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Topics in Solid Mechanics: Elasticity, Plasticity, Damage, Nano and Biomechanics

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Topics in Solid Mechanics: Elasticity, Plasticity, Damage, Nano and Biomechanics Topics in Solid Mechanics: Elasticity, Plasticity, Damage, Nano and Biomechanics Vlado A. Lubarda Montenegrin Academy of Sciences and Arts Division of Natural Sciencies OPN, CANU 2012 Contents Preface xi Part 1. LINEAR ELASTICITY 1 Chapter 1. Anisotropic Nonuniform Lam´eProblem 2 1.1. Elastic Anisotropy 2 1.2. Radial Nonuniformity 3 1.3. Governing Differential Equations 4 1.4. Stress and Displacement Expressions 5 1.5. Traction Boundary Conditions 6 1.6. Applied External Pressure 7 1.7. Plane Stress Approximation 8 1.8. Generalized Plane Stress 10 References 11 Chapter 2. Stretching of a Hollow Circular Membrane 12 2.1. Introduction 12 2.2. Basic Equations 12 2.3. Traction Boundary Conditions 13 2.4. Mixed Boundary Conditions 16 References 18 Chapter 3. Energy of Circular Inclusion with Sliding Interface 21 3.1. Energy Expressions for Sliding Inclusion 21 3.2. Energies due to Eigenstrain 22 3.3. Energies due to Remote Loading 23 3.4. Inhomogeneity under Remote Loading 25 References 26 Chapter 4. Eigenstrain Problem for Nonellipsoidal Inclusions 27 4.1. Eshelby Property 27 4.2. Displacement Expression 27 4.3. The Shape of an Inclusion 28 4.4. Other Shapes of Inclusions 30 References 32 Chapter 5. Circular Loads on the Surface of a Half-Space 33 5.1. Displacements due to Vertical Ring Load 33 5.2. Alternative Displacement Expressions 34 iii iv CONTENTS 5.3. Tangential Line Load 37 5.4. Alternative Expressions 39 5.5. Reciprocal Properties 43 References 45 Chapter 6. Elasticity Tensors of Anisotropic Materials 46 6.1. Elastic Moduli of Transversely Isotropic Materials 46 6.2. Elastic Compliances of Transversely Isotropic Materials 49 6.3. Elastic Moduli of Orthotropic Materials 50 6.4. Elastic Compliances of Orthotropic Materials 52 References 53 Chapter 7. Elastic Constants of Single Crystals and their Aggregates 54 7.1. Negative Poisson's Ratio 54 7.2. Effective Constants of Polycrystalline Aggregates 57 References 58 Chapter 8. Dual Conservation Integrals of Infinitesimal Elasticity 59 8.1. Dual J Integrals 59 8.2. Dual M Integrals 61 8.3. Dual L Integrals 62 8.4. Release Rate of Potential Energy 62 8.5. Rate of Complementary Energy 63 References 65 Part 2. MICROPOLAR ELASTICITY 67 Chapter 9. Circular Inclusion in Couple Stress Elasticity 68 9.1. Introduction 68 9.2. Governing Equations of Couple Stress Elasticity 69 9.3. Displacement Equations of Equilibrium 70 9.4. Correspondence Theorem 71 9.5. Plane Strain Problems 71 9.6. Governing Equations for Anti-Plane Strain 73 9.7. Circular Inclusion with Uniform Eigenstrain 75 References 79 Chapter 10. Noether's Theorem of Micropolar Elasticity 81 10.1. Basic Equations of Micropolar Elasticity 81 10.2. Noether's Theorem 83 10.3. Conservation Integrals 85 10.4. Conservation Laws for Plane Strain 86 10.5. M Integral 86 References 88 Chapter 11. Dual Energy Momentum Tensors of Micropolar Elasticity 89 11.1. Introduction 89 11.2. Energy Momentum Tensor 90 11.3. J integrals 91 CONTENTS v 11.4. M Integral 91 11.5. L Integrals 92 11.6. Energy Release Rate 93 11.7. Configurational Forces 94 11.8. Dual Energy Momentum Tensor 95 11.9. Dual Integrals 95 11.10. Complementary Potential Energy 97 References 98 Part 3. DISLOCATIONS 101 Chapter 12. Image Force on a Dislocation near Void 102 12.1. Stress Functions for a Dislocation near Void 102 12.2. Image Force on a Dislocation 104 12.3. Interaction Energy 106 12.4. Dislocation on an Inclined Slip Plane 108 References 111 Chapter 13. Dislocations Equilibrium Conditions 112 13.1. Elastic Strain Energy of a Dislocated Body 112 13.2. Equilibrium Conditions 114 13.3. Core Energy 116 13.4. Dislocations in a Loaded Body 118 13.5. Discussion 121 References 122 Chapter 14. Dislocation Arrays near a Bimaterial Interface 124 14.1. Introduction 124 14.2. Single Dislocation near a Bimaterial Interface 124 14.3. Dislocation Array near a Bimaterial Interface 127 14.4. Energy of a General Straight Dislocation Array 132 References 133 Chapter 15. Effects of Couple Stresses on Dislocation Strain Energy 135 15.1. Edge Dislocation in Couple Stress Elasticity 135 15.2. Edge Dislocation in a Hollow Cylinder 139 15.3. Screw Dislocation in Couple Stress Elasticity 141 References 143 Chapter 16. Dislocation in a Gravity Field 144 16.1. J Integral in the presence of Body Force 144 16.2. Relationship to the Energy Release Rate 145 16.3. Peach{Koehler Force on a Dislocation 146 References 149 Part 4. NONLINEAR ELASTICITY 151 Chapter 17. Analysis of Dilute Distribution of Inclusions 152 17.1. Spherical Inclusion in Second-Order Elasticity 152 vi CONTENTS 17.2. Volume Change due to Inclusions 154 17.3. Strain Energy of a Single Inclusion 155 17.4. Strain Energy of Two Distant Inclusions 156 17.5. Strain Energy of Dilute Distribution of Inclusions 158 References 159 Chapter 18. Third-Order Elastic Constants of Polycrystals 160 18.1. Introduction 160 18.2. Strain Energy Representation 161 18.3. Second-order Elastic Constants 163 18.4. Third-Order Elastic Constants 164 18.5. Voigt and Reuss Estimates 166 18.6. Semi-Consistent Estimates 169 Appendix: Derivation of Components Hijkl and Gijkl 172 References 174 Chapter 19. Finite-Strain Thermoelasticity 177 19.1. Introduction 177 19.2. Classical Theory of Finite-Strain Thermoelasticity 177 19.3. Thermoelasticity based on Multiplicative Decomposition 180 19.4. Entropy Expression 183 References 185 Chapter 20. Thermodynamic Potentials in Nonlinear Thermoelasticity 186 20.1. Introduction 186 20.2. Internal Energy 187 20.3. Helmholtz Free Energy 189 20.4. Enthalpy Function 191 20.5. Gibbs Energy 193 20.6. Applications 197 References 198 Part 5. PLASTICITY 201 Chapter 21. Phenomenological Plasticity 202 21.1. Multiplicative Decomposition 202 21.2. Decomposition of Strain Tensors 204 21.3. Velocity Gradient and Strain Rates 204 21.4. Objectivity Requirements 205 21.5. Jaumann Rate of Elastic Deformation Gradient 206 21.6. Partition of Rate of Deformation 207 21.7. Analysis of Elastic Rate of Deformation 208 21.8. Kinematic Hardening 211 References 212 Chapter 22. Elastoplasticity with Yield Surface in Strain Space 215 22.1. Kinematic and Kinetic Preliminaries 215 22.2. Partition of Strain and Stress Rates 217 22.3. Elaboration on Elastic and Plastic Strain Rates 220 CONTENTS vii 22.4. Constitutive Equations with Yield Surface in Strain Space 221 References 225 Chapter 23. Elastoplasticity based on a Reversed Decomposition 227 23.1. Reversed Multiplicative Decomposition 227 23.2. Uniqueness and Objectivity 228 23.3. Elastic Unloading 230 23.4. Polycrystalline Plasticity 230 23.5. Analysis of Strain Rates 232 23.6. Monocrystalline Plasticity 233 References 236 Chapter 24. Plasticity Postulates and Non-Associative Flow Rules 238 24.1. Introduction 238 24.2. Plasticity Postulates 239 24.3. Non-Associative Flow Rule with Drucker{Prager Yield Condition 241 24.4. Infinitesimal Cycles of Stress and Strain 242 References 246 Chapter 25. Rate-Dependent Plasticity 247 25.1. Introduction 247 25.2. Rate-Independent Plasticity 249 25.3. Viscoplasticity 251 25.4. Discussion 255 References 255 Chapter 26. Deformation Theory of Plasticity 257 26.1. Introduction 257 26.2. Rate-Type Deformation Theory 258 26.3. Application beyond Proportional Loading 259 26.4. J2 Corner Theory 260 References 262 Part 6. DAMAGE MECHANICS 263 Chapter 27. Damage Tensors and Crack Density Distribution 264 27.1. Introduction 264 27.2. Damage Tensors and Crack Distributions 264 27.3. Three-Dimensional Crack Distributions 269 27.4. Two-Dimensional Crack Distributions 272 References 277 Chapter 28. Analysis of Large Strain Damage-Elastoplasticity 279 28.1. Damage Variables 279 28.2. Description of Anisotropic Elastic Response 280 28.3. Rate-Type Analysis 281 28.4. Rates of Damage Tensors 282 28.5. Elastic, Plastic and Damage Strain Rate 284 References 285 viii CONTENTS Chapter 29. Rate-Theory of Damage-Elastoplasticity 287 29.1. Introduction 287 29.2. Anisotropic Elastic Response 289 29.3. Rate-Type Elasticity 291 29.4. Partition of Stress and Strain Rates 292 29.5. Damage and Plastic Stress and Strain Rates 292 29.6. Damage Potentials and Evolution Equations 294 29.7. Constitutive Equations for Inelastic Stress and Strain Rates 298 References 301 Chapter 30. Brittle Solids with Unequal Tensile and Compressive Strengths 303 30.1. Introduction 303 30.2. Physical Preliminaries 304 30.3. Unequal Tensile and Compressive Strengths 306 30.4. Damage Surface 307 30.5. Rate-Type Constitutive Equations 309 30.6. Applications 310 References 316 Part 7. NANOMECHANICS 317 Chapter 31. Effective Lattice Parameter of Binary Alloys 318 31.1. Introduction 318 31.2. Volume Change due to a Solute Atom 319 31.3. Volume Change of a Solid Solution 320 31.4. Effective Lattice Parameter of Binary Alloys 321 31.5. Apparent Radius of a Solute Atom 321 31.6. Deviations from Vegard's Law 322 31.7. Results based on the Apparent Atomic Radius 324 References 327 Chapter 32. Critical Size of a Twin Embryo 330 32.1. Introduction 330 32.2. Analytical Prediction of the Critical Twin Embryo 331 32.3. Grain-Size and the Length of Dislocation Pile-Up 334 References 335 Chapter 33. Emission of Dislocations from Nanovoids 337 33.1. Introduction 337 33.2. Dislocation Emission under Equal Biaxial Stress 337 33.3. Biaxial Loading Conditions 343 33.4. Lattice Orientation Effects 346 33.5. Discussion 349 References 352 Chapter 34. Strain Relaxation in Thin Films 354 34.1. Dislocation Array beneath a Free Surface 354 34.2. Energy of Dislocation Array 356 34.3. Strained-Layer Epitaxy 357 CONTENTS ix 34.4. Conditions for Array Formation 358 34.5. Frank and van der Merwe Energy Criterion 360 34.6.
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