Micromechanics of Heterogeneous Materials (eBook)

Preface 6
Contents 10
1 Introduction 20
1.1 Classification of Composites and Nanocomposites 20
1.2 Effective Material and Field Characteristics 27
1.3 Homogenization of Random Structure CM 31
1.4 Overview of the Book 34
2 Foundations of Solid Mechanics 36
2.1 Elements of Tensor Analysis 36
2.2 The Theory of Strains and Stresses 39
2.3 Basic Equations of Solid Mechanics 43
2.4 Basic Equations of Thermoelasticity and Electroelasticity 50
2.5 Symmetry of Elastic Properties 59
2.6 Basic Equations of Thermoelastoplastic Deformations 66
3 Green’s Functions, Eshelby and Related Tensors 70
3.1 Static Green’s Function 70
3.2 The Second Derivative of Green’s Function and Related Problems 73
3.3 Dynamic Green’s and Related Functions 77
3.4 Inhomogeneity in an Elastic Medium 81
3.5 Ellipsoidal Inhomogeneity in the Elastic Medium 86
3.6 Eshelby Tensor 90
3.7 Coated Ellipsoidal Inclusion 97
3.8 Related Problems for Ellipsoidal Inhomogeneity in an Infinite Medium 104
4 Multiscale Analysis of the Multiple Interacting Inclusions Problem: Finite Number of Interacting Inclusions 114
4.1 Description of Numerical Approaches Used for Analyses of Multiple Interacting Heterogeneities 114
4.2 Basic Equations for Multiple Heterogeneities and Numerical Solution for One Inclusion 117
4.3 Volume Integral Equation Method 128
4.4 Hybrid VEE and BIE Method for Multiscale Analysis of Interacting Inclusions ( Macro Problem) 139
4.5 Complex Potentials Method for 2-D Problems 149
5 Statistical Description of Composite Materials 156
5.1 Basic Terminology and Properties of Random Variables and Random Point Fields 157
5.2 Some Random Point Field Distributions 167
5.3 Ensemble Averaging of Random Structures 177
5.4 Numerical Simulation of Random Structures 195
6 Effective Properties and Energy Methods in Thermoelasticity of Composite Materials 204
6.1 Effective Thermoelastic Properties 204
6.2 Effective Energy Functions 213
6.3 Some General Exact Results 218
6.4 Variational Principle of Hashin and Shtrikman 228
6.5 Bounds of Effective Elastic Moduli 231
6.6 Bounds of Effective Conductivity 245
6.7 Bounds of Effective Eigenstrain 247
7 General Integral Equations of Micromechanics of Composite Materials 250
7.1 General Integral Equations for Matrix Composites of Any Structure 251
7.2 Random Structure Composites 253
7.3 Doubly and Triply Periodical Structure Composites 260
7.4 Random Structure Composites with Long-Range Order 263
7.5 Triply Periodic Particulate Matrix Composites with Imperfect Unit Cells 265
7.6 Conclusion 267
8 Multiparticle Effective Field and Related Methods in Micromechanics of Random Structure Composites 268
8.1 Definitions of Effective Fields and Effective Field Hypotheses 269
8.2 Analytical Representation of Effective Thermoelastic Properties 277
8.3 One-Particle Approximation of the MEFM and Mori- Tanaka Approach 283
9 Some Related Methods in Micromechanics of Random Structure Composites 302
9.1 Related Perturbation Methods 302
9.2 Effective Medium Methods 310
9.3 Differential Methods 317
9.4 Estimation of Effective Properties of Composites with Nonellipsoidal Inclusions 322
9.5 Numerical Results 325
9.6 Discussion 333
10 Generalization of the MEFM in Random Structure Matrix Composites 334
10.1 Two Inclusions in an Infinite Matrix 335
10.2 Composite Material 338
10.3 First-order Approximation of the Closing Assumption and Effective Elastic Moduli 343
10.4 Abandonment from the Approximative Hypothesis ( 10.26) 351
10.5 Some Particular Cases 353
10.6 Some Particular Numerical Results 361
11 Periodic Structures and Periodic Structures with Random Imperfections 366
11.1 General Analysis of Periodic Structures and Periodic Structures with Random Imperfections 366
11.2 Triply Periodical Particular Matrix Composites in Varying External Stress Field 370
11.3 Graded Doubly Periodical Particular Matrix Composites in Varying External Stress Field 380
11.4 Triply Periodic Particulate Matrix Composites with Imperfect Unit Cells 390
12 Nonlocal Effects in Statistically Homogeneous and Inhomogeneous Random Structure composites 404
12.1 General Analysis of Approaches in Nonlocal Micromechanics of Random Structure Composites 404
12.2 The Nonlocal Integral Equation 409
12.3 Methods for the Solution of the Nonlocal Integral Equation 411
12.4 Average Stresses in the Components and Effective Properties for Statistically Homogeneous Media 415
12.5 Effective Properties of Statistically Inhomogeneous Media 422
12.6 Concluding Remarks 433
13 Stress Fluctuations in Random Structure Composites 436
13.1 Perturbation Method 438
13.2 Method of Integral Equations 446
13.3 Elastically Homogeneous Composites with Randomly Distributed Residual Microstresses 453
13.4 Stress Fluctuations Near a Crack Tip in Elastically Homogeneous Materials with Randomly Distributed Residual Microstresses 459
13.5 Concluding Remarks 468
14 Random Structure Matrix Composites in a Half- Space 470
14.1 General Analysis of Approaches in Micromechanics of Random Structure Composites in a Half- space 470
14.2 General Integral Equation, Definitions of the Nonlocal Effective Properties, and Averaging Operations 474
14.3 Finite Number of Inclusions in a Half-Space 477
14.4 Nonlocal Effective Operators of Thermoelastic Properties of Microinhomogeneous Half- Space 481
14.5 Statistical Properties of Local Residual Microstresses in Elastically Homogeneous Half- Space 488
14.6 Numerical Results 493
15 Effective Limiting Surfaces in the Theory of Nonlinear Composites 500
15.1 Local Limiting Surface 501
15.2 Effective Limiting Surface 504
15.3 Numerical Results 511
15.4 Concluding Remarks 522
16 Nonlinear Composites 524
16.1 Nonlinear Elastic Composites 525
16.2 Deformation Plasticity Theory of Composite Materials 532
16.3 Power-Law Creep 536
16.4 Elastic–Plastic Behavior of Elastically Homogeneous Materials with Random Residual Microstresses 540
16.5 A Local Theory of Elastoplastic Deformations of Metal Matrix Composites 546
17 Some related problems 556
17.1 Conductivity 556
17.2 Thermoelectroelasticity of Composites 568
17.3 Wave Propagation in a Composite Material 580
18 Multiscale Mechanics of Nanocomposites 590
18.1 Elements of Molecular Dynamic (MD) Simulation 590
18.2 Bridging Nanomechanics to Micromechanics in Nanocomposites 597
18.3 Modeling of Nanofiber NCs in the Framework of Continuum Mechanics 601
18.4 Modeling of Clay NCs in the Framework of Continuum Mechanics 609
18.5 Some Related Problems in Modeling of NCs Reinforced with NFs and Nanoplates 621
19 Conclusion. Critical Analysis of Some Basic Concepts of Micromechanics 626
A Appendix 630
A.1 Parametric Representation of Rotation Matrix 630
A.2 Second and Fourth-Order Tensors of Special Structures 632
A.3 Analytical Representation of Some Tensors 638
References 642
Index 698