions to go beyond the representation of yield for metallic materials. In this model, the yield surface is located between those of Tresca and von Mises as shown in Figure 2-4.

Figure 2-4: Plane stress yield loci for Drucker, Tresca and von Mises (normalized by uniaxial tension) after Drucker [24].

Based on self-consistent polycrystal calculations, Hershey [25] introduced a higher order criterion later used by Hosford. Hosford [23] proposed a higher order, generalized form of an isotropic yield criterion that can be used to approximate experimental and theoretical results

15 more closely than either the von Mises or Tresca yield functions. Hosford’s yield function is expressed as

/ − + − + − (2.12) = 2 and 1 ≤ n ≤ ¶ ≥ ≥ where and are the principal stresses, is the yield stress from a uniaxial tension or compression, , , and the exponent n is material dependent exponent and need not be an integer. The Hosford yield function reduces to the von Mises and Tresca yield functions for n=2 and for n=1, respectively. For values of n above 2.767, the loci repeat shapes found at lower n-values. In fact n=4 and n=¶ reproduce the von Mises and Tresca yield functions. For values of the exponent greater than 4, this criterion predicts yield loci between those of Tresca and von Mises, as shown in Figure 2-5(a). Theoretically derived loci for randomly oriented FCC and BCC materials suggest an exponent n=6 for BCC and n=8 for FCC materials. These values fit experimental data well as shown in fig 2-5(b) (Hutchinson [29], Hill [30]). The ratio of yielding in pure shear to yielding in uni-axial tension can be adjusted using the exponent n in the Hosford model.

(a) (b)

Figure 2-5: Plane stress yield loci for Hosford, Tresca, von Mises (a) n=1 Tresca, n=2 von Mises and n=10 Hosford (b) plane stress yield loci predictions for different n values between 1 and 2.767 as shown by Hosford in 1972 [23].

16 Hosford’s formulation has been generalized by Karafillis and Boyce [26] in the following form:

(2.13) 3 ∅ = 1 − ∅ + ∅ = 2 2 + 1 where (2.14) ∅ = | − | + | − | + | − | = 2

and (2.15) ∅ = || + || + || = Here, and are the principal deviatoric stresses, c is a weighting coefficient, and 2k is an exponent , , having the same significance as the exponent n in Hosford’s criterion. For k = 1, and take the form given by von Mises, however, for k=∞ becomes the Tresca function,∅ and∅ gives an upper limit of the yield surface. Therefore ∅ describes the yield surfaces between∅ the lower bound (Tresca) and von Mises and describes∅ the yield surfaces between von Mises and the upper bound. In this work, k is∅ an integer with values varying from +1 to +¶. Thus, the exponent in and is always an even integer, thereby ensuring identical tensile and compressive yield∅ stress∅ for the yield function The value of the coefficient c is in the range [0, 1] and determines the weight of the functions∅. and in the yield function . As a consequence, there are two parameters k and c that may∅ be used∅ in order to ‘adjust’ the∅ shape of the yield locus, whereas the Hosford criteria use only one parameter (exponent n) for this purpose. A generic isotropic yield surface should be able to describe all yield surfaces lying between the lower bound (Tresca) and the upper bound yield surface. By varying the value of k and c, a family of yield surfaces swaps the space between lower bound and upper bound in Karafillis-Boyce criterion as shown in Figure 2-6 and Figure 2-7. On the other hand, the Hosford criterion swaps the space between lower bound and von Mises. Therefore the Karafillis and Boyce criterion is more flexible compared to Hosford’s model.

Figure 2-6: Different isotropic plane stress yield surfaces between the von Mises yield surface and the lower bound (Tresca) after Karafillis and Boyce [26].

Figure 2-7: Different isotropic plane stress yield surfaces between the von Mises yield surface and the upper bound after Karafillis and Boyce [26].

Vegter [31] in 1995 proposed an isotropic plane stress yield description which directly uses the experimental results at multi-axial stress states as follows:

(2.16) = −

(2.17) = 1 − + + 21 −

18 where is a kind of equivalent stress, and is the yield stress. and are reference points, and is hinge point for second order Bezier interpolation. In this model, a four point interpolation method has been developed based on the pure shear point, the uni-axial point, the plane strain point, and equi-biaxial point in principal stress space as shown in Figure 2-8. A yield surface is constructed using the four reference points and the gradient. Three Bezier splines using pure shear, uniaxial tension, plane strain tension, and equi-biaxial tension are used to describe a quarter of the yield function. This yield function is a multifaceted yield function. The advantage of using Bezier interpolation is that the normal of the yield function is continuous in the reference points. The Vegter model assumes tensile-compressive symmetry in yield stress, and it is limited to plane stress case, which is useful for sheet metal applications only.

Figure 2-8: The four reference points to construct the Vegter yield function in plane stress after [31].

2.1.3 Role of Third Deviatoric Stress Invariant ( ) in Modeling Asymmetry between Tensile and Compressive Yield

Higher order yield surfaces mentioned in the previous section are able to handle deviations from plasticity regarding yield strength in pure shear and characterization of the family of yield surfaces between the lower bound and the von Mises yield surface or the von Mises yield surface and the upper bound. On the other hand, these functions cannot handle tensile-compressive asymmetry in yielding, since they include only an even function of . A proposed yield function should be an odd function of or an odd function of

19 deviatoric principal stresses to be able to address the tension-compression asymmetry in yielding. Starting from the middle 1960s, experimental results were published which show that yield and subsequent flow stresses in tension and compression are different for some metals [32][33][34][35]. This phenomenon, known as the strength differential (SD) effect in yielding, is observed for a wide range of temperatures and strain rates. The strength differential (SD) effect, namely different yield stress in tension and compression, has been observed in many iron-based metals such as plain carbon or low alloy steels, cast iron, high strength steels, and also in some metals such as titanium, aluminum, magnesium and nickel- base super alloys, such as Inconel. The SD effect can be mathematically defined as [32]:

(2.18) − = + where and are yield strengths in uni-axial compression and uni-axial tension, respectively. According to this expression, the higher absolute-value of SD, the stronger strength differential effect is. For an isotropic material exhibiting SD, a yield function that only depends on or even function of will not give an accurate representation of the material behavior. BCC and FCC materials tend to deform by slip because of the large number of available slip systems. However, HCP materials, in which the number of potential slip systems is limited, also tend to twin as an alternate mechanism to accommodate an imposed deformation [36][37]. Thus twinning and slipping contribute to SD effect for HCP materials. As these metals are pressure insensitive, the dependence of the yield condition on the first stress invariant should be neglected, and the effect of the third stress deviator invariant becomes important. Plastic deformation of polycrystalline metals occurs by either slip or twinning (Figure 2-9 and 2-10). Whether slipping or twinning is the dominant deformation mechanism depends on which mechanism requires the least stress to initiate and sustain plastic deformation. In low symmetry materials such as HCP metals, which have too few slip systems to accommodate any shape change, twinning may become a dominant mechanism.

Figure 2-9: Elastic deformation corresponding to pure stretching of the crystal lattice (top) and plastic deformation trough dislocation slip (bottom) [36] [37].

Figure 2-10: Crystal lattice reorientation due to mechanical twinning [36][37].

Twinning, unlike slip, is sensitive to the sign of the applied stress; i.e ., if a particular twinning can be formed under a shear stress, it will not be formed by a shear stress of opposite sense. Because of the polar nature of twinning, HCP materials display a strong asymmetry between the yield in tension and compression. As HCP metals are pressure insensitive, the dependence of the yield condition on the first stress invariant should be neglected and the effect of the third stress deviator invariant becomes important. Studies by Raniecki and Mroz [38], Iyer and Lissenden [39], Cazacu and Barlat [40] provide formulations of yield conditions and flow potentials for magnesium alloys and nickel base alloys, assuming dependence on the third stress deviator invariant. Cazacu-Barlat [40] introduced a yield function for a pressure-insensitive metals exhibiting SD effect in the form of:

21 (2.19) / , = − = where is the yield stress in pure shear, and c is a material parameter. The constant c represents the SD effect in yielding in the form of:

(2.20) 3√3 − = 2 + where and are yield strengths in uni-axial compression and uni-axial tension, respectively. For equal yield stresses in tension and compression, the proposed criterion reduces to the von Mises yield criterion. The Cazacu-Barlat model was applied to magnesium and titanium. As shown in Figure 2-11, the shape of the yield locus is no longer elliptic in the plane stress case when the SD effect is taken into account in this model. It is more like a triangle with rounded corners when tensile-compressive asymmetry is present.

Figure 2-11: Plane stress yield loci corresponding to / =2/3, 3/3 ( von Mises), 4/3 [40].

According to many authors, the contribution of the effect of the third invariant is more severe than the contribution of pressure in the plastic flow (see Bai and Wierzbicki [41], and Mirone and Carolla [42], Gao [43]). Bai and Wierzbicki [41] in 2008 discussed a pressure and Lode dependent metal plasticity model and its application in failure analysis and proposed the following yield criteria:

(2.21) = 1 − − − − + 1 where is strain hardening function, is a material constant which needs to be calibrated and represents hydrostatic pressure effect on material plasticity, and is the reference value of stress triaxiality from the reference test. The parameters and are Lode angle related parameters, where for ≥ 0 and for ≤ 0. The ̅ ̅ parameter m is a nonnegative integer. Four material= constants, , , = and , need to be calibrated. The values of , , and are relative, and at least one of them , is equal to unity. This depends on which type of reference test is used for hardening For example if a . smooth round bar test is used, then =1; if a torsion test used ; if a cylinder specimen test is used then . Tensile-Compressive asymmetry is introduced = 1 in the model when ≠ . Bai and Wierzbicki = 1 worked on the experimental calibration based on the quasistatic test cases of Al2024-T351 material and concluded that percentage-wise, the correction for the hydrostatic pressure is small, and the magnitudes of correction due to the deviatoric state parameter (Lode angle parameter) is large, in some cases reaching 20% as shown in Figure 2-12.

Figure 2-12: A comparison of force displacement curves between experimental results and simulation results for flat grooved specimen (Bai & Wierzbicki [41]).

Gao [43] in 2009 noticed the plastic response of a 5083 aluminum alloy is stress-state dependent. Mirone and Corallo [42] in 2010 found that, for the metals they tested, the hydrostatic stress plays a significant role in accelerating failure but has negligible effect on the stress-plastic strain relationship, while the Lode angle has a considerable effect in modifying the stress–strain curves. These findings agree with the findings by Gao [43]. Recently, Gao [44] in 2010 proposed an elasto-plastic model, which is a function of the hydrostatic stress as well as the second and third invariants of the stress deviator. A simple form of the yield function is given as follows:

(2.22) / = + 27 + where and are material constants. This yield function is a first order homogeneous function of, stress,, and and can be found by manual calibration to have the best possible match compared to test data. The constant is dependent on and . For , this plasticity model turns into flow. The calibrated model constants for the= plas ticity= 0 behavior of the 5083 aluminum alloy are found to be and , which indicates no effect but significant effect on the plastic =response 0 of =this −60.75 material.

24 CHAPTER 3 - Methodology and Theoretical Framework

3.1 Fundamentals of continuum plasticity An enormous challenge in plasticity is to formulate a realistic mathematical model for describing the observed material behavior. The task is known as constitutive modeling, and the end product is a constitutive model or a set of constitutive equations. Various theories using different approaches and points of view have been proposed to formulate constitutive equations for metallic materials. Theories have also been formulated based on dislocation and slips. Although physically based polycrystal plasticity is emerging as a feasible method, the phenomenological (or continuum) approach is still the most practical computational mechanics method in industry. Presently, a lot of effort is being devoted to bridging the gap between mechanics at the microscopic and continuum levels. In spite of the interest in plasticity at the micro level, the continuum theory is still the backbone for the study of plasticity. In the continuum theory, the extension of one-dimensional plasticity into a multidimensional plasticity provides a great challenge both experimentally and theoretically. In the early development, two major theories were developed; the flow theory and deformation theory. The deformation theory was proposed by Hencky in 1924 [46]. Deformation theory is also known as the total strain theory since total strain is used rather than strain increment. On the other hand, the flow theory emphasizing the strain increment is also known as the incremental theory. The deformation theory is convenient for use, due to its mathematical simplicity, in solving problems with proportional loading. The final state of strain is determined by the final state of stress. However, it is known that equations of deformation theory are not suitable for non-proportional loading condition, therefore use of the flow theory of plasticity is necessary. The flow theory of plasticity describes the mechanical behavior of materials in the plastic range including energy dissipation, irreversible and history and path dependent process, the initial yield surface and its subsequent growth, the constitutive equation for plastic deformation, and criteria for loading and unloading. According to flow theory, the constitutive equations of plasticity consist of a yield condition, a flow rule, a strain hardening rule, and the loading-unloading conditions. The

25 yield criterion determines the stress state (elastic or plastic state). The flow rule describes the increment of plastic strain when yielding occurs. The hardening rule describes how the material is strain hardened as the plastic strain increases and determines the subsequent yield stress. The loading-unloading conditions specify the next move in the loading program. These are the major concepts of the flow theory. As plastic deformations are nonlinear and depend on the loading history, the constitutive equations are formulated in incremental form, where commonly strain rates are split into elastic and plastic parts, and this assumption is valid under small increments of plastic strain.

(3.1) = +

Let f be a yield function, then the yield criterion can be stated below as:

: Elastic deformation (3.2) − < 0 : Plastic deformation (3.3) − = 0 The yield criterion shows that the initiation of plastic deformation occurs at the yield point. If loading continues, the plastic flow rule with a normality condition enables us to determine the direction of the plastic strain increment as well as its magnitude, which is determined by the plastic multiplier λ. A flow rule obeying the normality condition is referred to as an associated flow rule. On the other hand, a flow rule in which the plastic strain increment is not normal to the yield surface is known as a non-associated flow rule. The non-associated flow rules have been used for geotechnical materials. For associated flow, the normality condition states that (i.e., the plastic flow potential coincides with the yield function) an increment in the plastic strain tensor is in a direction which is normal to the tangent to yield surface at load point. This can be written as in terms of yield function f :

(3.4) ∂ = λ ∂ (3.5) =

26 where is the plastic part of the strain rate tensor, the plastic multiplier rate, and σ is the Cauchy stress tensor. In non-associated flow, the plastic potential is proposed as an equation:

(3.6) = which forms the surface of plastic potential in the stress space. Then plastic strain increment is

(3.7) ∂ = λ ∂ When the plastic potential is the same as yield function, that is

(3.8) = then it reduces to equation 3.4. In this case the plastic strain increment is normal to the yield surface. However when the yield surface and the plastic potential are not equal, the plastic strain increment is normal to the surface of constant plastic potential but is not normal to the yield surface, and thus we have non-associated flow rule. There are three classes of materials: strain-hardening, perfectly-plastic, and strainsoftening. Generally, metals are strain hardening materials, and geotechnical materials may exhibit strain-softening under certain conditions. Several criteria have been proposed to classify materials. Generally, strain-hardening materials are regarded as stable materials by the well-known Drucker’s postulate [45]. In a multiaxial stress state, strain-hardening is considered in the form of hardening rules for subsequent yield surfaces. It has been experimentally observed that the yield surface, upon application of a deformation history, may undergo expansion, distortion, translation, and rotation. In the literature, expansion of the initial yield surface is called isotropic hardening, and translation of the initial yield surface is called kinematic hardening. While isotropic hardening assumes uniform expansion without translation of the yield

27 surface in the stress space, kinematic hardening assumes translation without expansion of the yield surface in the stress space, as plastic strain increases. In both strain-hardening models the yield surface shape remains unchanged during strain-hardening, as shown on Figure 3.1. Isotropic hardening is the most widely used strain-hardening concept, and kinematic hardening, either alone or in combination with isotropic hardening, is often introduced in material models while cyclic loadings are considered.

(a)

(b)

Figure 3-1: (a) Isotropic hardening and (b) kinematic hardening in plane stress yield loci [47].

Based on experimental observations, subsequent yield surfaces may distort due to the stress state dependency in the hardening behavior of the material. This means that hardening curves regarding tension, compression, and shear stress states can be different (See Figure 3- 2).

Figure 3-2: Hardening response of pure titanium according to theoretical yield surface and experimental data (symbols) corresponding to fixed values of the equivalent plastic strain after [48] (Stresses are in MPa).

The loading-unloading conditions (also known as the Kuhn-Tucker [49] conditions) are given as follows:

, , (3.9) ≥ 0 ≤ 0 ≥ 0 where the first expression states that the plastic multiplier rate is always positive or zero, the second expression states that the stress is always on or within the yield surface, and the last expression states that the stresses remain on the yield surface during plastic loading (i.e., when ). This last condition can also be stated as: > 0 (3.10) = 0 Since f is equal to zero during the plastic flow, this equation is known as the consistency condition, and it enables us to determine the plastic multiplier. The yield function f is

29 dependent on the components of the stress tensor and the yield stress of a material. When strain hardening is considered, the yield stress of the material can increase as a function of equivalent plastic strain for subsequent loading. Therefore, the yield function can be written as:

(3.11) , ̅

For an incremental change in the stress tensor ( ) and equivalent plastic strain ( ), ̅ consistency condition results in:

(3.12) ∂ ∂ = ∶ + ̅ = 0 ∂ ∂̅

(3.13) ∂ = = − = − λ ∂

(3.14) ∂ ∂ = ∶ − + ̅ = 0 ∂ ∂̅ where is a fourth order symmetric stiffness tensor. 3.1.1 Perspectives on Plastic Constitutive Equations: Principle of Maximum Plastic Resistance and Drucker’s Postulate

One perspective on the structure of constitutive laws for inelastic solids was developed by Drucker in the 1950s [45]. Drucker introduced the idea of a stable plastic material as follows. Consider the work done on a material element in equilibrium by an external mechanism which slowly applies a set of self-equilibrating forces and then slowly removes them. This external mechanism is to be understood as entirely separate and distinct from the mechanism which causes the existing state of stress. This process of application and removal of additional stress is called a stress cycle. Removing the additional stress enables the stress of body to return the original stress state, but the strain state can be different if plastic

30 deformation occurred during the stress cycle. Work hardening is then defined to mean that, for all such added sets of stresses, positive work is done by the external mechanism during the application of the stresses, and the net work performed by it over the cycle of application and removal is either zero or positive [45]. Consider a volume of material in which there is a homogeneous state of stress, , and strain, . Suppose that external mechanism applies a small surface traction that alters the stress at each point by and the strain by . On removal of the surface traction, returns to zero, and a strain is recovered. Then according to the definition above, the material is said to be work hardening if the following two conditions hold true:

• The work performed by the external mechanism during the application of incremental load is positive , upon loading (3.15) : > 0 • The work performed in the loading-unloading cycle is non-negative , on completing cycle (3.16) : − > 0

Let denote the plastic strain increment, which is not recovered by the process above. Then, since , the second condition may be written as = −

(3.17) : ≥ 0

This hypothesis can be satisfied only by materials whose subsequent yield strength increases with deformation (strain-hardening materials). Any material on which an external mechanism does positive work during an elastic-plastic stress cycle is called a hardening material. Otherwise it is called a non-hardening or softening material. As shown in Figure 3- 3, a strain-hardening material would be stable; whereas a strainsoftening one like a metal at high temperatures or soil-like material would be unstable according to the first statement.

Figure 3-3: Stable and unstable material behavior on stress–strain curve according to Drucker’s stable material postulate [45].

Consider that is a stress state on the yield surface, and is any possible stress state ∗ inside or on the yield surface, the statement:

(3.18) ∗ − : ≥ 0 is equivalent to a stress cycle process followed by the external mechanism and consists of a loading step to the yield surface, an incremental stress load that produces incremental strain , and an elastic unloading to initial state . Assuming that , the ∗ ∗ ≪ − work performed by the external mechanism is . Then the second statement of ∗ − : Drucker’s postulate implies that total work performed in the loading-unloading cycle is

(3.19) ∗ − : ≥ 0

Representing second order symmetric tensors as six columns vectors, we can write the same equation in vector form as the dot product of two vectors:

(3.20) ∗ { − }. { } ≥ 0 Since the dot product is non-negative, the angle between vector and the vector of ∗ { − } incremental plastic strains must be less than 90 o (see dashed vectors in Figure 3-4(a)). { }

32 If the incremental plastic strains vector is always normal to the yield surface, a vector cannot be found within 90 o of it. Therefore, the normality rule holds. For sharp ∗ { − } corners, it can be demonstrated that any incremental plastic strain tensor which lies inside the cone bounded by the normal vectors to the surface on the either side of the corner is valid (See Figure 3-4(b)). A curve is said to be convex if it always lies on one side of a tangent at any point. Following this argument, if there is any initial stress state lying on the other side of the tangent, as can be seen in Figure 3-4 (c), the inequality is violated. Therefore the yield surface should be convex to satisfy the inequality. Convexity and normality are significant implications of the second statement of Drucker’s postulate [45]. Bower [50] and Wu Han- Chin [51] have discussed the consequences of Drucker’s postulate regarding the convexity and normality rule.

Figure 3-4: Properties of a yield surface with associated flow rule (a) normality (b) corner condition (c) convexity [47][50][51].

The principle of maximum plastic resistance or work is a consequence of Drucker’s second statement. It is an interpreted from equation 3.19 as

(3.21) ∗ ∗ : − : ≥ 0 ⇒ : ≥ : Consider that the increment of the plastic work per unit volume for a given plastic strain increment can be written as

= : (3.22)

∗ ∗ = :

Then second statement of Drucker’s can be written in terms of the incremental plastic work:

(3.23) ∗ ∗ : ≥ : ; ≥

This is the maximum plastic resistance or work theorem, which states that the actual work done by (at which plastic strain increment is created) on a given plastic strain increment is greater than the fictitious work done by an arbitrary stress inside the ∗ yield surface. Equality holds when coincides with . Theorem signifies the fact that ∗ among all admissable stresses inside the yield surface, the actual stress (at which ∗ plastic strain increment is created) would produce the maximum plastic work. The principal of maximum plastic resistance or work is a mathematical statement of the convexity of the yield surface and the normality rule and generally holds for metal plasticity when compare with experimental evidence. For example, it can be shown that the stress field in a material that obeys the maximum plastic resitance or work principle is always unique [50].

3.1.2 Consequences of Drucker’s hypothesis

It is shown that, for a plastic material to be stable in Drucker’s sense, it must satisfy the following conditions:

• The initial yield surface and all subsequent yield surfaces must be convex: A yield function f is convex if its Hessian matrix is positive and semi-definite [52]. The property of the positive semi definiteness can be proved by showing that the eigenvalues of the Hessian matrix H are non-negative. • The plastic strain rate tensor must be normal to the yield surface: When applied to materials, Drucker's stability postulate results in the inelastic strain rate vector being

34 normal to the yield surface [45]. Thus, the yield function can be treated as a plastic potential and the inelastic strain rate can be determined using an associated flow law. On the other hand, a flow rule in which the plastic strain increment is not normal to the yield surface is known as a non-associated flow rule. A non-associated flow rule is used for geotechnical materials. • The rate of strain hardening must be positive or zero, and this holds for most the metals.

Furthermore, a material that is stable in the sense of Drucker must satisfy the principle of maximum plastic resistance or dissipation. The principle of maximum plastic dissipation is often used as a starting point to obtain the same consequences of convexity and normality on the yield surface. It should be noted that satisfying the principle of maximum plastic dissipation does not necessarily mean that the stability postulate is satisfied. That is, the stability postulate is a sufficient condition, not a necessary one. For convex yield surfaces, the principle of maximum plastic dissipation is a necessary and sufficient condition for normality. This is satisfied for strain hardening and softening plasticity. Nevertheless, a “stable” plastic material defined in terms of Drucker’s postulate requires strain hardening plasticity. If you try to solve boundary value problem for a material that violates Drucker’s stability criterion, you are likely to run into trouble. The problem will probably not have a unique solution; in addition you are likely to find that smooth curves on the undeformed solid develop kinks and may not even be continuous after the solid is deformed. This kind of deformation violates one of the fundamental assumptions of underlying continuum constitutive equations [50].

3.2 Development of proposed generalized yield surface (GYS) plasticity model involving stress invariants and . Material laws for the plastic behavior of isotropic materials can be formulated as a function of the three stress invariants ; pressure ( ), von Mises stress ( ) and Lode parameter ( ). Isotropic, , generalized plasticity/3 models introduc e dependency in addition27 to /2to be able to capture the accurate plastic stress-strain fields considering the impact of stress state on yielding and plastic flow. Especially HCP metals

35 show a strong strength differential effect (tension-compression asymmetry) in plastic flow, and the yield surface cannot be described by the von Mises plasticity. The main reason for the strength differential effect is the twinning deformation mechanism, which results in tensile-compressive asymmetry, and requires a yield surface with an odd power of to cover the strength differential effect. Volumetric deformation of metals is elastic and linear in the normal engineering range (up to pressures of roughly 10 GPa). Material laws for metals are independent of the pressure as the plastic deformation of metals happens at constant volume [53]. Thus assumption of a yield function depending upon the von Mises stress and the Lode angle without dependence on pressure or the first invariant of the stress tensor is valid considering metals [54]. Generalized yield surface can be obtained by multiplying the von Mises stress with a function of the Lode angle or Lode parameter:

( , ≤ ( , ≤ (3.24)

= 3 ≤ 3 = 0 ≤ → −1 ≤ 3 ≤ +1 where is the Lode angle, and is the lode angle parameter. The Lode angle is related to the third deviatoric stress invariant3 using the Lode parameter. Representation of the Lode angle ( ) and the corresponding Lode parameter for uni-axial tension, uni-axial compression and pure-shear stress states are shown in the deviatoric plane in Figure 3.5 (a). Comparison of Tresca, von Mises and GYS yield surface models are presented in the deviatoric plane in Figure 3.5 (b), and the effect of the tension-compression yielding asymmetry on GYS is shown in Figure 3.5 (c). All stress states or loading conditions can be characterized by the

Lode angle parameter and the stress triaxiality parameter , which 3 = give the ratio of pressure to von Mises stress. Various stress states encountered in test specimens used for plasticity and fracture testing can be uniquely characterized by these two parameters.

36 (a)

(b)

(c)

Figure 3-5: (a) Representation of Lode parameter and UT (uniaxial tension), UC (uniaxial compression) and PS (pure shear) stress states (b) Comparison of yield surface models in deviatoric plane (ᴨ–plane) (c) effect of tension ( σT) vs compression ( σC) yielding asymmetry on the GYS in deviatoric plane.

37 A new generalized yield surface (GYS) is proposed for pressure insensitive metal plasticity under the following assumptions: • Only isotropic behavior is considered. (No anisotropic effect) • A multiplicative decomposition is assumed for the and effect in the generalized yield function.

(3.25) ℎ 3, 3 − • Associated flow is preferred since associated rule is validated by the experiments for the yield surface of metals. Therefore the yield potential is equal to yield function. • Isotropic hardening is considered. • Isochoric plasticity is assumed (constant volume during plastic deformation). • Unique flow (strain rate) for any given state of stress on the yield surface. • Tension-compression asymmetry is allowed. • The yield surface can describe yield surfaces lying between Tresca and von Mises. • Any dependency on pressure or first stress invariant is excluded, thus generalized yield function is in the form of (3.26) ℎ 3, 3 − and this represents pressure independent flow, considering that term is not represented in the GYS yield function. • Yield stress, is dependent on the effective plastic strain ̅ , ̅ , ̅ (accumulated plastic strain), effective plastic strain rate , and current temperature ̅ . Considering the aforementioned assumptions, the following isotropic generalized yield function formulation is proposed;

(3.27) 27 729 , ̅ , ̅ , = + + − ̅ , ̅ , 2 4

38 here is an effective stress of the GYS model in terms of von Mises stress and .

(3.28) 27 729 = + + 2 4 The generalized yield function can also be written in terms of the von Mises stress parameter ) and Lode parameter as shown below: 3 (3.29) , ̅ , ̅ , = + 3 + 3 − ̅ , ̅ , where is the Lode angle, and the relation between the Lode angle and is given by a Lode angle parameter as shown below:

(3.30) 27 3 = 2 In this formulation, the yield surface in 3-D stress space is given by equation 3.31 as follows:

(3.31) , = + 3 + 3 − ̅ , ̅ , = 0

The condition can be called the yield criterion in the , ̅ , ̅ , = 0 multi-axial case. When the stress state is inside the surface ( , , ̅ , ̅ , < 0 then the current stress state is elastic, and the material will deforms elastically. When the stress state is on the surface ( , then the stress state becomes , ̅ , ̅ , = 0 plastic and the material deforms plastically. In the GYS formulation, , and are material dependent coefficients and vary with the strain hardening. Three experiments, uni- axial tension, uni-axial compression, and torsion (pure shear) (see Figure 3-6) are necessary to determine the coefficients as follows: Considering the uni-axial tension where , the GYS function takes the form: 3 = 1

(3.32) + + − = 0 where the von Mises stress is equal to the axial tensile stress ( ). We also assume that the yield stress is equal to the yield stress in tension ( = ). Then the sum of = coefficients , and must be equal to 1 to match the uni-axial tension test result. (3.33) + + − = 0 → + + = 1 Considering the pure shear where and the GYS function takes the form; 3 = 0

(3.34) − = 0 → √3 − = 0 → = √3 where and . The shear yield determines , and it is independent of = 3 = tensile-compressive√ asymmetry compared to the Cazacu-Barlat model where the shear yield relative to the tension yield is fixed by the tensile-compressive asymmetry. Considering the uni-axial compression case where cos( 3ϑ) = − 1 then the GYS function takes the form;

(3.35) − + = → − + = → − + = where the von Mises stress is equal to the axial compressive stress ( ), and the yield stress is equal to the yield stress in tension ( ). Since is dependent= on shear = yield, equation 3.35 shows that tensile-compressive asymmetry determines or in the GYS function. In summary, four material coefficients of the GYS model, , , and , are determined by four equations based on the three experiments, uniaxial tension, , uniaxial compression, and torsion (pure shear) as follows (See Figure 3-6):

40 (3.36) =

(3.37) + + = 1, − + = = √3

Figure 3-6: Uni-axial tension, uni-axial compression and pure-shear (torsion) tests required by GYS to determine model coefficients , , and . ,

As shown in Figure 3-6, yield stresses are obtained from tests under different stress states. The yield stress values used in the calculation must correspond to the same effective plastic strain, . Therefore plastic work for each stress state is taken as a reference point. ̅ The effective plastic work for each stress state should be the same since = : = (tensorial and scalar equivalence of work under any stress state). ̅ Consider the definition of effective stress for the GYS model:

and (3.38) = + 3 + 3 = ̅

In the case of uni-axial tension, , and , the effective stress is then: 3 = 1 = (for GYS model ) (3.39) = + + = = + + = 1

In the case of uni-axial compression, and , the effective stress is then: 3 = −1 =

(3.40) = − + = 1 − 2 = = (for GYS model ) + = 1 − In the case of pure shear, and , the effective stress is then: 3 = 0 = √3

(3.41) = = = √3 Thus in all cases then: = ̅ = ̅

thus for uniaxial tension = : = ̅ = = ̅ ̅ = for uniaxial compression (3.42) = : = ̅ = = ̅ ̅ = for pure shear = : = ̅ = = ̅ ̅ =

Therefore four curves are created internally to compute 4 material dependent hardening coefficients in terms of the effective plastic strain parameter of the GYS model as ̅ shown in Figure 3-7. Note that it is shown in the next chapter that the increment in is ̅ equivalent to the increment in plasticity parameter, .

Figure 3-7: (a) Four internal curves computed by GYS for initial yield and subsequent hardening (b) four equations for the determination of model coefficients , , and . ,

42 3.3 Associated flow rule, hardening rule, and definition of effective plastic strain for GYS plasticity model. One of the important assumptions of the GYS model is an associated flow. The associated flow rule can be interpreted as an “increment in the plastic strain tensor (or increment in the plastic strain vector in the case of principal stress space), this increment is in a direction which is normal to the tangent of the yield surface at the load point. This assumption is validated by the experiments for the yield surfaces of metals [51]. Consider the yield function proposed by the GYS model:

(3.43) 27 729 , = + + − 2 4 In the GYS terms, corresponds to of the plasticity, and the scale factor of is given by:

(3.44) 27 729 + + 2 4

(3.45) 27 729 = = + + 2 4 Thus the GYS effective plastic strain rate is defined under any stress state as shown below.

(3.46) 2 2 2 ̅ = : = ∶ = ∶ 3 3 3

(3.47) 2 ̅ = + ∶ + 3

(3.48) 2 ̅ = : + : + 2 3

(3.49) 2 ̅ = + : + 2: 3

In associated flow, flow potential (g) is equal to yield function (f), therefore the plastic strain rate tensor can be written as

(3.50) = And the yield function is given by: , (3.51) , = −

Then the plastic strain rate tensor is

(3.52) = = Considering the effective stress of the GYS model is given by equation 3.28, the plastic strain rate tensor takes a form:

27 729 (3.53) + + 2 4 = =

(3.54) = +

(3.55) 3 1 = + − 2 3

27 729 3 = − 2 − 5 2 4 2 (3.56)

27 729 1 + + 2 − 2 4 3

27 729 3 = − 2 − 5 2 4 2 (3.57) 27 729 1 + + 2 − 2 4 3 Then it can be shown that:

(3.58) = 0

This means that the GYS flow rule is clearly isochoric (no volume change during plastic deformation). In the GYS model, the effective plastic strain rate is defined as the energy conjugate to the effective stress. Therefore the plastic work rate can be written as

(3.59) = : = : = ̅ = : = : (3.60)

1 1 3 1 ̅ = = + − : 2 3 (3.61) 3 ̅ = + Also consider following relationships:

(3.62) = + , : = ;

(3.63) 3 = 2

(3.64) = =

(3.65) = = = =

(3.66) 1 1 1 = − = − = − 3 3 3

(3.67) 2 : = = : = 3 Then the definition of effective plastic strain rate takes the form:

(3.68) 27 729 27 729 ̅ = − 2 − 5 + + 2 3 2 4 2 4 (3.69) 27 729 27 729 ̅ = − 2 − 5 + 3 + 6 2 4 2 4 (3.70) 27 729 ̅ = + + = = 2 4 This result shows that the effective plastic strain rate is equal to the rate of the plastic consistency parameter itself for the GYS model. As a result, an increment in the effective plastic strain is equal to an increment in the plastic consistency parameter, which reduces the computational effort, and based which, the hardening rule of the GYS model can now be written as

46 (3.71) = : = ̅ = ⇒ ∆ = ∆

Considering the work hardening principle, the change in yield stress can be written as

(3.72) = = ̅ = = =

(3.73) = ̅ = ̅

Equation 3.73 shows that the increment in yield stress can be directly interpolated from the yield stress input curve (yield stress versus equivalent plastic strain) considering that the calculated increment in the plastic consistency parameter ( ) is equivalent to the increment in effective plastic strain of the GYS model. So, the effective∆ plastic strain increment and its slope can be read from the yield stress versus effective plastic strain curve, then the corresponding change in yield stress can be obtained. This load curve must be derived from test data for experiments in uni-axial tension, uni-axial compression and pure shear (torsion). In doing so, we will be determining the incremental change on four flow parameters: , , and , as plastic loading continues. Therefore, the hardening rule for the GYS model , can be written as

(3.74) = ̅ = ̅

(3.75) = ̅ = = 1,2,3 ̅

One consequence of this result is that the stress state dependent hardening is addressed using hardening parameters, , in the GYS model. Noting that because , the , = hardening rule for tension state can be written:

47 (3.76) = ̅ = = ̅

Since , , parameters depend on yield in uni-axial tension, uni-axial compression, and pure-shear (torsion),, we can write equation 3.75 in the form of tension yield, compression yield, and shear yield determined by the incremental change in shear yield and compression yield with respect to an incremental change in equivalent plastic strain or plastic parameter. We can then write the hardening rules for compression and shear states as:

(3.77) = = As a result, the hardening rule for the GYS model can be written in terms of model parameters as

(3.78) = ̅ = = ̅ = = 1,2,3 ̅ ̅ or, equivalently, in terms of stress states as

(3.79) = = = = where , , and are true plastic strain in uni-axial tension, uni-axial compression, and pure-shear. The relationships between the increment of plastic consistency parameter and the plastic strain increment of uni-axial tension, uni-axial compression, and pure-shear shear are dependent on model parameters , , and they are described below. Let’s consider the uni-axial tension , case where is a stress in the tension load direction ( ). Then the stress tensor of uni-axial tension case is given by: =

48 = (3.80) 0 0 2 0 0 = 0 0 0 ⇒ = − ⇒ = 0 −1 0 ⇒ 27 3 3 = 1 0 0 0 0 0 −1 2 Also note that:

(3.81) 3 1/2 0 0 1 1 0 0 = 0 −1 0 ⇒ − = 0 1 0 3 2 0 0 −1 0 0 1

(3.82) 3 2 3 = = 1 ℎ = 2 3 2

(3.83) 3 1 3 3 3 − − = + = 2 2 3 2 2 2 Then the plastic strain rate tensor is given by:

27 729 3 = − 2 − 5 2 4 2 (3.84) 27 729 1 + + 2 − 2 4 3

27 729 3 = − 2 − 5 2 4 2 (3.85) 3 27 729 2 1 + + 3 − 2 2 4 3 3 Then plastic strain in the tension load direction is = and given by:

(3.86) 3 2 = − 2 − 5 + + 3 2 2 3

49 (3.87) = + + = + + = 1 In the case of uni-axial compression where is the stress in the direction of the compression load ( ). Then the stress tensor− of the uni-axial compression case is given by: = −

= (3.88) 0 0 0 +1 0 0 = 0 0 0 ⇒ = ⇒ = 0 +1 0 ⇒ 27 = −1 3 3 0 0 − 0 0 −2 2 Also note that:

1 0 0 (3.89) 3 1 1 0 0 = 0 1 0 ⇒ − = − 0 1 0 3 2 0 0 −1/2 0 0 1

(3.90) 3 2 3 = = −1 ℎ = − 2 3 2−

(3.91) 3 1 3 3 3 − − = − + = −2 2 3 2 2 2 The plastic strain rate tensor is given by:

27 729 3 = − 2 − 5 2 4 2 (3.92) 3 27 729 2 1 + + 3 − 2 2 4 3 3 Then the plastic strain rate in the direction of the compressive load is = and given by: −

50 (3.93) 3 = − = − + 2 − 5 + − + 3 −2 2 (3.94) = − = − + − In the GYS formulation, the ratio of yield in tension to compression is given by:

(3.95) − + = Therefore the relation between the plastic consistency parameter increment and the compressive plastic strain increment can be written in rate form as

(3.96) = If there is no tensile-compressive asymmetry in the yielding, then and the plastic strain rate in the direction of the compressive load is given by: = 0

(3.97) = − = − − Since , can be written in terms of when : + + = 1 = 0 (3.98) = 1 − Then the plastic strain rate in the direction of the compressive load is given by:

(3.99) = − = − − 1 − = − In the case of pure shear, where is plastic shear stress, then the pure shear stress tensor is given by: = − =

51 0 0 +1 0 0 = √3 (3.100) = 0 0 0 ⇒ = 0 ⇒ = 0 0 0 ⇒ 27 = 0 0 0 − 0 0 −1 2

Since J3 related terms are zero in the plastic strain rate tensor, the shear plastic strain rate can be written as

27 729 3 = − 2 − 5 2 4 2 (3.101) 3 27 729 2 1 + + 3 − 2 2 4 3 3

(3.102) 3 3 √3 = = = = √3 2 2 2 √3 Considering

(3.103) = the plastic shear strain rate tensor can be written√3 as

(3.104) √3 = = 2 2 √3 Then the engineering plastic shear strain rate is given by:

(3.105) = 2 =

3.4 Perspectives on dependent yield surfaces: GYS, Hosford, Drucker, Cazacu- Barlat Working with the Lode angle dependence is another way of controlling the shape of the yield surface in a simple and elegant way. In general, the Lode angle parameter is

52 responsible for the shape of the yield surface. The determination of an adequate shape of the yield surface has been an important issue in the simulation of metallic plastic behavior for years. Separate hardening behaviors are possible for uni-axial tension, uni-axial compression, and pure-shear stress states. The GYS (generalized yield surface) model was developed to address these concerns. Consider uni-axial tension, uni-axial compression, and the equi- biaxial tension states for the same value of some positive stress. It can be seen that the deviatoric stress tensors for uni-axial compression and equi-biaxial tension are identical (Lode parameter = -1):

Uni-axial tension:

= (3.106) 0 0 2 0 0 = 0 0 0 ⇒ = − ⇒ = 0 −1 0 ⇒ 27 3 3 = 1 0 0 0 0 0 −1 2 Uni-axial compression:

= (3.107) 0 0 0 +1 0 0 = 0 0 0 ⇒ = ⇒ = 0 +1 0 ⇒ 27 3 3 = −1 0 0 − 0 0 −2 2 Equibiaxial tension:

= (3.108) 0 0 2 1 0 0 = 0 0 ⇒ = − ⇒ = 0 1 0 ⇒ 27 3 3 = −1 0 0 0 0 0 −2 2 Also the Lode parameter is zero for a pure shear state and the plain strain state under plane stress as shown below.

Pure Shear:

0 0 +1 0 0 = √3 (3.109) = 0 0 0 ⇒ = 0 ⇒ = 0 0 0 ⇒ 27 = 0 0 0 − 0 0 −1 2

53 Plain Strain:

(3.110) 0 0 1 0 0 = 2/√3 = 0 /2 0 ⇒ = − ⇒ = 0 0 0 ⇒ 27 2 2 = 0 0 0 0 0 0 −1 2 An important consequence of this result is that, following stress states will have fixed yield stress ratios in the GYS model since they have same value: • Biaxial tension and uni-axial compression (Lode parameter=-1) • Biaxial compression and uni-axial tension (Lode parameter=1) • Pure shear and plane strain (Lode parameter=0)

Table 3-1 provides degrees of freedom by each term in the GYS function in terms of yield stress prediction at different stress states. Therefore, the term with the odd function of f( , h( ), allows flexibility of tension-compression asymmetry into the GYS model. , 3

Table 3-1: Yield stress predictions of GYS at different stress states Stress state f ( f ( h( ) f ( 3 ℎ 3 3 -1 σc (uniaxial compression) -1 σb (biaxial tension) 0 σs (pure shear) √3 σ 2 2 0 p (plain strain) 2 √3

The von Mises [20], Drucker [24], Cazacu-Barlat[28] and Hosford [23] (up to order m=6) yield surfaces can be recovered using the GYS model. The GYS yield function is simpler to code than the Hosford yield function, and same type of behavior can be achieved

54 in both cases as long as the order m is less than or equal to 6. Table 3-2 compares the features of these yield surfaces in terms of uni-axial tension-compression asymmetry, ratio of yield in pure-shear to yield in uni-axial tension, number of yielding parameters, and order of the yield surface.

Table 3-2: Comparison of properties of yield surfaces Von Hosford Drucker Cazacu GYS Mises Number of yielding 1 1 2 2 3 input parameters Order of yield surface 2 m 6 3 6 Tension-Compression no no no yes yes asymmetry can be adjusted fixed to tension- can be fixed fixed to m(order Pure shear yield / using parameter compression adjusted using value of yield surface) tension yield c asymmetry via c parameter Pure shear yield / / / tension yield 0.577 1 1 4 1 2 1 / 1 − − 1 + 2 √3 27 3√3 27 √3

Consider the GYS yield function:

(3.111) 27 729 , = + + − = 3 2 4

The Drucker formulation can be recovered using , and similarly the Cazacu-Barlat formulation can be recovered using as shown = by 0 equation 3.113 and 3.114. = 0

(3.112) 27 729 = √3 + √3 3 + √3 3 − 108 2 108 2

(3.113) 3 3 27 = 0 ⇒ = 3 2 + − 2 4

55 (3.114) 3 3/2 27 = 0 ⇒ = 3 2 + − 2 4

Shear yield relative to tension yield in the Drucker model is given by:

/ (3.115) 1 4 = 1 − √3 27 if we specify a yield in the pure shear to yield in the uni-axial tension ratio equivalent to the Drucker model considering no tension-compression asymmetry , the and material coefficients of the GYS model corresponding to the Drucker =model 0 are given by:

1 (3.116) = / = 0 = 1 − √3 4 1 − 27 In the Cazacu-Barlat model, pure shear yield relative to uni-axial tension yield is fixed by tension-compression asymmetry via material parameter c and given by:

/ (3.117) 1 2 = − 3√3 27 Similarly the Cazacu-Barlat model can be recovered from the GYS model when we specify a yield in shear to yield in tension ratio equivalent to that in the Cazacu-Barlat model; so the GYS material coefficients corresponding to the Cazacu-Barlat model are given by:

1 (3.118) = = / = 0 = 1 − √3 1 2 √3 − 3√3 27 The Cazacu-Barlat yield surface for different tension/compression yield ratios is given in Figure 3-8. Comparison between the results from the GYS and Cazacu-Barlat models is

56 given in Figure 3-9 based on tension/compression yield ratios; and clearly the GYS model is able cover the same response as the Cazacu-Barlat model.

Figure 3-8: Cazacu-Barlat yield surface in the plane stress space corresponding to uni-axial tension/uni- axial compression yield ratios, 3/2, 3/3, 3/4.

Figure 3-9: Comparison between the Cazacu-Barlat and GYS yield surface models in the plane stress space corresponding to uni-axial tension/uni-axial compression yield ratios, 3/2, 3/3, 3/4.

57 The GYS yield function behaves similar to the Hosford yield function up to order ( m) six when (i.e., no tensile-compressive asymmetry). Let us formulate the Hosford criterion in terms of= stress 0 invariants to enable comparison of the Hosford and GYS models based on effect. The Hosford yield surface in terms stress invariants:

2 (3.119) 2 − = − + 3 3 3 2 + 3

2 (3.120) 2 − = − + 3 3 2 + 3 where is the Lode angle, and the relation between Lode angle and is given by . Then differences of principal deviatoric stress components can be written as 3 =

(3.121) 2 2 − = + − = − − 3 3 √3

(3.122) 2 2 − = − − = − 3 3 √3

(3.123) 2 2 2 2 − = − − + = 3 3 3 √3 The Hosford yield surface in terms of stress invariants is given by:

(3.124) | | | | | | − + − + − ≤ 2

58 (3.125) 2 + + − + ≤ 2 √3 √ 3 √3

/ (3.126) 1 2 = + + − + ≤ 2 √3 √3 √3 Multiplicative decomposition of the effective stress of the Hosford model ( ) in terms of the von Mises stress and is

/ (3.127) 2 = / + + − + ≤ 2 √3 √3 √3 Considering the trigonometric relationship:

+ (3.128) 6 6 − 6 = = − √3 6 6 can be reformulated as

/ (3.129) = 2 − + + + √3 6 6 So the related scale factor of the von Mises stress in the Hosford yield surface is

/ (3.130) 1 2 − + + + , 0 ≤ ≤ √3 6 6 3 The scale factor of the von Mises model in the GYS yield function is given by:

59 (3.131) 27 729 + 3 + 3 = + + 2 4 Figure 3-10 shows related factor of the von Mises stress considering different orders ( m) of the Hosford model corresponding to Tresca, von Mises, BCC, and FCC type materials.

Figure 3-10: related scale factor on von Mises stress for Hosford yield function

Let us consider the ratio of yield in uni-axial tension to yield in pure-shear based on the order ( m) of the Hosford yield surface:

= 1.732 ⇒ = 2 ,

=1.792⇒ =6 , (3.132) =1.835⇒ =8 ,

= 2⇒ =1 ∞ ,

60 Assume that the GYS model parameters have values for the (ratio of yield stress in the uni-axial tension to yield stress in the pure-shear) equivalent to the one given by equation 3.132 without tension-compression asymmetry . Figure 3-11 compares the GYS and Hosford formulations in terms of related scale = factors 0 of von Mises stress for order (m) of 2, 6, 8 and infinite. From Figure 3-11 we see that there is an exact match until the order ( m) of 6. The difference between two models starts to increase right after this order. In addition, the effect of tension-compression asymmetry on scale factor in the GYS model is shown corresponding to order 6 in Figure 3-12. Figure 3-12 compares the scale factors for the symmetric and asymmetric cases to prove the contribution of the GYS model compared to the Hosford model in the case of tension-compression asymmetry. Considering that the lode angle zero degree is the uni-axial tension meridian and the lode angle sixty degree is the uni-axial compression meridian, scale factors are the same for uni-axial tension and compression without tension-compression asymmetry. On the other hand, scale factors for the uni-axial tension and uni-axial compression are different in the case of tension- compression asymmetry in the GYS model. Hosford model can not account for the variation between the yield in the uni-axial tension and uni-axial compression thus scale factors at the uni-axial tension and uni-axial compression meridian are always the same.

Figure 3-11: Comparison of scale factor on von Mises stress for GYS and Hosford yield function.

Figure 3-12. Comparison of scale factor on von Mises stress for GYS with and without tension- compression asymmetry corresponding to order (m) 6.

Convexity and uniqueness are two important properties for yield surfaces. If a yield function is convex and smooth then the strain rate is unique for any given state of stress. Although the Hosford yield function is convex and smooth, the uniqueness is lost in practical (numerical) applications for m>10 due to the very ‘flat’ nature of the yield surface, as shown in Figure 3-13.

Figure 3-13: Hosford non-unique flow.

62 Although the GYS function is smooth and unique, it does not allow convexity for all cases, as shown in Figure 3-14. Therefore specific convexity conditions need to be determined for the GYS model. Convexity of the GYS model and a comparison between the GYS, Drucker and Cazacu-Barlat convexity regions will be discussed in detail in the next chapter.

Figure 3-14: GYS non-convex flow.

For a given state of (plastic) deformation multiple states of stress are allowed. Different states of deformation can be imposed in the test sample by creating unique geometries. However if the flow is non-unique the corresponding state of stress, and thus stress triaxiality (pressure/ von Mises) and Lode parameter (angle) are not uniquely determined. This is an important problem when test matrices are designed for the purpose of determining a stress state dependent yield point, hardening curve, or failure strain. Consider the example of a plane stress state. Assume a plain strain state with small elastic deformations compared to plastic deformation. Then

(3.133) 0 0 0 0 = 0 0 0 ≈ = = = 0 0 0 0 − 0 0 0

63 where a = . For von Mises/ plasticity, the plastic strain rate tensor takes a form

(3.134) 0 0 2 − 0 0 = 0 0 0 = = 0 2 − 1 0 ⇒ = = 0.5 3 0 0 − 0 0 −1 − We see that is the unique solution. Consequently the stress triaxiality (pressure/ von Mises) is 0.577, = the 0.5 Lode parameter is 0, and the Lode angle is 30 degrees. For the Hosford yield surface, with plain strain deformation leads to a = varying with m (order of yield surface) as shown in Figures 3-15 and 3-16. Consequently, / a = which will satisfies the condition for the plane strain deformation is not unique and deviates/ from 0.5 as the order of the Hosford yield function goes higher. In other words, the number of solutions increases as the order m is increased.

Figure 3-15: Example of the plane strain condition for the Hosford yield function and non-unique flow characteristic shown by the yield function as exponent m becomes higher.

Figure 3-16: a = value for plane strain condition of the Hosford yield function. /

Similarly consider the plastic strain rate tensor for the GYS function under plane strain deformation with the assumption of small elastic strain relative to plastic strain and plane stress state, we have:

(3.135) 0 0 0 0 = 0 0 0 ≈ = = = 0 0 0 0 − 0 0 0 If we consider no tension-compression asymmetry and shear yield relative to the tension yield equivalent to the Hosford yield function, = 0 then we also have and material coefficients in GYS defined as:

−1 1/ (3.136) 1 + 2 = = = 1 − √3 √3 Therefore, the corresponding plane strain condition for the GYS yield function has a unique solution at compared to the Hosford yield function for all orders ( m) as shown in Figure 3-17 = and 0.5 3-18.

Figure 3-17: Example of plane strain condition for the GYS function and unique flow characteristic shown by the yield function as order m gets higher.

Figure 3-18: a = value for plane strain condition of the GYS model. /

66 3.5 Convexity of the GYS model and comparison between the GYS, Drucker and Cazacu-Barlat convexity regions Convexity of the yield surface is a consequence of Drucker’s stability postulate. The principal of maximum plastic resistance is a mathematical statement of the convexity of yield surface and normality rule. In principal stress space, the product is represented by : the dot product of the stress and plastic strain rate vectors. The statement

(3.137) ∗ { − }. { } ≥ 0 is equivalent to the requirement that the angle between the vectors formed by and ∗ ( − ) must be less than 90 o for all stresses and strain rates. This is only possible if the yield surface is convex and the plastic strain rate vector is normal to the yield surface. The principle of maximum plastic resistance is important because it is the basis for a number of very important theorems concerning plastic deformation in solids. For example, it can be shown that the stress field in a material that obeys the principle is always unique. The basic requirement for convexity of a three dimensional surface is that its curvature must be non-negative everywhere on the surface. That is, the curvature of the function in two mutually perpendicular directions in the tangent plane at any given point on the surface must be non-negative. Mathematically, this requires that the curvature tensor be positive definite. The curvature in the tangential plane is given by:

(3.138) =

(3.139) = = where is the Hessian of f given by:

(3.140) =

where σi is the principal stress vector. U consists of two basis vectors that lie in the plane tangent to f in the principal stress space. After Rockafellar [52], to ensure convexity of the yield surface, the Hessian matrix of the yield function f must be positive semi-definite with respect to the principal stresses. The Hessian H of a function f (x 1, x2,…… xn) with respect to x1,

x2,… xn is the Jacobian matrix of the derivatives f ( ). , …… This means that the values of the GYS yield parameters , and are not completely arbitrary. Thus only those combinations that satisfy the convexity requirement of GYS are acceptable. Using the Hessian matrix of the GYS yield function, convexity conditions of the GYS yield surface can be determined based on model parameters. Basically, this condition will present a specific interval on the yield locus that is free from numerical issues at any stress state due to non-convexity of the yield surface. The four most prominent yield functions in the literature are von Mises, Hosford, Drucker-Prager, and Cazacu-Barlat; each of these have shown convexity range of the proposed yield stress functions in their studies at the very first step. Thus the convexity range or interval must be determined first to ensure that the proposed yield function is going to work out without any numerical issue at any stress state for a given convexity range. The GYS yield parameters , and are expressed in terms of the yield stress ratios the uni-axial tensile stress) / (the uni-axial compressive yield stress), and (the pure-shear yield stress) / the uni-axial tensile stress). For the yield function to be√3 convex, these parameters must be limited to a given numerical range. Thus there is going to be an interval on the yield function that is free from numerical issues at any stress state based on the yield stress ratios. Then convexity check is made by the GYS model to make sure yield stress ratios of given material is inside the interval otherwise projection is made onto the interval boundaries. The convexity of a yield surface in stress space can be described as the yield surface is convex if the Hessian matrix is positive semi-definite. The Hessian has 36 components, which reduces 21 due to symmetry of the stress tensor:

68 , (3.141) = , , , , ≤ 0

(3.142) = = Convexity in principal stress space can be described as

(3.143) , , ≤ 0

(3.144) = = = … … …

The yield surface is convex if the Hessian matrix is positive semi-definite in principal stress space. In this case, the Hessian has 9 components, which reduces 6 due to symmetry as shown in equation 3.144. Convexity in principal stress space is a sufficient condition for convexity in stress space. The properties of the eigenvalues of the Hessian matrix are

69 (3.145) = 1,2,3 : i = : + + = tr(HHHH) = + + (3.146) + + = + + + + = − + − + − = det(HHHH) All eigenvalues of a real symmetric matrix are real. The Hessian is positive semi-definite if all eigenvalues are positive or zero:

det ( HHH ) = 0 ⇒ = 0 then: + = tr( HHH) = + + (3.147) = − + − + − and + + ≥ 0 ⇒ ≥ 0 − + − + − If the determinant is zero then we only need to show that the trace and the sum of the principal minors are positive or zero. As a result, the three conditions described need to be fulfilled in order to obtain a physically meaningful yield surface. To prove convexity it is sufficient to show that:

+ + = 0 + + = 0 (3.148) + + = 0 + + ≥ 0 − + − + − ≥ 0

70 The GYS yield function in terms of J2 and J3 is described as

(3.149) 27 27 / (, ) = + + − = √3 2 2

/ 9 9 (3.150) (, ) = √3 + / + / − 2√3 2√3

(3.151) / 9 27 √3 (, ) = √3 + + / − 2 4 Then convexity conditions for GYS are shown below as follows:

The trace of the Hessian matrix is positive or zero:

(3.152) = () ≥ 0 The trace of a Hessian matrix for GYS model is described as follows:

(3.153) () = + +

(3.154) 2 2 = + + + + + 3 3

(3.155) 2 2 = + + + + + 3 3

71 (3.156) 2 2 = + + + + + 3 3 Then the trace of the Hessian matrix for the GYS yield function is given by:

() = + + = 2 + ( + + ) (3.157) + 2 + + + 2 ( + +) + + + Also note that the following relationships have been used to obtain the final form of equation 3.157: (3.158) + + = 0, + + = 2

(3.159) = + +

(3.160) 2 1 1 1 = − − = − ( + + ) = + 3 3 3 3 3

(3.161) = + , = + , = + 3 3 3

(3.162) + + = − ⇒ + + = 0 Necessary deviatoric invariant derivatives of the GYS yield function in equation 3.157

are and ; they are given as follows: , , , ,

72 (3.163) / 9 27 √3 = √3 + + / − 2 4

(3.164) √3 9 5 27 √3 = / − − / 2 2 2 4

(3.165) 9 27 √3 = + / 2 2

(3.166) 27 √3 = / 2

(3.167) 9 5 27 √3 = − − / 2 2 2

(3.168) √3 27 √3 = − / + 9 + 35 / 2 4 16 The first convexity condition is that trace of the Hessian matrix is positive or zero therefore:

() = + + ≥ 0 (3.169) √3 9√3 4√3 35 √3 = + − 3 − (3) ≥ 0 2 2

27 3 = 2 The second convexity condition is that sum of principal minors of Hessian matrix is positive or zero:

73 (3.170) + + ≥ 0

= − (3.171) = − = − The GYS stress derivatives are as follows for hessian matrix:

(3.172) 2 2 = + + + 2 + + + 3 3 3 3

(3.173) 2 2 = + + + 2 + + + 3 3 3 3

(3.174) 2 2 = + + + 2 + + + 3 3 3 3

1 2 5 = − + + + − 3 3 3 (3.175) + + − 9 3

1 2 5 = − + + + − 3 3 3 (3.176) + + − 9 3

74 1 2 5 = − + + + − 3 3 3 (3.177) + + − 9 3 See Appendix-A for the final form of the second convexity condition. The first and second conditions of convexity have been calculated using MATHEMATICA v8 symbolic calculation toolbox. Two convexity conditions have been programmed using symbolic language and the final form of these equations are obtained (details of symbolic equations in MATHEMATICA were given in appendix A). Programmed equations of the first and second convexity conditions were verified by matching the known convexity conditions of yield functions such as von Mises, Drucker and Cazacu-Barlat as shown in table 3-3. The program output has reproduced the convexity conditions of these yield functions, and these results were used as a verification case for the convexity equations programmed in the MATHEMATICA v8 symbolic calculation toolbox.

Table 3-3: Comparison of convexity ranges of von Mises, Cazacu-Barlat, Drucker models yield criterion yield function convexity range

von Mises No condition (convex at all points) / = Cazacu-Barlat < c < / √ √ − = Drucker < c < − = ( : yield stress in pure shear) For the second convexity condition it was found out that is equal to zero; thus the condition is always satisfied. Therefore + the+ only convexity condition for the GYS is the + first condition.+ ≥ 0 The first convexity condition can be rewritten in terms of GYS parameters , , and ; and deviatoric stress invariants and (or Lode parameter ). Also note that parameters , , and can be expressed in terms of 3

75 pure-shear yield relative to uni-axial tension yield and uni-axial compression yield relative to uni-axial tension yield as shown below: (3.178) 1 1 1 = , = 1 − , = 1 − − √3 2 If we assume that the yield stress in tension equals unity to be able to normalize stress invariant terms, then we can write:

(3.179) 1 = 1 = = 3 3 The the first convexity condition can be written in terms of three parameters: uni-axial compression yield relative to uni-axial tension yield , pure-shear yield relative to uni- axial tension yield , and Lode parameter : 3

(3.180) √3 9√3 4√3 35 √3 27 + − 3 − ( 3) ≥ 0 3 = 2 2 2

() = + +

√3 1 1 1 = + 27 1 − − − 2 2 √3 2 (3.181)

105 1 1 1 − (3) 1 − − − 2 2 √3 2

1 − 6 3 1 − ≥ 0

Equation 3.181 was plotted using MATHEMATICA in 3-d space of the Lode parameter, the yield in pure-shear/yield in uni-axial tension, the yield in uni-axial compression/yield in uni- axial tension and the convexity region of the GYS model was obtained as shown in Figure 3- 19 and 3-20.

Figure 3-19: Convexity region of the GYS model in terms of yield stress ratio of uni-axial compression to tension and pure-shear to uni-axial tension and Lode parameter.

Figure 3-20: Convexity region of the GYS model in terms of yield stress ratio of uni-axial compression to uni-axial tension and pure-shear to uni-axial tension and Lode parameter.

To be able to understand the effect of uni-axial tension-compression yielding asymmetry, the uni-axial compression/tension yield ratio was fixed and 2-d convexity region plots were obtained as presented through Figure 3-21 and 3-25. These plots are the 2-d cross section of 3-d convexity region shown in Figure 3-19 and 3-20 at specific uni-axial compression/tension yield ratio. As shown below, Figure 3-21(a) shows perfect symmetry with respect to Lode parameter value zero in the case of equal yield stress in uni-axial tension and uni-axial compression as expected. It also shows that continous path for the Lode parameter to transition from -1 to 1 is possible only when pure-shear to uni-axial tension yield ratio is between 0.545 and 0.611 (see dashed lines in Figure 3-21(b)). Note that in the von Mises plasticity, pure-shear to uni-axial tension yield ratio is fixed to a value 0.577. Therefore contribution of the GYS model in terms of pure-shear yield flexibility in the case of equal yield stress in uni-axial tension and uni-axial compression is around ±5 percent

78 compared to fixed pure-shear yield of the von Mises plasticity. In Figure 3-21, lower boundary value, 0.545 , corresponds to lode parameter 0 (pure-shear meridian) . Upper boundary value, 0.611, corresponds to maximum allowed point for the continous path of the lode parameter transition from -1 to 1 at lode parameter 1 (uni-axial tension meridian) or -1 (uni-axial compression meridian). Consider that lode parameter -1 (uni-axial compression meridian) and lode parameter 1 (uni-axial tension meridian) have the same upper boundary value for the continous path due to uni-axial tension-compression yield symmetry.

(a)

(b)

Figure 3-21: (a) Convexity region of GYS model with yield ratio uni-axial compression/uni-axial tension=1 (b) Lower and upper boundaries of the convexity region correspond to pure-shear yield to uni-axial tension yield ratio 0.545 and 0.611 with the yield ratio uni-axial compression/uni-axial tension=1.

79 Other plots from Figures 3-22 to 3-25 show asymmetric behavior based on the difference between uni-axial compression and uni-axial tension yield stresses as expected. In Figure 3- 22, continous path for the Lode parameter to transition from -1 to 1 is possible only when pure-shear to uni-axial tension yield ratio is between 0.531 and 0.588 when uni-axial compression to uni-axial tension yield ratio equals 0.95. In Figure 3-23, continous path for the Lode parameter to transition from -1 to 1 is possible only when pure-shear to uni-axial tension yield ratio is between 0.467 and 0.490 when uni-axial compression to uni-axial tension yield ratio equals 0.75. Both Figure 3-22 and 3-23 presents the case which uni-axial compression yield is less than uni-axial tension yield. In these figures, the area covered between the lode parameter 0 (pure-shear meridian) and -1 (uni-axial compression meridian) is less than the area covered between the lode parameter 0 (pure-shear meridian) and 1 (uni-axial tension meridian). Note that maximum point on the pure-shear/uni-axial tension yield axis is 0.602 at lode parameter 1 (uni-axial tension meridian) and 0.588 at lode parameter -1 (uni-axial compression meridian) in Figure 3-22. Therefore maximum allowed point on the pure-shear to uni-axial tension yield ratio axis for the continous path of the Lode parameter transition from -1 to 1 is 0.588 at lode parameter -1 in Figure 3-22. Also note that maximum point on the pure- shear/uni-axial tension yield axis is 0.561 at lode parameter 1 (uni-axial tension meridian) and 0.490 at lode parameter -1 (uni-axial compression meridian) in Figure 3-23. Therefore maximum allowed point on the pure-shear to uni-axial tension yield ratio axis for the continous path of the Lode parameter transition from -1 to 1 is 0.490 at lode parameter - 1.This means that upper limitof the pure-shear flexibility is determined by the maximum allowed value of the pure-shear to uni-axial tension yield ratio at uni-axial compression meridian (lode parameter -1) when uni-axial compression yield is less than uni-axial tension yield. As a conclusion, lower and upper boundary of the pure-shear flexiblity are determined by the value on the pure-shear (lode parameter 0) and uni-axial compressional meridian (lode parameter -1) respectively when uni-axial compression yield is less than uni-axial tension.

Figure 3-22: Convexity region of GYS model with yield ratio compression/tension=0.95.

Figure 3-23: Convexity region of GYS model with yield ratio compression/tension=0.75.

In Figure 3-24, continous path for the Lode parameter to transition from -1 to 1 is possible only when pure-shear to uni-axial tension yield ratio is between 0.571 and 0.626 when uni-axial compression to uni-axial tension yield ratio equals 1.1. In Figure 3-25, continous path for the Lode parameter to transition from -1 to 1 is possible only when pure- shear to uni-axial tension yield ratio is between 0.605 and 0.645 when uni-axial compression to uni-axial tension yield ratio equals 1.25. Note that maximum allowed point on the pure- shear/uni-axial tension yield axis is 0.626 at lode parameter 1 (uni-axial tension meridian) and 0.655 at lode parameter -1 (uni-axial compression meridian) in Figure 3-24. Therefore maximum allowed point on the pure-shear to uni-axial tension yield ratio axis for the continous path of the Lode parameter transition from -1 to 1 is 0.626 at lode parameter 1 in Figure 3-24. Also note that maximum allowed point on the pure-shear/uni-axial tension yield axis is 0.645 at lode parameter 1 (uni-axial tension meridian) and 0.716 at lode parameter -1 (uni-axial compression meridian) in Figure 3-25. Therefore maximum allowed point on the pure-shear to uni-axial tension yield ratio axis for the continous path of the Lode parameter transition from -1 to 1 is 0.645 at lode parameter 1.This means that upper limitof the pure- shear flexibility is determined by the maximum value of the pure-shear to uni-axial tension yield ratio at uni-axial tension meridian when uni-axial tension yield is less than uni-axial compression yield. As a conclusion, lower and upper boundary of the pure-shear flexiblity are determined by the value on the pure-shear (lode parameter 0) and uni-axial tension meridian (lode parameter -1) respectively when uni-axial tension yield is less than uni-axial compression yield. The range of pure-shear yield flexibility of the GYS model decreases with uni-axial tension-compression asymmetry (see Figure 3-29). In the GYS model, specific range in terms of the pure-shear yield flexibility converges to a fixed value at minimum and maximum allowed uni-axial compression to uni-axial tension yield ratio points of the convexity region in the GYS model (see Figure 3-29). In the case of maximum point when uni-axial compression to uni-axial tension yield ratio is equal to 1.693, times pure-shear to uni-axial tension yield ratio is fixed to value 1.187. In the case of minimum√3 point when

82 uni-axial compression to uni-axial tension yield ratio is equal to 0.590, times pure-shear to uni-axial tension yield ratio is fixed to value 0.701. √3

Figure 3-24: Convexity region of GYS model with yield ratio compression/tension=1.1.

Figure 3-25: Convexity region of GYS model with yield ratio compression/tension=1.25.

83 Consider the case, uni-axial compression to uni-axial tension yield ratio equals 2 which is above the maximum allowed ratio 1.693 in the convexity region of the GYS model. As shown in Figure 3-26 (a), there is no specific path for the Lode parameter to transition from - 1 to 1 when uni-axial compression to uni-axial tension yield ratio yield ratio equals 2. This means that uni-axial compression to uni-axial tension yield ratio is outside the convexity region whatever value of uni-axial tension to pure-shear ratio yield ratio of material is. In this case, uni-axial compression to uni-axial tension yield ratio is projected to maximum allowed value 1.693 in the convexity region (See Figure 3-29). Consider the case, uni-axial compression to uni-axial tension yield ratio equals 0.5 which is beyond the minimum ratio allowed, 0.590, in the convexity region of the GYS model. As shown in Figure 3-26 (b) , there is no specific path for the Lode parameter to transition from -1 to 1 when uni-axial compression to uni-axial tension yield ratio equals 0.5. This means that uni-axial compression to uni-axial tension yield ratio yield ratio is outside the convexity region whatever value of uni-axial tension to pure-shear ratio yield ratio of material is. In this case, uni-axial compression to uni-axial tension yield ratio is projected to minimum ratio allowed, 0.590, in the convexity region (see Figure 3-29).

(a)

(b)

Figure 3-26: (a) Convexity region of GYS model with yield ratio uni-axial compression/tension=2.00 (b) Convexity region of GYS model with yield ratio uni-axial compression/tension=0.50

Also the convexity region of the the GYS yield surface was shown in the space of uni- axial compression to tension versus pure-shear to uni-axial tension yield ratios (uni-axial compression/uni-axial tension is the ordinate and times pure-shear/uni-axial tension is the abscissa) using Lode parameter s 1, 0, -1 curves (see√3 Figure 3-27). The GYS convex ity region is the area inside the in tersection of Lode angle parameter ( ) curves as shown in Figure 3 -28 (a). The convexity region of the Cazacu, Drucker and von Mises yield surfaces was also shown on the same plot to enable comparison of the GYS model with the Cazacu-Barlat , Drucker, and von Mises models (see Figure 3 -28 (b)).

85 (a) (b)

(c)

Figure 3-27: Convexity curves of the GYS model corresponding to the Lode angle parameter (a) 0, (b) 1, (c) -1, respectively.

86 (a)

(b)

Figure 3-28: (a) Intersection of the Lode angle parameter ( ) curves corresponding to 0, 1, -1 for the GYS model (b) Comparison of the convexity regions of the GYS model and von Mises, Drucker, and Cazacu-Barlat models.

87 Numerical values of the minimum and maximum tip points of the GYS convexity region are given by Figure 3-29. Also min and max limits of pure-shear to uni-axial tension yield ratio under the condition of =1 (uni-axial tension yield is equal to uni-axial compression yield) are presented in Figure 3-29.

Figure 3-29: Min and Max points on the convexity region of GYS model in terms of yield stress ratios.

The von Mises convexity region is a point. The Drucker and Cazacu-Barlat convexity regions are line. The GYS convexity region is an area, and therefore provides more flexibility compared to other models while matching experimental data. Convexity conditions of the Cazacu Barlat, von Mises, and Drucker models are given as

• Cazacu-Barlat Convexity Condition: (3.180) / = −

(3.181) 3 / 3 / = , = − , = 0, = , = 0 2 4 • First Condition:

(3.182) () = + + ≥ 0

(3.183) 3 / / 2 = + − 4 3

(3.184) 3 / / 2 = + − 4 3

(3.185) 3 / / 2 = + − 4 3 (3.186)

/ 3 / 9 / () = 3 + 2 = ≥ 0 4 2 • Second Condition: (3.187) + + ≥ 0

(3.188) 4 9 − − √3( 3) + ≥ 0 (−1 ≤ 3 ≤ +1) 3 2

(3.189) 3√3 3√3 − ≤ ≤ + 4 4 • von Mises yield function: (3.190) / = √3

89 (3.191) √3 / √3 / = , = = = 0, = − 2 4 • First Condition:

(3.192) () = + + ≥ 0

(3.193) √3 / √3 / = − 3 4

(3.194) √3 / √3 / = − 3 4

(3.195) √3 / √3 / = − 3 4

/ √3 / / √3 / () = √3 − ( + + ) = √3 − (3.196) 4 2

√3 / = ≥ 0 ( + + = 2) 2 • Second condition: (3.197) + + = 0 (3.198) ℎ + + ≥ 0 • Drucker yield function: (3.199) = −

(3.200) = 3 , = −2 , = −2, = 0, = 6

90 • First condition:

() = + + (3.201) = 2 + 2 + + + 2 = 18 − 2 3 (3.202) 4 27 () = 18 − ≥ 0 ⇒ ≥ ( ) 3 2 • Second condition: (3.203) + + ≥ 0

12 40 12 45 − 20 + 180 ( 3) − ( 3) ≥ 0 81 3 ) 81 (0 ≤ (3) ≤ 1 (3.204)

−27 9 ≤ ≤ 8 4 ( ℎ ℎ ℎ ℎ ℎ ℎ )

3.6 Development of convexity algorithm for the GYS model The convexity region of the GYS model in terms of uni-axial compression to uni-axial tension and pure-shear to uni-axial tension yield ratios was presented in the √3 previous section. Based on the uni-axial tension, uni-axial compression, and pure shear

(torsion) test data, uni-axial compression to uni-axial tension and pure-shear to uni-axial tension yield ratios of material are computed at each time step. The location of the √3

91 corresponding points for these yield ratios should be checked out on the convexity region. If the point is outside the convexity region, it should be projected onto convexity boundary to satisfy the convexity requirement. Numerically this is done by the convexity algorithm in the GYS model. First, the location of the point is classified by the algorithm based on the uni-axial compression to uni-axial tension yield ratio minimum and maximum () ( ) points of the convexity region. Also note that the minimum and maximum () points of the convexity region are known since the Lode parameter 1, -1, 0 curves ( ) (blue, red, green curves) are available as mathematical functions (see Figure 3-27). If the data point is greater than the uni-axial compression to uni-axial tension yield ratio maximum point, it is projected to the maximum point (see light blue star in Figure 3-31). If the data point is less than the uni-axial compression to uni-axial tension yield ratio minimum point, then it is projected to the minimum point (see yellow star in Figure 3-30).

Figure 3-30: Min and Max tip points on the convexity region of GYS model.

92 If the data point is between compression to tension yield ratio minimum and () maximum points and outside the convexity region, it is projected parallel onto the ( ) convexity region (see red stars in Figure 3-31). Also note that the minimum and () maximum points of the convexity region are known since the Lode parameter 1, - ( ) 1, 0 curves (blue, red, green curves) are available as mathematical functions (see Figure 3- 27).

Figure 3-31: Projection operation of convexity algorithm of GYS model.

93 A summary of the convexity algorithm is presented in the box 3.1. Following variables are used in the algorithm:

: Uni-axial compression to uni-axial tension yield ratio calculated at each time step. : Pure-shear to uni-axial tension yield ratio calculated at each time step.

: Uni-axial compression to uni-axial tension yield ratio maximum point of ( ) . the convexity region.

: Uni-axial compression to uni-axial tension yield ratio minimum point of ( ) . the convexity region.

(1) , (2) : Projection points on the convexity boundary when is between √3 √3 and and ≥ 1 (see Figure 3-31). Note that in the special case when () ( ) = 1, √3 (1) = 0.944 and √3 (2) =1.0588 (see Figure 3-30).

√3 (3) , √3 (4) : Projection points on the convexity boundary when is between

() and () and < 1 (see Figure 3-31).

Convexity algorithm of the GYS model based on the variables given above is presented in the Box 3.1 below.

94 Box 3.1: Convexity algorithm of the GYS model

(i)) If is larger than () then it is projected to ()

≥ ( ) THEN → ( )(ii) If is less than IFIFIF

() then it is projected to ()

≤ ( ) THEN → () IFIFIF

(iii) If is between () and ( ) and ≥ 1 then compute

√3 (1) at point 1 and √3 (2) at point 2 (see Figure 3-30)

( ) < < ( ) AND ≥ 1 IFIFIF

√3 (1) , √3 (2)

(iii-a) If √3 is less than √3 (1) then it is projected to √3 (1)

√3 ≤ √3 (1) HHH N N N √3 → √3 (1)(iii-b) If √3 is IFIFIF TTT EEE

greater than √3 (2) then it is projected to √3 (2)

√3 ≥ √3 (2) HHH N N N √3 → √3 (2) IFIFIF TTT EEE

(iii-c) If √3 is between √3 (1) and √3 (2) then it is inside the convexity region and no projection is made .

√3 (1) < √3 < √3 (2) HHH N N N √3 → √3 IFIFIF TTT EEE

95 Box 3.1: Convexity algorithm of the GYS model (continues)

(iv) If is between ( ) and ( ) and < 1 then compute

√3 (3) at point 3 and √3 (4) at point 4 (see Figure 3-30)

( ) < < ( ) AND < 1 IFIFIF

√3 (3) , √3 (4)

(iv-a) If √3 is less than √3 (3) then it is projected to √3 (3)

√3 ≤ √3 (3) HHH N NNN √3 → √3 (3) IFIFIF TTT EEE

(iv-b) If √3 is greater than √3 (4) then it is projected to √3 (4)

√3 ≥ √3 (4) HHH N NNN √3 → √3 (4) IFIFIF TTT EEE

(iv-c) If √3 is between √3 (3) and √3 (4) then it is inside the convexity region and no projection is made .

√3 (3) < √3 < √3 (4) HHH N NNN √3 → √3 IFIFIF TTT EEE

Numerical verification cases for the projection onto convexity region are presented in appendix-B. Note that a compromise will be made on during the projection operation if point is located between min and max points of . As shown in Figure 3-31, stays the same at the convexity projection operation but value changes. Also if the point is less than or greater than min and max points of convexity region, a compromise will be made

96 on both and during the projection operation. It is assumed that experimental yield curves in uni-axial tension, uni-axial compression and pure-shear are available for a material which is the input data for the GYS material model. Thus it is important to check and data points of material before any finite element run to be able to understand if the given yield stress data set of a material is inside the convexity region or outside the convexity region. This will make it clear the compromise made on or data points due to projection on convexity boundary.

97 CHAPTER 4 - Numerical Implementation of GYS Yield Criterion In this chapter, the numerical implementation of GYS model is discussed. The general procedure to integrate the plastic rate equation is first reviewed. Two algorithms backward and semi-backward integration methods are explained. The procedure of implementing material constitutive models into a commercial finite element package LS-DYNA is presented. To verify the implementation, single element test cases at uni-axial tension, uni- axial compression, pure shear, biaxial tension and biaxial compression stress states were simulated and results were compared to the von Mises plasticity model. Single element test cases at different stress states were also run at different time steps within the limits of minimum allowable time step in LS-DYNA. The accuracy of the semi-implicit integration method used in the GYS stress update algorithm was shown to ensure that the step size used in LS-DYNA is small enough to achieve acceptable results.

4.1 Constitutive relation of the proposed GYS model An isotropic generalized yield surface (GYS) plasticity algorithm is utilized with isotropic strain hardening, stain rate hardening, and temperature softening features, where adiabatic heating due to the plastic work is considered. The flow stresses of the model can be formulated as a multiplicative decomposition of flow stress for strain rate and temperature dependency at uni-axial tension, uni-axial compression and pure shear as

(4.1) = ̅ , ̅ ̅ , , ̅ =

(4.2) = ̅ , ̅ ̅ , , ̅ = (4.3) = ̅ , ̅ ̅ , , ̅ = where is the effective plastic strain, is the effective plastic strain rate, and is the ̅ ̅ temperature. , , and are true plastic strain in uni-axial tension, uni-axial compression,

98 and pure-shear respectively. , , and are yield stress in uni-axial tension, uni-axial compression, and pure shear, respectively. Tabulated inputs of strain rate dependent isothermal curves and temperature dependent hardening curves can be provided directly as experimental curves true stress-true plastic strain (iflag=0) in uni-axial tension ( ) , uni-axial compression ( ), and pure shear ( ), or they can be provided as true stress-effective plastic strain ( ) ( (iflag=1) for uni-axial ̅ tension, uni-axial compression, and pure shear stress states, respectively. Adiabatic heating occurs due to plastic work and the temperature is calculated as

(4.4) = + ̅ where, is the room/current temperature, is the Taylor–Quinney coefficient that represents the proportion of plastic work converted into heat. Here is taken to be 0.9, which is assumed to be a proper value for metals [55]. is the specific heat at constant pressure, is the density, is the flow stress and, is the effective plastic strain. ̅ The elasto-plastic behavior of the material is assumed to be rate independent and elastically isotropic. In the small strain increment relative to the yield strain radius of material, the total strain rate tensor is decomposed into the elastic and plastic parts [56]:

(4.5) = + where and are the elastic and plastic components of the strain rate tensor, respectively. The plastic deformation is assumed as isochoric (i.e., volume preserving); therefore, plastic deformation is assumed to happen at constant volume and exhibit no volumetric strain. This is expressed as the trace of the plastic strain rate is equal to zero:

(4.6) = 0

99 Assuming that the elastic response of the material is isotropic, Hooke’s law relates the strain rate tensor to an objective rate of the Cauchy stress tensor as:

(4.7) = : = : − where is the fourth order positive definite isotropic stiffness tensor and is defined as:

(4.8) = + +

where, and are the Lamé constants given as functions of elastic constants, = − = is the bulk modulus, and is the shear modulus. = = The elastic modulus and Poisson’s ratio are assumed to remain unchanged through the plastic deformation. The plastic part of the deformation is treated by strain rate and temperature dependent tabulated hardening curves. The rate of the Cauchy stresses can be decomposed into deviatoric and spherical parts by using deviatoric stress rates ( ), and pressure rate ( ) as: = (4.9) = − Similarly, the strain rate tensor can be decomposed into deviatoric and volumetric parts by using deviatoric strain rate , and volumetric strain rate as:

(4.10) = + 3 And plastic strain rate is decomposed as

(4.11) = + 3

100 where, is the plastic deviatoric strain rate and is the plastic volumetric strain rate. The rate of the Cauchy stresses can then be written as:

(4.12) = 2 − + − The hypo-elastic pressure term reduces the material law further since volumetric plastic strain rate is zero ( ) and plastic strain rate is equal to its own derivative ( ): ≡ 0 =

(4.13) = 2 − + Therefore, the deviatoric stress rates can be written as:

(4.14) = 2 −

The yield criterion , which is defined as a convex function in stress space, determines the limit of the elastic range and due to the incompressible nature of the plastic deformation of metals: it is expected to be a function of deviatoric stresses only (no pressure dependency) and can be written as:

(4.15) c c c , , , ≤ 0 ⇒ , , , = , , , , ≤ 0 where, is the yield stress as a function of effective plastic strain, = ̅ , ̅ , effective plastic strain rate and temperature (note that are yield stresses at uniaxial , , tension, uniaxial compression, and pure shear stress states, respectively). Considering the GYS model, which depends on the invariants of deviatoric stress tensor and , the yield function can be written more explicitly as

(4.16) c c c 27 729 , , , = , , , , = + + − 2 4

101 where, and ( ) (s :deviatoric stress) and are stress state = : = , , dependent flow stress coefficients as a function of effective plastic strain and come naturally from the yield curves of the uni-axial tension, uni-axial compression and torsion (pure-shear) tests based on the following relationships:

c c c − + = c c c (4.17) + + = 1 c c c − + = √3 The change in shape of the yield loci during the deformation process occurs by letting the yield criterion coefficients be a function of the effective plastic strain. , , , The plastic strains are evaluated using the normality rule based on the associated flow. Plastic strain rate is defined by introducing a rate of plastic multiplier ( ) (multiplier of time independent plasticity) as

(4.18) = where is the plastic part of the strain rate tensor, is the plastic multiplier rate, is the Cauchy stress tensor, and is normal outward flow vector to the generalized yield surface as shown below and computed for each time step:

27 729 3 = − 2 − 5 2 4 2 (4.19)

27 729 1 + + 2 − 2 4 3

102 In the GYS model, equivalent plastic strain rate is defined as energy or work conjugate to the effective stress. The plastic rate parameter is found by defining the rate of work due to plastic dissipation ( ) and defining the rule for equivalent plastic strain rate work conjugate quantity to the effective stress of GYS. It was shown that the plastic rate parameter is equivalent to effective plastic strain rate (see chapter 3 for details): ̅

(4.20) = : = : = ̅ = : = : = ̅ The rate of plastic multiplier is used to obtain the evolution equation of the flow or hardening parameters of the GYS model as:

(4.21) = ̅ = = ̅ = = 1,2,3 ̅ ̅

The basic problem in elasto-plasticity is to obtain stresses that fulfill both the Kuhn- Tucker [49] conditions and the consistency condition. The Kuhn-Tucker conditions require that , where the first expression states that the plastic multiplier ≥ 0, rate , is always ≤ 0, positive , or zero, = 0 the second expression states that the stress is always on or within the yield surface, and the last expression states that the stresses remain on the yield surface during plastic loading (i.e., when >0). When >0, the last condition requires that and therefore the consistency condition requires that , = 0 , = the stress state remain on the yield surface during plastic loadings) which enables the 0 determination of plastic multiplier . The consistency condition for the GYS model was derived and the expression for plastic multiplier was obtained as shown below:

(4.22) c c c 27 729 , , , , = + + − 2 4

103 + + = 0 2 − + + = 0 (4.23) 2 − + + = 0 2 = ∑ 2 + −

In equation 4.23, term represents the hardening behaviour. In the case of ∑ − perfectly plastic material behaviour , this term equals to zero since material exhibits no strain hardening. In the GYS model, value of ∆λ is determined using a secant iteration, with for the first initial value and the second initial value for the increment of ∆λ the plastic= 0 multiplier determined( > 0 from) consistency equation 4.23 where the term ∆λ is taken as zero, meaning that the response is perfectly plastic. This ∑ − corresponds to the return of to the frozen yield surface of time as if the material were perfectly plastic with no hardening evolution. This means that solution interval is bounded using two initial values for the secant iteration, and corresponds to perfect plasticity in order to ensure that the stress∆λ =state 0 returns > 0 to the ∆λinterior of the yield surface, resulting in a negative value of the yield function ( ). < 0

4.2 Integration algorithms for elasto-plastic constitutive equations In any numerical scheme employed for the analysis of elasto-plastic problems, integrating the constitutive equations governing material behavior becomes necessary. This requires a stress update algorithm to calculate the stresses at each time increment based on the amount of plastic flow that has occurred in the increment. The precision with which constitutive relations are integrated has a direct impact on the overall accuracy of the

104 analysis. The importance of understanding and controlling this source of numerical error has motivated a number of research studies [57] [58] [59] [60] [61] [62] [63]. Before finite element calculations can be performed, the yield criterion must be implemented into a finite element framework (e.g., into the commercial finite element code LS-DYNA, ABAQUS), and the consistency condition must be satisfied. To implement a criterion into a finite element framework, a stress update algorithm must be used to calculate the stresses at each time increment based on the amount of plastic flow that has occurred in the increment. Return mapping schemes are widely-used for numerically implementing these stress updates. Return mapping schemes first predict the trial stresses based on the current total strain increment (an elastic predictor step) and then return the stresses back to the yield surface using an iterative procedure for the plastic deformation correction (see Figure 4-1). Two elements of the method are an integration scheme that transforms the set of constitutive equations into a set of nonlinear algebraic equations and a solution scheme for the nonlinear algebraic equations. The method can be based on different integration schemes such as the backward Euler, forward Euler, or Runge-Kutta methods.

Figure 4-1: Schematics of return mapping algorithm in stress space.

In the fully implicit method (backward Euler method), the increments in the plastic strain and plastic flow direction are calculated at the end of the step, and the yield condition is enforced at the end of the step (consistency condition is enforced at the end of the time step, i.e., , to avoid drift from the yield surface). The radial return method for von Mises = 0

105 plasticity is a special case of the implicit return mapping algorithms and was originally proposed by Wilkins [63]. The closest point algorithm is a generalization of the radial return rule to arbitrary convex yield surfaces (see fig 4-2-a). A summary of the principal methods is given by Belytschko [56] and Simo and Hughes [64]. The semi-implicit backward Euler method [56][57][64] is implicit in the plasticity parameter but explicit in the plastic flow direction; i.e. the increments in the plasticity parameter ( ∆λ) are calculated at the end of each step, but plastic flow direction is calculated at the beginning of the step. To avoid drift from the yield surface, the yield condition is enforced at the end of the step. The return mapping scheme iteratively updates the stresses such that the yield condition is satisfied.

Figure 4-2: Schematics of return mapping schemes (a) Backward Euler-fully implicit (b) Semi- implicit tangent cutting plane method.

In the return mapping procedure, the elastic predictor phase is driven by the increment in total strain while the plastic- corrector phase is driven by increment ∆λ plasticity parameter. Thus during the elastic predictor stage, the plastic strain and internal variables remain fixed, and during the plastic corrector stage, the total strain is fixed. The following set of equations can be used in order to determine this plastic correction to the stress tensor:

106 = + (4.24) = + ∆λ where is the plastic flow direction, and n indicates the increment number and should not be confused with the iterative counter. The converged stress tensor at step n+1 can now be expressed as follows:

= : = : − (4.25) = : + ∆ − − ∆ = + : ∆ − : ∆ such that:

= + : ∆ (4.26) = − : ∆ = − ∆λ: where is the elasticity tensor. An expression for the plastic corrector to the stress tensor, , needs to be determined for the iterative plastic correction ∆ = −∆λ: phase. Note that nothing has yet been said regarding whether the plastic flow direction, , is taken as the previously converged direction, , or as the current increment's direction, . In practice, either value can be used, although the simplicity and accuracy of the routine will depend on this choice. If the previously converged value for the plastic flow direction is used to update the stresses then the return mapping scheme presented below is referred to as a semi-implicit backward Euler scheme, since the algorithm is implicit in the plastic multiplier ( ) but explicit in the plastic flow direction . Alternatively, if the λ

107 current increment's value, , is used for the plastic flow direction in the plastic correction phase, then the return mapping algorithm presented here is referred to as a fully implicit backward Euler scheme, since it is implicit in both the plastic multiplier and the plastic flow direction. For associated plasticity, a geometric interpretation can be made regarding the difference between the two schemes. The semi-implicit routine projects the predicted stresses onto the yield surface along the normal to the previous yield surface while the fully implicit routine projects the predicted stresses onto the yield surface along the normal to the current yield surface (closest point projection). Obviously, as the step or increment size decreases, the two schemes converge to the same result, since the two normal will become approximately the same. The semi-implicit scheme is commonly used in explicit finite element methods, since the time step required is very small compared to to that required for implicit methods due to the Courant-Friedrichs-Lewy (CFL) condition [65] (for example, the CFL condition generally limits the time step such that a wave cannot pass through the smallest element in the mesh in less than one time increment). A semi-implicit scheme can also be used in implicit finite element methods, although care must be taken to ensure that the step size used is small enough to achieve acceptable results. Obviously, a semi-implicit scheme will require smaller step sizes for a given problem than a fully implicit scheme, since the plastic flow direction is explicit in the semi-implicit scheme.

4.2.1 Time integration and numerical implementation of GYS into LS-DYNA

The next few paragraphs outline the return mapping procedure used to implement the isotropic yield criterion of the equation into a finite element framework (in this case, into LS- DYNA, although the return mapping scheme is simply a numerical algorithm for integrating the constitutive equations and it is independent of the finite element code being used). In LS- DYNA [66] finite elastoplasticity implementation is based on the introduction of a hypo- elastic constitutive law and the additive elastic-plastic decomposition of the strain rate tensor (rate-additive hypo-elastic plasticity [67]). This approach hinges on the additive decomposition of the rate of deformation, hypoelasticity, and objective stress rates. Jaumann rate [68] of Cauchy stress is used as objective stress rate in LS-DYNA. LS-DYNA first

108 performs the stress rotation then call a constitutive subroutine to add incremental stress components [66]. The additive strain rate decomposition is a consistent approximation to multiplicative decomposition of the deformation gradient when the elastic strains are infinitesimal (negligible compared to unity) and the strain increment is small relative to the yield strain radius of material when the total strain rate measure used is the rate of deformation which is symmetric part of the velocity gradient with respect to current configuration ( where is the velocity gradient tensor with respect to current = + position [69] [70]). Typical dynamic stress wave codes like LS-DYNA solves problems with an explicit time-marching procedure. At the beginning of each computational cycle, the strain rate tensor is constructed from the material velocities. Next, this strain rate tensor and stresses at the beginning of cycle are then input to the elasto-plastic constitutive model, then updated stresses are calculated. The new stresses are subsequently used in the momentum equation to update the material velocities, completing the cycle. A strain rate dependent generalized yield surface material model with temperature softening was developed and implemented as a user material (UMAT) subroutine into the explicit finite element code LS-DYNA. Within the framework of rate independent plasticity, an elasto-plastic constitutive model with an efficient stress update algorithm was developed based on a semi-implicit integration (direction of the plastic flow is assumed to be coaxial with the elastic trial stress. Consequently, the stress at the end of the time step is in the same direction). The convexity requirement and associated law is a sufficient condition for the numerical computation to be stable. Assuming the isotropy of elastic deformation and the additive elastic-plastic decomposition of strain rate tensor based on the small strain increment relative to the yield strain radius of material, the problem of implicitly integrating the stresses is reduced to solving a nonlinear scalar equation which is a function of a plastic multiplier. This equation is solved using a secant method. In the solution step, all of the constitutive variables at the state as well as the total strain at the state are available, and the constitutive variables are updated at with the return mapping algorithm. The return mapping algorithm consists of an elastic predictor and a plastic correction, the elastic predictor is assumed to be completely elastic. Figure 4-3 presents input and output variables of the return mapping algorithm.

109

Figure 4-3: The role of elasto-plastic return-mapping algorithm.

Figure 4-4 shows the relationship between LS-DYNA and GYS subroutine and return mapping algorithm for the GYS.

Figure 4-4: Schematic illustration of return mapping algorithm for the GYS.

The main steps in the return mapping algorithm with semi-backward Euler (semi- implicit) integration are given as follows:

110 1. Initializing: the initial values of plastic strain and internal variables are set to the last converged values from previous time step. 2. Elastic prediction 3. Checking the yielding condition 4. Calculating the incremental plastic multiplier 5. Computing the incremental stress and internal variables 6. Updating the stress and internal variables

Assume that the stepping up from time to is done using time increment ∆ = . At the beginning of time step at , we know the stress and strain tensor. Also the total strain− at is known since the total strain increment for is provided by the finite element code. Therefore known values are: ∆

, , ∆, = + ∆ (4.27) , , ̅ , Using the total strain increment tensor corresponding to time increment , the increment in pressure is given by ∆ ∆

(4.28) ∆ = − ∆ The correspondingly hydrostatic increment is

∆ = −∆ = ∆ (4.29) = ∆ +

The hydrostatic stress tensor of elastic trial state is equivalent to the hydrostatic stress tensor at the end of the time increment at , since no volumetric strain occurs during plastic deformation : ∆

(4.30) =

The deviatoric strain increment is given by:

111

1 (4.31) ∆ = ∆ − ∆ 3 Then the trial stress tensor and the deviatoric trial stress tensor are given by:

= + 2∆ (4.32) = + = + ∆ + 2∆ Correspondingly the von Mises trial stress and trial Lode parameter is

(4.33) 3 = : 2

(4.34) = The yield surface is considered pressure independent and can be written solely in terms of the deviatoric stress tensor. Assuming at time step n that all variables are known, the objective is to compute stresses and strains for time step n+ 1. If the effective stress of the GYS model at step n+ 1, is inside the yield surface, the response is elastic and no return mapping is needed then trial values becomes final values at the end of the time step. When the goes beyond the yield surface, it is usually called trial stress and needs to be corrected back to the yield surface. The integration operator is composed of an elastic predictor and plastic correction step. In the elastic predictor, the incremental total strain vector, , is imposed while keeping internal variables constant and equal to their respective values∆ at . These conditions lead to an elastic trial state defined by the trial stress which is iteratively corrected by the plastic correction as shown in Figure 4-5. The direction of the plastic flow is assumed to be coaxial with the elastic trial stress. Consequently, the stress at the end of the time step is in the same direction with the elastic trial stress. This leads to a coaxial return in deviatoric stress space. In the case of von Mises plasticity, this return is radial in deviatoric space but in the case of tension-compression asymmetry, or shear to tension yield ratio different than 1/ √3 then

112 return is not radial since in these cases the yield surface is no longer a sphere in deviatoric stress space. The time increment is achieved through a central difference scheme and the incremental algorithm given by equation 4-35 was set up for the plastic correction term.

Figure 4-5: Stress update scheme using trial stress and plastic correction.

1 = − 3 = + 2 ∆ 3 = : 2 = det (4.35) 27 729 = + + 2 4 ≤ ⇒ ∆λ = 0 ⇒ = > ⇒ ∆λ > 0 ⇒ ∆λ = 0 ⇒ = − 2∆λ = − 2∆λ

113

Solving for the yield function at time a non-linear equation (considering nonlinear hardening) is derived and solved for =the increment of plastic multiplier (note that increment of plastic multiplier is equal to effective plastic strain increment =∆λ ) ∆λ ∆̅ as

∆λ ∆λ = λ − λ , , ∆ ∆λ ∆λ + ∆λ = λ + ∆λ − λ + ∆λ, , + ∆ 2 (4.36) 27 729 = + + 2 4 3 = det = : 2 where, is the current temperature, is the specific heat at constant pressure (assumed constant during the numerical simulation), is the Taylor-Quinney coefficient that represents the proportion of plastic work converted into heat, which is taken to be 0.9 in the current study. The incremental temperature increase caused by dissipation of plastic work due to adiabatic heating is calculated to incorporate temperature softening. Figure 4-6 (a) illustrates the geometric interpretation of the elastic predictor-return mapping radial-return algorithm in the principle stress space for von Mises plasticity, where the return to the yield surface is radial. In the GYS model, this return is coaxial with the trial state (see Figure 4-6 (b)), but not radial since yield surface is not a cylinder in 3-d principal stress space or a sphere in deviatoric stress space when tension-compression asymmetries exist or the shear to tension yield ratio is not fixed to 1/ √3. In the case of von Mises plasticity with radial return within the time interval, the normal to the yield surface doesn’t change, thus the number of steps used in the time interval does not have an impact on the accuracy of the calculation of the normal vector in the principal stress space. In the case of

114 GYS, the yield surface is not spherical in deviatoric stress space or not cylindrical in principal stress space therefore the return is not radial but the return direction still stays the same as that defined by the trial state at the beginning (see fig 4-6 (b)). It does not change during iterations within the time interval.

σ n+1 Prandtl-Reuss Surface vm , e 3 p N n+1 Old Yield Surface R = σ n = σ n σ σ n+1 y vm 3 vm 3G ∆ε p n Trial Yield Surface σvm σ n+1σ n+1 R = y,e = vm ,e Updated Yield Surface σ n+1σ n+1 R = y = vm

π -Plane Deviatoric Plane (σ+σ+σ=0) 1 2 3 σ 1

is x A 3 c σ i t =

a t 2 s σ

o = r d y σ 1 H (a)

(b)

Figure 4-6: (a) Geometric interpretation of the radial-return in principle stress space (b) GYS return mapping where mapping direction is the yield surface normal at the trial stress point B.

115 Figure 4-7 shows a clear interpretation of the return algorithm on the invariant plane of deviatoric stress versus pressure, where it is shown how the pressure remains unaffected by the GYS plasticity.

Figure 4-7: Geometric interpretation of the return on the invariant plane.

The high accuracy and efficiency of the selected return method lies within its simplicity which is reduced to finding the root of the nonlinear implicit scalar equation iteratively (see Figure 4-9 for more details). This root is the increment in plastic multiplier which corresponds to the increment in effective plastic strain. The scalar function might have more than one root some of which are negative (a solution with ∆λ <0). Since ∆λ is required to be positive, such a solution is physically meaningless and must be disregarded by the root finding method. Therefore estimating the first guess for the root of the function is very important. From the physical point of view, ∆λ can range from zero to large value of ∆λ. In the GYS model, secant method is used to find the correct root. Solution interval is bounded in the secant method using two initial guesses, and large value of (corresponds to Δ perfect plasticity). Note that nonlinear scalar function is monotonically Δdecreasing function

since the derivative of the function with respect to incremental plastic multiplier, , is ∆ negative (See Appendix-B for details) therefore it has only one positive root in the solution interval. Next, knowing the solution interval, and using the secant root finding method the correct root is estimated. The only assumption made by the algorithm is that the plastic flow direction is the same for the trial state and updated state. No iteration is done for the flow

116 direction and return to yield surface is coaxial with trial state. This assumption is valid for a small time step (small strain increment) relative to the yield strain radius of material. Considering the explicit finite element codes where a time step is limited by Courant law, the time steps are small enough (less than a microsecond) and this assumption is valid. The plastic strain increment (note that = ) in Equation (4.20) is solved through a ∆λ ∆̅ secant method by using the following initial values and iterative scheme [71] [72]:

(4.37) ∆λ ∆λ = λ − λ , , = 0 ∆

∆λ = 0 − ∆λ = 3 ∆ − ∆ (4.38) ∆λ = ∆ − ∆ ∆ − ∆ ⋮ ∆ − ∆ ∆λ = ∆ − ∆ ∆ − ∆ ∆λ − ∆λ ≤ = 10 ⇒ Representation of successive secant iterations are illustrated in Figures 4-8 and 4-9.

Figure 4-8: Schematics of the solution interval and successive secant iterations to find the increment in plastic strain assuming that root of the function (f=0) is found at 4 th iteration step [71].

117 The governing yield function f is a monotonically decreasing function as shown in Figure 4-9 [71]. This property of the function is very important for the practical implementation of a robust computational procedure. Namely starting with we have By assuming . a large value of to obtain the condition, ∆λwe can= 0proceed by >secant0 method to calculate which ∆λ is simply root finding < 0 for which within a tolerance. ∆λ = 0

(a)

(b)

Figure 4-9: Successive secant iterations and dependence of function f on the increment of equivalent plastic strain. Note that curvature of the function f could be both ways (a) and (b) without effecting monotonic decrease of the function [70][71].

Estimating the first guess for a root of the f unction is very important. The r oot should be in the solution interval covered by the first two initial value s of the secant method. In the

118 LS-DYNA implementation, two options are implemented for the numerical solution using secant iterations. These are SFIEPM and NITER. SFIEPM scales the solution interval determined by the two initial guess for plastic multiplier. NITER increases the number of secant iterations to be performed in the solution interval. SFIEPM scales the solution interval determined by the two initial guesses for plastic multiplier and NITER increases the number of secant iterations to be performed in the solution interval. Plastic deformation in metals can be strain rate dependent. A proper visco-plastic consideration of the rate effects is therefore important in the numerical treatment of the material law. von Mises plasticity assumes that strain rate sensitivity in all stress states is the same as in tension. But this may not be the case for some metals. Thus material tests at different strain rates for uni-axial tension, uni-axial compression and pure shear states have to be performed. If dynamic tests are available, the load curves defining the yield stress in uni-axial tension, uni-axial compression and pure shear are simply defined by the table definition in GYS material model. This table contains multiple load curves corresponding to different values of the plastic strain rate as illustrated in Figure 4-10. Input data for the table look up are the current uni-axial plastic strain and the current uni-axial plastic strain rate, the table look up delivers the yield value and the tangents with respect to plastic strain and plastic strain rate.

Figure 4-10: Rate dependent tensile hardening curve from dynamic tensile test.

The generalized yield surface model has been implemented into LS-DYNA explicit dynamic software as a material law called MAT224_GYS. In the LS-DYNA

119 implementation, a plasticity algorithm of the GYS model is coupled with the previously developed stress state dependent failure model (MAT224). Developing the failure model is not the focus of this work, and more information regarding failure models can be found at [73][74][75]. Prediction of failure requires a precise knowledge of the state of stress and strain in the component at every stage of the loading process. An accurate elasto-plastic material law is thus a prerequisite for a failure model to function adequately, as the states of stress and strain constitute the input to a failure model. In the MAT224_GYS model, the elasto-plastic material law and its function is to provide an accurate strain and stress field based on the stress state dependent plasticity for the stress-state-dependent failure model MAT224 as shown in Figure 4-11.

GYS

MAT224

MAT224_GYS

Figure 4-11: Coupling between GYS (plasticity model) and MAT224 (failure model).

The GYS material law is only available for solid elements. In LS-DYNA, material law MAT224 and MAT224_GYS use the same failure model but MAT224 uses classical plasticity (von Mises Plasticity) and MAT224_GYS uses a and dependent plasticity model. The input card for the MAT224_GYS model is given at Table 4-1.

120

Table 4-1: MAT224_GYS input card in LS-DYNA

RO: density E: Young Modulus PR: Poison ratio CP: Specific Heat, TR: Room temperature, BETA: Taylor–Quinney coefficient NITER: number of secant iterations to be performed IFLAG: if IFLAG=0, the compressive and shear yields are given as a function of plastic strain and if IFLAG=1, the compressive and shear yields are given as a function of effective plastic strain ATWEAK: scale factor on the initial estimate of the plastic multiplier LCK1: rate dependent table of load curves giving tensile yield stress as a function of plastic strain or effective plastic strain LCKT: temperature dependent table of load curves giving tensile yield stress as a function of plastic strain or effective plastic strain LCCT: temperature dependent table of load curves giving compressive yield stress as a function of plastic strain or effective plastic strain LCCR: rate dependent table of load curves giving compressive yield stress as a function of plastic strain or effective plastic strain LCCT: temperature dependent table of load curves giving compressive yield stress as a function of plastic strain or effective plastic strain LCSR: rate dependent table of load curves giving shear yield stress as a function of plastic strain or effective plastic strain LCST: temperature dependent table of load curves giving shear yield stress as a function of plastic strain or effective plastic strain

121 4.3 Verification of numerical results using single element test cases at different stress states. Single element FE models are used to test the stability, accuracy and robustness of the GYS model against the von Mises plasticity based material model. In the first part of this section, the new material model is reduced to behave similar to von Mises plasticity. The section proceeds with additional single element verification tests by incorporating stress states that have Lode-angle-parameter dependency, showing that significant improvement can be achieved by using GYS.

4.3.1 Single element analysis and verification of GYS material model

4.3.1.1 Benchmark-problem definition

Fully integrated, hexahedral finite elements with equal lengths are used for the analysis at different states of stress. Uni-axial tension, bi-axial tension, uni-axial compression, bi-axial compression, pure shear and plain strain tension and plain strain compression cases are selected for the first benchmark as illustrated in Figure 4-12. A displacement boundary condition, which provides a constant velocity, is defined for the nodes on the top surface of the finite elements. In LS-DYNA, material model MAT24 was selected to compare the results with GYS considering that MAT24 uses von Mises plasticity.

Figure 4-12: Single element models at different state of stresses.

122 The same generic rate independent stress-strain curves were used for both material models. In the GYS material model, as an input the uni-axial compression curve is the same as the tension curve, and the shear curve is 1/ √3 times uni-axial tension stress. This input set for GYS causes should match the von Mises results. The results are then compared with MAT24 model (see Figure 4-17), and it was shown that for a given von Mises tensile input curve, the GYS model produced accurate curves for tension, compression and shear. The two curves are compared and the factors are exactly the factors used in von Mises (see Figure 4- 12 through 4-17).

Figure 4-13: GYS-Uniaxial tension case. Axial stress (Z stress) versus Effective stress (von Mises).

Figure 4-14: GYS-Pure-shear case. Shear stress (YZ stress) versus Effective stress (von Mises).

123

Figure 4-15: GYS-Uniaxial compression case. Axial stress (Z stress) versus Effective stress (von Mises).

Figure 4-16: GYS-Plain strain tension case. Axial stress (Z stress) versus Effective stress (von Mises).

MAT24 and MAT224-GYS output are overlayed in figure 4-17 to show the exact match for each case.

124

Figure 4-17: GYS-Mat24 comparison under the von Mises plasticity assumption for both models.

Consider the generic temperature dependent yield curve in Figure 4-18, the effect of GYS on the response of single element test case for uni-axial tension and uni-axial-compression states have been studied and the stress state dependent temperature softening was addressed by the GYS model and results were compared to von Mises plasticity.

Figure 4-18: Temperature Dependent Yield Curves for Temperature Softening Case.

125 As shown in Figure 4-19 both models have the same response in uni-axial tension as expected.

Figure 4-19: Temperature Softening–Tension Single Element Case (von Mises Plasticity versus GYS).

As shown in Figure 4-20, in the case of uni-axial compression, the temperature softening effect has a different impact on effective stress of the GYS model compared to von Mises plasticity since GYS uses a different temperature curve in compression than tension as shown in Figure 4-18. Note that the von Mises plasticity model assumes that the temperature dependent tension curve is equal to the temperature dependent compression curve (see Figure 4-21 and 4-22 for temperature increase during the uni-axial compression and uni-axial tension loading). This extra capability of the the GYS model is advantageous for the simulating stress state dependent softening.

126

Figure 4-20: Temperature Softening - Compression Single element Case ( von Mises Plasticity versus GYS).

Figure 4-21: Temperature Softening – Temperature on Compression Single element Case (von Mises Plasticity versus GYS).

127

Figure 4-22: Temperature Softening–Temperature on Tension Single element Case (von Mises Plasticity versus GYS).

Similarly, the same type of single element tests have been run to show the effect of stress state on the strain rate dependency and contribution of GYS model compared to von Mises plasticity on this matter. Figure 4-18 shows the generic strain rate dependent yield curves for uni-axial tension and uni-axial compression states corresponding to strain rate of 1/s and 100/s.

Figure 4-23: Strain rate dependent yield curves for single element strain rate dependency case.

128 As presented in Figure 4-24, both models have same response at 1/s at uni-axial compression since tension and compression states have the same behavior at 1/s. On the other hand, GYS has higher flow stress at a strain rate of 100/s compared to von Mises plasticity as shown in Figure 4-25. This shows the effect of stress state on strain rate dependency at 100/s and which was predicted by the GYS model. von Mises plasticity underestimates the flow stress at this rate because of the assumption made by J 2 plasticity such that compression response is the same as tension and material has lower flow stress in tension compared to compression at 100/s as shown in Figure 4.23.

Figure 4-24: Strain Rate Single Element Test in compression [strain rate=1/sn].

Figure 4-25: Strain Rate Single Element Test in compression [strain rate=100/sn].

129 Numerical algorithms used for the stress integration in strain driven problem formulations utilize displacement based finite element formulations. Considering inelastic material deformations, the task is to calculate the stresses and inelastic strain at the end of the time step with a known stress and strain state at the start of the time step for given strain increments. Stress integration is a key step in inelastic incremental analysis. These calculations must be performed using an algorithm which is robust, accurate and efficient. Robustness of the algorithm means it provides a solution for stresses under any boundary conditions and stress state without experiencing any numerical instability. The solution of a nonlinear structural problem is obtained by incrementing strain in each time step ∆t. In general, there is going to be some numerical error in the stress calculations as a consequence of the numerical approximations used in the stress evaluations. The algorithm should provide reasonable accuracy for increments in strain and error should rapidly diminish as the strain increments are decreased. Therefore, an important factor affecting the accuracy of the result is the number of steps or the size of the strain increment used in the stress integration algorithm. In an implicit code, where larger strain increments can be used, the semi-implicit integration algorithms may not perform satisfactorily. However, in an explicit finite element analysis, where small step sizes are used, to ensure convergence and numerical stability, good accuracy could be expected. In Figure 4-26, effect of the time step and corresponding strain increment on the return mapping response is shown.

Figure 4-26: Stress increment and return mapping using a plastic strain direction estimated from the beginning of the time step for: a) a small strain increment and b) a strain increment more than an order of magnitude larger than the yield strain [60].

130 The stress increment and return mapping in figure 4-26 (a) is preferred where a plastic strain direction estimated from the beginning of the time step for a small strain increment (small time step) results in a small percentage change in stress level. In figure 4-26 (b) an exaggerated stress increment that is an order of magnitude larger than stress level (larger time step) is problematic as shown. To show the effect of time step (corresponding strain increment) on the integration method of GYS model, single element test cases at uni-axial compression, uni-axial tension and simple shear stress were run at different time steps allowed by LS-DYNA (see Figures 4- 27 through 4-29). Also non-proportional loading cases were run considering uni-axial tension–biaxial tension, plain strain tension-uniaxial tension stress state loading paths at different time steps (see Figure 4-30 through 4-33). To enforce larger magnitude strain increment at one time step, average strain rate of 1000/s is imposed on the single elements. This rougly corresponds to 20 percent plastic strain in 0.2 milliseconds. Due to high loading rate, inertial effects cause initial ringing in the stress plots for the uni-axial tension and compression. Also same type ringing occurs in the non-proportional loading cases both initially and during the transition phase where the stress state changes suddenly at 0.15 ms (see Figure 4-30 through 4-33). For all the cases, smooth stress response is obtained right after the inertial relaxation of the element. For all the cases, maximum time step corresponds to a strain increment where half of materials yield strain is consumed in a single increment.. In this order of the strain increment, the integration algorithm performs well on the prediction of stresses at different states both for proportional and non-proportional loading cases. There is a noticeable improvement in stability moving to smaller time steps.

131

Figure 4-27: Effect of time step on GYS response-uniaxial compression case.

Figure 4-28: Effect of time step on GYS response-uniaxial tension case.

132

Figure 4-29: Effect of time step on GYS response-simple shear case.

Figure 4-30: Effect of time step on GYS response non-proportional loading case-uniaxial tension-biaxial tension (Axial Stress).

133

Figure 4-31: Effect of time step on GYS response non-proportional loading case-uniaxial tension biaxial tension (Lateral Stress).

Figure 4-32: Effect of time step on GYS response non-proportional loading case-plain strain tension- uniaxial tension (Axial Stress).

134

Figure 4-33: Effect of time step on GYS response non-proportional loading case-plain strain tension- uniaxial tension (Lateral Stress).

135 CHAPTER 5 - Application and validations of GYS plasticity model

This chapter includes numerical simulations of the uni-axial tension, compression, and torsion (pure-shear) tests of Ti-6Al-4V and Al2024-T351 materials. Corresponding numerical simulations were run using LS-DYNA SMP double precision development version on Itanium 64 system. Force-deflection or torque-rotation prediction of GYS and von Mises plasticity models was compared to experimental results. First the quasi-static cases at room temperature were simulated for uni-axial tension, uni-axial compression and torsion (pure-shear) tests. For the influence of stress state on strain rate dependency, Ti-6Al- 4V uni-axial tension and compression tests at 1500/s were simulated using both GYS and von Mises plasticity models and the contribution of the GYS plasticity was shown using force-deflection prediction of these models. Influence of stress state on temperature dependent yield stress and hardening were studied by simulating Al2024-T351 uni-axial compression and tension tests at 150 oC and 300 oC. Finally, this chapter includes the effect of GYS plasticity model on the failure response of the material compared to von Mises plasticity based on single element test runs at different stress states.

5.1 Materials, specimen properties and finite element procedures for the numerical simulation of uni-axial tension and compression and torsion (pure-shear) test cases Ti-6Al-4V is a titanium alloy with aluminum and vanadium as the primary alloying element. Titanium alloys are often used in the aerospace industry because of their high strength to weight ratio, excellent thermal properties, and resistance to corrosion. Ti-6Al-4V accounts for 50% of all titanium alloys sold and is primarily used in aerospace applications. In commercial jet engines, traditionally fan blades and fan disk have been made of Ti-6Al- 4V. Al2024 is an aluminum alloy with copper as the primary alloying element. It is used in applications requiring high strength to weight ratio as well as good fatigue resistance. Al2024-T351 has been extensively used in fabricating aircraft parts such as fuselage skin, wing skin and engine nacelle skin. In flight safety, turbofan engines can pose a rare threat to aircraft safety since uncontained engine debris can cause damage to the fuselage skin, wings or control surfaces of commercial aircraft. Therefore accurate modeling of the plastic

136 deformation and failure of these components are critically important in terms of the safety of the flight [76]. Titanium and aluminum used in aerospace applications can experience various operating temperatures. At altitude, aircraft engine parts outside of the hot section can see subzero temperatures. Typically a blade-out event leads to large material deformation at high strain rates [77]. Adiabatic heating can take place under these conditions and local temperature increases of 200 °C are common at impact locations. A part operating at a temperature of 400 °C can reach local temperatures of 600 °C with adiabatic heating. Therefore, the availability of experimental data at the strain rates, temperatures and stress states present in the application is critical [76]. The material models used to simulate uncontained engine debris events can also be used in simulations of industrial applications to include metal forming and automotive crashworthiness. Since results from numerical simulations are dependent on the material models, they are also dependent on the experimental data used to calibrate these models. Therefore, there is a need for high quality experimental data at the strain rates, temperatures and stress states present in the eventual application of interest. Test data presented in this work has been provided by the Ohio State University Dynamic Mechanics of Materials Laboratory (OSU DMML). All the tests were conducted by OSU DMML. In this test program Digital Image Correlation (DIC) is used to measure surface strains of the specimens. DIC which is described in detail by Sutton, Orteu, and Schreier [78] is an optical method used to measure full field strains of a deforming specimen. This method of measurement provides good spatial resolution on the surface of a deforming body. Test specimens are all fabricated from 12.7 mm thick plate. All specimens are fabricated with the loading direction parallel to the rolled direction shown in Figure 5.1. The reported chemical compositions of Al2024-T351 and Ti-6Al-4V plate stocks that were used for the manufacturing of the specimens are presented in figure 5-2 and 5-3 respectively.

137

Figure 5-1: Al2024-T351 and Ti-6Al-4V specimen manufacturing orientations for the 12.7mm plate stock.

Figure 5-2: Reported composition of 12.7 mm thick Al2024-T351 aluminum plate [16].

Figure 5-3: Reported composition of 12.7 mm thick Ti-6Al-4V plate [76].

Quasi-static uni-axial tension, uni-axial compression, and torsion tests were conducted using the Instron 1321 bi-axial servo-hydraulic load frame. Elevated temperature tests were performed using a furnace attached to the hydraulic load frame. High strain rate tests were done using a tension and compression Split-Hopkinson bar apparatus (see Figures 5-5 and 5- 6).

138

Figure 5-4: (a) Instron 1321 servo-hydraulic biaxial load frame located in OSU DMML (b) Elevated temperature test set up.

Figure 5-5: Tension and torsion Split-Hopkinson bar located in OSU DMML.

Figure 5-6: Compression Split-Hopkinson bar located in OSU DMML.

139 Uni-axial tension tests were conducted on thin flat dog-bone specimens. Dimensions of a dog-bone specimen are shown in figure 5-7. Compression tests were conducted on a solid cylindrical specimen. Diameter and length of the compression specimen are both 3.81 mm. The length to diameter ratio of the compression specimens is kept close to one to limit buckling during testing. If buckling occurs, a complex stress state exists and the test is no longer uni-axial.

Figure 5-7: Dimensions of a thin flat dog-bone specimen for uni-axial tension test.

In the combined loading specimen, a pure shear state is obtained when a torsional moment is applied without axial force on the specimen. Therefore the combined loading specimen was used as the torsion test case for both materials. The geometry and specimen dimensions for the torsion case are presented in figure 5-8. In the torsion case, the only difference between Al2024-T351 and Ti-6Al-4V specimen is the gage length. Al2024-T351 specimen has 6.35 mm gage length whereas Ti-6Al-4V specimen has 3.18 mm gage length.

Figure 5-8: Geometry and specimen dimensions for torsion test case.

140 In tension and compression tests, engineering stress and strain data is converted to true stress and true strain data using the following formulas:

l (5.1) = n 1 +

(5.2) = 1 + where and are true strain and stress, respectively, and and are engineering strain and stress. These formulas make an assumption that volume is constant and deformation takes place uniformly within the specimen. These assumptions are no longer valid at the onset of the necking localization in the tension case therefore care must be taken when interpreting the true stress versus true strain response at large strains.

In the torsion test case, shear stress ( τ) and shear strain ( γ) data is converted from torque and rotation data using following formulas:

(5.3) =

(5.4) = where T is the torque, rm, is the mean gage radius, J is the polar moment of inertia, and Ls is the gage length. The need for a stress state dependent plasticity model is a consequence of experimental results obtained at OSU DMML during Al2024-T351 and Ti-6Al-4V test programs. Based on these results, figure 5-9 illustrates that there is variance between the tension and compression yield stress of Al2024-T351 material during hardening under different temperatures. The classical von Mises plasticity ( plasticity) cannot account for this variance, since calibration of the yield function parameters can be done using only one of either the tension, compression or shear data. Tension data is typically used in von Mises.

141

Figure 5-9: Tension (a) and Compression (b) temperature test series data for Al2024-T351 from OSU DMML [16].

142 Figure 5-11 and 5-12 show temperature and strain rate dependent yield stress data for Ti- 6Al-4V. Both figures illustrate that there is strong stress state dependency for both temperature and strain rate dependent yield stress data for Ti-6Al-4V for the range tested. Therefore the stress state variance on yield stress cannot be simulated using materials models based on classical von Mises or plasticity.

Figure 5-10: Tension (a) and Compression (b) strain rate series data for Al2024-T351 from OSU DMML [16].

143 Figure 5-11 and 5-12 show temperature and strain rate dependent yield stress data for Ti- 6Al-4V. Both figures illustrates that there is strong stress state dependency for both temperature and strain rate dependent yield stress data for Ti-6Al-4V for the range tested . Therefore the stress state variance on yield stress cannot be simulated using materials models based on classical von Mises or plasticity.

Figure 5-11: Tension (a) and Compression (b) temperature test series data for Ti-6Al-4V from OSU DMML [76].

144 (a)

(b)

Figure 5-12: Tension (a) and Compression (b) strain rate test series data for Ti-6Al-4V from OSU DMML [76].

5.2 Experimental results versus numerical simulations for uni-axial tension, compression and torsion (pure-shear) tests. This section includes the finite element simulation of uni-axial tests including tension, compression and torsion (pure-shear). At first, strain rate and temperature effects are not considered, test cases were simulated at room temperature and a strain rate of 1.0 s -1 for both Ti-6Al-4V and Al2024-T351 materials. All simulations were performed using both GYS and

145 von Mises plasticity models and results were compared to the experimental data to be able to show the contribution of GYS model in terms of stress state dependency of yield stress and hardening of material. In all simulation cases, the von Mises plasticity model yield stress data is calibrated data based solely on a tension test and this yield stress data is not changed from one test to another to match the experimental results. Therefore all simulation cases using the von Mises material model have exactly the same yield stress versus equivalent plastic strain input that is characterized based on the tension stress state. Failure strains are not included in these simulations to be able to address the influence of the plasticity models rather than failure models. To show the effect of stress state on temperature dependent yield stress and hardening of Al2024-T351, tension and compression tests at 150 °C and 300 °C and at a strain rate of 1.0 s -1 were simulated using GYS and von Mises plasticity models and results were compared to the experimental data collected at these temperatures. The effect of stress state on strain rate dependency of Ti-6Al-4V was shown using simulation of tension and compression tests at 1500 s-1.

Selected experiments from those listed above are simulated using the explicit finite element code LS-DYNA using the GYS and von Mises plasticity models, and results are compared to experimental data in terms of force-deflection or torque-rotation output.

Three dimensional, eight-node, constant stress, solid elements are used in all simulations. The simulation geometry and setup are illustrated in Figure 5-13. Figure 5-13 (a) presents the tension simulation mesh and setup. There are five elements through the thickness of the tensile specimen. In the specimen gage section, elements have an aspect ratio of 1.0. The load is applied to the specimen by constraining one end and applying a displacement history, u(t) at constant velocity to the other, closely mimicking the actual mechanical experiment. Figure 5-13 (b) presents the compression simulation mesh and setup. There are twenty elements across both the length and diameter of the compression specimen. A translating, rigid platen with displacement history, u(t), is used to compress the specimen against a constrained, rigid platen. Friction between the specimen and the platens is considered in the simulation, using a static friction coefficient of 0.01. The torsion test simulation setup is illustrated in fig 5-14. There are five elements through the gage thickness of the torsion specimen. In the specimen gage section, these elements have an aspect ratio of nearly 1.0.

146 Torque is applied to the specimen by constraining one end and applying a rotation history, θ(t) at constant rotational velocity to the other.

Figure 5-13: Finite element simulation model and set up: (a) Tension simulation and (b) Compression simulation

Figure 5-14: Finite element simulation model and set up: Torsion simulation.

Simulations are conducted using yield stress versus equivalent plastic strain input curves. Three yield curves based on tension, compression and pure shear stress states are provided for GYS material model input. For von Mises material model input, only one yield curve

147 based on the tension stress state is used. The results of these simulations are compared to actual test data in terms of force versus deflection and torque versus rotation curves and these curves are extracted from the simulation results using the method below. For tension simulations, strain is measured using a “numerical extensometer” that parallels the actual extensometer record from the experiment. The history of the length between two nodes on the surface of the specimen in the gage section is recorded. The initial length between these two nodal points is nearly identical to the gage length of the extensometer used for the experiments. Force is extracted across a cross-sectional plane normal to the direction of loading. This record parallels that of the force recorded by the load cell in the actual experiment. For compression simulations, strains are measured using a “numerical LVDT” record, which tracks the length between the moving compression platen and the stationary platen. This directly parallels the LVDT measurement from the fixture described in test. The contact force between the specimen and the stationary platen provides a direct comparison to the experimental load cell record. In torsion simulations, rotation of the top section of specimen is conducted using rigid links which have a master node at the center of the cross section and are connected to other nodes of the top section. Simulated torque is extracted across a cross-section plane normal to the axis of symmetry of the specimen. This provides parallel data to that recorded by the torque cell in the actual experiment. Figure 5-15 and 5-17 show uni-axial tension and compression test at room temperature versus numerical results including GYS and von Mises plasticity models. Both numerical models match the test data for uni-axial tension test as shown in figure 5-15. Also figure 5-16 illustrates that the maximum principal strain contours in simulation and DIC strain measurement agree well. On the other hand, von Mises plasticity underestimates the compression test results while GYS matches test data as shown in figure 5-17.

148

Figure 5.15: Al2024-T351 uni-axial tension test and simulation at room temperature.

Figure 5.16: Maximum principal strain contours of Al2024-T351 tension test specimen at room temperature (a) simulation using GYS (b) Test (DIC).

149

Figure 5-17: Al2024-T351 uni-axial compression test and simulation at room temperature.

Figure 5-18 and 5-19 show the uniaxial tension and compression test at 150 °C versus numerical results including GYS and von Mises plasticity models. Both numerical models match the test data for the uni-axial tension test as shown in figure 5-18. On the other hand, von Mises plasticity overestimates the compression test results until 0.2mm displacement and then underestimates between 0.2 and 0.8 mm displacement while GYS matches test data as shown in figure 5-19.

Figure 5-18: Al2024-T351 uni-axial tension test and simulation at 150 °C.

150

Figure 5-19: Al2024-T351 uni-axial compression test and simulation at 150 °C.

Figure 5-20 and 5-21 show uniaxial tension and compression test at 300 °C versus numerical results including GYS and von Mises plasticity models. Both numerical models match the test data for the uniaxial tension test as shown in figure 5-20. On the other hand, von Mises plasticity overestimates the compression test by about 10 percent while GYS matches the test data as shown in figure 5-21. The largest difference between test results and von Mises plasticity occurs at this temperature which is important for impact problems where localized temperature softening occurs.

Figure 5-20: Al2024-T351 uni-axial tension test and simulation at 300 °C.

151

Figure 5-21: Al2024-T351 uniaxial compression test and simulation at 300 °C.

Figure 5-22 shows the torsion test curve at room temperature versus numerical results including GYS and von Mises plasticity models. In this case, the von Mises plasticity model overestimates the torsion test by approximately 5 percent while GYS almost matches test data as shown in figure 5-22. The small difference between test results and GYS can be due to the non-isotropic nature of the Al2024-T351 torsion specimen. Figure 5-23 presents the maximum principal strain distribution of a specimen in torsion simulation and a test case which are in good correlation.

Figure 5-22: Al2024-T351 torsion test at room temperature.

152

Figure 5-23: Al2024-T351 torsion test at room temperature (a) simulation using GYS (b) Test (DIC data).

Figure 5-24 and 5-27 show the Ti-6Al-4V uniaxial tension and compression tests at a strain rate of 1 s -1 (quasi-static) at room temperature versus simulation results using GYS and von Mises plasticity models. Both numerical models match the test data for the uni-axial tension test as shown in figure 5-24. Also the maximum principal strain and axial strain distribution on a tension specimen are presented in figure 5-25 and 5-26 and it shows that simulation agrees well with DIC test data. On the other hand, von Mises plasticity underestimates the compression test result by approximately 8 percent while GYS matches the test data as shown in figure 5-27.

Figure 5-24: Ti-6Al-4V uniaxial tension test and simulation at strain rate of 1/s.

153

Figure 5-25: Ti-6Al-4V uni-axial tension test and simulation at strain rate of 1/s (a) simulation using GYS (b) Test (DIC data).

Figure 5-26: Ti-6Al-4V uni-axial tension test and simulation at strain rate of 1/s (a) simulation using GYS (b) Test (DIC data).

Figure 5-27: Ti-6Al-4V uniaxial compression test and simulation at strain rate of 1/s.

154 These numerical results related with GYS and von Mises models predict similar trends for both Al2024-T351 and Ti-6Al-4V uni-axial tension and uni-axial compression cases at room temperature. However figure 5-28 shows a difference between GYS and DIC test data on the axial strain distribution of a Ti-6Al-4V compression specimen. In the simulation, deformation is localized in the diagonal bands. Similar band is observed in the DIC test data but strain field is inhomogeneous compared to simulation.

Figure 5-28: Axial strain distribution of Ti-6Al-4V uni-axial compression test and simulation at strain rate of 1/s (a) simulation using GYS (b) Test (DIC data).

Figure 5-29 shows a Ti-6Al-4V torsion test at room temperature versus simulation results using GYS and von Mises plasticity models. In this case, von Mises plasticity model overestimates the torsion test by approximately 20 percent while GYS overestimates the test data by approximately 10 percent. As presented in figure 30, the shear strain distribution on a torsion specimen in simulation roughly match the DIC test data but shear strain distrubution is stronly non-uniform circumferentially compared to simulation. Furthermore, maximum principal strain distribution is also circumferentially non-uniform in the test result compared to simulation as shown in figure 5-31. For a specimen cut from anisotropic Ti-6Al-4V metal sheet, for example shear deformations are no longer uniform around the circumference (because the shear moduli and plastic behaviour are different in the rolling, through- thickness and tranverse directions). GYS still provides a 10 percent improvement compared to von Mises plasticity in terms of torque prediction but this test result supports the need for

155 an anisotropic or orthotropic plasticity model for better correlation between the test and simulation for Ti-6Al-4V material.

Figure 5-29: Ti-6Al-4V torsion test and simulation at strain rate of 1/s.

Figure 5-30: Shear strain distribution on Ti-6Al-4V torsion test specimen at strain rate of 1/s and room temperature (a) simulation using GYS (b) Test (DIC data).

Figure 5-31: Maximum principal strain distribution on Ti-6Al-4V torsion test specimen at a strain rate of 1/s and room temperature (a) simulation using GYS (b) Test (DIC data).

156

Figure 5-32 and 5-33 show Ti-6Al-4V uni-axial tension and compression test at a strain rate of 1500 s -1 versus numerical results including GYS and von Mises plasticity models. Both numerical models match the test data in the uni-axial tension test as shown in figure 5- 32. On the other hand, von Mises plasticity overestimates the compression test results by approximately 10 percent while GYS almost matches the compression test data right after the inertial stress wave effect disappears in the test data as shown in figure 5-33. Figure 5-34 shows the axial strain distribution on a tensile specimen and figure 5-35 shows the correlation between simulation and DIC test data.

Figure 5-32: Ti-6Al-4V uniaxial tension test and simulation at strain rate of 1500/s.

Figure 5-33: Ti-6Al-4V uniaxial compression test and simulation at strain rate of 1500/s.

157

Figure 5-34: Axial strain distribution on Ti-6Al-4V tension test specimen at strain rate of 1500/s.

Figure 5-35: Axial strain distribution on Ti-6Al-4V tension specimen (a) simulation using GYS and (b) DIC test data at strain rate of 1500/s.

158 5.3 Influence of the GYS plasticity model on the failure response compared to the von Mises plasticity model

Consideration of yield strength differential in uni-axial tension, uni-axial compression and pure shear stress states for GYS plasticity model leads to differences in numerical failure prediction compared to von Mises plasticity. Single element test cases in tension, compression and pure shear stress states have been run using the same failure surface and differences in failure prediction of GYS and von Mises plasticity have been shown. Von Mises plasticity assumes that yield in tension equals yield in compression and yield in pure shear equals 0.577 times yield in tension. Figure 5-36 and 5-37 show uniaxial tension and compression single element numerical test cases with yield in compression/tension ratio is 0.9 for GYS and von Mises plasticity models. Both numerical models predict similar failure stress at identical time for the uniaxial tension test as shown in figure 5-36. On the other hand, von Mises plasticity overestimates the failure stress by approximately 9 percent and premature failure occurs compared to GYS model since yield in tension is equal to yield in compression for the von Mises model. Figure 5-38 shows the difference in the equivalent plastic strain path to failure for both models. Although equivalent plastic strain to failure is the same, the path to reach failure strain is different due to yielding asymmetry.

Figure 5-36: Axial stress for yield in Compression/Tension=0.9 (uni-axial tension case).

159

Figure 5-37: Axial stress for yield in Compression/Tension=0.9 (uni-axial compression case).

Figure 5-38: Equivalent plastic strain for yield in Compression/Tension=0.9 (uni-axial compression case).

Figure 5-39 shows show the uni-axial compression single element numerical test case with yield in compression/tension ratio is 1.1 for the GYS and von Mises plasticity models. Both numerical models predict similar failure stress at identical time for uniaxial tension test. On the other hand, the von Mises plasticity underestimates the failure stress by approximately 10 percent and post-mature failure occurs compared to GYS model since yield in tension is equal to yield in compression for the von Mises model. Figure 5-40 shows

160 the difference in the equivalent plastic strain path to failure for both models. Although equivalent plastic strain to failure is the same, the path to reach failure strain is different due to yielding asymmetry.

Figure 5-39: Axial stress for yield in Compression/Tension=1.1 (uni-axial compression case).

Figure 5-40: Equivalent Plastic Strain for Yield in Compression/Tension=1.1 (uni-axial compression case).

Figure 5.41 shows the uniaxial compression single element numerical test case with uni- axial compression to uni-axial tension yield curve input ratio varying linearly from 0.9 to 1.1. To obtain variational ratio, uni-axial compression yield curve input is scaled by the constant

161 that linearly changes from 0.9 to 1.1 at each data point of the uni-axial compression yield curve . Then this scaled curve is used as the uni-axial tension yield curve input. Von Mises plasticity underestimates the failure stress and premature failure occurs compared to GYS model since yield in tension is equal to yield in compression for von Mises model. On the other hand, failure stress and failure time is at the point between compression/tension yield ratio 0.9 and 1.1 as shown in figure 5-42.

Figure 5-41: Axial stress for yield ratio Uni-axial Compression/Tension=0.9-1.1 (uni-axial compression case).

Figure 5-42: Effect of the Uni-axial Compression/Tension yield ratio on failure with the GYS uni- axial compression case.

162 Figure 5.43 shows the effect of shear/tension yield ratio on failure using GYS plasticity model. Three different shear/tension yield ratios, 0.545, 0.577 (von Mises) and 0.610 have been tested under pure shear loading using GYS. As expected, failure stress increases with higher shear/yield ratio and earlier failure occurs compared to the von Mises case for the ratio of 0.610. Von Mises predicts post-mature failure in this case. Although equivalent strain to failure is the same for all cases, paths to the failure are different in each case as shown in figure 5.44.

Figure 5-43: Effect of the Pure-Shear/Uni-axial Tension yield ratio on failure using GYS.

Figure 5-44: Effect of the Pure-Shear/Uni-axial Tension yield ratio on failure using GYS.

163 Similar effects of GYS model on failure can be also seen for the case when there is a transition from pure-shear state to uni-axial tension state as presented in figure 5.45 and 5.46.

Figure 5-45: Effect of the Pure-Shear/Uni-axial Tension yield ratio on failure using GYS (Non- proportional loading case Pure Shear-Uniaxial Tension).

Figure 5-46: Effect of the Pure-Shear/Uni-axial Tension yield ratio on failure using GYS. (Non- proportional loading case Pure Shear-Uni-axial Tension).

164 Figure 5.47 and 5.48 present the influence of loading sequence on failure using GYS. Pure-shear, pure-shear-uniaxial tension and uniaxial tension-pure-shear loading sequences have been investigated and different equivalent plastic strain to failure values with different failure times have been obtained as expected using GYS model. Also it can be noted that change in the maximum principal stress during the transition from one stress state to another is addressed well as shown in figure 5.47. This makes it possible to predict failure under the non-proportional loading cases using GYS.

Figure 5-47: Effect of loading sequence on failure using GYS.

Figure 5-48: Effect of loading sequence on failure using GYS.

165 CHAPTER 6 - Concluding remarks and Future Work

A stress state dependent numerical plasticity model with strain rate dependency and temperature softening has been developed. The convexity region of the newly developed GYS model has been compared to other alternative plasticity models and contribution of newly developed material model was shown. The effect of the GYS model on the numerical simulation of uni-axial tension, uni-axial compression and torsion tests has been investigated by comparing GYS and von Mises plasticity results. DIC test data from the Ti-6Al-4V and Al2024-T351 test programs have been compared to simulation results using GYS and von Mises plasticity. Uni-axial tension-compression asymmetry in yielding and pure-shear/uni- axial tension yield stress ratios different than classical von Mises plasticity have been simulated successfully using the GYS model. Also the effect of stress state dependency on strain rate dependent plastic flow and temperature dependent plastic flow have been addressed in the simulation of experiments in the form of force–deflection and torque- rotation curves. Considering Al2024-T351 plastic deformation test data, prediction of the GYS plasticity model has exactly matched the test data for uni-axial tension, uni-axial compression and torsion (pure-shear) tests. The results were compared to von Mises plasticity. At room temperature and 150 °C, the von Mises plasticity model underestimated the compression forces since there is no uni-axial tension-compression asymmetry in yield in classical von Mises plasticity and where the yield stress in uni-axial compression is always equal to uni- axial tension. Considering that the yield stress in uni-axial compression is higher than yield stress in uni-axial tension at these temperatures for Al2024-T351, under prediction by von Mises model is expected. On the other hand at 300 °C, the von Mises plasticity model has overestimated the compressive force since the uni-axial tensile yield stress is higher than uni- axial compressive yield stress in this case. In Al2024-T351 torsion test, von Mises plasticity has over predicted the torque by five percent compared to the GYS and the test data as expected due to fixed yield stress in shear (the von Mises model assumes that yield stress in pure-shear is 0.577 times the yield stress in uni-axial tension). Considering that Al2024-T351 yield stress in pure-shear is less than 0.577 times the yield stress in uni-axial tension, this over prediction is expected.

166 Considering Ti-6Al-4V plastic deformation test data, of the GYS plasticity model predictions are roughly matched the test data for uni-axial tension, uni-axial compression and torsion (pure-shear) tests. Although force-deflection predictions of the GYS model were successful in uni-axial tension and compression tests compared to the von Mises plasticity model and GYS also addressed the tensile-compressive asymmetry in yield stress. However, axial strain distribution is non-homogenus in the DIC test data compared to simulation in the uni-axial compression case. The stress state effect on strain rate dependency of Ti-6Al-4V yield stress has been addressed by the GYS by matching uni-axial tension and compression test forces at a strain rate of 1500/s compared to von Mises plasticity. The maximum principal strain and axial strain distribution matched the DIC test data well in the tension test. Von Mises plasticity underestimated the compressive force since yield stress in uni-axial tension is less than yield stress in uni-axial compression at 1500/s. In Ti-6Al-4V torsion test, both GYS and von Mises plasticity over predicted the torque from the test. GYS made a 10 percent overestimation whereas von Mises made a 20 percent overestimation compared to test. The GYS model is a 10 percent better estimate compared to von Mises. The GYS model improvement comes from flexibility in pure-shear yield stress. But this is also limited by the convexity region of the GYS. Therefore if pure-shear to uni-axial tensile yield ratio is less than 0.544 (minimum ratio possible with GYS model) as it is the case for Ti-6Al-4V (pure- shear to uni-axial tensile yield stress ratio is around 0.5), the GYS contribution is limited. At the same time, shear strain and maximum principal strain distributions show that there is a difference between simulation using GYS and DIC test data. The contribution of the GYS plasticity model compared to J2 dependent plasticity in the von Mises model is promising but also limited by the convexity region. Results show that the GYS model can be used in all applications to address the uni-axial tensile-compressive asymmetry in yielding. Also, yield in pure-shear stress is not fixed and this can provide better estimation in local stress-strain fields where the shear state is dominant. The shear mode is an important failure mode in many impact problems, so the flexibility to cover a range of pure-shear/uni-axial tension yield stress ratios different than 0.577 has a significant effect on simulation results. To be able to address this point, plate impact tests with different projectile geometries should be studied. Stress state dependent failure modes of these tests can be predicted both using GYS and von Mises plasticity. The comparison of the analytical

167 simulations with to the test results is expected to show a solid contribution of the GYS model. Another important assumption of the GYS model is isotropic plasticity. This means that there is no directional dependence in plastic flow no matter what material is used. As it was observed in Ti-6Al-4V tests [76] which was supported by the plate anisotropy tests conducted by the OSU DMML, this assumption is not valid for some materials. Materials with strong anisotropic or orthotropic nature need to be simulated with anisotropic or orthotropic plasticity models [79][80] to be able to match the test data. Therefore development of orthotropic or anisotropic plasticity model is a necessary step for predictive analysis. Presently, this work should be considered as the first step of validation. Since the model is intended to cover dynamic impact and deformation applications, it should be extensively checked with available test data to assess its benefits and limitations. After the verification of stress state, strain rate, and temperature dependent failure surfaces of Al-2024 and Ti-6Al-4V materials, direct comparison of simulation with GYS material model versus tests can be done based on the projectile exit velocity and ballistic limits considering high speed plate impact problems. This should be an essential step to confirm the validity of the model in the high speed impact field [81][82].

168 References

[1] Borvik T., Hopperstad O., Berstad T., and Langseth M., "Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with flat, hemispherical and conical noses, part I: experimental study," International Journal of Impact Engineering , vol. 27, pp. 19-35, 2002. [2] Hallquist J.O., "LS-DYNA Keyword User’s Manual," Livermore Software Technology Corporation (LSTC), Volumes I and II 2009. [3] "Aircraft accident report," National Transportation Safety Board, NTSB/AAR-90/06 1990. [4] "Aircraft accident report," National Transportation Safety Board, NTSB/AAR-98/01 1998. [5] Frankenberger C.E., "Large engine uncontained debris analysis," Federal Aviation Administration, Report #DOT/FAA/AR-99/11, 1999. [6] "Part 33.94 of Title 14 of the Code of Federal Regulations (14 CFR part 33.94, Blade Containment and Rotor Unbalance Tests)," US Government Printing Office, 2009. [7] Wilkins M.L., "Mechanics of penetration and perforation," International Journal of Engineering Science , vol. 16, pp. 793-807, 1978. [8] Zukas J.A., Impact dynamics .: John Wiley & Sons, 1982. [9] Zukas J.A., "Numerical simulation of high rate behavior," in Computational Mechanics Publications - Shock and Impact on Structures , Brebbia C.A. and Sanchez-Galvez V., Eds. Southampton , UK, 1994, pp. 1-26. [10] Johnson G.R. and Cook W.H., "A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures," in 7th International Symposium on Ballistics , Hague, Netherlands, 1983. [11] Johnson G.R. and Cook W.H., "Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures," Engineering Fracture Mechanics , vol. 21, no. 1, pp. 31- 48, 1985. [12] Backman M.E., "Terminal Ballistics," Naval Weapons Center, NWC TP 578 1976. [13] Backman M.E. and Goldsmith W., "The mechanics of penetration of projectiles into targets," International Journal of Engineering Science , vol. 16, pp. 1-99, 1978. [14] Perzyna P., "Fundamental problems in viscoplasticity," Advances in Applied Mechanics , vol. 9, pp. 243–377, 1966. [15] Bai Y. and Wierzbicki T., "A new model of metal plasticity and fracture with pressure and Lode dependence," International Journal of Plasticity , vol. 24, pp. 1071-1096, 2008. [16] Seidt J.D., "Plastic deformation and ductile fracture of 2024-t351 aluminum under various loading conditions," The Ohio State University, PhD Thesis 2010. [17] Lode von W., "Versuche uber den Einfluss der mittleren Hauptspannung auf das Fliessen der Metalle Eisen, Kupfer und Nickell," Zeitschrift fur Physik , vol. 36, pp. 913-939 (In German), 1926. [18] Tresca H., "On the yield of solids at high pressures," Comptes Rendus Academie des Sciences 59:754 (in French),1864. [19] Huber M., "Speciﬁc work of strain as a measure of material e ort," Towarzystwo Politechniczne , Czas. Techniczne, Lwów,1903. ﬀ [20] von Mises R ., "Mechanik der festen Körper im plastisch deformablen Zustand ." Nachr. Math. Phys ., vol. 1, pp. 582–592 ,1913 . [21] Taylor G.I. and Quinney H., "The plastic distortion of metals," Philosophical Transactions of the Royal Society of London, Series A , vol. 230, pp. 323-362, 1932. [22] Mendelson A., Plasticity Theory and applications.: Springer, 1968. [23] Hosford, W. F., "A generalized isotropic yield criterion, " J. Appl. Mech . 39, pp.607-609, 1972. [24] Drucker, D.C., "Relation of experiments to mathematical theories of plasticity," J. Appl. Mech . 16, 349-357, 1949. [25] Hershey A. V., "The plasticity of an isotropic aggregate of anisotropic face centered cubic crystals,"

169 J. Appl Mech Trans ASME , 21: 241–249,1954. [26] Karafillis, A.P., Boyce, M.C., "A general anisotropic yield criterion using bounds and a transformation weighting tensor," J. Mech. Phys. Solids , vol. 41, pp.1859-1886,1993. [27] Cazacu, O., Barlat, F., "Generalization of Drucker's yield criterion to orthotropy." Mathematics and Mechanics of Solids , vol. 6, pp.613-630, 2001. [28] Cazacu O. and Barlat, F., "A criterion for description of anisotropy and yield differential effects in pressure-insensitive metals," International Journal of Plasticity, vol . 20, pp.2027-2045, 2004. [29] Hutchinson, J.W.,Jun S., Journal ofMechanics and Physcicsof solids , vol.12, pp. 25-33, 1964. [30] Hill R., Philosphical Magazine , vol. 41, pp. 733-744, 1950. [31] Vegter H., Drent P., Huetnik J., "A planar isotropic yield criterion based on mechanical testing at multi-axial stress states, " Proc. of Numiform Conf ., pp. 345-350, 1995. [32] Spitzig, W.A., Sober, R.J., Richmond, O., "The effect of hydrostatic pressure on the deformation behavior of Maraging and HY-80 steels and its implication for plasticity theory," Metallurgical Transactions , vol. 7, pp.1703–1710, 1976. [33] Rauch, G.C., Leslie, W.C.,"The extent and nature of the strength differential effects in steels," Metallurgical Transactions, vol. 3, pp.373, 1972. [34] Chait, R., "Factors influencing the strength differential of high strength steels," Metallurgical Transactions , vol. 3, pp. 365, 1972. [35] Spitzig, R.J., Richmond, O., "The effect of pressure on the flow stress of metals,"Acta Metallurgica , vol. 32, pp. 457–463, 1984. [36] Hosford, W. F. and Allen, T. J., "Twining and directional slip as a cause for strength differential effect.," Metallurgical Transactions , vol. 4, pp.1424-1425, 1973. [37] Hosford W.F., Mechanical behavior of materials, Cambridge University Press.: 2005 [38] Raniecki B., Mro´z., Z. " Yield or martensitic phase transformation conditions and dissipation functions for isotropic, pressure-insensitive alloys exhibiting SD effect", Acta Metallurgica , vol. 195, pp.81–102 2008. [39] Iyer, S.K., Lissenden, C.J., "Multiaxial constitutive model accounting for the strength-differential in Inconnel 718," International Journal of Plasticity , vol.19, pp. 2055–2081, 2003. [40] Cazacu O. and Barlat, F., "A criterion for description of anisotropy and yield differential effects in pressure-insensitive metals," International Journal of Plasticity, vol. 20, pp.2027-2045, 2004. [41] Bai Y. and Wierzbicki T., "A new model of metal plasticity and fracture with pressure and Lode dependence," International Journal of Plasticity , vol. 24, pp. 1071-1096, 2008. [42] Mirone, G., Corallo, D.," A local viewpoint for evaluating the influence of stress triaxiality and Lode angle on ductile failure and hardening," International Journal of Plasticity, vol. 26, pp.348– 371, 2010. [43] Gao, X., Zhang, T., Hayden, M., Roe, C., "Effects of the stress state on plasticity and ductile failure of an aluminum 5083 alloy," International Journal of Plasticity, vol. 25, pp.2366–2382, 2009. [44] Gao, X., Zhang, G., Zhou J.,Graham M.S., Hayden M.,Roe, C., "On stress-state dependent plasticity modelling: significance of the hydrostatic stress, the third invariant of stress deviator and the non- associated flow rule, " International Journal of Plasticity , vol.27, pp.217-231, 2010. [45] Drucker, D.C., "A definition of stable inelastic material," Report to the Office of Naval Research. NR -064-424, 1950. [46] Hencky, H., "Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen." ZaMM 4, 323–335,1924. [47] Dunne F., Petrinic N., Introduction to computational plasticity .: Oxford University Press, 2005. [48] Michael E. Nixon, Oana Cazacu , Ricardo A. Lebensohn "Anisotropic response of high-purity a- titanium: Experimental characterization and constitutive modeling," International Journal of Plasticity, vol.26, pp.516–532, 2010.

170 [49] Kuhn H., Tucker A., " Nonlinear programming," Proceedings of 2nd Berkeley Symposium. Berkeley: University of California Press. pp. 481–492, 1951. [50] Bower F.A., Applied Mechanics of Solids .: CRC Press, 2010. [51] Wu Han-Chin, Continuum Mechanics and Plasticity .: CRC Press, 2005. [52] Rockafellar, R.T., Convex Analysis .: Princeton University Press,1972. [53] Hill R., The mathematical theory of plasticity .: Oxford: The University Press, 1950. [54] Bridgman P.W., Studies in large plastic flow and fracture .: NewYork: McGraw-Hill, 1952. [55] Jones N., "Dynamic inelastic failure of structures," Transactions of the Japan Society of Mechanical Engineers , vol. 12, pp. 21-31, 1997. [56] Belytschko T., Liu W.K., and Moran B., Nonlinear finite elements for continua and structures .: John Wiley & Sons, 2000. [57] M. Ortiz and J.C. Simo. An Analysis of a New Class of Integration Algorithms for Elastoplastic Constitutive Relations. International Journal for Numerical Methods in Engineering , vol.23, pp.353- 366, 1986. [58] Kojic´ M.., "Stress integration procedures for inelastic material models within the finite element method." Appl. Mech. Rev ., vol. 55, pp.389–414, 2002. [59] Hughes T.J.R., "Analysis of transient algorithms with particular reference to stability behavior," in Computational methods for transient analysis , Belytschko T. and Hughes T.J.R., Eds. Amsterdam: North-Holland, 1983. [60] Becker, R., "An alternative approach to integrating plasticity relations," Int. J. Plasticity ,vol.27, pp.1224-1238, 2011. [61] Simo J.C., Taylor R.L., A Return Mapping Algorithm for Plane Stress Elastoplasticity, International Journal for Numerical Engineering , vol. 22, pp.649-670, 1986. [62] Krieg R.D. and Krieg D.B., "Accuracies of numerical solution methods for the elastic-perfectly plastic model," Journal of pressure vessel technology , vol. 99, pp. 510-515, 1977. [63] Wilkins M.L., "Calculation of elastic-plastic flow," in Methods of computational physics , Adler B. et al., Ed. New York: Academic Press, 1964, vol. 3, pp. 211-263. [64] Simo J.C. and Hughes T.J.R., Computational Inelasticity .: Springer-Verlag, 1998. [65] Courant R., Friedrichs K., and Lewy H., "On the partial differential equations of mathematical physics," IBM Journal , pp. 215-234, March 1967, English translation and republication of the 1928 German original "Mathematische Annalen 100, 32-74 (1928). [66] Hallquist J.O., "LS-DYNA Theory Manual," Livermore Software Technology Corporation (LSTC), 1993. [67] Akhtar S. Khan., Sujian Huang ., Continuum theory of plasticity.: John Wiley & Sons, 1995. [68] Jaumann, G.: Geschlossenes System physikalischer und chemischer Differentialgesetze. Akad.Wiss. Wien, Sitzber.IIa , pages 385–530, 1911. [69] "ABAQUS Theory Manual Version 6.7," ABAQUS Inc, 2009. [70] Du Bois P.A., "Lecture notes for plasticity in LS-DYNA", 2012. [71] Kojic M.,. and Bathe K.J., Inelastic analysis of solids and structures .: Springer, 2005. [72] Berstad T., Hopperstad O.S., and Langseth M., "Elasto-viscoplastic constitutive models in the explicit finite element code LS-DYNA3D," in 2nd International LS-DYNA3D Conference , St. Francis, 1994, [73] Buyuk M., Kan S., and Loikkanen M.J., "Explicit finite-element analysis of 2024-T3/T351 Aluminum material under impact loading for airplane engine containment and fragment shielding," Journal of Aerospace Engineering , vol. 22, pp. 287-295, 2009. [74] Buyuk M.,"Development of A Tabulated Thermo-Viscoplastic Material Model with Regularized Failure for Dynamic Ductile Failure Prediction of Structures under Impact Loading." The George Washington University, Phd thesis, 2013.

171 [75] Carney K.S., Du Bois P.A., Buyuk M., and S. Kan, "Generalized, three-dimensional definition, description, and derived limits of the triaxial failure of metals," Journal of Aerospace Engineering , vol. 22, pp. 280-286, 2009. [76] Hammer J.T., "Plastic deformation and ductile fracture of Ti-6Al-4V aluminum under various loading conditions," The Ohio State University, MSc Thesis 2012. [77] Carney K., Pereira M., Revilock D., and Matheny P., "Jet engine fan blade containment using an alternate geometry," International Journal of Impact Engineering , vol. 36, pp. 720-728, 2009. [78] Sutton M., Orteu J.J., and Schreier H.W., Image correlation for shape, motion and deformation measurements.: Springer, 2009. [79] Stewart J.B., "Development of yield criteria for describing the behavior of porous metals with tension-compression asymmetry," University of Florida, PhD Thesis 2009. [80] Plunkett B.W., "Plastic anisotropy of hexagonal close packed metals," University of Florida, PhD Thesis 2005. [81] Echavarri, B.E., Flow and fracture behaviour of high performance alloys, PhD Thesis, Universida Politechinca de Madrid, 2012. [82] Chocron S., Erice B., Anderson C., "A new plasticity and failure model for ballistic application," International Journal of Impact Engineering , vol. 38, pp. 755-764, 2011.

172 Appendix-A: Second condition of convexity for the GYS model

Second conditions of convexity has been calculated using MATHEMATICA v8 symbolic calculation toolbox Second condition of convexity is that sum of principal minors of Hessian matrix should be non-negative:

(A.1) + + ≥ 0 = − (A.2) = − = − The GYS stress derivatives are as follows for hessian matrix:

(A.3) 2 2 = + + + 2 + + + 3 3 3 3

(A.4) 2 2 = + + + 2 + + + 3 3 3 3

(A.5) 2 2 = + + + 2 + + + 3 3 3 3

1 2 5 = − + + + − 3 3 3 (A.6) + + − 9 3

173 1 2 5 = − + + + − 3 3 3 (A.7) + + − 9 3

1 2 5 = − + + + − 3 3 3 (A.8) + + − 9 3 Helpful relationships for the principal deviatoric stresses in terms of the deviatoric stress invariants and are as follows: , , (A.9) + + = 0 (A.10) + + = − (A.11) + + = 2 (A.12) + + = 3 (A.13) + + = 2 (A.14) + + = 5 (A.15) + + = 3 + 2 (A.16) + + = (A.17) + + + + + = −3 (A.18) + + + + + = (A.19) +( + + + + = 2 − 3 (A.20) + + + + + = −2 (A.21) + + + + + = −3 Using helpful relationships for the principal deviatoric stresses in terms of the deviatoric stress invariants and following equation is obtined for, the, sum of minors of Hessian matrix for the GYS model: ,

174 2 4 + + = + + 8 + + 12 − 3 − 2 3 3 10 20 − 4 − 10 − 6 − + + 4 3 3 2 8 2 2 − + + + + 2 − 3 9 3 3 3 2 4 4 + 4 + − − − 2 − 2 (A.22) 9 3 3 4 8 8 + 4 + − 2 + + 8 − 3 3 9 40 2 4 4 2 50 − − − + + − 9 3 9 9 9 9 4 20 8 4 8 + − − − − 27 3 3 27 3

4 8 4 8 − − − − + 4 9 9 9 9 Invariant derivatives of the GYS model, are given as follows: , , , , (A.23) / 9 27 √3 = √3 + + / − 2 4 (A.24) √3 9 5 27 √3 = = / − − / 2 2 2 4 (A.25) 9 27 √3 = = + / 2 2 (A.26) √3 9 27 √3 = = − / + + 35 / 4 16 (A.27) 27 √3 = = / 2 (A.28) 9 5 27 √3 = = − − / 2 2 2 (A.29) + + = 0 (A.30) + + ≥ 0

175 Appendix-B: Derivative of GYS yield function with respect to the increment of the plastic strain multiplier

Derivative of the yield function with respect to the increment of the plastic multiplier ( is an important property in terms of finding a correct root in the solution interval. Following∆ equations prove that derivative is negative. This leads to monotonically decreasing function ( ) in the solution interval thus only one positive root exists for the solution of ( ) = 0. ∆ ∆ (B.1) , = − (B.2)

= = − ∆ ∆ ∆ (B.3)

= − 2∆ (B.4)

= (B.5) = = − : 2 − : = ∆ ∆ (B.6) = = − 2 + ∆ ∆

Consider that term is always positive for the strain hardening material therefore ∆ derivative of the yield function with respect to the increment of the plastic multiplier is always negative. This validates that nonlinear scalar function ( ) is monotonically decreasing function for the positive values of ∆ ∆.

176 Appendix-C: Single element verification cases of convexity projection algorithm

The convexity algorithm of the GYS model based on the yield ratio of uni-axial compression to uni-axial tension was presented in chapter 3. Figure B-1 shows the projection principle. In the algorithm, data projection onto convexity boundary is verified by comparing with cases presented in Figure C-2. Details of verification case-2, case-3, case-4 and case-8 are presented below.

Figure C-1: The GYS convexity region and data point projection for non-convex cases.

Figure C-2: Verification cases of projection algorithm for the data set out of convexity region.

177 Figure C-3 shows the case-2 where the yield ratio of uni-axial compression to uni-axial tension is 1.8 which is above the maximum tip point (blue point) of the convexity region. Effective stress-effective plastic strain response for the single element tests in uni- axial tension and uni-axial compression is given by Figure C-4 (a). Due to projection of the data set to the maximum tip point , the updated uni-axial compression to uni-axial tension yield ratio is expected to be 1.693. This response is checked using scale factor of 1.693 on the tension curve as shown in Figure C-4 (b) and exact match with compression curve is obtained.

Figure C-3: Verification case-2 ( σc/ σt =1.8) and projection point on the convexity region.

178 (a)

(b)

Figure C-4: Verification case-2 ( σc/ σt =1.8) (a) tension and compression effective stresses without scale factor (b) scale factor of 1.693 on the tension curve.

Figure C-5 shows case-3 where the ratio of uni-axial compression to uni-axial tension yield curve is 0.5 which is below the minimum tip point (yellow point) of the GYS convexity region. Effective stress-effective plastic strain response for the single element tests in uni- axial tension and uni-axial compression is given by Figure C-6 (a). Due to projection of data set to minimum tip point, updated ratio of uni-axial compression to uni-axial tension yield curve is expected to be 0.59. This response is checked using scale factor of 0.59 on the uni- axial tension curve as shown in Figure C-6 (b) and an exact match with uni-axial compression yield curve is obtained.

179

Figure C-5: Verification case-3 ( σc/ σt =1.8) and projection point on the convexity region.

(a)

180 (b)

Figure C-6: Verification case-3 ( σc/σt =0.5). (a) Uni-axial tension and uni-axial compression effective stresses without scale factor. (b) Scale factor of 0.59 on the uni-axial tension curve.

Figure C-7 shows case-8, where √3 times pure-shear to uni-axial tension yield ratio is 1.3 which is outside the convexity region. Effective stress-effective plastic strain response for the single element tests in uni-axial tension and pure shear is given by Figure C-8 (a). Due to projection of data set, updated √3 times pure-shear to uni-axial tension yield ratio is expected to be 1.0588. This response is checked using scale factor of 1.0588 on the tension curve as shown in Figure C-8(b) and exact match with pure shear curve is obtained.

Figure C-7: Verification case-4 √3σs/ σt =1.3 and projection point on the convexity region.

181 (a)

(b)

Figure C-8: Verification case-4 √3σs/ σt =1.3 (a) Uni-axial tension and pure shear effective stresses without scale factor. (b) Scale factor of 1.0588 on the uni-axial tension curve.

Figure C-9 (a) shows case-8 where yield ratio of compression to tension is varying from 0.3 to 2.29, and 3 times shear to tension yield ratio is 1. It is partially outside the convexity region. The effective stress-effective plastic strain response for the single element tests in uni-axial tension and compression is given by Figure C-9 (b). Projection is done for the ratio below the minimum (0.59) and above the maximum 1.69 points of convexity region (Figure C-9 (c-d)). Between these two ratios no projection is done, and compression curve stays as it is. Due to projection of the data set above the maximum, compression to tension yield ratio is expected to be 1.69. This response is checked using scale factor of 1.69 on the tension curve as shown in figure C-9 (c) and an exact match with compression curve is obtained. Due to projection of data set below the minimum, compression to tension yield ratio is expected to

182 be 0.59. This response is checked using scale factor of 0.59 on the tension curve as shown in Figure C-9 (d) and an exact match with uni-axial compression curve is obtained.

(a)

(b)

(c)

183 (d)

Figure C-9. Verification case-8 σc/ σt =0.3-2.29 for data point projection for non-convex cases. (a) Uni- axial tension and uni-axial compression effective stresses without scale factor. (b) Scale factor of 1.0588 on the uni-axial tension curve. (c) Uni-axial tension and pure shear effective stresses without scale factor. (d) Scale factor of 1.059 on the uni-axial tension curve.

184 Appendix-D: Comparison of MAT224 and MAT224_GYS models under the von Mises plasticity assumption considering proportional (uni-axial tension and plain strain tension) and non-proportional loading (plain strain tension-uni-axial tension) case.

Single element test cases at uni-axial tension, plain strain tension and plain strain tension- uni-axial tension (non-proportional loading) have been run for both MAT224 and MAT224_GYS using the same failure surface under the von MisesMises plasticity assumption. Figure D-1 presents loading cases and evolution of the stress-state values. Effective strain to failure, damage, triaxiality, and effective stress outputs were checked-out and it was shown that MAT224 and MAT224_GYS model outputs match as expected under the von Mises plasticity assumption (see Figure D-2 through D-4).

Free-Free Confined-Confined Confined-Free

Figure D-1. Loading cases and evolution of the stress-state values.

185

Figure D-2: Uni-axial tension case – Plastic strain to failure, damage, triaxiality, effective stress.

Figure D-3: Plain strain tension case – Plastic strain to failure, damage, triaxiality, effective stress.

186

Figure D-4: Non-proportional loading case (Uni-axial tension-plain strain tension) case – Plastic strain to failure, damage, triaxiality, effective stress.

187