Nonlinear Continuum Mechanics and Large Inelastic Deformations

1. Auflage 2010
ISBN: 978-94-007-0034-5
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 721 Seiten

The book provides a rigorous axiomatic approach to continuum mechanics under large deformation. In addition to the classical nonlinear continuum mechanics – kinematics, fundamental laws, the theory of functions having jump discontinuities across singular surfaces, etc. - the book presents the theory of co-rotational derivatives, dynamic deformation compatibility equations, and the principles of material indifference and symmetry, all in systematized form. The focus of the book is a new approach to the formulation of the constitutive equations for elastic and inelastic continua under large deformation. This new approach is based on using energetic and quasi-energetic couples of stress and deformation tensors. This approach leads to a unified treatment of large, anisotropic elastic, viscoelastic, and plastic deformations. The author analyses classical problems, including some involving nonlinear wave propagation, using different models for continua under large deformation, and shows how different models lead to different results. The analysis is accompanied by experimental data and detailed numerical results for rubber, the ground, alloys, etc. The book will be an invaluable text for graduate students and researchers in solid mechanics, mechanical engineering, applied mathematics, physics and crystallography, as also for scientists developing advanced materials.

Autoren/Hrsg.

Inhaltsverzeichnis

1;Preface;6
2;Contents;10
3;Chapter 1:Introduction: Fundamental Axioms of Continuum Mechanics;26
4;Chapter 2:Kinematics of Continua;30
4.1;2.1 Material and Spatial Descriptions of Continuum Motion;30
4.1.1;2.1.1 Lagrangian and Eulerian Coordinates: The Motion Law;30
4.1.2;2.1.2 Material and Spatial Descriptions;34
4.1.3;2.1.3 Local Bases in K and K;34
4.1.4;2.1.4 Tensors and Tensor Fields in Continuum Mechanics;36
4.1.5;2.1.5 Covariant Derivatives in K and K;38
4.1.6;2.1.6 The Deformation Gradient;39
4.1.7;2.1.7 Curvilinear Spatial Coordinates;41
4.2;2.2 Deformation Tensors and Measures;49
4.2.1;2.2.1 Deformation Tensors;49
4.2.2;2.2.2 Deformation Measures;50
4.2.3;2.2.3 Displacement Vector;50
4.2.4;2.2.4 Relations Between Components of Deformation Tensors and Displacement Vector;51
4.2.5;2.2.5 Physical Meaning of Components of the Deformation Tensor;53
4.2.6;2.2.6 Transformation of an Oriented Surface Element;55
4.2.7;2.2.7 Representation of the Inverse Metric Matrix in terms of Components of the Deformation Tensor;58
4.3;2.3 Polar Decomposition;61
4.3.1;2.3.1 Theorem on Polar Decomposition;61
4.3.2;2.3.2 Eigenvalues and Eigenbases;65
4.3.3;2.3.3 Representation of the Deformation Tensors in Eigenbases;67
4.3.4;2.3.4 Geometrical Meaning of Eigenvalues;69
4.3.5;2.3.5 Geometric Picture of Transformation of a Small Neighborhood of a Point of a Continuum;70
4.4;2.4 Rate Characteristics of Continuum Motion;74
4.4.1;2.4.1 Velocity;74
4.4.2;2.4.2 Total Derivative of a Tensor with Respect to Time;75
4.4.3;2.4.3 Differential of a Tensor;78
4.4.4;2.4.4 Properties of Derivatives with Respect to Time;79
4.4.5;2.4.5 The Velocity Gradient, the Deformation Rate Tensor and the Vorticity Tensor;81
4.4.6;2.4.6 Eigenvalues of the Deformation Rate Tensor;83
4.4.7;2.4.7 Resolution of the Vorticity Tensor for the Eigenbasis of the Deformation Rate Tensor;84
4.4.8;2.4.8 Geometric Picture of Infinitesimal Transformation of a Small Neighborhood of a Point;85
4.4.9;2.4.9 Kinematic Meaning of the Vorticity Vector;87
4.4.10;2.4.10 Tensor of Angular Rate of Rotation (Spin);88
4.4.11;2.4.11 Relationships Between Rates of Deformation Tensors and Velocity Gradients;90
4.4.12;2.4.12 Trajectory of a Material Point, Streamlineand Vortex Line;98
4.4.13;2.4.13 Stream Tubes and Vortex Tubes;100
4.5;2.5 Co-rotational Derivatives;102
4.5.1;2.5.1 Definition of Co-rotational Derivatives;102
4.5.2;2.5.2 The Oldroyd Derivative (hi = ri);104
4.5.3;2.5.3 The Cotter--Rivlin Derivative (hi = ri);105
4.5.4;2.5.4 Mixed Co-rotational Derivatives;106
4.5.5;2.5.5 The Derivative Relative to the Eigenbasis pi of the Right Stretch Tensor;106
4.5.6;2.5.6 The Derivative in the Eigenbasis (hi =pi) of the Left Stretch Tensor;107
4.5.7;2.5.7 The Jaumann Derivative (hi = qi);108
4.5.8;2.5.8 Co-rotational Derivatives in a Moving Orthonormal Basis;108
4.5.9;2.5.9 Spin Derivative;109
4.5.10;2.5.10 Universal Form of the Co-rotational Derivatives;110
4.5.11;2.5.11 Relations Between Co-rotational Derivatives of Deformation Rate Tensors and Velocity Gradient;110
5;Chapter 3:Balance Laws;114
5.1;3.1 The Mass Conservation Law;114
5.1.1;3.1.1 Integral and Differential Forms;114
5.1.2;3.1.2 The Continuity Equation in Lagrangian Variables;115
5.1.3;3.1.3 Differentiation of Integral over a Moving Volume;116
5.1.4;3.1.4 The Continuity Equation in Eulerian Variables;117
5.1.5;3.1.5 Determination of the Total Derivatives with respect to Time;118
5.1.6;3.1.6 The Gauss--Ostrogradskii Formulae;119
5.2;3.2 The Momentum Balance Law and the Stress Tensor;120
5.2.1;3.2.1 The Momentum Balance Law;120
5.2.2;3.2.2 External and Internal Forces;122
5.2.3;3.2.3 Cauchy's Theorems on Properties of the Stress Vector;123
5.2.4;3.2.4 Generalized Cauchy's Theorem;126
5.2.5;3.2.5 The Cauchy and Piola--Kirchhoff Stress Tensors;127
5.2.6;3.2.6 Physical Meaning of Components of the Cauchy Stress Tensor;128
5.2.7;3.2.7 The Momentum Balance Equation in Spatial and Material Descriptions;132
5.3;3.3 The Angular Momentum Balance Law;134
5.3.1;3.3.1 The Integral Form;134
5.3.2;3.3.2 Tensor of Moment Stresses;135
5.3.3;3.3.3 Differential Form of the Angular Momentum Balance Law;136
5.3.4;3.3.4 Nonpolar and Polar Continua;137
5.3.5;3.3.5 The Angular Momentum Balance Equation in the Material Description;138
5.4;3.4 The First Thermodynamic Law;139
5.4.1;3.4.1 The Integral Form of the Energy Balance Law;139
5.4.2;3.4.2 The Heat Flux Vector;141
5.4.3;3.4.3 The Energy Balance Equation;142
5.4.4;3.4.4 Kinetic Energy and Heat Influx Equation;143
5.4.5;3.4.5 The Energy Balance Equation in LagrangianDescription;144
5.4.6;3.4.6 The Energy Balance Law for Polar Continua;146
5.5;3.5 The Second Thermodynamic Law;149
5.5.1;3.5.1 The Integral Form;149
5.5.2;3.5.2 Differential Form of the Second Thermodynamic Law;151
5.5.3;3.5.3 The Second Thermodynamic Law in the Material Description;152
5.5.4;3.5.4 Heat Machines and Their Efficiency;153
5.5.5;3.5.5 Adiabatic and Isothermal Processes. The Carnot Cycles;157
5.5.6;3.5.6 Truesdell's Theorem;161
5.6;3.6 Deformation Compatibility Equations;166
5.6.1;3.6.1 Compatibility Conditions;166
5.6.2;3.6.2 Integrability Condition for Differential Form;167
5.6.3;3.6.3 The First Form of Deformation CompatibilityConditions;167
5.6.4;3.6.4 The Second Form of Compatibility Conditions;168
5.6.5;3.6.5 The Third Form of Compatibility Conditions;170
5.6.6;3.6.6 Properties of Components of the Riemann--Christoffel Tensor;171
5.6.7;3.6.7 Interchange of the Second Covariant Derivatives;173
5.6.8;3.6.8 The Static Compatibility Equation;173
5.7;3.7 Dynamic Compatibility Equations;174
5.7.1;3.7.1 Dynamic Compatibility Equations in Lagrangian Description;174
5.7.2;3.7.2 Dynamic Compatibility Equations in Spatial Description;176
5.8;3.8 Compatibility Equations for Deformation Rates;177
5.9;3.9 The Complete System of Continuum Mechanics Laws;180
5.9.1;3.9.1 The Complete System in Eulerian Description;180
5.9.2;3.9.2 The Complete System in Lagrangian Description;181
5.9.3;3.9.3 Integral Form of the System of Continuum Mechanics Laws;182
6;Chapter 4:Constitutive Equations;185
6.1;4.1 Basic Principles for Derivation of Constitutive Equations;185
6.2;4.2 Energetic and Quasienergetic Couples of Tensors;186
6.2.1;4.2.1 Energetic Couples of Tensors;186
6.2.2;4.2.2 The First Energetic Couple (TI,);188
6.2.3;4.2.3 The Fifth Energetic Couple (TV , C);189
6.2.4;4.2.4 The Fourth Energetic Couple (TIV, (U -E));190
6.2.5;4.2.5 The Second Energetic Couple (TII, (E-U-1));191
6.2.6;4.2.6 The Third Energetic Couple (TIII, B);191
6.2.7;4.2.7 General Representations for Energetic Tensors of Stresses and Deformations;192
6.2.8;4.2.8 Energetic Deformation Measures;197
6.2.9;4.2.9 Relationships Between Principal Invariants of Energetic Deformation Measures and Tensors;199
6.2.10;4.2.10 Quasienergetic Couples of Stress and Deformation Tensors;200
6.2.11;4.2.11 The First Quasienergetic Couple (SI, A);201
6.2.12;4.2.12 The Second Quasienergetic Couple (SII, (E -V-1));202
6.2.13;4.2.13 The Third Quasienergetic Couple (Y, TS);202
6.2.14;4.2.14 The Fourth Quasienergetic Couple (SIV, (V-E));203
6.2.15;4.2.15 The Fifth Quasienergetic Couple (SV, J);204
6.2.16;4.2.16 General Representation of Quasienergetic Tensors;204
6.2.17;4.2.17 Quasienergetic Deformation Measures;206
6.2.18;4.2.18 Representation of Rotation Tensor of Stresses in Terms of Quasienergetic Couples of Tensors;207
6.2.19;4.2.19 Relations Between Density and Principal Invariants of Energetic and Quasienergetic Deformation Tensors;209
6.2.20;4.2.20 The Generalized Form of Representation of the Stress Power;210
6.2.21;4.2.21 Representation of Stress Power in Terms of Co-rotational Derivatives;211
6.2.22;4.2.22 Relations Between Rates of Energetic and Quasienergetic Tensors and Velocity Gradient;212
6.3;4.3 The Principal Thermodynamic Identity;220
6.3.1;4.3.1 Different Forms of the Principal Thermodynamic Identity;220
6.3.2;4.3.2 The Clausius--Duhem Inequality;222
6.3.3;4.3.3 The Helmholtz Free Energy;222
6.3.4;4.3.4 The Gibbs Free Energy;223
6.3.5;4.3.5 Enthalpy;225
6.3.6;4.3.6 Universal Form of the Principal Thermodynamic Identity;226
6.3.7;4.3.7 Representation of the Principal Thermodynamic Identity in Terms of Co-rotational Derivatives;227
6.4;4.4 Principles of Thermodynamically Consistent Determinism, Equipresence and Local Action;229
6.4.1;4.4.1 Active and Reactive Variables;229
6.4.2;4.4.2 The Principle of Thermodynamically Consistent Determinism;230
6.4.3;4.4.3 The Principle of Equipresence;232
6.4.4;4.4.4 The Principle of Local Action;232
6.5;4.5 Definition of Ideal Continua;233
6.5.1;4.5.1 Classification of Types of Continua;233
6.5.2;4.5.2 General Form of Constitutive Equations for Ideal Continua;234
6.6;4.6 The Principle of Material Symmetry;237
6.6.1;4.6.1 Different Reference Configurations;237
6.6.2;4.6.2 H-indifferent and H-invariant Tensors;239
6.6.3;4.6.3 Symmetry Groups of Continua;243
6.6.4;4.6.4 The Statement of the Principle of Material Symmetry;244
6.7;4.7 Definition of Fluids and Solids;245
6.7.1;4.7.1 Fluids and Solids;245
6.7.2;4.7.2 Isomeric Symmetry Groups;246
6.7.3;4.7.3 Definition of Anisotropic Solids;250
6.7.4;4.7.4 H-indifference and H-invariance of Tensors Describing the Motion of a Solid;252
6.7.5;4.7.5 H-invariance of Rate Characteristics of a Solid;255
6.8;4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations for Ideal Continua;260
6.8.1;4.8.1 Corollary of the Principle of Material Symmetry for Models An of Ideal (Elastic) Solids;260
6.8.2;4.8.2 Scalar Indifferent Functions of Tensor Argument;261
6.8.3;4.8.3 Producing Tensors of Groups;263
6.8.4;4.8.4 Scalar Invariants of a Second-Order Tensor;264
6.8.5;4.8.5 Representation of a Scalar Indifferent Function in Terms of Invariants;267
6.8.6;4.8.6 Indifferent Tensor Functions of Tensor Argument and Invariant Representation of Constitutive Equations for Elastic Continua;268
6.8.7;4.8.7 Quasilinear and Linear Models An of Elastic Continua;273
6.8.8;4.8.8 Constitutive Equations for Models Bn of Elastic Continua;277
6.8.9;4.8.9 Corollaries to the Principle of Material Symmetry for Models Cn and Dn of Elastic Continua;278
6.8.10;4.8.10 General Representation of Constitutive Equations for All Models of Elastic Continua;286
6.8.11;4.8.11 Representation of Constitutive Equations of Isotropic Elastic Continua in Eigenbases;289
6.8.12;4.8.12 Representation of Constitutive Equations of Isotropic Elastic Continua `in Rates';295
6.8.13;4.8.13 Application of the Principle of Material Symmetry to Fluids;300
6.8.14;4.8.14 Functional Energetic Couples of Tensors;307
6.9;4.9 Incompressible Continua;311
6.9.1;4.9.1 Definition of Incompressible Continua;311
6.9.2;4.9.2 The Principal Thermodynamic Identity for Incompressible Continua;312
6.9.3;4.9.3 Constitutive Equations for Ideal Incompressible Continua;313
6.9.4;4.9.4 Corollaries to the Principle of Material Symmetry for Incompressible Fluids;315
6.9.5;4.9.5 Representation of Constitutive Equations for Incompressible Solids in Tensor Bases;316
6.9.6;4.9.6 General Representation of Constitutive Equations for All the Models of Incompressible Ideal Solids;319
6.9.7;4.9.7 Linear Models of Ideal Incompressible Elastic Continua;320
6.9.8;4.9.8 Representation of Models Bn and Dn of Incompressible Isotropic Elastic Continua in Eigenbasis;322
6.10;4.10 The Principle of Material Indifference;324
6.10.1;4.10.1 Rigid Motion;324
6.10.2;4.10.2 R-indifferent and R-invariant Tensors;325
6.10.3;4.10.3 Density and Deformation Gradient in Rigid Motion;326
6.10.4;4.10.4 Deformation Tensors in Rigid Motion;327
6.10.5;4.10.5 Stress Tensors in Rigid Motion;328
6.10.6;4.10.6 The Velocity in Rigid Motion;330
6.10.7;4.10.7 The Deformation Rate Tensorand the Vorticity Tensor in Rigid Motion;330
6.10.8;4.10.8 Co-rotational Derivatives in Rigid Motion;331
6.10.9;4.10.9 The Statement of the Principle of Material Indifference;336
6.10.10;4.10.10 Material Indifference of the Continuity Equation;337
6.10.11;4.10.11 Material Indifference for the MomentumBalance Equation;337
6.10.12;4.10.12 Material Indifference of the Thermodynamic Laws;340
6.10.13;4.10.13 Material Indifference of the Compatibility Equations;342
6.10.14;4.10.14 Material Indifference of Models An and Bn of Ideal Continua;343
6.10.15;4.10.15 Material Indifference for Models Cn and Dn of Ideal Continua;344
6.10.16;4.10.16 Material Indifference for Incompressible Continua;346
6.10.17;4.10.17 Material Indifference for Models of Solids `in Rates';346
6.11;4.11 Relationships in a Moving System;348
6.11.1;4.11.1 A Moving Reference System;348
6.11.2;4.11.2 The Euler Formula;350
6.11.3;4.11.3 The Coriolis Formula;351
6.11.4;4.11.4 The Nabla-Operator in a Moving System;353
6.11.5;4.11.5 The Velocity Gradient in a Moving System;354
6.11.6;4.11.6 The Continuity Equation in a Moving System;354
6.11.7;4.11.7 The Momentum Balance Equation in a Moving System;355
6.11.8;4.11.8 The Thermodynamic Laws in a Moving System;355
6.11.9;4.11.9 The Equation of Deformation Compatibility in a Moving System;356
6.11.10;4.11.10 The Kinematic Equation in a Moving System;359
6.11.11;4.11.11 The Complete System of Continuum Mechanics Laws in a Moving Coordinate System;359
6.11.12;4.11.12 Constitutive Equations in a Moving System;359
6.11.13;4.11.13 General Remarks;362
6.12;4.12 The Onsager Principle;363
6.12.1;4.12.1 The Onsager Principle and the Fourier Law;363
6.12.2;4.12.2 Corollaries of the Principle of Material Symmetry for the Fourier Law;365
6.12.3;4.12.3 Corollary of the Principle of Material Indifference for the Fourier Law;367
6.12.4;4.12.4 The Fourier Law for Fluids;368
6.12.5;4.12.5 The Fourier Law for Solids;369
7;Chapter 5:Relations at Singular Surfaces;371
7.1;5.1 Relations at a Singular Surface in the Material Description;371
7.1.1;5.1.1 Singular Surfaces;371
7.1.2;5.1.2 The First Classification of Singular Surfaces;371
7.1.3;5.1.3 Axiom on the Class of Functions across a Singular Surface;374
7.1.4;5.1.4 The Rule of Differentiation of a Volume Integral in the Presence of a Singular Surface;376
7.1.5;5.1.5 Relations at a Coherent Singular Surface in K;379
7.1.6;5.1.6 Relation Between Velocities of a Singular Surface in K and K;381
7.2;5.2 Relations at a Singular Surface in the Spatial Description;382
7.2.1;5.2.1 Relations at a Coherent Singular Surface in the Spatial Description;382
7.2.2;5.2.2 The Rule of Differentiation of an Integral over a Moving Volume Containing a Singular Surface;384
7.3;5.3 Explicit Form of Relations at a Singular Surface;386
7.3.1;5.3.1 Explicit Form of Relations at a Surface of a Strong Discontinuity in a Reference Configuration;386
7.3.2;5.3.2 Explicit Form of Relations at a Surface of a Strong Discontinuity in an Actual Configuration;387
7.3.3;5.3.3 Mass Rate of Propagation of a Singular Surface;387
7.3.4;5.3.4 Relations at a Singular Surface Without Transition of Material Points;389
7.4;5.4 The Main Types of Singular Surfaces;390
7.4.1;5.4.1 Jump of Density;390
7.4.2;5.4.2 Jumps of Radius-Vector and Displacement Vector;391
7.4.3;5.4.3 Semicoherent and Completely Incoherent Singular Surfaces;393
7.4.4;5.4.4 Nondissipative and Homothermal Singular Surfaces;393
7.4.5;5.4.5 Surfaces with Ideal Contact;394
7.4.6;5.4.6 On Boundary Conditions;396
7.4.7;5.4.7 Equation of a Singular Surface in K;396
7.4.8;5.4.8 Equation of a Singular Surface in K;398
8;Chapter 6:Elastic Continua at Large Deformations;400
8.1;6.1 Closed Systems in the Spatial Description;400
8.1.1;6.1.1 RUVF-system of Thermoelasticity;400
8.1.2;6.1.2 RVF-, RUV-, and UV-Systems of Dynamic Equations of Thermoelasticity;404
8.1.3;6.1.3 TRUVF-system of Dynamic Equations of Thermoelasticity;407
8.1.4;6.1.4 Component Form of the Dynamic Equation System of Thermoelasticity in the Spatial Description;408
8.1.5;6.1.5 The Model of Quasistatic Processes in Elastic Solids at Large Deformations;411
8.2;6.2 Closed Systems in the Material Description;413
8.2.1;6.2.1 UVF-system of Dynamic Equations of Thermoelasticity in the Material Description;413
8.2.2;6.2.2 UV- and U-systems of Thermoelasticity in the Material Description;417
8.2.3;6.2.3 TUVF-system of Thermoelasticity in the Material Description;418
8.2.4;6.2.4 The Equation System of Thermoelasticity for Quasistatic Processes in the Material Description;419
8.3;6.3 Statements of Problems for Elastic Continua at LargeDeformations;422
8.3.1;6.3.1 Boundary Conditions in the Spatial Description;422
8.3.2;6.3.2 Boundary Conditions in the Material Description;425
8.3.3;6.3.3 Statements of Main Problems of Thermoelasticity at Large Deformations in the Spatial Description;428
8.3.4;6.3.4 Statements of Thermoelasticity Problems in the Material Description;431
8.3.5;6.3.5 Statements of Quasistatic Problems of Elasticity Theory at Large Deformations;433
8.3.6;6.3.6 Conditions on External Forces in Quasistatic Problems;435
8.3.7;6.3.7 Variational Statement of the Quasistatic Problem in the Spatial Description;436
8.3.8;6.3.8 Variational Statement of Quasistatic Problem in the Material Description;440
8.3.9;6.3.9 Variational Statement for Incompressible Continua in the Material Description;440
8.4;6.4 The Problem on an Elastic Beam in Tension;444
8.4.1;6.4.1 Semi-Inverse Method;444
8.4.2;6.4.2 Deformation of a Beam in Tension;444
8.4.3;6.4.3 Stresses in a Beam;445
8.4.4;6.4.4 The Boundary Conditions;447
8.4.5;6.4.5 Resolving Relation 1 k1;447
8.4.6;6.4.6 Comparative Analysis of Different Models An;448
8.5;6.5 Tension of an Incompressible Beam;453
8.5.1;6.5.1 Deformation of an Incompressible Elastic Beam;453
8.5.2;6.5.2 Stresses in an Incompressible Beam for Models Bn;454
8.5.3;6.5.3 Resolving Relation 1(k1);455
8.5.4;6.5.4 Comparative Analysis of Models Bn;455
8.5.5;6.5.5 Stresses in an Incompressible Beam for Models An;459
8.6;6.6 Simple Shear;461
8.6.1;6.6.1 Deformations in Simple Shear;461
8.6.2;6.6.2 Stresses in the Problem on Shear;462
8.6.3;6.6.3 Boundary Conditions in the Problem on Shear;464
8.6.4;6.6.4 Comparative Analysis of Different Models An for the Problem on Shear;465
8.6.5;6.6.5 Shear of an Incompressible Elastic Continuum;466
8.7;6.7 The Lamé Problem;469
8.7.1;6.7.1 The Motion Law for a Pipe in the Lamé Problem;469
8.7.2;6.7.2 The Deformation Gradient and Deformation Tensors in the Lamé Problem;471
8.7.3;6.7.3 Stresses in the Lamé Problem for Models An;472
8.7.4;6.7.4 Equation for the Function f;473
8.7.5;6.7.5 Boundary Conditions of the Weak Type;474
8.7.6;6.7.6 Boundary Conditions of the Rigid Type;476
8.8;6.8 The Lamé Problem for an Incompressible Continuum;477
8.8.1;6.8.1 Equation for the Function f;477
8.8.2;6.8.2 Stresses in the Lamé Problem for an Incompressible Continuum;477
8.8.3;6.8.3 Equation for Hydrostatic Pressure p;479
8.8.4;6.8.4 Analysis of the Problem Solution;479
9;Chapter 7:Continua of the Differential Type;484
9.1;7.1 Models An and Bn of Continua of the Differential Type;484
9.1.1;7.1.1 Constitutive Equations for Models An of Continua of the Differential Type;484
9.1.2;7.1.2 Corollary of the Onsager Principle for Models An of Continua of the Differential Type;486
9.1.3;7.1.3 The Principle of Material Symmetry for Models An of Continua of the Differential Type;488
9.1.4;7.1.4 Representation of Constitutive Equations for Models An of Solids of the Differential Type in Tensor Bases;490
9.1.5;7.1.5 Models Bn of Solids of the Differential Type;493
9.1.6;7.1.6 Models Bn of Incompressible Continua of the Differential Type;494
9.1.7;7.1.7 The Principle of Material Indifference for Models An and Bn of Continua of the Differential Type;495
9.2;7.2 Models An and Bn of Fluids of the Differential Type;496
9.2.1;7.2.1 Tensor of Equilibrium Stresses for Fluids of the Differential Type;496
9.2.2;7.2.2 The Tensor of Viscous Stresses in the Model AI of a Fluid of the Differential Type;497
9.2.3;7.2.3 Simultaneous Invariants for Fluids of the Differential Type;499
9.2.4;7.2.4 Tensor of Viscous Stresses in Model AV of Fluids of the Differential Type;501
9.2.5;7.2.5 Viscous Coefficients in Model AV of a Fluid of the Differential Type;502
9.2.6;7.2.6 General Representation of Constitutive Equations for Fluids of the Differential Type;503
9.2.7;7.2.7 Constitutive Equations for Incompressible Viscous Fluids;504
9.2.8;7.2.8 The Principle of Material Indifference for Models An and Bn of Fluids of the Differential Type;504
9.3;7.3 Models Cn and Dn of Continua of the Differential Type;505
9.3.1;7.3.1 Models Cn of Continua of the Differential Type;505
9.3.2;7.3.2 Models Cnh of Solids with Co-rotational Derivatives;507
9.3.3;7.3.3 Corollaries of the Principle of Material Symmetry for Models Cnh of Solids;508
9.3.4;7.3.4 Viscosity Tensor in Models Cnh;511
9.3.5;7.3.5 Final Representation of Constitutive Equations for Model Cnh of Isotropic Solids;512
9.3.6;7.3.6 Models Dnh of Isotropic Solids;514
9.4;7.4 The Problem on a Beam in Tension;515
9.4.1;7.4.1 Rate Characteristics of a Beam;515
9.4.2;7.4.2 Stresses in the Beam;515
9.4.3;7.4.3 Resolving Relation (k1, 1);516
9.4.4;7.4.4 Comparative Analysis of Creep Curves for Different Models Bn;516
9.4.5;7.4.5 Analysis of Deforming Diagrams for Different Models Bn of Continua of the Differential Type;519
10;Chapter 8:Viscoelastic Continua at Large Deformations;520
10.1;8.1 Viscoelastic Continua of the Integral Type;520
10.1.1;8.1.1 Definition of Viscoelastic Continua;520
10.1.2;8.1.2 Tensor Functional Space;521
10.1.3;8.1.3 Continuous and Differentiable Functionals;522
10.1.4;8.1.4 Axiom of Fading Memory;526
10.1.5;8.1.5 Models An of Viscoelastic Continua;528
10.1.6;8.1.6 Corollaries of the Principle of Material Symmetry for Models An of Viscoelastic Continua;529
10.1.7;8.1.7 General Representation of Functional of Free Energy in Models An;530
10.1.8;8.1.8 Model An of Stable Viscoelastic Continua;533
10.1.9;8.1.9 Model An of a Viscoelastic Continuum with Difference Cores;534
10.1.10;8.1.10 Model An of a Thermoviscoelastic Continuum;536
10.1.11;8.1.11 Model An of a Thermorheologically Simple Viscoelastic Continuum;537
10.2;8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua;539
10.2.1;8.2.1 Principal Models An of Viscoelastic Continua;539
10.2.2;8.2.2 Principal Model An of an Isotropic Thermoviscoelastic Continuum;541
10.2.3;8.2.3 Principal Model An of a Transversely Isotropic Thermoviscoelastic Continuum;542
10.2.4;8.2.4 Principal Model An of an Orthotropic Thermoviscoelastic Continuum;543
10.2.5;8.2.5 Quadratic Models An of Thermoviscoelastic Continua;545
10.2.6;8.2.6 Linear Models An of Viscoelastic Continua;545
10.2.7;8.2.7 Representation of Linear Models An in the Boltzmann Form;548
10.2.8;8.2.8 Mechanically Determinate Linear Models An of Viscoelastic Continua;551
10.2.9;8.2.9 Linear Models An for Isotropic Viscoelastic Continua;553
10.2.10;8.2.10 Linear Models An of Transversely Isotropic Viscoelastic Continua;554
10.2.11;8.2.11 Linear Models An of Orthotropic Viscoelastic Continua;555
10.2.12;8.2.12 The Tensor of Relaxation Functions;556
10.2.13;8.2.13 Spectral Representation of Linear Models An of Viscoelastic Continua;558
10.2.14;8.2.14 Exponential Relaxation Functions and Differential Form of Constitutive Equations;562
10.2.15;8.2.15 Inversion of Constitutive Equations for Linear Models of Viscoelastic Continua;565
10.3;8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids;572
10.3.1;8.3.1 Models An of Incompressible Viscoelastic Continua;572
10.3.2;8.3.2 Principal Models An of Incompressible Isotropic Viscoelastic Continua;572
10.3.3;8.3.3 Linear Models An of Incompressible Isotropic Viscoelastic Continua;574
10.3.4;8.3.4 Models Bn of Viscoelastic Continua;575
10.3.5;8.3.5 Models An and Bn of Viscoelastic Fluids;577
10.3.6;8.3.6 The Principle of Material Indifference for Models An and Bn of Viscoelastic Continua;579
10.4;8.4 Statements of Problems in Viscoelasticity Theory at Large Deformations;581
10.4.1;8.4.1 Statements of Dynamic Problems in the Spatial Description;581
10.4.2;8.4.2 Statements of Dynamic Problems in the Material Description;585
10.4.3;8.4.3 Statements of Quasistatic Problems of Viscoelasticity Theory in the Spatial Description;587
10.4.4;8.4.4 Statements of Quasistatic Problems for Models of Viscoelastic Continua in the Material Description;589
10.5;8.5 The Problem on Uniaxial Deforming of a Viscoelastic Beam;591
10.5.1;8.5.1 Deformation of a Viscoelastic Beam in Uniaxial Tension;591
10.5.2;8.5.2 Viscous Stresses in Uniaxial Tension;592
10.5.3;8.5.3 Stresses in a Viscoelastic Beam in Tension;592
10.5.4;8.5.4 Resolving Relation 1(k1) for a Viscoelastic Beam;593
10.5.5;8.5.5 Method of Calculating the Constants B() and ();594
10.5.6;8.5.6 Method for Evaluating the Constants , l1, l2 and , m;597
10.5.7;8.5.7 Computations of Relaxation Curves;598
10.5.8;8.5.8 Cyclic Deforming of a Beam;600
10.6;8.6 Dissipative Heating of a Viscoelastic Continuum Under Cyclic Deforming;602
10.6.1;8.6.1 The Problem on Dissipative Heating of a Beam Under Cyclic Deforming;602
10.6.2;8.6.2 Fast and Slow Times in Multicycle Deforming;603
10.6.3;8.6.3 Differentiation and Integration of Quasiperiodic Functions;603
10.6.4;8.6.4 Heat Conduction Equation for a Thin Viscoelastic Beam;604
10.6.5;8.6.5 Dissipation Function for a Viscoelastic Beam;605
10.6.6;8.6.6 Asymptotic Expansion in Terms of a Small Parameter;606
10.6.7;8.6.7 Averaged Heat Conduction Equation;607
10.6.8;8.6.8 Temperature of Dissipative Heating in a Symmetric Cycle;608
10.6.9;8.6.9 Regimes of Dissipative Heating Without Heat Removal;608
10.6.10;8.6.10 Regimes of Dissipative Heating in the Presence of Heat Removal;609
10.6.11;8.6.11 Experimental and Computed Data on Dissipative Heating of Viscoelastic Bodies;611
11;Chapter 9:Plastic Continua at Large Deformations;613
11.1;9.1 Models An of Plastic Continua at Large Deformations;613
11.1.1;9.1.1 Main Assumptions of the Models;613
11.1.2;9.1.2 General Representation of Constitutive Equations for Models An of Plastic Continua;616
11.1.3;9.1.3 Corollary of the Onsager Principle for Models An of Plastic Continua;619
11.1.4;9.1.4 Models An of Plastic Yield;620
11.1.5;9.1.5 Associated Model of Plasticity An;621
11.1.6;9.1.6 Corollary of the Principle of Material Symmetry for the Associated Model An of Plasticity;625
11.1.7;9.1.7 Associated Models of Plasticity An for Isotropic Continua;627
11.1.8;9.1.8 The Huber--Mises Model for Isotropic Plastic Continua;629
11.1.9;9.1.9 Associated Models of Plasticity An for Transversely Isotropic Continua;632
11.1.10;9.1.10 Two-Potential Model of Plasticity for a Transversely Isotropic Continuum;634
11.1.11;9.1.11 Associated Models of Plasticity An for Orthotropic Continua;636
11.1.12;9.1.12 The Orthotropic Unipotential Huber--Mises Model for Plastic Continua;638
11.1.13;9.1.13 The Principle of Material Indifference for Models An of Plastic Continua;640
11.2;9.2 Models Bn of Plastic Continua;645
11.2.1;9.2.1 Representation of Stress Power for Models Bn of Plastic Continua;645
11.2.2;9.2.2 General Representation of Constitutive Equations for Models Bn of Plastic Continua;649
11.2.3;9.2.3 Corollaries of the Onsager Principle for Models Bn of Plastic Continua;651
11.2.4;9.2.4 Associated Models Bn of Plastic Continua;653
11.2.5;9.2.5 Corollary of the Principle of Material Symmetry for Associated Model Bn of Plasticity;654
11.2.6;9.2.6 Associated Models of Plasticity Bn with Proper Strengthening;657
11.2.7;9.2.7 Associated Models of Plasticity Bn for Isotropic Continua;657
11.2.8;9.2.8 Associated Models of Plasticity Bn for Transversely Isotropic Continua;659
11.2.9;9.2.9 Associated Models of Plasticity Bn for Orthotropic Continua;660
11.2.10;9.2.10 The Principle of Material Indifference for Models Bn of Plastic Continua;661
11.3;9.3 Models Cn and Dn of Plastic Continua;662
11.3.1;9.3.1 General Representation of Constitutive Equations for Models Cn of Plastic Continua;662
11.3.2;9.3.2 Constitutive Equations for Models Cn of Isotropic Plastic Continua;665
11.3.3;9.3.3 General Representation of Constitutive Equations for Models Dn of Plastic Continua;668
11.3.4;9.3.4 Constitutive Equations for Models Dn of Isotropic Plastic Continua;672
11.3.5;9.3.5 The Principles of Material Symmetry and Material Indifference for Models Cn and Dn;673
11.4;9.4 Constitutive Equations of Plasticity Theory `in Rates';674
11.4.1;9.4.1 Representation of Models An of Plastic Continua `in Rates';674
11.5;9.5 Statements of Problems in Plasticity Theory;677
11.5.1;9.5.1 Statements of Dynamic Problems for Models An of Plasticity;677
11.5.2;9.5.2 Statements of Quasistatic Problems for Models An of Plasticity;679
11.6;9.6 The Problem on All-Round Tension--Compression of a Plastic Continuum;681
11.6.1;9.6.1 Deformation in All-Round Tension--Compression;681
11.6.2;9.6.2 Stresses in All-Round Tension--Compression;682
11.6.3;9.6.3 The Case of a Plastically Incompressible Continuum;683
11.6.4;9.6.4 The Case of a Plastically Compressible Continuum;684
11.6.5;9.6.5 Cyclic Loading of a Plastically Compressible Continuum;686
11.7;9.7 The Problem on Tension of a Plastic Beam;688
11.7.1;9.7.1 Deformation of a Beam in Uniaxial Tension;688
11.7.2;9.7.2 Stresses in a Plastic Beam;689
11.7.3;9.7.3 Plastic Deformations of a Beam;690
11.7.4;9.7.4 Change of the Density;692
11.7.5;9.7.5 Resolving Equation for the Problem;693
11.7.6;9.7.6 Numerical Method for the Resolving Equation;694
11.7.7;9.7.7 Method for Determination of Constants H0, n0, and s;697
11.7.8;9.7.8 Comparison with Experimental Data for Alloys;698
11.7.9;9.7.9 Comparison with Experimental Data for Grounds;699
11.8;9.8 Plane Waves in Plastic Continua;701
11.8.1;9.8.1 Formulation of the Problem;701
11.8.2;9.8.2 The Motion Law and Deformation of a Plate;702
11.8.3;9.8.3 Stresses in the Plate;703
11.8.4;9.8.4 The System of Dynamic Equations for the Plane Problem;704
11.8.5;9.8.5 The Statement of Problem on Plane Waves in Plastic Continua;706
11.8.6;9.8.6 Solving the Problem by the Characteristic Method;707
11.8.7;9.8.7 Comparative Analysis of the Solution for Different Models An;711
11.8.8;9.8.8 Plane Waves in Models AIV and AV;713
11.8.9;9.8.9 Shock Waves in Models AI and AII;715
11.8.10;9.8.10 Shock Adiabatic Curves for Models AI and AII;717
11.8.11;9.8.11 Shock Adiabatic Curves at a Given Rate of Impact;719
11.9;9.9 Models of Viscoplastic Continua;721
11.9.1;9.9.1 The Concept of a Viscoplastic Continuum;721
11.9.2;9.9.2 Model An of Viscoplastic Continua of the Differential Type;721
11.9.3;9.9.3 Model of Isotropic Viscoplastic Continua of the Differential Type;723
11.9.4;9.9.4 General Model An of Viscoplastic Continua;724
11.9.5;9.9.5 Model An of Isotropic Viscoplastic Continua;725
11.9.6;9.9.6 The Problem on Tension of a Beam of Viscoplastic Continuum of the Differential Type;725
12;References;729
13;Basic Notation;729
14;Index;734