Continuum Mechanics in the Earth Sciences

✍ Scribed by William I. Newman

• Publisher: Cambridge University Press
• Year: 2012
• Tongue: English
• Leaves: 196
• Edition: 1
• Category: Library

✦ Synopsis

Continuum mechanics underlies many geological and geophysical phenomena, from earthquakes and faults to the fluid dynamics of the Earth. This interdisciplinary book provides geoscientists, physicists and applied mathematicians with a class-tested, accessible overview of continuum mechanics. Starting from thermodynamic principles and geometrical insights, the book surveys solid, fluid and gas dynamics. In later review chapters, it explores new aspects of the field emerging from nonlinearity and dynamical complexity and provides a brief introduction to computational modeling. Simple, yet rigorous, derivations are used to review the essential mathematics. The author emphasizes the full three-dimensional geometries of real-world examples, enabling students to apply this in deconstructing solid earth and planet-related problems. Problem sets and worked examples are provided, making this a practical resource for graduate students in geophysics, planetary physics and geology and a beneficial tool for professional scientists seeking a better understanding of the mathematics and physics within Earth sciences.

✦ Table of Contents

Cover
......Page 1
CONTINUUM MECHANICS IN THE EARTH SCIENCES......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 11
Acknowledgements......Page 14
1.1 Scalars, vectors, and Cartesian tensors......Page 15
1.2 Matrices and determinants......Page 21
1.3 Transformations of Cartesian tensors......Page 23
1.4 Eigenvalues and eigenvectors......Page 26
1.5 Simplified approach to rotation......Page 30
1.6 Curvature, torsion, and kinematics......Page 32
Exercises......Page 37
2.1 Body and surface forces......Page 41
2.2 Cauchy stress principle......Page 43
2.3 Stress tensor......Page 45
2.4 Symmetry and transformation laws......Page 47
2.5 Principal stresses and directions......Page 48
2.6 Solving the cubic eigenvalue equation problem......Page 51
2.7 Maximum and minimum stress values......Page 53
2.8 Mohr's circles......Page 56
2.9 Plane, deviator, spherical, and octahedral stress......Page 0
Exercises......Page 60
3.1 Coordinates and deformation......Page 63
3.2 Strain tensor......Page 67
3.3 Linearized deformation theory......Page 68
3.4 Stretch ratios......Page 72
3.5 Velocity gradient......Page 73
3.6 Vorticity and material derivative......Page 75
Exercises......Page 78
4.1 Terminology and material derivatives......Page 81
4.2 Conservation of mass and the continuity equation......Page 85
4.3 Linear momentum and the equations of motion......Page 87
4.4 Piola–Kirchhoff stress tensor......Page 88
4.5 Angular momentum principle......Page 89
4.6 Conservation of energy and the energy equation......Page 90
4.7 Constitutive equations......Page 93
4.8 Thermodynamic considerations......Page 96
Exercises......Page 100
5.1 Elasticity, Hooke's law, and free energy......Page 103
5.2 Homogeneous deformations......Page 108
5.3 Role of temperature......Page 111
5.4 Elastic waves for isotropic bodies......Page 114
5.5 Helmholtz's decomposition theorem......Page 116
5.6 Statics for isotropic bodies......Page 118
5.7 Microscopic structure and dislocations......Page 120
Exercises......Page 123
6 Classical fluids......Page 126
6.1 Stokesian and Newtonian fluids: Navier–Stokes equations......Page 127
6.2 Some special fluids and flows......Page 131
Exercises......Page 143
7.1 Dimensional analysis and dimensionless form......Page 148
7.2 Dimensionless numbers......Page 152
Exercises......Page 157
8 Computation in continuum mechanics......Page 161
8.1 Review of partial differential equations......Page 162
8.2 Survey of numerical methods......Page 167
9 Nonlinearity in the Earth......Page 173
9.1 Friction......Page 177
9.2 Fracture......Page 179
9.3 Percolation and self-organized criticality......Page 182
9.4 Fractals......Page 185
References......Page 189
Index......Page 194