Fundamental solutions in elastodynamics: a compendium

Half-title......Page 2
Title......Page 4
Copyright......Page 5
Content......Page 6
Preface......Page 10
1.1 Notation and table of symbols......Page 12
a) Three-dimensional space (Fig. 1.1a)......Page 15
1.3.2 Cylindrical coordinates......Page 17
1.3.3 Spherical coordinates......Page 20
Strains......Page 24
Stresses......Page 25
Stresses in principal surfaces......Page 26
Strains......Page 27
Stresses......Page 28
Stresses in principal surfaces......Page 29
Wave equation......Page 31
Strains and stresses......Page 32
Stresses in principal surfaces......Page 34
Wave equation......Page 36
2.1 Point dipoles or doublets: single couples and tensile crack sources......Page 38
Dipoles in a homogeneous space......Page 39
2.2 Line dipoles......Page 40
2.4 Seismic moments (double couples with no net resultant)......Page 41
2.5 Blast loads (explosive line and point sources)......Page 42
2.6 Dipoles in cylindrical coordinates......Page 43
Time domain, impulse response......Page 46
3.3 SH line load in an orthotropic space......Page 47
Frequency domain......Page 49
Time domain, impulse response......Page 50
3.5 Dipoles in plane strain......Page 51
Single dipoles (Fig. 2.2)......Page 52
Double couple (Fig. 2.4)......Page 53
3.6 Line blast source: suddenly applied pressure......Page 54
3.7 Cylindrical cavity subjected to pulsating pressure......Page 55
Time domain......Page 56
Frequency domain......Page 59
Impulse response......Page 62
4.3 Tension cracks......Page 63
Frequency domain......Page 64
Cartesian coordinates......Page 65
4.5 Torsional point source......Page 66
Spherical coordinates......Page 67
Time domain, arbitrary causal variation of source with time......Page 68
4.7 Point blast source......Page 69
Frequency domain......Page 70
Time domain......Page 71
Time domain......Page 72
4.9 Spatially harmonic line source (2½-D problem)......Page 74
Green\'s functions......Page 75
Strain components (l = x, y, z = direction of load)......Page 77
The Bn functions......Page 78
Mixed wavenumber–time domain......Page 80
5.2 SH line source in an orthotropic half-plane......Page 81
5.3 Half-plane, SV-P source and receiver at surface (Lamb\'s problem)......Page 82
5.4 Half-plane, SV-P source on surface, receiver at interior, or vice versa......Page 84
5.5 Half-plane, line blast load applied in its interior (Garvin\'s problem)......Page 87
6.1 3-D half-space, suddenly applied vertical point load on its surface (Pekeris-Mooney\'s problem)......Page 89
b) v > 0.2631…......Page 90
6.2 3-D half-space, suddenly applied horizontal point load on its surface (Chao\'s problem)......Page 92
6.3 3-D half-space, buried torsional point source with vertical axis......Page 94
Time domain, arbitrary causal variation of source with time......Page 96
7.1.2 Normal mode solution......Page 98
7.2 Stratum subjected to SH line source......Page 99
7.2.1 Solution using the method of images......Page 100
7.3 Plate with mixed boundary conditions subjected to SV-P line source......Page 101
7.3.1 Solution using the method of images......Page 102
a) FC-FC......Page 103
c) FC-CF......Page 104
d) CF-FC......Page 105
Read me first......Page 108
Cartesian coordinates......Page 109
Cylindrical coordinates......Page 110
8.2 Scalar Helmholtz equation in Cartesian coordinates......Page 112
8.3 Vector Helmholtz equation in Cartesian coordinates......Page 113
8.4 Elastic wave equation in Cartesian coordinates......Page 115
8.4.1 Horizontally stratified media, plane strain......Page 116
8.5 Scalar Helmholtz equation in cylindrical coordinates......Page 119
8.6 Vector Helmholtz equation in cylindrical coordinates......Page 120
8.7 Elastic wave equation in cylindrical coordinates......Page 121
8.7.1 Horizontally stratified media......Page 123
8.7.2 Radially stratified media......Page 125
8.8 Scalar Helmholtz equation in spherical coordinates......Page 128
8.9 Vector Helmholtz equation in spherical coordinates......Page 129
8.10 Elastic wave equation in spherical coordinates......Page 131
9.1 Cartesian coordinates......Page 136
9.2.1 Horizontally stratified media......Page 141
9.2.2 Cylindrically stratified media......Page 147
9.3 Spherical coordinates......Page 148
10 Stiffness matrix method for layered media......Page 151
10.1 Summary of method......Page 152
10.2 Stiffness matrix method in Cartesian coordinates......Page 153
10.2.1 Analytic continuation in the layers......Page 159
10.2.2 Numerical computation of stiffness matrices......Page 161
10.2.3 Summary of computation......Page 162
10.3 Stiffness matrix method in cylindrical coordinates......Page 170
10.3.1 Horizontally layered system......Page 171
a) Single layer......Page 175
b) Unbounded external region......Page 177
c) Layered system......Page 178
d) Solid core......Page 179
10.4 Stiffness matrix method for layered spheres......Page 186
10.4.2 Asymmetry......Page 191
10.4.3 Expansion of source and displacements into spherical harmonics......Page 192
10.4.4 Rigid body spheroidal modes......Page 193
11.1.2 Recurrence relations......Page 196
First condition......Page 197
11.2.1 Differential equation......Page 198
11.2.2 Trigonometric representations......Page 199
11.3.1 Differential equation......Page 200
11.3.5 Orthogonality condition......Page 201
11.4.1 Differential equation......Page 202
11.4.3 Orthogonality conditions......Page 203
11.4.6 Leibniz rule for the derivative of a product of two functions......Page 204
a) Transforms in time–frequency......Page 205
b) Wavenumber–space integrals......Page 206
12.2 Hankel transforms......Page 207
12.3 Spherical Hankel transforms......Page 208
13 MATLAB programs......Page 209