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Cambridge2012Continuum Mechanics in the Earth Sciences
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Preface page ix Acknowledgements xii 1 Some mathematical essentials 1 1.1 Scalars, vectors, and Cartesian tensors 1 1.2 Matrices and determinants 7 1.3 Transformations of Cartesian tensors 9 1.4 Eigenvalues and eigenvectors 12 1.5 Simplified approach to rotation 16 1.6 Curvature, torsion, and kinematics 18 Exercises 23 2 Stress principles 27 2.1 Body and surface forces 27 2.2 Cauchy stress principle 29 2.3 Stress tensor 31 2.4 Symmetry and transformation laws 33 2.5 Principal stresses and directions 34 2.6 Solving the cubic eigenvalue equation problem 37 2.7 Maximum and minimum stress values 39 2.8 Mohr’s circles 42 2.9 Plane, deviator, spherical, and octahedral stress 44 Exercises 46 3 Deformation and motion 49 3.1 Coordinates and deformation 49 3.2 Strain tensor 53 3.3 Linearized deformation theory 54 3.4 Stretch ratios 58 3.5 Velocity gradient 59 3.6 Vorticity and material derivative 61 Exercises 64 vii viii Contents 4 Fundamental laws and equations 67 4.1 Terminology and material derivatives 67 4.2 Conservation of mass and the continuity equation 71 4.3 Linear momentum and the equations of motion 73 4.4 Piola–Kirchhoff stress tensor 74 4.5 Angular momentum principle 75 4.6 Conservation of energy and the energy equation 76 4.7 Constitutive equations 79 4.8 Thermodynamic considerations 82 Exercises 86 5 Linear elastic solids 89 5.1 Elasticity, Hooke’s law, and free energy 89 5.2 Homogeneous deformations 94 5.3 Role of temperature 97 5.4 Elastic waves for isotropic bodies 100 5.5 Helmholtz’s decomposition theorem 102 5.6 Statics for isotropic bodies 104 5.7 Microscopic structure and dislocations 106 Exercises 109 6 Classical fluids 112 6.1 Stokesian and Newtonian fluids: Navier–Stokes equations 113 6.2 Some special fluids and flows 117 Exercises 129 7 Geophysical fluid dynamics 134 7.1 Dimensional analysis and dimensionless form 134 7.2 Dimensionless numbers 138 Exercises 143 8 Computation in continuum mechanics 147 8.1 Review of partial differential equations 148 8.2 Survey of numerical methods 153 9 Nonlinearity in the Earth 159 9.1 Friction 163 9.2 Fracture 165 9.3 Percolation and self-organized criticality 168 9.4 Fractals 171 References 175 Index 180 |
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