Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

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6.1 3-D half-space, suddenly applied vertical point load on its surface 79 which has three roots [ξ1 2,ξ2 2 ,ξ2 3 ], the first two of which are non-physical solutions of the rationalized Rayleigh function, while ξ 3 ≡ β/C R is the actual true root. When ν< ν 0 = 0.2631, all three roots are real and satisfy 0
80 Three-dimensional problems in homogeneous half-spaces 1.0 µru rz 0.6 0.2 Figure 6.1a: Radial displacement at surface due to suddenly applied vertical point load, ν = 0.25 (Lamb–Pekeris– Mooney problem). tβ τ = r -0.2 0 0.5 1.0 1.5 2.0 in which A 3 is as before, and √ Q 1 (τ) = 1 + 2z + z 2 + z, z = a2 − ξ1 2 (6.12) τ 2 − a 2 provided that |Q 1 | < 1. If |Q 1 | > 1, replace Q 1 → 1/Q 1 . Observe that ξ1 2 is complex and that Q 2 (τ) = 1 + 2z − √ z 2 + z satisfies Q 1 Q 2 = 1, that is, Q −1 1 = Q 2 . Hence, the test |Q 1 | < 1 implicitly assigns the proper sign to the difference Q 1 − Q −1 1 . However, it is not enough to simply always add the square root term, because the proper sign of the square root depends also on the complex quadrant that z lies in, which may change with time τ. Thus, it behooves one to check the absolute value of Q 1 , and replace it with its reciprocal as needed. 0.2 µru zz -0.2 -0.6 Figure 6.1b: Vertical displacement at surface due to suddenly applied vertical point load, ν = 0.25 (Lamb–Pekeris– Mooney problem). tβ τ = r -1.0 0 0.5 1.0 1.5 2.0
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FUNDAMENTAL SOLUTIONS IN ELASTODYNA
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Fundamental Solutions in Elastodyna
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Contents Preface page ix SECTION I:
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Contents vii SECTION V: ANALYTICAL
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Preface We present in this work a c
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SECTION I: PRELIMINARIES 1 Fundamen
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1.1 Notation and table of symbols 3
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1.3 Coordinate systems and differen
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1.4 Strains, stresses, and the elas
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2 Dipoles In most cases, we provide
Page 40 and 41: 2.2 Line dipoles 29 z z z y y y x x
Page 42 and 43: 2.5 Blast loads (explosive line and
Page 44 and 45: 2.6 Dipoles in cylindrical coordina
Page 46 and 47: SECTION II: FULL SPACE PROBLEMS 3 T
Page 48 and 49: 3.3 SH line load in an orthotropic
Page 50 and 51: 3.4 In-plane line load (SV-P waves)
Page 52 and 53: 3.5 Dipoles in plane strain 41 0.6
Page 54 and 55: 3.6 Line blast source: suddenly app
Page 56 and 57: 3.7 Cylindrical cavity subjected to
Page 58 and 59: 3.7 Cylindrical cavity subjected to
Page 60 and 61: 4.2 Point load (Stokes problem) 49
Page 62 and 63: 4.2 Point load (Stokes problem) 51
Page 64 and 65: 4.3 Tension cracks 53 given earlier
Page 66 and 67: 4.5 Torsional point source 55 Spher
Page 68 and 69: 4.6 Torsional point source with ver
Page 70 and 71: 4.8 Spherical cavity subjected to a
Page 72 and 73: 4.8 Spherical cavity subjected to a
Page 74 and 75: 4.9 Spatially harmonic line source
Page 80 and 81: SECTION III: HALF-SPACE PROBLEMS 5
Page 82 and 83: 5.3 Half-plane, SV-P source and rec
Page 84 and 85: 5.4 Half-plane, SV-P source on surf
Page 86 and 87: 5.4 Half-plane, SV-P source on surf
Page 88 and 89: 5.5 Half-plane, line blast load app
Page 92 and 93: 6.2 3-D half-space, suddenly applie
Page 94 and 95: 6.3 3-D half-space, buried torsiona
Page 96 and 97: 6.3 3-D half-space, buried torsiona
Page 98 and 99: SECTION IV: PLATES AND STRATA 7 Two
Page 100 and 101: 7.2 Stratum subjected to SH line so
Page 102 and 103: 7.3 Plate with mixed boundary condi
Page 108 and 109: SECTION V: ANALYTICAL AND NUMERICAL
Page 110 and 111: 8.1 Summary of results 99 Plane str
Page 112 and 113: 8.2 Scalar Helmholtz equation in Ca
Page 114 and 115: 8.3 Vector Helmholtz equation in Ca
Page 116 and 117: 8.4 Elastic wave equation in Cartes
Page 118 and 119: 8.4 Elastic wave equation in Cartes
Page 120 and 121: 8.6 Vector Helmholtz equation in cy
Page 122 and 123: 8.7 Elastic wave equation in cylind
Page 128 and 129: 8.8 Scalar Helmholtz equation in sp
Page 130 and 131: 8.9 Vector Helmholtz equation in sp
Page 132 and 133: 8.10 Elastic wave equation in spher
Page 134 and 135: 8.10 Elastic wave equation in spher
Page 136 and 137: 9 Integral transform method The int
Page 138 and 139: 9.1 Cartesian coordinates 127 In pa
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9.1 Cartesian coordinates 129 Writi
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9.2 Cylindrical coordinates 131 Fou
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9.2 Cylindrical coordinates 133 Hen
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9.2 Cylindrical coordinates 135 Oth
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9.3 Spherical coordinates 137 √
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9.3 Spherical coordinates 139 ∫
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10.1 Summary of method 141 special
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10.2 Stiffness matrix method in Car
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10.3 Stiffness matrix method in cyl
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10.4 Stiffness matrix method for la
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SECTION VI: APPENDICES 11 Basic pro
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11.2 Spherical Bessel functions 187
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11.3 Legendre polynomials 189 1.0 P
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11.4 Associated Legendre functions
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11.4 Associated Legendre functions
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12.1 Fourier transforms 195 b) Wave
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12.3 Spherical Hankel transforms 19
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function [ ] = SH2D Full( ) 199 fun
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function [ ] = SVP2D Full(x, z, poi
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function [ ] = SVP2D Full(x, z, poi
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function [ ] = Blast2D(pois) 205 Ur
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function [ ] = Cavity2D(r, r0, pois
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function [ ] = Point Full(pois) 209
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function Torsion Full(x, y, z, td,
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function [ ] = Cavity3D(r, pois) 21
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function [ ] = SH2D Half(xs, zs, xr
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function [T, Ux, Uz] = Garvin(x, z,
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function [T, Uxx, Uxz, Uzz] = lamb2
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function [T, Uxx, Utx, Uzz, Urz] =
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function [T, Uxx, Utx, Uzz, Urz] =
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function [ ] = Torsion Half(r, z, z
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function [ ] = Torsion Half(r, z, z
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function [ ] = SH Plate(x, z, z0) 2
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function [ ] = SH Stratum(x, z, z0)
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function [ ] = SH Stratum(x, z, z0)
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function [ ] = SVP Plate(x, z, z0,
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function [ztors, zspher] = spheroid
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function [ztors, zspher] = spheroid
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function [si] = cisib(x) 249 functi
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function [el3] = ellipint3(phi, N,
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Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

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