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Cambridge2010Principles of Continuum Mechanics
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Contents Preface page xi 1 Introduction 1 1.1 Continuum mechanics 1 1.2 Objective of the study 7 1.3 Summary 8 2 Vectors and Tensors 10 2.1 Motivation 10 2.2 Definition of a vector 10 2.3 Vector algebra 11 2.3.1 Unit vector 11 2.3.2 Zero vector 12 2.3.3 Vector addition 12 2.3.4 Multiplication of a vector by a scalar 14 2.3.5 Scalar product of vectors 14 2.3.6 Vector product 15 2.3.7 Triple products of vectors 18 2.3.8 Plane area as a vector 20 2.3.9 Components of a vector 22 2.4 Index notation and summation convention 24 2.4.1 Summation convention 24 2.4.2 Dummy index 25 2.4.3 Free index 25 2.4.4 Kronecker delta and permutation symbols 26 2.4.5 Transformation law for different bases 29 2.5 Theory of matrices 31 2.5.1 Definition 31 2.5.2 Matrix addition and multiplication of a matrix by a scalar 32 2.5.3 Matrix transpose and symmetric and skew symmetric matrices 33 2.5.4 Matrix multiplication 34 2.5.5 Inverse and determinant of a matrix 36 vii viii Contents 2.6 Vector calculus 39 2.6.1 The del operator 39 2.6.2 Divergence and curl of a vector 41 2.6.3 Cylindrical and spherical coordinate systems 43 2.6.4 Gradient, divergence, and curl theorems 45 2.7 Tensors 46 2.7.1 Dyads 46 2.7.2 Nonion form of a dyad 48 2.7.3 Transformation of components of a dyad 49 2.7.4 Tensor calculus 49 2.8 Summary 51 Problems 51 3 Kinematics of a Continuum 55 3.1 Deformation and configuration 55 3.2 Engineering strains 56 3.2.1 Normal strain 56 3.2.2 Shear strain 57 3.3 General kinematics of a solid continuum 61 3.3.1 Configurations of a continuous medium 61 3.3.2 Material and spatial descriptions 62 3.3.3 Displacement field 65 3.4 Analysis of deformation 66 3.4.1∗Deformation gradient tensor 66 3.4.2∗Various types of deformations 69 3.4.2.1 Pure dilatation 70 3.4.2.2 Simple extension 70 3.4.2.3 Simple shear 71 3.4.2.4 Nonhomogeneous deformation 71 3.4.3 Green strain tensor 72 3.4.4 Infinitesimal strain tensor 77 3.4.5 Principal values and principal planes of strains 79 3.5 Rate of deformation and vorticity tensors 81 3.5.1 Velocity gradient tensor 81 3.5.2 Rate of deformation tensor 81 3.5.3 Vorticity tensor and vorticity vector 82 3.6 Compatibility equations 84 3.7 Summary 86 Problems 87 4 Stress Vector and Stress Tensor 93 4.1 Introduction 93 4.2 Stress vector, stress tensor, and Cauchy’s formula 94 4.3 Transformations of stress components and principal stresses 102 ix Contents 4.3.1 Transformation of stress components 102 4.3.2 Principal stresses and principal planes 104 4.4 Summary 107 Problems 107 5 Conservation of Mass, Momentum, and Energy 111 5.1 Introduction 111 5.2 Conservation of mass 112 5.2.1 Preliminary discussion 112 5.2.2 Conservation of mass in spatial description 112 5.2.3 Conservation of mass in material description 117 5.2.4 Reynolds transport theorem 119 5.3 Conservation of momenta 119 5.3.1 Principle of conservation of linear momentum 119 5.3.2 Principle of conservation of angular momentum 134 5.4 Thermodynamic principles 136 5.4.1 Introduction 136 5.4.2 Energy equation for one-dimensional flows 136 5.4.3 Energy equation for a three-dimensional continuum 140 5.5 Summary 142 Problems 143 6 Constitutive Equations 149 6.1 Introduction 149 6.2 Elastic solids 150 6.2.1 Introduction 150 6.2.2 Generalized Hooke’s law for orthotropic materials 151 6.2.3 Generalized Hooke’s law for isotropic materials 153 6.3 Constitutive equations for fluids 156 6.3.1 Introduction 156 6.3.2 Ideal fluids 157 6.3.3 Viscous incompressible fluids 157 6.4 Heat transfer 158 6.4.1 General introduction 158 6.4.2 Fourier’s heat conduction law 158 6.4.3 Newton’s law of cooling 159 6.4.4 Stefan–Boltzmann law 159 6.5 Summary 160 Problems 160 7 Applications in Heat Transfer, Fluid Mechanics, and Solid Mechanics 162 7.1 Introduction 162 7.2 Heat transfer 162 7.2.1 Governing equations 162 x Contents 7.2.2 Analytical solutions of one-dimensional heat transfer 165 7.2.2.1 Steady-state heat transfer in a cooling fin 165 7.2.2.2 Steady-state heat transfer in a surface-insulated rod 167 7.2.3 Axisymmetric heat conduction in a circular cylinder 169 7.2.4 Two-dimensional heat transfer 170 7.3 Fluid mechanics 172 7.3.1 Preliminary comments 172 7.3.2 Summary of equations 173 7.3.3 Inviscid fluid statics 174 7.3.4 Parallel flow (Navier–Stokes equations) 175 7.3.4.1 Steady flow of viscous incompressible fluid between parallel plates 176 7.3.4.2 Steady flow of viscous incompressible fluid through a pipe 177 7.3.5 Diffusion processes 179 7.4 Solid mechanics 182 7.4.1 Governing equations 182 7.4.2 Analysis of bars 184 7.4.3 Analysis of beams 188 7.4.3.1 Principle of superposition 195 7.4.4 Analysis of plane elasticity problems 196 7.4.4.1 Plane strain and plane stress problems 196 7.4.4.2 Plane strain problems 196 7.4.4.3 Plane stress problems 198 7.4.4.4 Solution methods 199 7.4.4.5 Airy stress function 202 7.5 Summary 204 Problems 205 Answers to Selected Problems 215 References and Additional Readings 225 Subject Index 227 |
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