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

1.4 Strains, stresses, and the elastic wave equation 25
⎧ ⎫ ⎧ ⎫ ⎧
⎫
⎨ 0 λ 0 ⎬ ⎨ 0 0 0 ⎬ ⎨ 0 0 0 ⎬
D R2 = 0 0 0
⎩ ⎭ , D φ2 = 0 λ 0
⎩ ⎭ , D θ2 = 0 0 −µ
⎩
⎭
0 0 0
0 0 −µ
0 λ + 2µ 0
(1.101d)
⎧
⎫
⎧
⎫
⎧
⎫
⎨ 4(λ + µ) 0 0⎬
⎨ 0 2(λ + µ) 0⎬
⎨ 0 0 0⎬
D 11 = 0 µ 0
⎩
⎭ , D 12 = 0 0 0
⎩
⎭ , D 22 = 0 λ + 2µ 0
⎩
⎭
0 0 µ
0 0 0
0 0 µ
(1.101e)
Wave equation
The dynamic equilibrium equation can be written as
L T σ DL εu + b = ρ ü (1.102)
For any arbitrary, spherically anisotropic material with D αβ = L T α DL β, the wave
equation is
∂ 2
ρü = b + D RR
∂ R + D 1 ∂ 2
2 φφ
R 2 ∂φ + D 1 ∂ 2
2 θθ
R 2 sin 2 φ ∂θ + (D 2 Rφ + D 1 ∂ 2
φR)
R ∂ R∂φ
1
+ (D Rθ + D θR)
R sin φ ∂ R∂θ + (D 1
φθ + D θφ)
R 2 sin φ ∂φ∂θ
∂ 2
+ (2D RR + D R1 − D 1 ∂
1R)
R ∂ R + (D φ R + D R2 − D cot φ ∂
2R)
R ∂ R
+ (D Rφ + D φ1 − D 1 ∂
1φ)
R 2 ∂φ + (D φφ + D φ2 − D cot φ ∂
2φ)
R 2 ∂φ
1 ∂
+ (D Rθ + D θ1 − D 1θ)
R 2 sin φ ∂θ + (D cot φ ∂
θ2 − D 2θ)
R 2 sin φ ∂θ
+ (D R1 − D 11) 1 R 2 + (D R2 + D φ1 − D 12 − D 21) cot φ
R 2
− D φ2
1
R 2 sin 2 φ + (D φ2 − D 22) cot2 φ
R 2 (1.103)
∂ 2