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

82 Three-dimensional problems in homogeneous half-spaces
0.2
µru rx
0.1
0
-0.1
Figure 6.2a: Radial displacement at
surface due to suddenly applied horizontal
point load, ν = 0.25 (Chao’s
problem). Varies as cos θ.
-0.2
tβ
τ =
r
-0.3
0 0.5 1.0 1.5 2.0
Displacementsonthe epicentral axis (r= 0, z ≠ 0)
At all times, u xx (0, z, t) = u rx (0, z, t), u θ x (0, z, t) =−u rx (0, z, t), u z (0, z, t) = 0. Define the
auxiliary functions
( )√
τ τ 2 − 1 τ
3
2 + 2 3
f (τ) = ( ) 2 ( )√
2τ 2 + 1 3 − 4τ τ 2 − 1 τ
3
2 + 2 3
( √
)
τ 2 (τ 2 − 1) 2τ τ 2 − 2 3 − (2τ 2 − 1)
g(τ) =
√
(2τ 2 − 1) 2 − 4τ(τ 2 − 1) τ 2 − 2 3
(6.17)
(6.18)
0.07
µru θx
0
-0.07
Figure 6.2b: Tangential displacement at
surface due to suddenly applied horizontal
point load, ν = 0.25 (Chao’s
problem). Varies as −sin θ.
tβ
τ =
r
-0.14
0 0.5 1.0 1.5 2.0