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)
u =−∇
(
∇· G 1 ) ,
(1.446)
and the displacement field is found to have the components
A
x 1
ξ 1
Q 3 , u 2
A
x 2
ξ 2
Q 3 , u 3
A
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
,
(1.448)
1
Q 3
ξ 2 ) 2
Q 5
u 2
x 2 =
A
3 ( x 2
,
(1.449)
1
Q 3
+ ξ 3 ) 2
Q 5
u 3
A
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
∇·
u
=
=
0.
(1.451)
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