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Cambridge2010The Geometrical Language of Continuum Mechanics
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Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi PART 1 MOTIVATION AND BACKGROUND 1 1 The Case for Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Classical Space-Time and Fibre Bundles 4 1.2 Configuration Manifolds and Their Tangent and Cotangent Spaces 10 1.3 The Infinite-dimensional Case 13 1.4 Elasticity 22 1.5 Material or Configurational Forces 23 2 Vector and Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1 Vector Spaces: Definition and Examples 24 2.2 Linear Independence and Dimension 26 2.3 Change of Basis and the Summation Convention 30 2.4 The Dual Space 31 2.5 Linear Operators and the Tensor Product 34 2.6 Isomorphisms and Iterated Dual 36 2.7 Inner-product Spaces 41 2.8 Affine Spaces 46 2.9 Banach Spaces 52 3 Tensor Algebras and Multivectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.1 The Algebra of Tensors on a Vector Space 57 3.2 The Contravariant and Covariant Subalgebras 60 3.3 Exterior Algebra 62 3.4 Multivectors and Oriented Affine Simplexes 69 3.5 The Faces of an Oriented Affine Simplex 71 3.6 Multicovectors or r-Forms 72 3.7 The Physical Meaning of r-Forms 75 3.8 Some Useful Isomorphisms 76 vii viii Contents PART 2 DIFFERENTIAL GEOMETRY 79 4 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.1 Introduction 81 4.2 Some Topological Notions 83 4.3 Topological Manifolds 85 4.4 Differentiable Manifolds 86 4.5 Differentiability 87 4.6 Tangent Vectors 89 4.7 The Tangent Bundle 94 4.8 The Lie Bracket 96 4.9 The Differential of a Map 101 4.10 Immersions, Embeddings, Submanifolds 105 4.11 The Cotangent Bundle 109 4.12 Tensor Bundles 110 4.13 Pull-backs 112 4.14 Exterior Differentiation of Differential Forms 114 4.15 Some Properties of the Exterior Derivative 117 4.16 Riemannian Manifolds 118 4.17 Manifolds with Boundary 119 4.18 Differential Spaces and Generalized Bodies 120 5 Lie Derivatives, Lie Groups, Lie Algebras . . . . . . . . . . . . . . . . . . . . 126 5.1 Introduction 126 5.2 The Fundamental Theorem of the Theory of ODEs 127 5.3 The Flow of a Vector Field 128 5.4 One-parameter Groups of Transformations Generated by Flows 129 5.5 Time-Dependent Vector Fields 130 5.6 The Lie Derivative 131 5.7 Invariant Tensor Fields 135 5.8 Lie Groups 138 5.9 Group Actions 140 5.10 One-Parameter Subgroups 142 5.11 Left- and Right-Invariant Vector Fields on a Lie Group 143 5.12 The Lie Algebra of a Lie Group 145 5.13 Down-to-Earth Considerations 149 5.14 The Adjoint Representation 153 6 Integration and Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.1 Integration of Forms in Affine Spaces 155 6.2 Integration of Forms on Chains in Manifolds 160 Contents ix 6.3 Integration of Forms on Oriented Manifolds 166 6.4 Fluxes in Continuum Physics 169 6.5 General Bodies and Whitney’s Geometric Integration Theory 174 PART 3 FURTHER TOPICS 189 7 Fibre Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.1 Product Bundles 191 7.2 Trivial Bundles 193 7.3 General Fibre Bundles 196 7.4 The Fundamental Existence Theorem 198 7.5 The Tangent and Cotangent Bundles 199 7.6 The Bundle of Linear Frames 201 7.7 Principal Bundles 203 7.8 Associated Bundles 206 7.9 Fibre-Bundle Morphisms 209 7.10 Cross Sections 212 7.11 Iterated Fibre Bundles 214 8 Inhomogeneity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8.1 Material Uniformity 220 8.2 The Material Lie groupoid 233 8.3 The Material Principal Bundle 237 8.4 Flatness and Homogeneity 239 8.5 Distributions and the Theorem of Frobenius 240 8.6 Jet Bundles and Differential Equations 242 9 Connection, Curvature, Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 9.1 Ehresmann Connection 245 9.2 Connections in Principal Bundles 248 9.3 Linear Connections 252 9.4 G-Connections 258 9.5 Riemannian Connections 264 9.6 Material Homogeneity 265 9.7 Homogeneity Criteria 270 AppendixA A Primer in Continuum Mechanics . . . . . . . . . . . . . . . . . . 274 A.1 Bodies and Configurations 274 A.2 Observers and Frames 275 A.3 Strain 276 A.4 Volume and Area 280 A.5 The Material Time Derivative 281 x Contents A.6 Change of Reference 282 A.7 Transport Theorems 284 A.8 The General Balance Equation 285 A.9 The Fundamental Balance Equations of Continuum Mechanics 289 A.10 A Modicum of Constitutive Theory 295 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 |
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