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by Michael Slawinski

TOC

Chapt 1. Deformations
1.1. Notion of continuum
1.2. Rudiments of continuum mechanics
1.2.1. Axiomatic format
1.2.2. Primitive concepts of continuum mechanics
1.3. Material and spatial descriptions
1.4. Strain 18
1.4.1. Introductory comments
1.4.2. Derivation of strain tensor
1.4.3. Physical meaning of strain tensor
1.5. Rotation tensor and rotation vector

Chapt 2. Forces and balance principles
2.1. Conservation of mass
2.2. Time derivative of volume integral
2.3. Stress
2.3.1. Stress as description of surface forces
2.3.2. Traction
2.4. Balance of linear momentum
2.5. Stress tensor
2.6. Cauchy’s equations of motion
2.7. Balance of angular momentum
2.7.1. Introductory comments
2.7.2. Integral equation
2.7.3. Symmetry of stress tensor
2.8. Fundamental equations

Chapt 3. Stress-strain equations
3.1. Rudiments of constitutive equations
3.2. Formulation of stress-strain equations: Hookean solid
3.2.1. Introductory comments
3.2.2. Tensor form
3.2.3. Matrix form
3.3. Determined system
3.4. Anelasticity: Example
3.4.1. Introductory comments
3.4.2. Viscosity: Stokesian fluid
3.4.3. Viscoelasticity: Kelvin-Voigt model

Chapt 4. Strain energy
4.1. Strain-energy function
4.2. Strain-energy function and elasticity-tensor symmetry
4.2.1. Fundamental considerations
4.2.2. Elasticity parameters
4.2.3. Matrix form of stress-strain equations
4.2.4. Coordinate transformations
4.3. Stability conditions
4.4. System of equations for elastic continua

Chapt 5. Material symmetry
5.1. Orthogonal transformations
5.2. Transformation of coordinates
5.2.1. Introductory comments
5.2.2. Transformation of stress-tensor components
5.2.3. Transformation of strain-tensor components
5.2.4. Stress-strain equations in transformed coordinates
5.3. Condition for material symmetry
5.4. Point symmetry
5.5. Generally anisotropic continuum
5.6. Monoclinic continuum
5.6.1. Elasticity matrix
5.6.2. Vanishing of tensor components
5.6.3. Natural coordinate system
5.7. Orthotropic continuum
5.8. Trigonal continuum
5.9. Tetragonal continuum
5.10. Transversely isotropic continuum
5.11. Cubic continuum
5.12. Isotropic continuum
5.13. Relations among symmetry classes

Chapt 6. Equations of motion: Isotropic homogeneous continua
6.1. Wave equations
6.1.1. Equation of motion
6.2. Plane waves
6.3. Displacement potentials
6.3.1. Helmholtz’s decomposition
6.3.2. Gauge transformation
6.3.3. Equation of motion
6.3.4. P and S waves
6.4. Solutions of wave equation for single spatial dimension
6.4.1. d’Alembert’s approach
6.4.2. Directional derivative
6.4.3. Well-posed problem
6.4.4. Causality, finite propagation speed and sharpness of signals
6.5. Solution of wave equation for two and three spatial dimensions
6.6. On evolution equation
6.7. Solutions of wave equation for one-dimensional scattering 168
6.8. On weak solutions of wave equation
6.9. Reduced wave equation
6.9.1. Harmonic-wave trial solution
6.9.2. Fourier’s transform of wave equation
6.10. Extensions of wave equation

Chapt 7. Equations of motion: Anisotropic inhomogeneous continua
7.1. Formulation of equations
7.2. Formulation of solutions
7.3. Eikonal equation

Chapt 8. Hamilton’s ray equations
8.1. Method of characteristics
8.2. Time parametrization of characteristic equations
8.3. Physical interpretation of Hamilton’s ray equations and solutions
8.4. Relation between p and _x
8.4.1. General formulation
8.4.2. Phase and ray velocities
8.4.3. Phase and ray angles
8.4.4. Geometrical illustration
8.5. Example: Elliptical anisotropy and linear inhomogeneity
8.5.1. Introductory comments
8.5.2. Eikonal equation
8.5.3. Hamilton’s ray equations
8.5.4. Initial conditions
8.5.5. Physical interpretation of equations and conditions
8.5.6. Solution of Hamilton’s ray equations
8.5.7. Solution of eikonal equation
8.5.8. Physical interpretation of solutions
8.6. Example: Isotropy and inhomogeneity

Chapt 9. Christoffel’s equations
9.1. Explicit form of Christoffel’s equations
9.2. Christoffel’s equations and anisotropic continua
9.2.1. Introductory comments
9.2.2. Monoclinic continua
9.2.3. Transversely isotropic continua
9.3. Phase-slowness surfaces

Chapt 10. Reflection and transmission
10.1. Angles at interface
10.1.1. Phase angles
10.1.2. Ray angles
10.1.3. Example: Elliptical velocity dependence
10.2. Amplitudes at interface
10.2.1. Kinematic and dynamic boundary conditions
10.2.2. Reflection and transmission amplitudes

Chapt 11. Lagrange’s ray equations
11.1. Legendre’s transformation of Hamiltonian
11.2. Formulation of Lagrange’s ray equations
11.3. Beltrami’s identity

Chapt 12. Euler’s equations
12.1. Mathematical background
12.2. Formulation of Euler’s equation
12.3. Beltrami’s identity
12.4. Generalizations of Euler’s equation
12.5. Special cases of Euler’s equation
12.6. First integrals 333
12.7. Lagrange’s ray equations as Euler’s equations

Chapt 13. Variational principles
13.1. Fermat’s principle
13.1.1. Statement of Fermat’s principle
13.1.2. Properties of Hamiltonian H
13.1.3. Variational equivalent of Hamilton’s ray equations
13.1.4. Properties of Lagrangian L
13.1.5. Parameter-independent Lagrange’s ray equations
13.1.6. Ray velocity
13.1.7. Proof of Fermat’s principle
13.2. Hamilton’s principle: Example
13.2.1. Introductory comments
13.2.2. Action
13.2.3. Lagrange’s equations of motion
13.2.4. Wave equation

Chapt 14. Ray parameters
14.1. Traveltime integrals
14.2. Ray parameters as first integrals
14.3. Example: Elliptical anisotropy and linear inhomogeneity
14.3.1. Introductory comments
14.3.2. Rays
14.3.3. Traveltimes
14.4. Rays in isotropic continua
14.5. Lagrange’s ray equations in xz-plane
14.6. Conserved quantities and Hamilton’s ray equations

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