Page 8 - Introduction to Continuum Mechanics

Contents vii

3.9 Principal Strain 105
3.10 Dilatation 105
3.11 The Infinitesimal Rotation Tensor 106
3.12 Time Rate of Change of a Material Element 106
3.13 The Rate of Deformation Tensor 108
3.14 The Spin Tensor and the Angular Velocity Vector 111
3.15 Equation of Conservation Of Mass 112
3.16 Compatibility Conditions for Infinitesimal Strain Components 114
3.17 Compatibility Conditions for the Rate of Deformation Components 119
3.18 Deformation Gradient 120
3.19 Local Rigid Body Displacements 121
3.20 Finite Deformation 121
3.21 Polar Decomposition Theorem 124
3.22 Calculation of the Stretch Tensor from the Deformation Gradient 126
3.23 Right Cauchy-Green Deformation Tensor 128
3.24 Lagrangian Strain Tensor 134
3.25 Left Cauchy-Green Deformation Tensor 138
3.26 Eulerian Strain Tensor 141
3.27 Compatibility Conditions for Components of Finite Deformation Tensor 144
3.28 Change of Area due to Deformation 145
3.29 Change of Volume due to Deformation 146
3.30 Components of Deformation Tensors in other Coordinates 149
3.31 Current Configuration as the Reference Configuration 158
Problems 160

Chapter 4 Stress 173
4.1 Stress Vector 173
4.2 Stress Tensor 174
4.3 Components of Stress Tensor 176
4.4 Symmetry of Stress Tensor - Principle of Moment of Momentum 178
4.5 Principal Stresses 182
4.6 Maximum Shearing Stress 182
4.7 Equations of Motion - Principle of Linear Momentum 187
4.8 Equations of Motion in Cylindrical and Spherical Coordinates 190
4.9 Boundary Condition for the Stress Tensor 192
4.10 Piola Kirchhoff Stress Tensors 195