Peridynamic Theory for Progressive Failure Prediction in Homogeneous and Heterogeneous Materials
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Peridynamic Theory for Progressive Failure Prediction in Homogeneous and Heterogeneous Materials Item Type text; Electronic Dissertation Authors Kilic, Bahattin Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 02/10/2021 05:00:26 Link to Item http://hdl.handle.net/10150/193658 PERIDYNAMIC THEORY FOR PROGRESSIVE FAILURE PREDICTION IN HOMOGENEOUS AND HETEROGENEOUS MATERIALS by Bahattin Kilic _____________________ A Dissertation Submitted to the Faculty of the DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINEERING In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY WITH A MAJOR IN MECHANICAL ENGINEERING In the Graduate College THE UNIVERSITY OF ARIZONA 2008 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Bahattin Kilic entitled Peridynamic Theory for Progressive Failure Prediction in Homogeneous and Heterogeneous Materials and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy _______________________________________________________________________ Date: 09/18/2008 Erdogan Madenci _______________________________________________________________________ Date: 09/18/2008 Pierre A. Deymier _______________________________________________________________________ Date: 09/18/2008 Louis Rene Corrales _______________________________________________________________________ Date: 09/18/2008 Steward A. Silling _______________________________________________________________________ Date: 09/18/2008 Alexander Tessler Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. ________________________________________________ Date: 09/18/2008 Dissertation Director: Erdogan Madenci 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: Bahattin Kilic 4 ACKNOWLEDGEMENTS I would like to express the deepest appreciation to the following persons for their continued assistance and patience during the research and compilation of this study - Erdogan Madenci for his help and support. - Steward Silling for bringing my attention to the paper on adaptive dynamic relaxation. - Pierre Deymier, Rene Corrales, Steward Silling and Alexander Tessler for serving on my committee. - Molly Moore and Tom Perry for teaching me a lot about the American culture and language. - All of my friends for their help and support. - My uncle, Faruk Guder for helping me to come to Arizona. I also appreciate for his support and guidance while learning English in Chicago. - Finally, my parents, Cevriye and Sebahattin Kilic for their support and encouragement during my education. 5 DEDICATION Dedicated to my mother, Cevriye Kilic 6 TABLE OF CONTENTS LIST OF FIGURES…………………………………………………………………….10 LIST OF TABLES……………………………………………………………………...20 ABSTRACT……………………………………………………………………………..21 1. INRODUCTION……………………………………………………………………..22 2. PERIYNAMIC THEORY…………………………………………………………...34 2.1. Introduction ……………………………………………………………………...34 2.2. Response Function ………………………………………………………………36 2.2.1. Derivation of the Response Function ………………………………………...44 2.2.2. Damage ………………………………………………………………………47 2.2.3. Damping ……………………………………………………………………...48 2.2.4. Selection of the Material Constants ……………………………………….…50 2.3. Boundary Conditions in Peridynamics ……………………………………...…52 2.4. Solution Method …………………………………………………………………54 2.5. Remarks ………………………………………………………………………….55 3. NUMERICAL SOLUTION METHOD ………………………………………….…57 3.1. Introduction ………………………………………………………………..…….57 3.2. Discretization …………………………………………………………………….59 3.3. Spatial Partitioning ……………………………………………………………...62 3.4. Binary Space Decomposition …………………………………………………...64 3.5. Surface Effects …………………………………………………………………..68 7 TABLE OF CONTENTS - Continued 3.6. Adaptive Dynamic Relaxation ……………………………………………….…76 3.6.1. Introduction ………………………………………………………………..…76 3.6.2. Solution Method ……………………………………………………………..78 3.7. Benchmark Solutions ……………………………………………………………83 3.7.1. Beam under Tension and Bending …………………………………………...83 3.7.2. Rod Subjected to Tension ……………………………………………………91 3.7.3. Plate Subjected to Tension ………………………………………………...…94 3.7.3.1. Effective Poisson’s ratio ………………………………………...………95 3.7.3.2. Effect of order of Gaussian integration technique …………………..…100 3.7.3.3. Effect of internal length and cutoff radius …………………………..…102 3.7.4. Plate with a Circular Cutout ………………………………………………...106 3.8. Remarks ………………………………………………………………………...117 4. MODELING OF UNIDIRECTIONAL COMPOSITE MATERIALS……….…120 4.1. Introduction …………………………………………………………………….120 4.2. Modeling of Laminae …………………………………………………………..123 4.3. Numerical Results ……………………………………………………………...126 4.3.1. Geometry, Discretization, and Material Properties …………………………126 4.3.2. Off-Axis Modulus Prediction ………………………………………………129 4.3.3. Damage Prediction in Center-Cracked Laminates ……………………….…132 4.3.4. Effect of Fiber Orientation ……………………………………………….…133 8 TABLE OF CONTENTS – Continued 4.3.5. Effect of Fiber Distribution …………………………………………………136 4.3.6. Damage Progression ………………………………………………………..139 4.3.7. Effect of Stacking Sequence ………………………………………………..141 4.4. Remarks ……………………………………………………………………...…146 5. THERMAL PROBLEMS ……………………………………………………….…147 5.1. Introduction …………………………………………………………………….147 5.2. Bimaterial Strip Subjected to Uniform Temperature …………….…………148 5.3. Rectangular Plate Subjected to Temperature Gradient ………………….…155 5.4. Plate with Pre-Existing Crack ………………………………………………...159 5.5. Bonded Dissimilar Plates ………………………………………………………162 5.6. Remarks ………………………………………………………………………...169 6. CONTROLLED CRACK GROWTH …………………………………………….170 6.1. Introduction …………………………………………………………………….170 6.2. Numerical Results ……………………………………………………………...176 6.2.1. Single Crack ………………………………………………………………...178 6.2.2. Multiple Cracks ……………………………………………………………..188 6.6. Remarks ………………………………………………………………………...195 7. STABILTY ANALYSIS ……………………………………………………………196 7.1. Introduction ……………………………………………………………...……..196 7.2. Column Subjected to Uniaxial Compression ……………………………...…198 9 TABLE OF CONTENTS – Continued 7.3. Rectangular Plate Subjected to Uniform Temperature Load ………………212 7.4. Remarks ………………………………………………………………………...216 8. COUPLING WITH THE FINITE ELEMENT METHOD ……………………...217 8.1. Introduction …………………………………………………………………….217 8.2. Finite Element Equations ……………………………………………………...219 8.3. Coupled Equations ………………………………………………………..……221 8.4. Numerical Results ……………………………………………………………...226 8.4.1. Beam Subjected to Tension ………………………………………………...226 8.4.2. Plate with a Circular Cutout ……………………………………………...…230 8.5. Remarks ………………………………………………………………………237 9. FINAL REMARKS AND FUTURE WORK …………………………………..…238 APPENDIX A. INTEGRATION POINTS AND THEIR WEIGHTS …………..…248 APPENDIX B. FINITE ELEMENT FORMULATION ……………………………251 REFERENCES …………………………………………………………………...……255 10 LIST OF FIGURES Figure 1.1. Relationship among scales ………………..……………………………….27 Figure 2.1. Pairwise interaction of a material point with its neighboring point…….….35 Figure 2.2. The positions of two material points in the initial and deformed states ……………………………………………………….……37 Figure 2.3. Generalized function representing Dirac delta function ..………………….39 Figure 2.4. Interaction failure based on critical stretch ………………………………...41 Figure 2.5. Damaged and undamaged regions for a point ……………………………..41 Figure 2.6. Effect of internal length …………………………………………………....43 Figure 2.7. Bond force as a function of stretch ………………………………………...44 Figure 2.8. Isotropic extension of homogeneous body ………………………………...45 Figure 2.9. Interactions among material points ………………………………………...51 Figure 2.10. Boundary conditions: (a) domain of interest; (b) tractions in classical continuum mechanics; (c) interaction of a point in domain Ω+ with domain Ω− ; (d) force densities acting on domain Ω+ due to the domain Ω− ……………………………………………………..……....53 Figure 3.1. Discretization of the domain of interest …………………………………....60 Figure 3.2. Subdomain shapes ………………………………………………………….61 Figure 3.3. Uniform grid overlaid on problem domain ………………………...............63 Figure 3.4. Interaction of collocation points located in uniform grid ..………………...63 Figure 3.5. Processor distribution of problem domain ………………………………...65 Figure 3.6. Tree structure to construct decomposition …………………….…………..67 11 LIST OF FIGURES