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

158 Stiffness matrix method for layered media
a)
1
N
b)
1
−sN, uN
N
sN, uN
Figure 10.4: Amplification of plane SV-P waves using stiffness
matrices. a) Actual layered system with seismic source. b) Layers
and half-space as free bodies. c) Half-space alone (either a
rock outcrop or a divided full space).
c)
Upper half-space
-s*, u*
s*, u*
Outcrop
another, despite the fact that the lower substructures are identical and the source is exactly
the same. This is ostensibly the result of the interface stresses acting between the layers
and the half-space, so the difference in motions is solely the result of the differences in
stresses, which act as secondary “sources” applied at the half-space interface. Since the
stress imbalance and deviations in displacements are observed on the same horizon, they
must satisfy the dynamic equilibrium equation
s N − s ∗ = K half (u N − u ∗ ) , so that − s N =−s ∗ + K half u ∗ − K half u N (10.69)
In the case of a rock outcrop, s ∗ = 0 is the stress-free condition at the surface of the halfspace,
and u ∗ is the (presumably known) outcropping motion elicited by the source. For