Generalized continua from the theory to engineering applications

Cover......Page 1
Preface
......Page 6
Contents
......Page 8
1 Introduction......Page 16
2.1 Overview......Page 20
2.2 Manifolds......Page 21
2.3 Connection Partial Derivatives......Page 24
2.4 Parallel Transport......Page 27
2.5 Torsion Transformation of Connection......Page 28
2.6 Curvature......Page 33
2.7 Metric......Page 41
2.8 Metric Curvature......Page 47
3.1 Kinematics......Page 51
3.2 Distortion First-Order Distortions......Page 55
3.3 Integrability......Page 56
3.4 Elasticity......Page 59
3.5 Elastoplasticity Plastic Part of the Deformation......Page 62
Bibliography......Page 76
1 Introduction......Page 80
1.1 Elements of Rigid Body Dynamics......Page 82
1.2 Elements of Mechanics of Elastic Rods......Page 89
2 Kinematics of Cosserat Continuum......Page 93
3.1 Forces and Couples......Page 95
3.2 Euler’s Laws of Motion......Page 97
3.3 Stress Tensor and Couple Stress Tensor......Page 98
3.4 Principal Stresses in Micropolar Continua......Page 102
3.5 Equations of Motion......Page 104
3.6 Boundary-Value Problems......Page 105
4.1 General Principles Restricting the Constitutive Equations......Page 108
4.2 Natural Lagrangian Strain Measures of Cosserat Continuum......Page 110
4.3 Vectorial Parameterizations of Strain Measures......Page 116
4.4 Kinetic Constitutive Equations......Page 117
4.5 Material Symmetry Group......Page 119
4.6 Non-Linear Micropolar Isotropic Solids......Page 123
4.7 Physically Linear Micropolar Solids......Page 125
4.8 Linear Micropolar Isotropic Solids......Page 126
4.9 Constraints......Page 128
4.10 Constitutive Inequalities......Page 130
4.11 Micropolar Fluid......Page 134
4.12 Some Sources of Cosserat’s Constitutive Equations......Page 135
Bibliography......Page 137
1 Introduction......Page 146
2.1 Kinematics......Page 152
2.2 Strain Energy Density of an Elastic Cosserat Surface......Page 153
2.3 Principle of Virtual Work and the Equilibrium Conditions......Page 154
3 Micropolar Shells......Page 155
3.1 Kinematics......Page 156
3.2 Principle of Virtual Work and Boundary-Value Problems......Page 158
3.3 On the Constitutive Equations......Page 160
3.4 Compatibility Conditions......Page 161
3.6 Linear Theory of Micropolar Shells......Page 164
3.7 Constitutive Restrictions for Micropolar Shells......Page 167
3.8 Strong Ellipticity Condition and Acceleration Waves......Page 170
3.9 Principle Peculiarities of the Micropolar Shell Theory......Page 172
4.1 Basic Equations of Three-dimensional Linear Cosserat Continuum......Page 173
4.2 Transition to the Two-Dimensional Equilibrium Equations: Eringen’s Approach......Page 175
4.3 Transition to the Two-Dimensional Equilibrium Equations: Other Reduction Procedures......Page 177
5 Conclusions and Discussion......Page 178
Bibliography......Page 180
1 Introduction......Page 194
2 Kinematical Model of Directed Curves......Page 196
3 Governing Equations of the Non-Linear Theory......Page 199
4 Constitutive Equations for Thermoelastic Porous Rods......Page 204
4.2 Structure of Constitutive Tensors......Page 205
5 Linearized Equations of Directed Rods......Page 207
5.1 Boundary-Initial-Value Problems......Page 208
5.2 Uniqueness of Solution......Page 209
5.3 Existence Results in the Dynamical Theory......Page 211
6 Statical Theory for Rods......Page 215
6.1 Inequalities of Korn-Type for Cosserat Rods......Page 216
6.2 Existence of Solution......Page 219
6.3 Analysis of Pure Traction Problems......Page 221
7 Equations for Straight Rods......Page 223
7.1 Decoupling of the Problem......Page 224
7.2 Solution of Simple Problems......Page 225
8 Derivation of Rods Equations from the Three-Dimensional Equations......Page 228
9 Identification of Constitutive Coefficients for Thermo-Elastic Porous Orthotropic Rods......Page 231
9.1 Bending and Extension of Orthotropic Rods......Page 232
9.2 Torsion of Orthotropic Rods......Page 233
9.3 Shear Vibrations of an Orthotropic Rod......Page 234
9.4 Problem of Thermal Deformation......Page 236
9.5 Extension of Porous Thermo-Elastic Rods......Page 237
10.1 Basic Laws......Page 240
10.2 Constitutive Equations......Page 242
11 Non-Homogeneous Rods and Composite Beams Analyzed by the Direct Approach......Page 247
11.1 Constitutive Coefficients for Non-Homogeneous Rods......Page 250
11.2 Effective Stiffness Properties of Composite Beams......Page 255
Bibliography......Page 259
1.1 Scope of this Chapter......Page 264
1.2 Notations......Page 265
2.1 Kinematics of Micromorphic Media......Page 266
2.2 Principle of Virtual Power......Page 268
2.3 Elastoviscoplasticity of Micromorphic Media Elastic-Plastic Decomposition of the Generalised Strain Measures......Page 270
3 From a Heterogeneous Cauchy Material to a Homogeneous Equivalent Micromorphic Medium......Page 273
3.1 Definition of the Micromorphic Degrees of Freedom......Page 275
3.2 Higher Order Strain Measures......Page 277
3.3 Polynomial Ansatz......Page 278
3.4 Identification of Generalised Effective Elastic Moduli......Page 280
3.5 Validation of the Extended Homogenisation Method......Page 288
4.1 Multiscale Asymptotic Expansion Method......Page 289
4.2 Application to Polycrystalline Plasticity......Page 305
Bibliography......Page 311
1 Introduction and Historical Perspective......Page 316
2.1 Maxwell’s Equations......Page 321
2.2 Ponderomotive Force and Couple in a Continuum......Page 324
3.1 Model of Interactions......Page 327
3.2 Statement of Global Balance Laws......Page 330
3.3 Approach via the Principle of Virtual Power......Page 334
3.4 Hamiltonian Variational Formulation......Page 336
3.5 Ferrimagnetic and Antiferromagnetic Materials......Page 338
3.7 Reduction to a Model without Microstructure (Paramagnetic and Soft-Ferromagnetic Bodies)......Page 340
4.1 Model of Interactions......Page 341
4.2 Approach via the Principle of Virtual Power......Page 343
4.3 Hamiltonian Variational Principle......Page 344
4.5 Analogy with Cosserat Continua......Page 345
4.7 Remark on Electric Quadrupoles......Page 346
5.1 Introductory Note: Resonance Coupling between Wave Modes A. General Features......Page 347
5.2 The Case of Magnetoelasticity in Ferromagnets A. Magnon Magnons......Page 350
5.3 The Case of Electroelasticity in Ferroelectrics A. Polaritons Polaritons......Page 358
6.1 Definition......Page 360
6.2 Reminder of a Purely Mechanical Case......Page 361
6.3 The Ferroelectric Case......Page 364
6.4 The Ferromagnetic Case......Page 366
A Reminder of Basic Equations of Generalized Mechanical Continua......Page 368
Bibliography......Page 370
1 Introduction......Page 376
2 Isotropic Elasticity-Based Damage......Page 377
3.1 Stability and Ellipticity......Page 381
3.2 Mesh Sensitivity......Page 385
4.1 Non-Local Damage Models......Page 387
4.2 Gradient Damage Models......Page 388
5.1 Cosserat Elasticity......Page 392
5.2 Cosserat Plasticity......Page 394
5.3 A Return-Mapping Algorithm......Page 397
6.1 Non-Local Plasticity......Page 398
6.2 Gradient plasticity......Page 400
Bibliography......Page 402