Non-Classical Continuum Mechanics (eBook)

Gérard A. Maugin (born December 2, 1944 in Angers) is a French engineering scientist. Maugin acquired his engineering degree in mechanical engineering in 1966 at the Ecole Nationale Supérieure d'Arts et Métiers (Ensam) and he continued his studies at the school of aeronautics Sup Aéro in Paris until 1968. In 1966 he worked for the French Ministry of Defence on ballistic missiles. In 1968 he received his (DEA) degree in hydrodynamics in Paris. In 1969, he earned his master's degree from Princeton University, where he graduated in 1971 (Ph.D.). He was a NASA International Fellow between 1968 and 1970. In 1971/72 he was an officer in the French Air Force. In 1975 he received his doctorate in mathematics (Doctorat d'Etat) at the University of Paris VI (Pierre et Marie Curie), where he also taught and directed a team at the Laboratoire de Mécanique Théorique conducting research since 1985 on Continuum mechanics and Theoretical Mechanics. After its name change to the Laboratoire de Modélisation en Mécanique (LMM), he headed this from 1998. From 1979 he was Director of Research at CNRS. He was a visiting professor and visiting scientist at Princeton, Belgrade, Warsaw, Istanbul, at the Royal Institute of Technology in Stockholm, at the TU Berlin, Rome, Tel Aviv, the Lomonosov University, Kyoto, Darmstadt and Berkeley. His work deals with continuum mechanics, including relativistic continuum mechanics, micro magnetism, electrodynamics of continua, thermo mechanics, surface waves and nonlinear waves in continua, lattice dynamics, material equations and biomechanical applications (tissue growth). In 2001 he received the Max Planck Research Award, was the 1991/92 Fellow of the Berlin Institute for Advanced Study, and in 2001 received an honorary doctorate from the Technical University of Darmstadt . In 1982 he received the mechanics Prize of French Academy of Sciences and in 1977 the Medal of the CNRS in physics and engineering. He is a member of the Polish Academy of Sciences (1994) and the Estonian Academy of Sciences and has an honorary professorship at the Moscow State University. In 2003, he received the A. Cemal Eringen Medal.

Foreword 7
Preface 9
Contents 12
Prerequisites 17
1 What Is Classical Continuum Mechanics? 18
Introduction 18
Balance Equations 19
Reminder: The Most Classical Behaviours of Classical Continuum Thermo-Mechanics 23
Finite-Strain Thermoelasticity 23
Linear Homogeneous Isotropic Elasticity 24
Linear Elastic Crystals 25
Eulerian Fluids 25
Newtonian-Stokesian Fluids 25
Fourier Heat Conduction and Linear Thermoelasticity (Duhamel-Neumann) 26
Linear Piezoelectricity 27
References 28
2 What Is Generalized Continuum Mechanics (GCM)? 29
Introduction 29
Asymmetric Stress 29
Surface Couples 30
Eringen-Mindlin Micromorphic Model of Microstructured Continua 31
Weakly Nonlocal Modelling 33
Strongly Nonlocal Modelling 34
The Loss of Euclidean Structure 35
The Dictionary in Alphabetic Order 38
3 A–B: From “Aifantis E.C” to “Biot’s Poro-Elasticity” 39
Aifantis E.C 39
Anisotropic Fluids 40
Asymmetric Elasticity 44
Auxetic Materials 45
Biot’s Theory of Poro-elasticity 46
4 C: From “Capillarity” to “Couple Stress (in Medium with Constrained Rotation)” 49
Capillarity 49
Cellular Materials as Generalized Continua 50
Configurational Mechanics 52
Connection and Torsion 56
Contiguity 58
Continua with Latent Microstructure 59
Continuously Defective Materials 60
Cosserat Continua 60
Cosserat Continua (Experimental Confrontation) 62
Cosserat Eugène and François 63
Cosserat Point 64
Couple Stress 65
Couple Stress (in Medium with Constrained Rotation) 66
5 D: From “Defects in GCM” to “Duhem Pierre” 69
Defects in GCM 69
Density-Gradient Fluids 71
Differential Geometry in Nonclassical Continuum Mechanics 71
Dilatational Elasticity 72
Dipolar Continua 72
Directors’ Theory 73
Dislocations and Disclinations 74
Double Force 76
Duhem Pierre 78
6 E: From “Edelen D.G.B.” to “Extra-Entropy Flux” 80
Edelen D.G.B 80
EDGE FORCES 81
Electric Quadrupoles 81
Electromagnetic Continua 81
Ericksen J.L 82
Eringen A. Cemal 83
Eringen-Mindlin Medium 84
Extended Thermodynamics 84
Extra-Entropy Flux 85
7 F: From “Ferroelectric Crystals (Elasticity of)” to “Fractal Continua” 87
Ferroelectric Crystals (Elasticity of) 87
Modelling 87
Approach via the Principle of Virtual Power 90
Analogy with Cosserat Continua 92
Reduction to a Model Without Microstructure 92
Antiferroelectric Materials 93
Ferroic States 94
Fractal Continua 95
8 G: From “Generalized Continuum Mechanics” to “Green A.E.” 99
Generalized Continuum Mechanics (GCM) 99
Generalized Internal Forces 99
Generalized Thermo-Elasticity 99
Gradient Elasticity 102
Gradient Plasticity 108
Granular Materials as Generalized Continua 111
Green A.E 116
9 H–I: From “Higher-Order Gradient Theories” to “Ionic Crystals (Elasticity of)” 117
Higher-Order Gradient Theories 117
Homogenization 120
Hyperstress (Notion of) 121
Implicit Gradient Elasticity Models 122
Internal Degrees of Freedom 122
Internal Variables of State 124
Interstitial Working 128
Ionic Crystals (Elasticity of) 130
Remark on electric quadrupoles 133
10 K–L: From “Kelvin Continuum” to “Long-Range Interactions” 136
Kelvin Continuum 136
Kondo K 137
Korteweg Fluids 138
Kröner Ekkehart 140
Kunin I.A 141
Lattice Dynamics 142
Le Roux Elasticity 144
Liquid Crystals as Continua 146
Liquid Crystals (Ericksen-Leslie Theory) 148
Interaction With Electromagnetic Fields 152
Liquid Crystals (Eringen-Lee theory) 153
Liquid Crystals (Landau-De Gennes theory) 154
Long-Range Interactions 156
11 M: From “Material Growth (Theory of)” to “Micromagnetism in Elastic Solids” 158
Material Growth (theory of) 158
Material Inhomogeneities (Theory of) 162
Materials with Voids 162
Mesoscopic Theory of Complex Continua 163
Metamaterials 166
Micromagnetism in Elastic Solids 167
Continuum Modelling 168
Global Balance Laws 170
Local Balance Laws 171
Approach via the Principle of Virtual Power 174
Hamiltonian Variational Formulation 177
Ferrimagnetic and Antiferromagnetic Materials 177
Analogy with Cosserat Continua 177
Reduction to a Model Without Microstructure (Paramagnetic and Soft-ferromagnetic Bodies) 178
Micromorphic Continua 180
Micromorphic Fluids 180
Micropolar Continua (Cf. Cosserat Continua) 183
Linear Strain Measures 184
Micropolar Elasticity 185
Theory for Small Strains and Small Internal Rotation Angles 188
Micropolar Fluids 192
Microstretch Continua 195
Constitutive Equations 197
Microstretch Elasticity 197
Microstretch Fluids 199
Microstructure 201
Microstructured Continuum Theory (Eringen) 202
Special cases 204
Microsctructured Continuum Theory (Mindlin) 206
Field Equations 207
Microstructured Fluids 208
Mindlin R.D 208
Mixtures (Mechanics of) 209
Multipolar Continua (Green-Rivlin) 211
12 N: From “Naghdi P.M.” to “Nowacki W.” 214
Naghdi P.M 214
Non-euclidean Geometry of Defective Materials 215
Non-holonomic Continua 215
Nonlinear Waves in Generalized Continua 217
Nonlocal Damage 219
Weak Nonlocality 221
Nonlocality (as Opposed to Contiguity) 223
Nonlocality (Strong) 224
Nonlocality (Weak) 228
Nowacki W 228
13 O–P: From “Oriented Media (with Directors)” to “Porous Media as Seen in GCM” 230
Oriented Media (with Directors) 230
Peridynamics 233
Introduction 233
The Main Idea 233
Polarization Gradient 235
Ponderomotive Couple 235
Porous Media (as Seen in GCM) 237
Porous Media and the Theory of Mixtures 240
14 Q–R: From “Quasi-crystals (Elasticity of)” to “Rogula R.D.” 244
Quasi-crystals (Elasticity of) 244
Introduction 244
General Field Equations 245
Nonlinearity and Plasticity of Quasicrystals 248
Conclusion 249
Relaxed Micromorphic Continua 251
Rivlin R.S 252
Rogula D 252
15 S–T: From “Solitons (in on-Classical Continua)” to “Truesdell C.A.” 254
Solitons (in Non-classical Continua) 254
Solutions of Macromolecules 255
Introduction 255
Microstructure and Conformation 256
Constitutive Relations 257
Superfluids 261
Two-Fluid Model and Internal Momentum 262
Surface Tension 265
Toupin R.A 266
Truesdell C.A 267
Conclusion 268