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

176 Stiffness matrix method for layered media
Table 10.6. Stiffness matrices in spherical coordinates
Note: Stiffness matrices given below are not symmetric. See Section 10.4.2 on how to
make them symmetric. Also, for greater computational efficiency, solve separately for
the SH and SVP degrees of freedom in K when solving for ũ, since they are uncoupled.
For definition of the submatrices below, see Tables 10.7 and 10.8.
R e = external radius R i = internal radius R = generic radius
˜p = Kũ K = K(R e , R i ,λ,µ,ω,m) (not a function of n)
F (i)
mj
≡ F m (i) (R j ) H (i)
mj
≡ H m (i) (R j )
{ }
Kee K ei
Spherical layer: (6 × 6) K =
K ie K ii
=
{
R
2
e F (1)
me
−R 2 i F(1) mi
Re 2F(2)
me
−Ri 2F(2)
mi
}{
H
(1)
me H (2) } −1
me
H (1)
mi
H (2)
mi
Solid sphere: (3 × 3) K = R 2 F m (H m) −1 Core (use j m in F m , H m )
( )
Cavity: (3 × 3) K =−R 2 F (2)
m H (2) −1
m Infinite external space (use h (2)
m for
matrices)
Wavenumber–space transform:
˜p(R, m, n,ω) = J ∫ { −1 π ∫ }
0 sin φ 2π
Ln m 0
T n p(R,φ,θ,ω) dθ dφ traction per steradian
ũ = K −1 ˜p
∑
u(R,φ,θ,ω) = ∞ m∑
T n L n mũ(R, m, n,ω) = ∑ ∞ m∑
ū(R, m, n,ω)
m=0 n=0
m=0 n=0
with
⎧ ⎫ ⎧ ⎫
⎨ u R ⎬ ⎨ p R ⎬
u = u
⎩ φ
⎭ = displacements, p = p
⎩ φ = tractions per steradian
⎭
u θ p
⎧ θ
⎫
J = π(1 + δ n0)(n + m)! ⎨ 1 0 0 ⎬
(m + 1 2 )(m − n)! 0 m(m + 1) 0
⎩
⎭
0 0 m(m + 1)
The negative sign in the second term comes from the fact that external tractions are
opposite in direction to the internal stresses at the inner surface. Considering that the wave
field in the layer is composed of both outgoing and ingoing waves, and with reference to
Tables 10.6, 10.7 and the results in Section 9.3, we write the displacement and traction
vectors for specific indices m, n as
(
)
ū(R,φ,θ,ω) = T n L n m H (1)
m a 1 + H (2)
m a 2
= T n L n m
{ } { }
H (1)
m H (2) a 1
m
a 2
(10.145)