Geology Reference
In-Depth Information
From relation (1.335),
λ
+
2μ
λ
+
μ
∇
2
G
2
2μ
u
=
−∇
(
∇·
G
2
),
(1.464)
giving a displacement field with components
−
ξ
1
)
x
3
+
ξ
3
Q
3
B
2μ
1
u
1
=
(
x
1
−
,
(1.465)
Q
(
Q
+
x
3
+
ξ
3
)
(
x
2
−
ξ
2
)
x
3
B
2μ
+
ξ
3
Q
3
1
u
2
=
−
,
(1.466)
Q
(
Q
+
x
3
+
ξ
3
)
(
x
3
+
ξ
3
)
2
Q
3
B
2μ
B
λ
+
μ
1
Q
,
u
3
=
+
(1.467)
and derivatives
∂
u
1
∂
x
1
=
x
3
+
ξ
3
Q
3
B
2μ
1
−
(1.468)
Q
(
Q
+
x
3
+
ξ
3
)
−
ξ
1
)
2
+
ξ
3
)
2
B
2μ
3
x
3
+
ξ
3
Q
5
1
1
+
(
x
1
−
+
+
ξ
3
)
+
,
Q
3
(
Q
+
x
3
Q
2
(
Q
+
x
3
x
3
∂
u
2
∂
x
2
=
B
2μ
+
ξ
3
Q
3
1
−
(1.469)
Q
(
Q
+
x
3
+
ξ
3
)
−
ξ
2
)
2
+
ξ
3
)
2
B
2μ
3
x
3
+
ξ
3
Q
5
1
1
+
(
x
2
−
+
+
ξ
3
)
+
,
Q
3
(
Q
+
x
3
Q
2
(
Q
+
x
3
+
ξ
3
)
1
+
ξ
3
)
2
Q
5
∂
u
3
∂
x
3
=
B
2μ
x
3
+
ξ
3
Q
3
B
2μ
3
(
x
3
B
λ
+
μ
x
3
+
ξ
3
Q
3
. (1.470)
+
(
x
3
Q
3
−
−
Summing the derivatives, the cubical dilatation is found to be
B
λ
+
μ
x
3
+
ξ
3
Q
3
.
∇·
u
=−
(1.471)
This displacement field has the associated normal stresses
μ
λ
+
μ
−
−
ξ
1
)
2
Q
2
B
x
3
+
ξ
3
Q
3
3
(
x
1
τ
11
=
(
x
1
−
ξ
1
)
2
(2
Q
1
B
+
x
3
+
ξ
3
)
+
−
,
(1.472)
Q
(
Q
+
x
3
+
ξ
3
)
Q
2
(
Q
+
+
ξ
3
)
x
3
μ
λ
+
μ
−
3
(
x
2
−
ξ
2
)
2
Q
2
B
x
3
+
ξ
3
Q
3
τ
22
=
(
x
2
1
−
ξ
2
)
2
(2
Q
B
+
x
3
+
ξ
3
)
+
+
ξ
3
)
−
,
(1.473)
Q
(
Q
+
x
3
+
ξ
3
)
Q
2
(
Q
+
x
3
λ
λ
+
μ
−
+
ξ
3
)
2
Q
2
B
x
3
+
ξ
3
Q
3
3
(
x
3
τ
33
=
,
(1.474)
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