DEVELOPMENT OF YIELD CRITERIA FOR DESCRIBING THE BEHAVIOR OF POROUS METALS WITH TENSION-COMPRESSION ASYMMETRY By JOEL B. STEWART A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 1 c 2009 Joel B. Stewart 2 ACKNOWLEDGMENTS I would like to thank the members of my supervisory committee for their suggestions and support. I would like to especially thank Dr. Oana Cazacu for her support, her patience and her enthusiasm throughout this study. Her comments and direction have been invaluable to the completion of this research. I would also like to thank the Air Force Research Laboratory for providing the oppor- tunity to pursue a graduate education. Special recognition is extended to Dr. Lawrence Lijewski and Dr. Kirk Vanden for enthusiastically supporting my graduate research goals from the outset. I would also like to acknowledge the many helpful conversations and generous moral support received from Dr. Michael Nixon, Dr. Martin Schmidt and Dr. Brian Plunkett. Their steadfast support and honest critiques were always appreciated and a source of encouragement. I would like to extend special gratitude to Dr. Michael Nixon for reading through the dissertation and making a number of insightful comments and suggestions. Dr. Stefan Soare was of great assistance while he and I were studying for the qualifying exam; his helpful suggestions and engaging conversation were greatly appreciated. Finally, I would like to thank my friends and family who supported me throughout both this current research effort and earlier academic endeavors. Special appreciation is extended to my wife, Kelly, for supporting me throughout my academic journey. The process would have been much more painful without her unfailing support and encouragement. 3 TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................3 LIST OF TABLES.....................................7 LIST OF FIGURES....................................8 ABSTRACT........................................ 11 CHAPTER 1 INTRODUCTION.................................. 13 2 HOMOGENIZATION APPROACH......................... 27 2.1 Kinematic Homogenization Approach of Hill and Mandel.......... 28 2.2 Yield Criterion for the Matrix Material.................... 30 2.3 Plastic Multiplier Rate Derivation....................... 33 2.3.1 Plastic multiplier rate when J3 ≤ 0.................. 35 2.3.2 Plastic multiplier rate when J3 ≥ 0.................. 38 2.3.3 General plastic multiplier rate expression............... 41 3 PLASTIC POTENTIAL FOR HCP METALS WITH SPHERICAL VOIDS... 46 3.1 Limit Solutions................................. 48 3.1.1 Zero porosity and equal yield strengths limiting cases........ 48 3.1.2 Exact solution for a hydrostatically-loaded hollow sphere...... 49 3.1.2.1 Strain-displacement relations................ 49 3.1.2.2 Strain compatibility..................... 50 3.1.2.3 Equations of motion..................... 51 3.1.2.4 Elastic constitutive relation: Hooke's law.......... 52 3.1.2.5 Ultimate pressure....................... 52 3.2 Choice of Trial Velocity Field......................... 57 3.3 Calculation of the Local Plastic Dissipation................. 62 3.4 Development of the Macroscopic Plastic Dissipation Expressions...... 66 4 NUMERICAL IMPLEMENTATION OF THE MATRIX YIELD CRITERION. 78 4.1 Return Mapping Procedure.......................... 78 4.2 First Derivatives................................ 85 4.2.1 Isotropic CPB06 first derivatives: general loading.......... 87 4.2.2 Isotropic CPB06 first derivatives: biaxial loading........... 88 4.3 Second Derivatives............................... 89 4.3.1 Von mises second derivatives...................... 89 4.3.2 Isotropic CPB06 second derivatives: general loading......... 92 4.3.3 Isotropic CPB06 second derivatives: biaxial loading......... 96 4 5 ASSESSMENT OF THE PROPOSED SPHERICAL VOID MODEL BY FI- NITE ELEMENT CALCULATIONS........................ 97 5.1 Modeling Procedure.............................. 97 5.2 Finite Element Results............................. 102 5.3 Concluding Remarks.............................. 104 6 PLASTIC POTENTIALS FOR HCP METALS WITH CYLINDRICAL VOIDS 122 6.1 Limit Solutions................................. 123 6.1.1 Zero porosity and von mises material limiting cases......... 123 6.1.2 Analysis of a hydrostatically-loaded hollow cylinder......... 124 6.1.2.1 Strain-displacement relations................ 125 6.1.2.2 Strain compatibility..................... 126 6.1.2.3 Equations of motion..................... 127 6.1.2.4 Elastic constitutive relation: Hooke's law.......... 128 σ 6.1.2.5 Relation between J3 and Σm ................ 128 6.2 Choice of Trial Velocity Field......................... 133 6.3 Parametric Representation of the Porous Aggregate Yield Locus for Ax- isymmetric Loading............................... 137 6.3.1 β ≥ 1: the matrix yield strength in tension is greater than in com- pression................................. 138 Σ 6.3.1.1 Σm > 0 and J3 < 0...................... 138 6.3.2 Discussion................................ 144 6.4 Proposed Closed-Form Expression for a Plane Strain Yield Criterion.... 145 6.4.1 Calculation of the local plastic dissipation............... 145 6.4.2 Development of the macroscopic plastic dissipation expressions... 148 7 ASSESSMENT OF THE PROPOSED CYLINDRICAL VOID MODEL BY FI- NITE ELEMENT CALCULATIONS........................ 162 7.1 Modeling Procedure.............................. 162 7.2 Finite Element Results............................. 166 7.3 Concluding Remarks.............................. 168 8 ANISOTROPIC PLASTIC POTENTIAL FOR HCP METALS CONTAINING SPHERICAL VOIDS................................. 180 8.1 Kinematic Homogenization Approach of Hill and Mandel.......... 181 8.2 Yield Criterion for the Matrix Material.................... 183 8.3 Choice of Trial Velocity Field......................... 187 8.4 Calculation of the Local Plastic Dissipation................. 188 8.5 Development of the Macroscopic Plastic Dissipation Expression...... 189 8.6 Assessment of the Proposed Anisotropic Criterion through Comparison with Finite Element Calculations....................... 192 8.7 Concluding Remarks.............................. 197 5 9 CONCLUSIONS................................... 210 APPENDIX A PARAMETRIC REPRESENTATION DERIVATION OF THE AXISYMMET- RIC YIELD LOCUS................................. 213 A.1 General Form of Equations........................... 213 A.1.1 Plastic multiplier rate branches.................... 213 A.1.2 Macroscopic plastic dissipation and derivatives............ 217 A.2 β ≥ 1: The Matrix Yield Strength in Tension is Greater than in Compres- sion........................................ 220 Σ A.2.1 J3 < 0.................................. 220 A.2.1.1 Σm > 0............................ 220 A.2.1.2 Σm < 0............................ 221 Σ A.2.2 J3 > 0.................................. 223 A.2.2.1 Σm > 0............................ 223 A.2.2.2 Σm < 0............................ 226 A.3 β ≤ 1: The Matrix Yield Strength in Tension is Less than in Compression. 229 Σ A.3.1 J3 < 0.................................. 230 A.3.1.1 Σm > 0............................ 230 A.3.1.2 Σm < 0............................ 233 Σ A.3.2 J3 > 0.................................. 236 A.3.2.1 Σm > 0............................ 236 A.3.2.2 Σm < 0............................ 237 B RELATIONSHIP BETWEEN HILL48 AND CPB06 COEFFICIENTS..... 240 B.1 Determine The Hill48 Coefficients Given The CPB06 Coefficients..... 245 B.2 Determine The CPB06 Coefficients Given The Hill48 Coefficients..... 246 REFERENCES....................................... 247 BIOGRAPHICAL SKETCH................................ 251 6 LIST OF TABLES Table page 2-1 k-relations....................................... 44 2-2 z-parameters...................................... 44 5-1 k = 0, J3 > 0 and f0 = 0:01 spherical void computational test matrix....... 108 5-2 k = 0, J3 < 0 and f0 = 0:01 spherical void computational test matrix....... 108 5-3 k = −0:3098, J3 > 0 and f0 = 0:01 spherical void computational test matrix... 108 5-4 k = −0:3098, J3 < 0 and f0 = 0:01 spherical void computational test matrix... 109 5-5 k = 0:3098, J3 > 0 and f0 = 0:01 spherical void computational test matrix.... 109 5-6 k = 0:3098, J3 < 0 and f0 = 0:01 spherical void computational test matrix.... 109 5-7 k = 0, J3 > 0 and f0 = 0:04 spherical void computational test matrix....... 111 5-8 k = 0, J3 < 0 and f0 = 0:04 spherical void computational test matrix....... 111 5-9 k = −0:3098, J3 > 0 and f0 = 0:04 spherical void computational test matrix... 111 5-10 k = −0:3098, J3 < 0 and f0 = 0:04 spherical void computational test matrix... 112 5-11 k = 0:3098, J3 > 0 and f0 = 0:04 spherical void computational test matrix.... 112 5-12 k = 0:3098, J3 < 0 and f0 = 0:04 spherical void computational test matrix.... 112 5-13 k = 0, J3 > 0 and f0 = 0:14 spherical void computational test matrix....... 114 5-14 k = 0, J3 < 0 and f0 = 0:14 spherical void computational test matrix....... 114 5-15 k = −0:3098, J3 > 0 and f0 = 0:14 spherical void computational test matrix... 114 5-16 k = −0:3098, J3 < 0 and f0 = 0:14 spherical void computational test matrix... 115 5-17 k = 0:3098, J3 > 0 and f0 = 0:14 spherical void computational test matrix.... 115 5-18 k = 0:3098, J3 < 0 and f0 = 0:14 spherical void computational test matrix.... 115 7-1 k = 0 plane strain cylindrical void computational test matrix..........