| Inhomogeneity in Continuum Mechanics | |
| An overview of inhomogeneity theory | p. 3 |
| The constitutive equation of a material body | p. 3 |
| Configurations, deformations and their gradient | p. 3 |
| Locality, simplicity, elasticity | p. 4 |
| Material uniformity | p. 7 |
| The notion of material isomorphism | p. 7 |
| Material symmetries and the non-uniqueness of material isomorphisms | p. 10 |
| The material archetype | p. 12 |
| Local material parallelisms | p. 16 |
| Non-uniqueness of the (local) material connection | p. 20 |
| The material G-structure and the material groupoid | p. 20 |
| The material G-structure | p. 20 |
| The material groupoid and its associated G-structures | p. 26 |
| Homogeneity | p. 30 |
| Uniformity and homogeneity | p. 30 |
| Homogeneity in terms of a material connection | p. 31 |
| Homogeneity in terms of a material G-structure | p. 33 |
| Homogeneity in terms of the material groupoid | p. 36 |
| Homogeneity criteria | p. 36 |
| Solids | p. 36 |
| Fluids | p. 38 |
| Fluid crystals | p. 39 |
| Uniformity of second-grade materials | p. 41 |
| An intuitive picture | p. 41 |
| The second-grade constitutive law | p. 43 |
| Jets of maps | p. 43 |
| Composition of jets | p. 44 |
| Second-grade materials | p. 47 |
| Second-grade uniformity | p. 47 |
| Material isomorphisms | p. 47 |
| Second-grade material archetypes | p. 48 |
| Second-grade symmetries | p. 49 |
| An example of a nontrivial second-grade symmetry | p. 50 |
| The material second-order G-structures and groupoid | p. 52 |
| The second-order frame bundle | p. 52 |
| The material G-structures | p. 52 |
| The material groupoid | p. 54 |
| The subgroups of G[superscript 2](n) | p. 54 |
| The generic subgroup | p. 54 |
| Toupin subgroups | p. 57 |
| The subgroups {I, [Sigma subscript I]} and their conjugates | p. 58 |
| Second-grade homogeneity | p. 59 |
| The second-order frames induced by a coordinate system | p. 59 |
| Homogeneity | p. 59 |
| Coordinate expressions | p. 60 |
| Homogeneity in terms of a material G-structure | p. 63 |
| Homogeneity in terms of the material groupoid | p. 65 |
| Uniformity of Cosserat media | p. 67 |
| Kinematics of a Cosserat body | p. 67 |
| The constitutive law of a simple elastic Cosserat body | p. 71 |
| Material isomorphisms and uniformity | p. 74 |
| Material isomorphisms in a Cosserat body | p. 74 |
| Uniformity and the Cosserat archetype | p. 75 |
| Cosserat symmetries | p. 76 |
| Changing coordinates | p. 77 |
| Changing the archetype | p. 78 |
| Homogeneity conditions | p. 78 |
| Homogeneity of a Cosserat body | p. 78 |
| The three kinds of material connections of a uniform Cosserat body | p. 79 |
| Homogeneity conditions | p. 80 |
| The Cosserat material G-structures and groupoid | p. 81 |
| Frames, and frames of frames | p. 81 |
| Non-holonomic, semi-holonomic and holonomic frames | p. 85 |
| The Cosserat material G-structures | p. 90 |
| The Cosserat material groupoid | p. 91 |
| Homogeneity, flatness and integrable prolongations | p. 91 |
| Sections of F[superscript 2]B | p. 92 |
| Invariant sections and linear connections | p. 93 |
| Prolongations | p. 94 |
| Functionally graded bodies | p. 97 |
| The extended notion of material isomorphism | p. 97 |
| Non-uniqueness of symmetry isomorphisms | p. 98 |
| The material N-structure | p. 99 |
| Homosymmetry | p. 100 |
| Unisymmetric homogeneity of elastic solids | p. 101 |
| The reduced N-structure | p. 103 |
| Algebraic preliminaries | p. 103 |
| The N-structure of a solid functionally-graded unisymmetric body | p. 104 |
| Examples | p. 106 |
| The isotropic solid | p. 106 |
| The transversely isotropic solid | p. 106 |
| The n-agonal solids | p. 106 |
| Orthotropic materials | p. 107 |
| Summary | p. 107 |
| Material Evolution | |
| On energy, Cauchy stress and Eshelby stress | p. 111 |
| Preliminary considerations | p. 111 |
| The Cauchy stress revisited | p. 112 |
| Eshelby's tensor as Cauchy's dual | p. 114 |
| Complete expressions of hyperelastic uniformity | p. 115 |
| The Eshelby and Mandel Stresses in the Context of Material Uniformity | p. 116 |
| Eshelby-stress identities | p. 118 |
| Consequences of balance of angular momentum | p. 118 |
| Consequences of a continuous symmetry group | p. 118 |
| Consequences of the balance of linear momentum | p. 119 |
| Inhomogeneity with compact support and the J-integral | p. 120 |
| The Eshelby stress in thermoelasticity | p. 122 |
| Thermoelastic uniformity | p. 122 |
| The Eshelby stress identity | p. 123 |
| Thermal stresses | p. 124 |
| The material heat conduction tensor | p. 127 |
| On stress, hyperstress and Eshelby stress in second-grade bodies | p. 128 |
| On stress, microstress and Eshelby stress in Cosserat bodies | p. 130 |
| Equilibrium equations | p. 130 |
| Eshelby stresses | p. 131 |
| Eshelby stress identities | p. 133 |
| An overview of the theory of material evolution | p. 135 |
| What is material evolution? | p. 135 |
| A geometric picture | p. 137 |
| Evolution equations | p. 138 |
| General form | p. 138 |
| Reduction to the archetype | p. 139 |
| The principle of actual evolution | p. 141 |
| Material symmetry consistency | p. 143 |
| The field equations of remodelling and bulk growth | p. 145 |
| Balance of mass | p. 146 |
| Balance of linear momentum | p. 147 |
| Balance of angular momentum | p. 148 |
| Balance of energy | p. 148 |
| The Clausius-Duhem inequality and its consequences | p. 149 |
| An alternative approach | p. 156 |
| Example: Visco-elasto-plastic theories | p. 162 |
| A simple non-trivial model | p. 162 |
| Some computational considerations | p. 163 |
| Creep of a bar under uniaxial loading | p. 165 |
| Evolution, rheological models and the Eshelby stress | p. 167 |
| Example: Bulk growth | p. 168 |
| Exercise stimulates growth | p. 169 |
| A challenge to Wolff's law? | p. 170 |
| Example: Self-driven evolution | p. 174 |
| Introduction | p. 174 |
| A solid crystal body | p. 175 |
| An isotropic solid | p. 178 |
| Second-grade evolution | p. 183 |
| Introduction | p. 183 |
| Reduction to the archetype | p. 184 |
| Actual evolution | p. 186 |
| Material symmetry consistency | p. 186 |
| An example | p. 187 |
| Concluding remarks | p. 188 |
| Mathematical Foundations | |
| Basic geometric concepts | p. 193 |
| Differentiable manifolds | p. 193 |
| Lie groups | p. 202 |
| Fibre bundles | p. 204 |
| Principal fibre bundles | p. 206 |
| Associated fibre bundles | p. 208 |
| Sections of fibre bundles | p. 211 |
| Theory of connections | p. 213 |
| Connections on principal G-bundles | p. 213 |
| Parallelism in a principal G-bundle | p. 216 |
| Reduction of a connection | p. 217 |
| Structure equation, curvature and holonomy | p. 218 |
| Flat connections | p. 222 |
| Linear connections | p. 222 |
| Connections in an associated bundle | p. 226 |
| G-structures | p. 230 |
| Examples of G-structures | p. 232 |
| Bundles of linear frames | p. 235 |
| Jet prolongations of fibre bundles | p. 235 |
| Local coordinates on prolongations | p. 237 |
| Lie groups of jets of diffeomorphisms | p. 239 |
| Higher-order frame bundles | p. 240 |
| Connections of higher order | p. 243 |
| Fundamental form | p. 243 |
| [epsi]-connection | p. 245 |
| Second-order (holonomic) connection | p. 248 |
| Simple connections | p. 252 |
| Groupoids | p. 257 |
| Introduction | p. 257 |
| Groupoids | p. 257 |
| Transitive Lie groupoids and principal bundles | p. 260 |
| References | p. 263 |
| Index | p. 269 |
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