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

Recommendations

Info

10.3 Stiffness matrix method in cylindrical coordinates 171 Torsional point source (n = 0) A torsional point source of unit intensity can be obtained as the limit of a tangential ring load p θ acting on a ring of radius r 1 → 0, which elicits a torsional moment M t = 2π ˜p θ r 1 = 1 (observe that ˜p θ is the traction per radian, so the total ring load is 2π times larger, and multiplication by the radius gives the moment). Hence, the load vector in the frequency– axial-wavenumber domain for a torsional point source of unit intensity is ⎧ ⎫ ⎧ ⎫ ⎨ ˜p r ⎬ p 1 = ˜p ⎩ θ ⎭ = 1 ⎨ 0 ⎬ 1 (10.125) 2πr −i ˜p 1 ⎩ ⎭ z 0 Hence, ⎧ ( ) 1 π ⎪⎨ lim r1→0 (r1F11)−1 p 1 = ( ( ) ) 2 2πr1 4µ 1 + kz kβ ⎪⎩ that is, − kα kβ ( ( ) ) 2 1 − kz ⎫ zβ 0 −2 kα kz kβ kβ kβ ( ( ) ) ⎫⎪ ⎬ ⎧⎪ ⎨ 0 ⎪⎬ 2 0 − 1 + kz zβ 0 1 kβ ⎪ −2 kz zβ 0 2 ⎪ ⎩ ⎭ 0 ⎪⎭ kβ (10.126) ⎧ ⎫ lim (r 1F 11) −1 p 1 = k ⎨ 0 ⎬ β 1 r1→0 8µ ⎩ ⎭ (use to form p 2eq) (10.127) 0 Axial point load (n = 0) This load can be visualized as the limit of an axial ring traction per radian, p z , acting on a ring of radius r 1 → 0, which implies an axial load P z = 2π ˜p z = 1. Hence, the load vector in the frequency–axial-wavenumber domain for a unit axial load is ⎧ ⎫ ⎧ ⎫ ⎨ ˜p r ⎬ p 1 = ˜p ⎩ θ ⎭ = −i ⎨ 0 ⎬ 0 (10.128) 2π ⎩ ⎭ −i ˜p z 1 lim (r 1F 11) −1 p 1 = r1→0 ( ) −i 2π ⎧ ⎪⎨ × ⎪⎩ − kα kβ ( 4µ π ( ) ) 2 1 + kz kβ ( ( ) ) 2 1 − kz z kβ β 0 −2 kα kβ ( ( ) ) 2 0 − 1 + kz z β 0 −2 kz kβ z β 0 2 kβ kz kβ ⎫ ⎧ ⎫ ⎪⎬ ⎨ 0 ⎬ 0 ⎩ ⎭ 1 ⎪⎭ (10.129)
172 Stiffness matrix method for layered media that is, lim r1→0 (r 1F 11) −1 p 1 = ( i ( ) ) 2 kz 4µ 1+ kβ ⎧ ⎪⎨ ⎪⎩ ( ) 2 kα kz kβ 0 −1 kβ ⎫ ⎪⎬ ⎪⎭ (use to form p 2eq) (10.130) Lateral point load (n = 1) A lateral load of unit intensity can be visualized as the combination of radial and tangential ring loads per radian of equal strength acting on a ring of radius r 1 → 0: ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎨ p r ⎬ p = p ⎩ θ ⎭ = 1 ⎨ cos θ 0 0 ⎬ ⎨ 1 ⎬ 0 − sin θ 0 1 2π ⎩ ⎭ ⎩ ⎭ = 1 ⎨ 1 ⎬ 2π T 1 1 (10.131) ⎩ ⎭ p z 0 0 cos θ 0 0 Projecting this onto the direction of the load and integrating over the ring, we obtain a unit load in the x direction: ∫ 2π 0 (p r cos θ − p θ sin θ) r 1 dθ = 1 2πr1 ∫ 2π 0 ( ) cos 2 θ + sin 2 θ r 1 dθ = 1 (10.132) The load vector in the azimuthal wavenumber domain follows from the Fourier transform ⎧ ⎫ ∫ 2π ˜p = 1 T π 1 p(r 1 ,θ,k z ,ω)dθ = 1 ⎨ 1 ⎬ 1 (10.133) 0 4π ⎩ ⎭ 0 (the additional factor 1/2 arises from the integral 1 π ⎧ ⎫ ⎧ ⎫ ⎨ ˜p r ⎬ p 1 = ˜p ⎩ θ ⎭ = 1 ⎨ 1 ⎬ 1 4π ⎩ ⎭ −i ˜p z 0 ∫ 2π 0 T 1 T 1 dθ = 1 I). Hence, 2 (10.134) ( ) 1 lim (r 1F 11) −1 p 1 = r1→0 4π [ 8µ π ( ) ] 2 1 + kz kβ ⎧ ⎪⎩ ( kα kβ ) 2 ( ) 2 kα −3 kβ ( ) 2 kα kz z β kβ kβ ⎪⎨ ( ) 2 ( ) [ 2 ( ) ] 2 × 1 + kz 1 + kz − 1 + kz kβ kβ kz kβ kz kβ kβ kβ kz kβ z β [ ( ) ] 2 2 − kz z β ⎧ ⎫ ⎫⎪ ⎬ ⎨ 1 ⎬ 1 ⎩ ⎭ 0 ⎪ ⎭ (10.135) that is, lim r1→0 (r 1F 11) −1 p 1 = [ 1 ( ) ] 2 kz 16µ 1+ kβ ⎧ ⎪⎨ ( ) 2 kα kβ ( 1 + kz kβ ⎪⎩ kz kβ ) 2 ⎫ ⎪⎬ ⎪⎭ (use to form p 2eq ) (10.136)
Page 2 and 3:
FUNDAMENTAL SOLUTIONS IN ELASTODYNA
Page 4 and 5:
Fundamental Solutions in Elastodyna
Page 6 and 7:
Contents Preface page ix SECTION I:
Page 8 and 9:
Contents vii SECTION V: ANALYTICAL
Page 10 and 11:
Preface We present in this work a c
Page 12 and 13:
SECTION I: PRELIMINARIES 1 Fundamen
Page 14 and 15:
1.1 Notation and table of symbols 3
Page 16 and 17:
1.3 Coordinate systems and differen
Page 18 and 19:
Page 20 and 21:
Page 22 and 23:
Page 24 and 25:
1.4 Strains, stresses, and the elas
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
Page 38 and 39:
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 76 and 77:
Page 78 and 79:
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 90 and 91:
6.1 3-D half-space, suddenly applie
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 104 and 105:
Page 106 and 107:
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 124 and 125:
Page 126 and 127:
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
Page 140 and 141: 9.1 Cartesian coordinates 129 Writi
Page 142 and 143: 9.2 Cylindrical coordinates 131 Fou
Page 144 and 145: 9.2 Cylindrical coordinates 133 Hen
Page 146 and 147: 9.2 Cylindrical coordinates 135 Oth
Page 148 and 149: 9.3 Spherical coordinates 137 √
Page 150 and 151: 9.3 Spherical coordinates 139 ∫
Page 152 and 153: 10.1 Summary of method 141 special
Page 154 and 155: 10.2 Stiffness matrix method in Car
Page 170 and 171: 10.3 Stiffness matrix method in cyl
Page 186 and 187: 10.4 Stiffness matrix method for la
Page 196 and 197: SECTION VI: APPENDICES 11 Basic pro
Page 198 and 199: 11.2 Spherical Bessel functions 187
Page 200 and 201: 11.3 Legendre polynomials 189 1.0 P
Page 202 and 203: 11.4 Associated Legendre functions
Page 204 and 205: 11.4 Associated Legendre functions
Page 206 and 207: 12.1 Fourier transforms 195 b) Wave
Page 208 and 209: 12.3 Spherical Hankel transforms 19
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 232 and 233:
function [T, Uxx, Uxz, Uzz] = lamb2
Page 234 and 235:
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 252 and 253:
Page 254 and 255:
Page 256 and 257:
function [ztors, zspher] = spheroid
Page 258 and 259:
function [ztors, zspher] = spheroid
Page 260 and 261:
function [si] = cisib(x) 249 functi
Page 262:
function [el3] = ellipint3(phi, N,
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?