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

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12.3 Spherical Hankel transforms 197 condition is satisfied), in which case the transform is self-reciprocating, namely f (r) = ∫ ∞ 0 k ν F n (k) J n (kr) dk We list below some Hankel transforms using ν =1. f (r) = ∫ ∞ 0 F n(k) J n (kr) kdk F n (k) = ∫ ∞ 0 f (r) J n(kr) rdr δ(r) ←→ n=0 1 ∫ ∞ 0 ∫ ∞ 0 ∫ ∞ 0 r δ(r − a) r e −αr 2 ←→ n ←→ n=0 J n (ka) 1 2α e−k 2 /4α ⎧ π k ⎪⎨ 2i J ν(k 0 r) H ν (2) (k 0 R), r ≤ R k 2 − k0 2 J ν (kR)J ν (kr) dk = ⎪⎩ π Im(k 0 ) < 0 2i J ν(k 0 R) H ν (2) (k 0 r), R ≤ r ⎧ π 1 ⎪⎨ J 0 (k 0 r) H (2) 1 (k 0 R) − 1 k 2 − k0 2 J 1 (kR)J 0 (kr) dk = 2i k 0 Rk 2 , r ≤ R 0 Im(k π 0 ) < 0 ⎪⎩ J 1 (k 0 R) H (2) 0 (k 0 r), R ≤ r 2i k 0 ⎧ π 1 ⎪⎨ k(k 2 − k0 2) J 2i k 2 J 1 (k 0 r) H (2) 1 (k 0 R) − r 0 2Rk 2 , r ≤ R 0 1(kR)J 1 (kr) dk = π ⎪⎩ 2i k0 2 J 1 (k 0 R) H (2) 1 (k 0 r) − R Im(k 0 ) < 0 2r k0 2 , R ≤ r 12.3 Spherical Hankel transforms Starting from the expression j n (kr) = √ π/2kr J n+1/2 (kr), which relates the spherical Bessel function of the first kind to the half-integer order conventional Bessel function, we conclude that an appropriate spherical Hankel transform is f (r) = ∫ ∞ 0 k ν+ 1 2 F n (k) j n (kr) dk ⇔ F n (k) = ∫ ∞ 0 r ν+ 1 2 f (r) j n (kr) dr provided that the integrals ∫ ∞ ∣ 0 r ν f (r) ∣ ∫ ∞ dr and/or ∣ 0 k ν F(k) ∣ dk exist. A convenient value is ν = 1/2, in which case f (r) = ∫ ∞ 0 kF n (k) j n (kr) dk ⇔ F n (k) = ∫ ∞ 0 r f (r) j n (kr) dr
13 MATLAB programs The following Matlab programs are listed herein: SH2D FULL SH line load in full space SVP2D FULL SVP line load in full space BLAST2D Blast line source in full space CAVITY2D 2-D cavity in full space POINT FULL Point source in full 3-D space TORSION FULL Torsional point source in full space CAVITY3D 3-D cavity in full space SH2D HALF SH line source in half-space GARVIN Blast line source in half-space LAMB2D SVP line source in half-space LAMB3D Point source in half-space (horizontal and vertical) TORSION HALF Torsional point source in half-space SH PLATE SH line load in homogeneous plate SH STRATUM SH line source in homogeneous stratum SVP PLATE SVP line source in plate with mixed boundary conditions SPHEROIDAL Spheroidal and torsional modes of homogeneous sphere In addition, the following functions are called by some of the programs EASYPLOT Writes text files for plotting (most programs) CISIB Sine and cosine integrals (cavity 2-D and 3-D) ELLIPINT3 Elliptic integrals (Lamb 3-D) Note: Some programs have same-named functions embedded within, but they are not identical to each other, even if functionally related. Thus, they are not interchangeable. 198
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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
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2.2 Line dipoles 29 z z z y y y x x
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2.5 Blast loads (explosive line and
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2.6 Dipoles in cylindrical coordina
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SECTION II: FULL SPACE PROBLEMS 3 T
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3.3 SH line load in an orthotropic
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3.4 In-plane line load (SV-P waves)
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3.5 Dipoles in plane strain 41 0.6
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3.6 Line blast source: suddenly app
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3.7 Cylindrical cavity subjected to
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3.7 Cylindrical cavity subjected to
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4.2 Point load (Stokes problem) 49
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4.2 Point load (Stokes problem) 51
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4.3 Tension cracks 53 given earlier
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4.5 Torsional point source 55 Spher
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4.6 Torsional point source with ver
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4.8 Spherical cavity subjected to a
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4.8 Spherical cavity subjected to a
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4.9 Spatially harmonic line source
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SECTION III: HALF-SPACE PROBLEMS 5
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5.3 Half-plane, SV-P source and rec
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5.4 Half-plane, SV-P source on surf
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5.4 Half-plane, SV-P source on surf
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5.5 Half-plane, line blast load app
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6.1 3-D half-space, suddenly applie
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6.2 3-D half-space, suddenly applie
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6.3 3-D half-space, buried torsiona
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6.3 3-D half-space, buried torsiona
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SECTION IV: PLATES AND STRATA 7 Two
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7.2 Stratum subjected to SH line so
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7.3 Plate with mixed boundary condi
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SECTION V: ANALYTICAL AND NUMERICAL
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8.1 Summary of results 99 Plane str
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8.2 Scalar Helmholtz equation in Ca
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8.3 Vector Helmholtz equation in Ca
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8.4 Elastic wave equation in Cartes
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8.4 Elastic wave equation in Cartes
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8.6 Vector Helmholtz equation in cy
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8.7 Elastic wave equation in cylind
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8.8 Scalar Helmholtz equation in sp
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8.9 Vector Helmholtz equation in sp
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8.10 Elastic wave equation in spher
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8.10 Elastic wave equation in spher
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9 Integral transform method The int
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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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10.2 Stiffness matrix method in Car
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Page 210 and 211: function [ ] = SH2D Full( ) 199 fun
Page 212 and 213: function [ ] = SVP2D Full(x, z, poi
Page 214 and 215: function [ ] = SVP2D Full(x, z, poi
Page 216 and 217: function [ ] = Blast2D(pois) 205 Ur
Page 218 and 219: function [ ] = Cavity2D(r, r0, pois
Page 220 and 221: function [ ] = Point Full(pois) 209
Page 222 and 223: function Torsion Full(x, y, z, td,
Page 224 and 225: function [ ] = Cavity3D(r, pois) 21
Page 226 and 227: function [ ] = SH2D Half(xs, zs, xr
Page 228 and 229: function [T, Ux, Uz] = Garvin(x, z,
Page 230 and 231: function [T, Uxx, Uxz, Uzz] = lamb2
Page 236 and 237: function [T, Uxx, Utx, Uzz, Urz] =
Page 238 and 239: function [T, Uxx, Utx, Uzz, Urz] =
Page 240 and 241: function [ ] = Torsion Half(r, z, z
Page 242 and 243: function [ ] = Torsion Half(r, z, z
Page 244 and 245: function [ ] = SH Plate(x, z, z0) 2
Page 246 and 247: function [ ] = SH Stratum(x, z, z0)
Page 248 and 249: function [ ] = SH Stratum(x, z, z0)
Page 250 and 251: function [ ] = SVP Plate(x, z, z0,
Page 256 and 257: 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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