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

76 Two-dimensional problems in homogeneous half-spaces
4
µru x
2
x=1
x=10
x=100
0
-2
tβ
τ =
r
2
0 0.5 1.0 1.5 2.0
µru z
Figure 5.4: Horizontal (top) and vertical
(bottom) displacements due to line blast
source at depth h = 1,ν = 0.25 (Garvin’s
problem).
-2
-6
x=1
x=10
x=100
tβ
τ =
r
-10
0 0.5 1.0 1.5 2.0
5.5 Half-plane, line blast load applied in its interior (Garvin’s problem) 4
A line blast source acts within an infinitesimally small cavity at x = 0 and some depth
h = |z| below the free surface. The singularly large pressure source is a step load in time
and of magnitude such that in the limit of a vanishingly small cylindrical cavity, the integral
of the pressure over the cross section is unity. Using the same notation as for the line source
in the Section 5.4, it is found that the displacements on the surface are
{
u x (x, 0, t) = 1
√ }
πµr Im 2qα 1 + q
2 α ∂q α
R(qα 2)
sgn(x) H (τ − a)
(5.27a)
∂τ
{(
u z (x, 0, t) = −1
) }
1 + 2q
2
πµr Re α ∂q α
R(qα 2)
sgn(z) H (τ − a)
(5.27b)
∂τ
4 Garvin, W. W. (1956), Exact transient solution of the buried line source problem, Proceedings of the Royal
Society ofLondon, Series A, Mathematical and Physical Sciences, Vol. 234, No. 1199 (Mar.), pp. 528–541.