1. Introduction.- 2. Medium of Simple Structure.- 2.1 Quasicontinuum.- 2.2 Equations of Motion.- 2.3 Elastic Energy Operator.- 2.4 Symmetric Stress Tensor and Energy Density.- 2.5 Homogeneous Media.- 2.6 Approximate Models.- 2.7 Cubic Lattice.- 2.8 Isotropic Homogeneous Medium.- 2.9 Debye Quasicontinuum.- 2.10 Boundary-Value Problems and Surface Waves.- 2.11 Notes.- 3. Medium of Complex Structure.- 3.1 Equations of Motion.- 3.2 Energy Operator.- 3.3 Approximate Models and Comparison with Couple-Stress Theories.- 3.4 Exclusion of Internal Degrees of Freedom in the Acoustic Region.- 3.5 Cosserat Model.- 3.6 Notes.- 4. Local Defects.- 4.1 General Scheme.- 4.2 Impurity Atom in a Lattice.- 4.3 Point Defects in a Quasicontinuum.- 4.4 System of Point Defects.- 4.5 Local Inhomogeneity in an Elastic Medium.- 4.6 Homogeneous Elastic Medium.- 4.7 The Interface of Two Media.- 4.8 Integral Equations for an Inhomogeneous Medium.- 4.9 Ellipsoidal Inhomogeneity.- 4.10 Ellipsoidal Crack and Needle.- 4.11 Crack in a Homogeneous Medium.- 4.12 Elliptic Crack.- 4.13 Interaction Between Ellipsoidal Inhomogeneities.- 4.14 Notes.- 5. Internal Stress and Point Defects.- 5.1 Internal Stress in the Nonlocal Theory.- 5.2 Geometry of a Medium with Sources of Internal Stress.- 5.3 Green´s Tensors for Internal Stress.- 5.4 Isolated Point Defect.- 5.5 System of Point Defects.- 5.6 Notes.- 6. Dislocations.- 6.1 Elements of the Continuum Theory of Dislocations.- 6.2 Some Three-Dimensional Problems.- 6.3 Two-Dimensional Problems.- 6.4 Screw Dislocations.- 6.5 Influence of Change of the Force Constants in Cores of Screw Dislocations.- 6.6 Edge Dislocations.- 6.7 Notes.- 7. Elastic Medium with Random Fields of Inhomogeneities.- 7.1 Background.- 7.2 Formulation of the Problem.- 7.3 The Effective Field.- 7.4 Several Mean Values of Homogeneous Random Fields.- 7.5 General Scheme for Constructing First Statistical Moments of the Solution.- 7.6 Random Field of Ellipsoidal Inhomogeneities.- 7.7 Regular Structures.- 7.8 Fields of Elliptic Cracks.- 7.9 Two-Dimensional Systems of Rectilinear Cuts.- 7.10 Random Field of Point Defects.- 7.11 Correlation Functions in the Approximation by Point Defects.- 7.12 Conclusions.- 7.13 Notes.- Appendices.- A 1. Fourth-Order Tensors of Special Structure.- A 2. Green´s Operators of Elasticity.- A 4. Calculation of Certain Conditional Means.- References.
Crystals and polycrystals, composites and polymers, grids and multibar systems can be considered as examples of media with microstructure. A characteristic feature of all such models is the existence of scale parameters which are con nected with microgeometry or long-range interacting forces. As a result the cor responding theory must essentially be a nonlocal one. This treatment provides a systematic investigation of the effects of micro structure, inner degrees of freedom and non locality in elastic media. The prop agation of linear and nonlinear waves in dispersive media, static, deterministic and stochastic problems, and the theory of local defects and dislocations are considered in detail. Especial attention is paid to approximate models and lim iting transitions to classical elasticity. The book forms the second part of a revised and updated edition of the author's monograph published under the same title in Russian in 1975. The first part (Vol. 26 of Springer Series in Solid-State Sciences) presents a self contained theory of one-dimensional models. The theory of three-dimensional models is considered in this volume. I would like to thank E. Kroner and A. Seeger for supporting the idea of an English edition of my original Russian book. I am also grateful to E. Borie, H. Lotsch and H. Zorski who read the manuillegalscript and offered many sugges tions. Houston, Texas Isaak A. Kunin January, 1983 Contents 1. Introduction ............................................ .
1. Introduction.- 2. Medium of Simple Structure.- 2.1 Quasicontinuum.- 2.2 Equations of Motion.- 2.3 Elastic Energy Operator.- 2.4 Symmetric Stress Tensor and Energy Density.- 2.5 Homogeneous Media.- 2.6 Approximate Models.- 2.7 Cubic Lattice.- 2.8 Isotropic Homogeneous Medium.- 2.9 Debye Quasicontinuum.- 2.10 Boundary-Value Problems and Surface Waves.- 2.11 Notes.- 3. Medium of Complex Structure.- 3.1 Equations of Motion.- 3.2 Energy Operator.- 3.3 Approximate Models and Comparison with Couple-Stress Theories.- 3.4 Exclusion of Internal Degrees of Freedom in the Acoustic Region.- 3.5 Cosserat Model.- 3.6 Notes.- 4. Local Defects.- 4.1 General Scheme.- 4.2 Impurity Atom in a Lattice.- 4.3 Point Defects in a Quasicontinuum.- 4.4 System of Point Defects.- 4.5 Local Inhomogeneity in an Elastic Medium.- 4.6 Homogeneous Elastic Medium.- 4.7 The Interface of Two Media.- 4.8 Integral Equations for an Inhomogeneous Medium.- 4.9 Ellipsoidal Inhomogeneity.- 4.10 Ellipsoidal Crack and Needle.- 4.11 Crack in a Homogeneous Medium.- 4.12 Elliptic Crack.- 4.13 Interaction Between Ellipsoidal Inhomogeneities.- 4.14 Notes.- 5. Internal Stress and Point Defects.- 5.1 Internal Stress in the Nonlocal Theory.- 5.2 Geometry of a Medium with Sources of Internal Stress.- 5.3 Green's Tensors for Internal Stress.- 5.4 Isolated Point Defect.- 5.5 System of Point Defects.- 5.6 Notes.- 6. Dislocations.- 6.1 Elements of the Continuum Theory of Dislocations.- 6.2 Some Three-Dimensional Problems.- 6.3 Two-Dimensional Problems.- 6.4 Screw Dislocations.- 6.5 Influence of Change of the Force Constants in Cores of Screw Dislocations.- 6.6 Edge Dislocations.- 6.7 Notes.- 7. Elastic Medium with Random Fields of Inhomogeneities.- 7.1 Background.- 7.2 Formulation of the Problem.- 7.3 The Effective Field.-7.4 Several Mean Values of Homogeneous Random Fields.- 7.5 General Scheme for Constructing First Statistical Moments of the Solution.- 7.6 Random Field of Ellipsoidal Inhomogeneities.- 7.7 Regular Structures.- 7.8 Fields of Elliptic Cracks.- 7.9 Two-Dimensional Systems of Rectilinear Cuts.- 7.10 Random Field of Point Defects.- 7.11 Correlation Functions in the Approximation by Point Defects.- 7.12 Conclusions.- 7.13 Notes.- Appendices.- A 1. Fourth-Order Tensors of Special Structure.- A 2. Green's Operators of Elasticity.- A 4. Calculation of Certain Conditional Means.- References.
Inhaltsverzeichnis
1. Introduction.- 2. Medium of Simple Structure.- 2.1 Quasicontinuum.- 2.2 Equations of Motion.- 2.3 Elastic Energy Operator.- 2.4 Symmetric Stress Tensor and Energy Density.- 2.5 Homogeneous Media.- 2.6 Approximate Models.- 2.7 Cubic Lattice.- 2.8 Isotropic Homogeneous Medium.- 2.9 Debye Quasicontinuum.- 2.10 Boundary-Value Problems and Surface Waves.- 2.11 Notes.- 3. Medium of Complex Structure.- 3.1 Equations of Motion.- 3.2 Energy Operator.- 3.3 Approximate Models and Comparison with Couple-Stress Theories.- 3.4 Exclusion of Internal Degrees of Freedom in the Acoustic Region.- 3.5 Cosserat Model.- 3.6 Notes.- 4. Local Defects.- 4.1 General Scheme.- 4.2 Impurity Atom in a Lattice.- 4.3 Point Defects in a Quasicontinuum.- 4.4 System of Point Defects.- 4.5 Local Inhomogeneity in an Elastic Medium.- 4.6 Homogeneous Elastic Medium.- 4.7 The Interface of Two Media.- 4.8 Integral Equations for an Inhomogeneous Medium.- 4.9 Ellipsoidal Inhomogeneity.- 4.10 Ellipsoidal Crack and Needle.- 4.11 Crack in a Homogeneous Medium.- 4.12 Elliptic Crack.- 4.13 Interaction Between Ellipsoidal Inhomogeneities.- 4.14 Notes.- 5. Internal Stress and Point Defects.- 5.1 Internal Stress in the Nonlocal Theory.- 5.2 Geometry of a Medium with Sources of Internal Stress.- 5.3 Green's Tensors for Internal Stress.- 5.4 Isolated Point Defect.- 5.5 System of Point Defects.- 5.6 Notes.- 6. Dislocations.- 6.1 Elements of the Continuum Theory of Dislocations.- 6.2 Some Three-Dimensional Problems.- 6.3 Two-Dimensional Problems.- 6.4 Screw Dislocations.- 6.5 Influence of Change of the Force Constants in Cores of Screw Dislocations.- 6.6 Edge Dislocations.- 6.7 Notes.- 7. Elastic Medium with Random Fields of Inhomogeneities.- 7.1 Background.- 7.2 Formulation of the Problem.- 7.3 The Effective Field.- 7.4 Several Mean Values of Homogeneous Random Fields.- 7.5 General Scheme for Constructing First Statistical Moments of the Solution.- 7.6 Random Field of Ellipsoidal Inhomogeneities.- 7.7 Regular Structures.- 7.8 Fields of Elliptic Cracks.- 7.9 Two-Dimensional Systems of Rectilinear Cuts.- 7.10 Random Field of Point Defects.- 7.11 Correlation Functions in the Approximation by Point Defects.- 7.12 Conclusions.- 7.13 Notes.- Appendices.- A 1. Fourth-Order Tensors of Special Structure.- A 2. Green's Operators of Elasticity.- A 4. Calculation of Certain Conditional Means.- References.
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