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

142 Stiffness matrix method for layered media
This matrix is block-tridiagonal (i.e., narrowly banded) and symmetric, and in general
its elements are complex. Damping is incorporated via complex moduli.
Solve the system of equations p = Ku by standard methods, and obtain the displacements
in the frequency–wavenumber domain.
Carry out an inverse transform into the spatial–temporal domain, which yields the
desired response.
Omitting the inverse Fourier transform over frequencies, the requisite inverse integral
transforms needed to obtain the displacement vector u in the spatial domain are of the
following form:
Cartesian coordinates, horizontal layers:
∫ +∞
u(x, z,ω) = 1 ũ(k, z,ω) e −i kx dk (10.1)
2π −∞
Cylindrical coordinates, horizontal layers:
∞∑
∫ ∞
u(r,θ,z,ω) = T n kC n ũ n (k, z,ω) dk (10.2)
n=0
0
Cylindrical coordinates, cylindrical layers:
u(r,θ,z,ω) = 1 ∫ { }
+∞ ∞∑
T n ũ n (r, k z ,ω) e −ikzz dk z (10.3)
2π −∞ n=0
Spherical coordinates, spherical layers:
∞∑ m∑
u(R,φ,θ,ω) = T n L n mũmn(R,ω) (10.4)
m=0 n=0
Details and examples are given in the pages that follow.
10.2 Stiffness matrix method in Cartesian coordinates
Consider a homogeneous medium subjected to waves in the x, z plane, i.e., SV-P and SH
waves in plane strain. In the absence of external sources (i.e., b = 0), the elastic wave
equation in the transformed frequency– wavenumber space ω, k is (see Section 9.1)
k 2 ∂u
D xx u + kB xz
∂z − D ∂ 2 u
zz
∂z − 2 ρω2 u = 0 (10.5)
where for simplicity we have written the horizontal wavenumber as k x ≡ k. Making an
ansatz u(k,ω,z) = v(k,ω) e nz and substituting it into the above expression, we obtain
(
k 2 D xx − n 2 D zz + n kB xz − ρω 2 I ) v = 0 (10.6)
This equation is an eigenvalue problem in n for plane waves with vertical wavenumber k z =
±i n and eigenvectors v. Nontrivial solutions exist if the determinant of the 3 × 3 matrix
in parenthesis vanishes, a condition that leads to a sixth order equation, which for an
isotropic medium reduces to a quadratic and a biquadratic equation for SH and SV-P