Geology Reference
In-Depth Information
The last Galerkin vector of the group,
G
4
, represents a Kelvin problem generated
by rotating a lower half-space solution by 180
◦
into the upper half-space, as an
image Kelvin solution. All four of the Galerkin vectors represent solutions singular
at the image point. The arbitrary constants
A
,
B
,
C
,
D
of the linear combination
remain to be determined.
For their determination, three conditions arise from the vanishing of the normal
stress τ
33
and the shear stresses τ
13
, τ
23
on the surface
x
3
0 for it to be a free
surface, and a fourth condition arises to ensure that the total force on a sphere
surrounding the image point balances the point force in the lower half-space.
The displacement field corresponding to each of the four Galerkin vectors is
given by relation (1.335).
For the first Galerkin vector
G
1
,wehave
=
e
1
A
(
x
1
−
ξ
1
)
Q
3
e
2
A
(
x
2
−
ξ
2
)
Q
3
e
3
A
(
x
3
+
ξ
3
)
Q
3
(
G
1
)
∇
∇·
=−
−
−
,
(1.444)
and
∂
x
j
∂
x
j
log(
Q
+
ξ
3
)
2
A
e
3
∂
2
G
1
∇
=
+
x
3
=
0.
(1.445)
Then, from (1.335)
2μ
u
=−∇
(
∇·
G
1
)
,
(1.446)
and the displacement field is found to have the components
A
2μ
x
1
−
ξ
1
Q
3
,
u
2
A
2μ
x
2
−
ξ
2
Q
3
,
u
3
A
2μ
x
3
+
ξ
3
Q
3
.
u
1
=
=
=
(1.447)
Taking derivatives, we find that
1
Q
3
−
3
(
x
1
−
ξ
1
)
2
Q
5
∂
u
1
∂
x
1
=
A
2μ
,
(1.448)
1
Q
3
−
−
ξ
2
)
2
Q
5
∂
u
2
∂
x
2
=
A
2μ
3
(
x
2
,
(1.449)
1
Q
3
−
+
ξ
3
)
2
Q
5
∂
u
3
A
2μ
3
(
x
3
∂
x
3
=
.
(1.450)
Thus, the cubical dilatation, the divergence of the displacement field, is
3
Q
3
−
Q
5
3
Q
2
A
2μ
∇·
u
=
=
0.
(1.451)
Search WWH ::
Custom Search