CONTENTS PREFACE 1 INTRODUCTION 1 1.1 Rational Continuum Mechanics 1 1.2 Rational Continuum Mechanics and Rheology 1 1.3 System of Rational Mechanics 2 1.3.1 Primitive elements 2 1.3.2 Basic laws 2 1.3.3 Constitutive relations 2 2 KINEMATICS 4 2.1 Bodies, Configurations and Motions 4 2.1.1 Bodies 4 2.1.2 Configurations 5 2.1.3 Motions and coordinates 5 2.1.4 Base vectors, metric tensors, shifters, displacements 7 2.2 Description of Motion 9 2.2.1 The material description or Lagrangean description 9 2.2.2 The spatial description or Eulerian description 10 2.3 Material and Spatial Coordinate Systems, Line Elements,Strain Ellipsoids 10 2.3.1 Material coordinate systems 11 2.3.2 Spatial coordinate systems 11 2.3.3 Line elements 11 2.3.4 Strain ellipsoids 11 2.4 Deformation Gradients, Deformation and Strain Tensors 12 2.4.1 Deformation gradients 12 2.4.2 Deformation tensors 14 2.4.3 Strain tensors 15 2.5 Transformations of Line, Area and Volume Elements 16 2.5.1 Length and angle change 18 2.6 Principal Stretches, Principal Directions, Invariants 20 2.7 Fundamental Theorem of Deformation, Polar Decomposition Theorem 23 2.8 Compatibility Conditions 26 2.9 Change of Reference Configuration, Relative Description 27 2.9.1 Change of reference configuration 27 2.9.2 Relative description 28 2.10 Velocity, Accelerate and Material Time Derivatives 29 2.10.1 Velocity 29 2.10.2 Material time rate (material derivative) and acceleration 30 2.10.3 Spatial time rate (spatial derivative or local derivative) 30 2.10.4 Acceleration 30 2.11 Velocity Gradient, Material Derivative of Deformation Gradient 34 2.11.1 Velocity gradient, material derivative of deformation gradient 34 2.11.2 Material derivatives of line, area and volume elements 34 2.12 Deformation Rate and Spin 36 2.12.1 Deformation rate and spin 36 2.12.2 Physical significance of deformation rate, rate of length, area and volume 37 2.12.3 Physical significance of spin, vorticity 38 2.12.4 Physical significance of decomposition of velocity gradient 39 2.13 Material Derivatives of Strain Tensors 40 2.14 Transport Theorem 41 2.15 Cartesian Coordinates, Some Simple Finite Deformations 42 2.15.1 Cartesian coordinates 42 2.15.2 Some simple finite deformations 43 2.16 Total Absolute Derivatives and Total Covariant Derivatives 50 3 BASIC BALANCE LAWS 52 3.1 Scope of This Chapter 52 3.2 Global Balance Laws 52 3.2.1 Conservation of mass 53 3.2.2 Balance of momentum 53 3.2.3 Balance of moment of momentum 54 3.2.4 Conservation of energy 54 3.2.5 Balance of entropy, entropy inequality 54 3.3 Master Law for Local Balance 54 3.4 Local Balance Laws 56 3.4.1 Conservation of mass 56 3.4.2 Balance of momentum 56 3.4.3 Balance of moment of momentum 56 3.4.4 Conservation of energy 57 3.4.5 Balance of entropy, entropy inequality 57 3.5 Balance Equations in Reference Configuration 58 3.5.1 Local conservation law of mass 59 3.5.2 Local balance law of momentum 59 3.5.3 Local balance law of moment of momentum 60 3.5.4 Local conservation law of energy 60 3.5.5 Local inequality of the rate of entropy production 60 4 CONSTITUTIVE THEORY 62 4.1 Admissible Thermodynamic Processes 62 4.2 Constitutive Axioms 63 4.2.1 Principle of determinism 63 4.2.2 Principle of equipresence 63 4.2.3 Principle of local action 63 4.2.4 Principle of material frame-indifference (Principle of material objectivity) 64 4.2.5 Principle of dissipation 65 4.3 Simple Thermodynamic Materials 65 4.4 Simple Materials 65 5 INFINITESIMAL THERMOVISCOELASTIC MECHANICS 68 5.1 Infinitesimal Strain 68 5.2 Governing Equations for Infinitesimal Thermoviscoelastic Mechanics 69 6 EXERCISES 72 7 REFERENCE 76 APPENDEX A TENSORS 77 A.1 Curvilinear Coordinate Systems, Base Vectors, Metric Tensor 77 A.2 Coordinate Transformations, Tensors 79 A.3 Tensor Algebra 80 A.4 Second-Order Tensors and Linear Transformations 81 A.5 Principal Direction and Invariants of A symmetrical Tensor 83 A.6 Covariant Differentiation, Christoffel Symbols 84 A.6.1 Absolute derivatives of vectors 84 A.6.2 Christoffel symbols 86 A.7 Absolute Differentials and Absolute Derivatives of Tensor Fields 87 A.8 Riemann-Christoffel Curvature Tensor, Riemannian Spaces 88 A.9 Manifolds 90 A.10 Torsion Tensors, Non-Riemannian Spaces 91 A.11 Curvilinear Orthogonal Coordinates and Physical Components of Tensors 95 A.12 Cylindrical Coordinates and Spherical Coordinates 97 A.12.1 Cylindrical coordinates 97 A.12.2 Spherical coordinates 99 A.13 Integral Theorems 101 A.14 Some Usual Formulae 102 A.14.1 Rules for differentiation 102 A.14.2 Gradients 102 A.14.3 Divergences 102 A.14.4 Curls 103 INDEX 104
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