Fundamentals of Continuum Mechanics

A concise introductory course text on continuum mechanics Fundamentals of Continuum Mechanics focuses on the fundamentals of the subject and provides the background for formulation of numerical methods for large deformations and a wide range of material behaviours. It aims to provide the foundations for further study, not just of these subjects, but also the formulations for much more complex material behaviour and their implementation computationally.  This book is divided into 5 parts, covering mathematical preliminaries, stress, motion and deformation, balance of mass, momentum and energy, and ideal constitutive relations and is a suitable textbook for introductory graduate courses for students in mechanical and civil engineering, as well as those studying material science, geology and geophysics and biomechanics.   A concise introductory course text on continuum mechanics Covers the fundamentals of continuum mechanics Uses modern tensor notation Contains problems and accompanied by a companion website hosting solutions Suitable as a textbook for introductory graduate courses for students in mechanical and civil engineering

Table of Contents:
Preface xiii Nomenclature xv Introduction 1 Part One Mathematical Preliminaries 3 1 Vectors 5 1.1 Examples 9 1.1.1 9 1.1.2 9 Exercises 9 Reference 11 2 Tensors 13 2.1 Inverse 15 2.2 Orthogonal Tensor 16 2.3 Principal Values 16 2.4 Nth-Order Tensors 18 2.5 Examples 18 2.5.1 18 2.5.2 18 Exercises 19 3 Cartesian Coordinates 21 3.1 Base Vectors 21 3.2 Summation Convention 23 3.3 Tensor Components 24 3.4 Dyads 25 3.5 Tensor and Scalar Products 27 3.6 Examples 29 3.6.1 29 3.6.2 29 3.6.3 29 Exercises 30 Reference 30 4 Vector (Cross) Product 31 4.1 Properties of the Cross Product 32 4.2 Triple Scalar Product 33 4.3 Triple Vector Product 33 4.4 Applications of the Cross Product 34 4.4.1 Velocity due to Rigid Body Rotation 34 4.4.2 Moment of a Force P about O 35 4.5 Non-orthonormal Basis 36 4.6 Example 37 Exercises 37 5 Determinants 41 5.1 Cofactor 42 5.2 Inverse 43 5.3 Example 44 Exercises 44 6 Change of Orthonormal Basis 47 6.1 Change of Vector Components 48 6.2 Definition of a Vector 50 6.3 Change of Tensor Components 50 6.4 Isotropic Tensors 51 6.5 Example 52 Exercises 53 Reference 56 7 Principal Values and Principal Directions 57 7.1 Example 59 Exercises 60 8 Gradient 63 8.1 Example: Cylindrical Coordinates 66 Exercises 67 Part Two Stress 69 9 Traction and Stress Tensor 71 9.1 Types of Forces 71 9.2 Traction on Different Surfaces 73 9.3 Traction on an Arbitrary Plane (Cauchy Tetrahedron) 75 9.4 Symmetry of the Stress Tensor 76 Exercise 77 Reference 77 10 Principal Values of Stress 79 10.1 Deviatoric Stress 80 10.2 Example 81 Exercises 82 11 Stationary Values of Shear Traction 83 11.1 Example: Mohr–Coulomb Failure Condition 86 Exercises 88 12 Mohr’s Circle 89 Exercises 93 Reference 93 Part Three Motion and Deformation 95 13 Current and Reference Configurations 97 13.1 Example 102 Exercises 103 14 Rate of Deformation 105 14.1 Velocity Gradients 105 14.2 Meaning of D 106 14.3 Meaning of W 108 Exercises 109 15 Geometric Measures of Deformation 111 15.1 Deformation Gradient 111 15.2 Change in Length of Lines 112 15.3 Change in Angles 113 15.4 Change in Area 114 15.5 Change in Volume 115 15.6 Polar Decomposition 116 15.7 Example 118 Exercises 118 References 120 16 Strain Tensors 121 16.1 Material Strain Tensors 121 16.2 Spatial Strain Measures 123 16.3 Relations Between D and Rates of EG and U 124 16.3.1 Relation Between Ė and D 124 16.3.2 Relation Between D and U 125 Exercises 126 References 128 17 Linearized Displacement Gradients 129 17.1 Linearized Geometric Measures 130 17.1.1 Stretch in Direction N 130 17.1.2 Angle Change 131 17.1.3 Volume Change 131 17.2 Linearized Polar Decomposition 132 17.3 Small-Strain Compatibility 133 Exercises 135 Reference 135 Part Four Balance of Mass, Momentum, and Energy 137 18 Transformation of Integrals 139 Exercises 142 References 143 19 Conservation of Mass 145 19.1 Reynolds’ Transport Theorem 148 19.2 Derivative of an Integral over a Time-Dependent Region 149 19.3 Example: Mass Conservation for a Mixture 150 Exercises 151 20 Conservation of Momentum 153 20.1 Momentum Balance in the Current State 153 20.1.1 Linear Momentum 153 20.1.2 Angular Momentum 154 20.2 Momentum Balance in the Reference State 155 20.2.1 Linear Momentum 156 20.2.2 Angular Momentum 157 20.3 Momentum Balance for a Mixture 158 Exercises 159 21 Conservation of Energy 161 21.1 Work-Conjugate Stresses 163 Exercises 165 Part Five Ideal Constitutive Relations 167 22 Fluids 169 22.1 Ideal Frictionless Fluid 169 22.2 Linearly Viscous Fluid 171 22.2.1 Non-steady Flow 173 Exercises 175 Reference 176 23 Elasticity 177 23.1 Nonlinear Elasticity 177 23.1.1 Cauchy Elasticity 177 23.1.2 Green Elasticity 178 23.1.3 Elasticity of Pre-stressed Bodies 179 23.2 Linearized Elasticity 182 23.2.1 Material Symmetry 183 23.2.2 Linear Isotropic Elastic Constitutive Relation 185 23.2.3 Restrictions on Elastic Constants 186 23.3 More Linearized Elasticity 187 23.3.1 Uniqueness of the Static Problem 188 23.3.2 Pressurized Hollow Sphere 189 Exercises 191 Reference 194 Index 195