Introduction .................................................... 1
1 Kinematics ................................................... 7
1.1 Material Bodies ......................................... 7
1.2 Material and Spatial Representation .................... 19
1.3 Deformation Gradient ................................... 23
1.4 Strain Tensors ......................................... 32
1.5 Convective Coordinates ................................. 38
1.6 Velocity Gradient ...................................... 42
1.7 Strain Rate Tensors .................................... 46
1.8 Strain Rates in Convective Coordinates ................. 50
1.9 Geometric Linearisation ................................ 53
1.10 Incompatible Configurations ............................ 57
1.10.1 Euclidean Space ................................. 58
1.10.2 Non-Euclidean Spaces ............................ 63
1.10.3 Conditions of Compatibility ..................... 64
2 Balance Relations of Mechanics .............................. 75
2.1 Preliminary Remarks .................................... 75
2.2 Mass ................................................... 78
2.2.1 Balance of Mass: Global Form .................... 78
2.2.2 Balance of Mass: Local Form ..................... 79
2.3 Linear Momentum and Rotational Momentum ................ 84
2.3.1 Balance of Linear Momentum and Rotational
Momentum: Global Formulation .................... 84
2.3.2 Stress Tensors .................................. 90
2.3.3 Stress Tensors in Convective Coordinates ........ 95
2.3.4 Local Formulation of the Balance of Linear
Momentum and Rotational Momentum ................ 95
2.3.5 Initial and Boundary Conditions ................ 101
2.4 Conclusions from the Balance Equations of Mechanics ... 104
2.4.1 Balance of Mechanical Energy ................... 105
2.4.2 The Principle of d'Alembert .................... 109
2.4.3 Principle of Virtual Work ...................... 114
2.4.4 Incremental Form of the Principle of
d'Alembert ..................................... 115
3 Balance Relations of Thermodynamics ........................ 119
3.1 Preliminary Remarks ................................... 119
3.2 Energy ................................................ 120
3.3 Temperature and Entropy ............................... 125
3.4 Initial and Boundary Conditions ....................... 130
3.5 Balance Relations for Open Systems .................... 132
3.5.1 Transport Theorem .............................. 132
3.5.2 Balance of Linear Momentum for Systems with
Time-Dependent Mass ............................ 135
3.5.3 Balance Relations: Conservation Laws ........... 137
3.5.4 Discontinuity Surfaces and Jump Conditions ..... 140
3.5.5 Multi-Component Systems (Mixtures) ............. 144
3.6 Summary: Basic Relations of Thermomechanics ........... 153
4 Objectivity ................................................ 155
4.1 Frames of Reference ................................... 155
4.2 Affine Spaces ......................................... 156
4.3 Change of Frame: Passive Interpretation ............... 159
4.4 Change of Frame: Active Interpretation ................ 162
4.5 Objective Quantities .................................. 164
4.6 Observer-Invariant Relations .......................... 171
5 Classical Theories of Continuum Mechanics .................. 177
5.1 Introduction .......................................... 177
5.2 Elastic Fluid ......................................... 178
5.3 Linear-Viscous Fluid .................................. 182
5.4 Linear-Elastic Solid .................................. 185
5.5 Linear-Viscoelastic Solid ............................. 188
5.6 Perfectly Plastic Solid ............................... 227
5.7 Plasticity with Hardening ............................. 231
5.8 Viscoplasticity with Elastic Range .................... 243
5.9 Remarks on the Classical Theories ..................... 249
6 Experimental Observation and Mathematical Modelling ........ 251
6.1 General Aspects ....................................... 251
6.2 Information from Experiments .......................... 255
6.2.1 Material Properties of Steel XCrNi 18.9 ........ 255
6.2.2 Material Properties of Carbon-Black-Filled
Elastomers ..................................... 263
6.3 Four Categories of Material Behaviour ................. 269
6.4 Four Theories of Material Behaviour ................... 271
6.5 Contribution of the Classical Theories ................ 273
7 General Theory of Mechanical Material Behaviour ............ 275
7.1 General Principles .................................... 275
7.2 Constitutive Equations ................................ 279
7.2.1 Simple Materials ............................... 279
7.2.2 Reduced Forms of the General Constitutive
Equation ....................................... 283
7.2.3 Simple Examples of Material Objectivity ........ 288
7.2.4 Frame-Indifference and Observer-Invariance ..... 289
7.3 Properties of Material Symmetry ....................... 293
7.3.1 The Concept of the Symmetry Group .............. 293
7.3.2 Classification of Simple Materials into
Fluids and Solids .............................. 298
7.4 Kinematic Conditions of Internal Constraint ........... 305
7.4.1 General Theory ................................. 305
7.4.2 Special Conditions of Internal Constraint ...... 308
7.5 Formulation of Material Models ........................ 311
7.5.1 General Aspects ................................ 311
7.5.2 Representation by Means of Functionals ......... 312
7.5.3 Representation by Means of Internal
Variables ...................................... 313
7.5.4 Comparison ..................................... 315
8 Dual Variables ............................................. 317
8.1 Tensor-Valued Evolution Equations ..................... 317
8.1.1 Introduction ................................... 317
8.1.2 Objective Time Derivatives of Objective
Tensors ........................................ 319
8.1.3 Example: Maxwell Fluid ......................... 322
8.1.4 Example: Rigid-Plastic Solid with Hardening .... 325
8.2 The Concept of Dual Variables ......................... 329
8.2.1 Motivation ..................................... 329
8.2.2 Strain and Stress Tensors (Summary) ............ 331
8.2.3 Dual Variables and Derivatives ................ 334
9 Elasticity ................................................. 345
9.1 Elasticity and Hyperelasticity ........................ 345
9.2 Isotropic Elastic Bodies .............................. 352
9.2.1 General Constitutive Equation for Elastic
Fluids and Solids .............................. 352
9.2.2 Isotropic Hyperelastic Bodies .................. 358
9.2.3 Incompressible Isotropic Elastic Materials ..... 363
9.2.4 Constitutive Equations of Isotropic
Elasticity (Examples) .......................... 365
9.3 Anisotropic Hyperelastic Solids ....................... 376
9.3.1 Approximation of the General Constitutive
Equation ....................................... 376
9.3.2 General Representation of the Strain Energy
Function ....................................... 379
9.3.2 Physical Linearisation ......................... 388
10 Viscoelasticity ............................................ 397
10.1 Representation by Means of Functionals ................ 397
10.1.1 Rate-Dependent Functionals with Fading Memory
Properties ..................................... 398
10.1.2 Continuity Properties and Approximations ....... 410
10.2 Representation by Means of Internal Variables ......... 419
10.2.1 General Concept ................................ 419
10.2.2 Internal Variables of the Strain Type .......... 426
10.2.3 A General Model of Finite Viscoelasticity ...... 433
11 Plasticity ................................................. 435
11.1 Rate-Independent Functionals .......................... 435
11.2 Representation by Means of Internal Variables ......... 444
11.3 Elastoplasticity ...................................... 450
11.3.1 Preliminary Remarks ............................ 450
11.3.2 Stress-Free Intermediate Configuration ......... 454
11.3.3 Isotropic Elasticity ........................... 459
11.3.4 Yield Function and Evolution Equations ......... 460
11.3.5 Consistency Condition .......................... 463
12 Viscoplasticity ............................................ 475
12.1 Preliminary Remarks ................................... 475
12.2 Viscoplasticity with Elastic Domain ................... 477
12.2.1 A General Constitutive Model ................... 477
12.2.2 Application of the Intermediate
Configuration .................................. 480
12.3 Plasticity as a Limit Case of Viscoplasticity ......... 484
12.3.1 The Differential Equation of the Yield
Function ....................................... 484
12.3.2 Relaxation Property ............................ 489
12.3.3 Slow Deformation Processes ..................... 491
12.3.4 Elastoplasticity and Arclength
Representation ................................. 497
12.4 A Concept for General Viscoplasticity ................. 499
12.4.1 Motivation ..................................... 499
12.4.2 Equilibrium Stress and Overstress .............. 500
12.4.3 An Example of General Viscoplasticity .......... 501
12.4.4 Conclusions Regarding the Modelling of
Mechanical Material Behaviour .................. 507
13 Constitutive Models in Thermomechanics ..................... 509
13.1 Thermomechanical Consistency .......................... 509
13.2 Thermoelasticity ...................................... 514
13.2.1 General Theory ................................. 514
13.2.2 Thermoelastic Fluid ............................ 520
13.2.3 Linear-Thermoelastic Solids .................... 527
13.3 Thermoviscoelasticity ................................. 530
13.3.1 General Concept ................................ 530
13.3.2 Thermoelasticity as a Limit Case of
Thermoviscoelasticity .......................... 537
13.3.3 Internal Variables of Strain Type .............. 541
13.3.4 Incorporation of Anisotropic Elasticity
Properties ..................................... 545
13.3.5 Incompressible Materials: An Extension of
the Mooney-Rivlin Model to
Thermoviscoelasticity .......................... 545
13.4 Thermoviscoplasticity with Elastic Domain ............. 554
13.4.1 Uniaxial Viscoplasticity ....................... 554
13.4.2 General Concept ................................ 560
13.4.3 Application of the Intermediate
Configuration .................................. 564
13.4.4 Thermoplasticity as a Limit Case of
thermoviscoplasticity .......................... 568
13.5 General Thermoviscoplasticity ......................... 577
13.5.1 Small Deformations ............................. 578
13.5.2 Finite Deformations ............................ 581
13.5.3 Conclusion ..................................... 585
13.6 Anisotropic Material Properties ....................... 586
13.6.1 Motivation ..................................... 586
13.6.2 Axes of Elastic Anisotropy ..................... 587
13.7 Anisotropic Viscoplasticity ........................... 591
13.7.1 General Considerations ......................... 591
13.7.2 Free Energy Function ........................... 593
13.7.3 Evolution Equations ............................ 596
13.7.4 Lattice Spin ................................... 603
13.7.5 Summary: A Constitutive Model of Anisotropic
Viscoplasticity ................................ 606
13.7.6 Numerical Simulations ............................... 608
13.7.7 Closing Remark ...................................... 618
References ................................................. 619
Index ......................................................... 635
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