Micromechanics of Materials, with Applications (eBook)

Mark Kachanov is a Professor of Mechanical Engineering and Editor-in-Chief of the International Journal of Engineering Science. Igor Sevostianov is a Dwight L. and Aubrey Chapman Distinguished Professor of Mechanical Engineering at New Mexico State University, USA. He serves on the editorial boards of several international journals.

Preface 6
Contents 8
1 Background Results on Elasticity and Conductivity 17
1.1 Basic Equations of Linear Elasticity. Elastic Symmetries 17
1.2 Energy Principles of Elasticity 25
1.2.1 Virtual Changes of State 26
1.2.2 The Principle of Virtual Displacements 27
1.2.3 The Principle of Virtual Forces 28
1.2.4 The Principle of Stationarity of Potential Energy of an Elastic Solid 29
1.2.5 The Principle of Stationarity of Complementary Energy of an Elastic Solid 31
1.3 Approximate Symmetries of the Elastic Properties 32
1.4 A Summary of Algebra of Fourth-Rank Tensors 38
1.4.1 Isotropic Fourth-Rank Tensors 40
1.4.2 Anisotropic Fourth-Rank Tensors 41
1.4.3 Transversely Isotropic Tensors 42
1.4.4 Averaging of Tensors nn and nnnn Over Orientations in Simplest Cases of Orientation Distribution 44
1.4.5 Orthotropic Tensors 47
1.5 Thermal and Electric Conductivity: Fourier and Ohm’s Laws 48
1.6 Green’s Tensors in Elasticity and Conductivity and Their Derivatives 50
1.6.1 General Representation of Green’s Tensor in Elasticity 50
1.6.2 Isotropic Elastic Material 53
1.6.3 Transversely Isotropic Elastic Material 54
1.6.4 Green’s Tensor for a Monoclinic Material, in the Plane of Elastic Symmetry and in the Direction Normal to It 61
1.6.5 Cubic Symmetry 64
1.6.6 Two-Dimensional Anisotropic Elastic Material 68
1.6.7 Derivatives of Green’s Tensor 72
1.6.8 Green’s Function in the Conductivity Problem 74
1.7 Dipoles, Moments, and Multipole Expansions in Elasticity and Conductivity 76
1.7.1 System of Forces Distributed in Small Volume 77
1.7.2 Dipole 78
1.7.3 Center of Dilatation 79
1.7.4 Force Couple 80
1.7.5 Center of Rotation 81
1.7.6 Multipole Expansion 82
1.8 Stress Intensity Factors 85
1.9 General Thermodynamics Framework for Transition from Microscale to Macroscopic Constitutive Equations (Rice’s Formalism) 88
1.10 Mathematical Analogies Between Elastostatics and Steady-State Heat Flux. Conductivity Analogues of Stress Intensity Factors 95
1.11 Discontinuities of the Elastic and Thermal Fields at Interfaces of Two Different Materials 99
1.11.1 Stress Discontinuities in the Elasticity Problem 99
1.11.2 Flux Discontinuities in the Conductivity Problem 102
2 Quantitative Characterization of Microstructures in the Context of Effective Properties 105
2.1 Representative Volume Element (RVE) and Related Issues 106
2.1.1 Hill’s Condition. Homogeneous Boundary Conditions 106
2.1.2 Averages Over Volume and Their Relation to Quantities Accessible on Its Boundary 109
2.1.3 Volumes Smaller than RVE 114
2.2 The Concept of Proper Microstructural Parameters 119
2.3 The Simplest Microstructural Parameters and Their Limitations 122
2.4 Microstructural Parameters Are Rooted in the Non-interaction Approximation 125
2.5 Property Contribution Tensors of Inhomogeneities 126
2.6 Hill’s Comparison (Modification) Theorem and Its Implications 130
2.7 Microstructural Parameters Are Different for Different Physical Properties 134
2.8 Benefits of Identifying the Proper Microstructural Parameters 136
2.9 On the “Fabric” Tensor Approach 137
2.10 Summary on Microstructural Characterization 141
3 Inclusion and Inhomogeneity in an Infinite Space (Eshelby Problems) 143
3.1 The First and the Second Eshelby Problems 143
3.1.1 The Eigenstrain Problem (The First Eshelby Problem) 144
3.1.2 The Inhomogeneity Problem (The Second Eshelby Problem) 149
3.1.3 Eshelby Theorem for the Ellipsoidal Domain 150
3.1.4 Extension of the Eshelby Theorem to Nonlinear Ellipsoidal Inhomogeneities 154
3.2 Elastic Fields Outside Inhomogeneities and Inclusions 157
3.2.1 Stress Concentrations at Boundary of an Inhomogeneity 157
3.2.2 External Fields in the Inclusion (Eigenstrain) Problem 159
3.2.3 Connection Between Inclusion- and Inhomogeneity-Generated Elastic Fields 161
3.2.4 Far-Field Asymptotics of Elastic Fields of Inhomogeneities and Its Relation to the Effective Elastic Properties. The Multipole Expansion 162
3.2.5 Shape Dependence of the Far-Field: Inhomogeneity Versus Inclusion. Far-Field of an Inclusion of Arbitrary Shape 166
3.3 Ellipsoidal Inhomogeneities and Inclusions in the Isotropic Matrix: Special Cases of Ellipsoid Geometry 171
3.4 Spheroidal Inhomogeneity Embedded in a Transversely Isotropic Matrix 179
3.5 Non-ellipsoidal Inclusion in Isotropic Material (First Eshelby Problem) 185
3.5.1 Eshelby Tensor for a Cuboid (Rectangular Parallelepiped) 186
3.5.2 Eshelby Tensor for Polyhedra 189
3.5.3 Eshelby Tensor for a Supersphere 190
3.5.4 Eshelby Tensor for a Torus 192
3.6 Eshelby Problem for Conductivity 194
3.6.1 Formulation of the Problem 195
3.6.2 Ellipsoidal Inhomogeneity 197
3.6.3 Ellipsoidal Inhomogeneity in an Isotropic Matrix 198
3.6.4 Ellipsoidal Inhomogeneity Arbitrarily Oriented in an Orthotropic Matrix 199
4 Property Contribution Tensors of Inhomogeneities 205
4.1 General Representations 205
4.2 Ellipsoidal Inhomogeneity 207
4.2.1 General Ellipsoid 207
4.2.2 Special Cases of the Spheroid Geometry 216
4.2.3 Fluid-Filled Pore 225
4.3 Non-ellipsoidal Inhomogeneities 228
4.3.1 Bounds for Property Contribution Tensors Implied by Hill’s Modification Theorem 229
4.3.2 First-Order Approximations 230
4.3.3 Concave Pores 232
4.3.4 Inhomogeneities of Polyhedral Shapes 247
4.3.5 Helical Inhomogeneities 249
4.3.6 Toroidal Inhomogeneity 252
4.4 Cracks 260
4.4.1 Flat (Planar) Cracks 261
4.4.2 Application of Rice’s Internal Variables Technique to Calculation of Property Contribution Tensors of 3-D Cracks 265
4.4.3 Intersecting 3-D Cracks 268
4.4.4 Non-flat Cracks 269
4.4.5 Fluid-Filled Crack Closed Sliding Crack
4.5 Inhomogeneity in a Three-Dimensional Anisotropic Material 281
4.5.1 Compliance and Stiffness Contribution Tensors for a Spheroidal Inhomogeneity Aligned with the Axis of Transverse Isotropy of the Matrix 282
4.5.2 Applications of the Concept of Approximate Elastic Symmetry 285
4.5.3 Resistivity and Conductivity Contribution Tensors of Ellipsoidal Inhomogeneities Arbitrarily Oriented in the Orthotropic Matrix 290
4.6 Inhomogeneity with Interphase Zone 291
4.6.1 Homogeneous Inclusion that Is Equivalent to the “Stiff Core–Graded Interface” System 293
4.6.2 On the Relative Importance of Various Interface Parameters 297
4.7 Two-Dimensional Inhomogeneities 300
4.7.1 Elliptical Inhomogeneity in the Isotropic Matrix 301
4.7.2 Holes and Inhomogeneities of Non-elliptical Shapes in 2-D Isotropic Matrix 303
4.7.3 Cracks of Complex and Intersected Shapes in 2-D Isotropic Matrix 307
4.7.4 Effect of Matrix Anisotropy 311
4.8 Other Property Contribution Tensors 318
4.8.1 Diffusivity Contribution Tensor 319
4.8.2 Thermal Expansion Contribution Tensor 321
4.9 Replacement Relations Between Property Contribution Tensors of Inhomogeneities Having the Same Shape but Different Properties 325
4.10 Summary: Tensors Used in Micromechanics Analyses and Relations Between Them 327
5 Effective Properties of Heterogeneous Materials 331
5.1 Bounds for the Effective Elastic Constants 332
5.1.1 Stress and Strain Concentration Tensors for Phase Averages 332
5.1.2 Voigt–Reuss–Hill Bounds 333
5.1.3 Polarization Tensors 336
5.1.4 Extremum Principles and Walpole Theorem 337
5.1.5 Estimate for Weighted Average Stiffness Tensor {/bar{{/varvec C}}} for Isotropic Microstructures 340
5.1.6 Hashin–Shtrikman Bounds in the Case of Isotropic Constituents 343
5.1.7 Matrix with Spheroidal Pores 345
5.1.8 Isotropic Matrix with Transversely Isotropic Randomly Oriented Spheroidal Inhomogeneities 348
5.1.9 Composite with Aligned Transversely Isotropic Phases 350
5.2 Bounds for Conductivity 352
5.2.1 Weakly Inhomogeneous Media 353
5.2.2 Auxiliary Results 355
5.2.3 Wiener Bounds 357
5.2.4 Hashin–Shtrikman Bounds 358
5.2.5 Shape-Specific Bounds for Conductivity 359
5.3 The Non-interaction Approximation and Its Relation to the “Dilute Limit” 360
5.3.1 Two Dual Versions of the NIA 360
5.3.2 The “Dilute Limit” and Counterproductive Linearizations 366
5.3.3 Microstructural Parameters Are Identified in the Framework of the Non-interaction Approximation 367
5.3.4 The Non-interaction Approximation for Microcracked Materials 368
5.3.5 Nonrandomly Oriented Spheroidal Inhomogeneities 373
5.3.6 Cracks Filled with Compressible Fluid 375
5.3.7 Cracks Undergoing Frictional Sliding Under Compression 378
5.3.8 The NIA as the Basic Building Block for Various Approximate Schemes 392
5.4 The Self-consistent (Effective Matrix) Scheme 393
5.4.1 Effective Conductive Properties 393
5.4.2 Effective Elastic Properties 395
5.5 The Differential Scheme 397
5.5.1 Effective Conductive Properties 398
5.5.2 Effective Elastic Properties 400
5.6 The Mori–Tanaka–Benveniste Scheme 402
5.6.1 Effective Conductive Properties 402
5.6.2 Effective Elastic Properties 405
5.6.3 Problems Encountered in the Mori–Tanaka–Benveniste Scheme. The Symmetrized Version 407
5.7 The Kanaun–Levin Scheme 410
5.8 The Maxwell Scheme and Its Extension to Materials Containing Inhomogeneities of Diverse Shapes 413
5.8.1 The Original Maxwell Scheme 414
5.8.2 Maxwell Scheme in Terms of Property Contribution Tensors 417
5.8.3 The Choice of Shape of the Effective Inhomogeneity 422
5.8.4 Maxwell Scheme for Anisotropic Multiphase Composites 426
5.8.5 Maxwell Scheme as One of Effective Field Methods 429
5.9 Comparison of Approximate Schemes. Effects of Interactions and of Inhomogeneity Shapes on the Overall Properties 431
5.9.1 Comparison of Various Approximate Schemes 431
5.9.2 Pair Interactions and Their Effect on Property Contribution Tensors 434
5.9.3 On the Relative Importance of Inhomogeneity Shapes and of Interaction Effects 439
5.10 Yield Condition for Anisotropic Porous Metals, in Relation to Pore Shapes and Effective Elastic Properties 442
5.10.1 The Concept 443
5.10.2 Enhancement of the Deviatoric Strain by Pores 446
5.10.3 Basic Equations 448
5.10.4 Cases of Overall Isotropy 450
5.10.5 Transversely Isotropic Mixtures of Pores 455
5.10.6 Equations of Plastic Flow 458
5.11 The Concept of “Average Shape” for a Mixture of Inhomogeneities of Diverse Shapes 460
5.11.1 Formulation of the Problem 461
5.11.2 Two-Dimensional Holes of Diverse Shapes 462
5.11.3 Implications for General 2-D Shapes 468
5.11.4 Three-Dimensional Pores 469
5.11.5 The “Average Shape” in the Context of Conductivity 473
5.12 On the Possibility to Represent Effective Properties in Terms of Concentration Parameters of Inhomogeneities. An Alternative to Concentration Parameters 474
5.12.1 Cases When Simple Concentration Parameters Can Be Identified 475
5.12.2 On Applying the Volume Fraction and Crack Density Parameters to Complex Microstructures 477
5.12.3 An Alternative to Concentration Parameters 478
6 Connections Between Elastic and Conductive Properties of Heterogeneous Materials. Other Cross-Property Relations 484
6.1 History of Cross-Property Connections 485
6.1.1 Bristow’s Elastic–Conductive Properties Connection for a Microcracked Material 485
6.1.2 Cross-Property Connections Involving the Bulk Modulus 487
6.1.3 Cross-Property Bounds 490
6.1.4 Empirical Observations on Cross-Property Relations 502
6.2 Explicit Approximate Elastic–Conductive Properties Connections for Two-Phase Composites 504
6.2.1 Approximate Representations of the Compliance and Stiffness Contribution Tensors of a Spheroidal Inhomogeneity 504
6.2.2 Elasticity–Conductivity Connections: General Case 514
6.2.3 Cases of Overall Isotropy and Transverse Isotropy 520
6.2.4 Materials with Cracks or Rigid Disks 522
6.2.5 On the Sensitivity of the Connection to Shapes of Inhomogeneities 524
6.2.6 Connection Between the Degrees of the Elastic and Conductive Anisotropies Is Insensitive to Inhomogeneity Shapes 524
6.2.7 On the Effect of Interactions and of Non-spheroidal Inhomogeneity Shapes on the Cross-Property Connections 525
6.2.8 Connection Between the Electric and the Thermal Conductivities 528
6.3 Cross-Property Connections that Are Exact in the Noninteraction Approximation 528
6.3.1 Cross-Property Connections for Materials with Parallel Anisotropic Inhomogeneities 529
6.3.2 Moderate Orientation Scatter 533
6.3.3 Two or Three Families of Approximately Parallel Inhomogeneities 534
6.3.4 Nonlinear Connections for Parallel Isotropic Inhomogeneities of Unknown Orientation 537
6.3.5 General Constraints on the Elasticity–Conductivity Connections 541
6.4 Plastic Yield Surfaces of Anisotropic Porous Materials in Terms of Effective Conductivities 543
6.4.1 Approximate Constancy of Macroscopic Strain at Yield 545
6.4.2 Plastic Yield in Terms of Effective Conductivities 547
6.4.3 Cases of Overall Isotropy 549
6.5 On the Effective Stiffness–Fracture Connections: Loss of Stiffness May Not Be a Reliable Monitor of Fracture Processes 554
6.5.1 Do Quantitative Correlations Exist Between Fracture Processes and Loss of Stiffness, in Brittle-Elastic Solids? 555
6.5.2 Comments on Damage Models 559
6.5.3 Clusters of Microcracks and Their Detection Via Cross-Property Connection 561
7 Applications to Specific Materials 564
7.1 Plasma-Sprayed Ceramic Coatings: Elastic and Conductive Properties in Relation to Microstructure 566
7.1.1 Quantitative Characterization of the Microstructure in the Context of Elastic and Conductive Properties 569
7.1.2 Effective Elastic Properties 571
7.1.3 The Conductive Properties and Cross-Property Connection 572
7.1.4 Modeling of YSZ Coatings: Case Studies 573
7.2 Micromechanics of Geomaterials 578
7.2.1 Fontainebleau Sandstone: Micromechanical Modeling Versus Digitization and Finite Elements 578
7.2.2 Effective Elastic Properties of Oolitic Limestone 579
7.2.3 Inelasticity of Rocks Under Compression, and Its Micromechanics Interpretation 585
7.3 Micromechanics of Cortical Bone 589
7.3.1 Microstructure of Cortical Bone and Its Modeling 589
7.3.2 Effective Elastic Properties 592
7.3.3 Effective Electric Conductivity and Cross-Property Connections 596
7.4 Short-Fiber-Reinforced Composites 605
7.4.1 Cross-Property Connections for Short-Fiber-Reinforced Thermoplastics 605
7.4.2 Changes in Properties of Short Glass Fiber-Reinforced Plastics Due to Damage 608
7.4.3 Stress Partition Between Phases in Aluminum Alloy Reinforced with Short Alumina Fibers 613
7.5 Closed-Cell Aluminum Foams: Elasticity, Electric Conductivity, and Cross-Property Connection 619
7.5.1 Experiments 623
7.5.2 Micromechanics Modeling: The Electric Resistivity 625
7.5.3 Micromechanics Modeling: Young’s Modulus 629
7.5.4 Cross-Property Connection 634
7.6 Radiation Damage in Austenitic Steel 636
7.6.1 Microstructural Changes in Irradiated Steel 637
7.6.2 Changes in the Effective Elastic and Conductive Properties Due to Radiation-Induced Swelling 639
7.6.3 Cross-Property Connections 641
7.7 Porous Microcracked Ceramics 644
7.7.1 Microstresses and Microcracking Generated by Cooling of Polycrystalline Ceramics 644
7.7.2 Estimation of Strength of Intergranular Interfaces 652
7.7.3 Nonlinear Behavior Under Compression 655
Appendix A: Components of Eshelby Tensor for Various Ellipsoidal Shapes 667
Appendix B: Details of Calculations in Section 4.3.6 687
Appendix C: Hypergeometric Functions and Quantities Entering Solutions (4.6.9) and (4.6.10) 692
Appendix D: Components of Collective Property Contribution Tensors for Two Spherical Pores 695
References 698