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

4 Three-dimensional problems in full,
homogeneous spaces
4.1 Fundamental identities and definitions
t P = R α , t S = R β , R = √ x 2 + y 2 + z 2 (4.1)
P = ωR
α ,
S = ωR
β ,
√
a = β
α = 1 − 2ν
2(1 − ν)
(4.2)
γ i = cos θ i = x i
R ,
γ i,k = ∂ cos θ i
∂x k
= 1 R (δ ik − γ i γ k),
∂ f (R)
∂x k
= γ k
∂ f
∂ R
(4.3)
4.2 Point load (Stokes problem)
Unit impulsive point sources act at the origin, in any coordinate direction, within an infinite,
homogeneous space. The receiver is at x 1 , x 2 , x 3 .
Frequency domain 1
g i j (R,ω) = 1
4πµR {ψδ i j + χγ i γ j }, i, j = 1, 2, 3 (or x, y, z) (4.4)
∂g i j
= 1 { [( ∂ψ
γ k
∂x k 4πµR ∂ R − ψ ) ( ∂χ
δ i j +
R ∂ R − 3χ ) ]
γ i γ j + χ }
R R [δ ik γ j + δ jk γ i ]
( ) β 2 { i
ψ = e −iP + 1 } {
α P 2 + e −iS 1 − i − 1 }
P
S 2 S
χ = e −iP ( β
α
) 2 {
1 − 3i − 3 } {
P 2 − e −iS 1 − 3i − 3 }
P
S 2 S
(4.5)
(4.6)
(4.7)
1 Dominguez, J. and Abascal, R., 1984, On fundamental solutions for the boundary integral equations in static
and dynamic elasticity, Engineering Analysis, Vol. 1, No. 3, pp. 128–134, eqs. 20, 21.
48