Geology Reference
In-Depth Information
while the associated shear stresses of this displacement field are
1
5
(
x
3
+
ξ
3
)
2
Q
2
3
C
(
x
1
−
ξ
1
)
x
3
Q
5
3
C
λ
λ
+
μ
(
x
1
−
ξ
1
)(
x
3
+
ξ
3
)
Q
5
=−
−
−
τ
13
, (1.494)
1
5
(
x
3
+
ξ
3
)
2
Q
2
3
C
(
x
2
−
ξ
2
)
x
3
Q
5
3
C
λ
λ
+
μ
(
x
2
−
ξ
2
)(
x
3
+
ξ
3
)
Q
5
τ
23
=−
−
−
, (1.495)
1
3
C
(
x
1
−
ξ
1
)(
x
2
−
ξ
2
)
5
x
3
(
x
3
+
ξ
3
)
Q
2
τ
12
=−
−
.
(1.496)
Q
5
The stress field generated by the Galerkin vector
G
3
produces the traction per unit
area, on a small sphere of radius
a
surrounding the image point, with components
(
x
3
3λ
+
,
+
ξ
3
)
2
a
2
3
C
x
1
−
ξ
1
a
4
3
x
3
(
x
3
+
ξ
3
)
a
2
1
3
2μ
F
1
=
+
−
(1.497)
λ
+
2μ
(
x
3
3λ
+
,
+
ξ
3
)
2
a
2
3
C
x
2
−
ξ
2
a
4
3
x
3
(
x
3
+
ξ
3
)
a
2
1
3
2μ
F
2
=
+
−
(1.498)
λ
+
2μ
F
3
=
3
C
x
3
(
x
3
3λ
+
a
4
+
ξ
3
)
2
a
2
+
ξ
3
a
4
+
3
x
3
(
x
3
+
ξ
3
)
a
2
2
3
2μ
λ
+
μ
+
ξ
3
−
. (1.499)
The components of the total traction on the small sphere are found to be
T
1
=
T
2
=
0and
a
2
2π
0
π
T
3
=
F
3
sinθ
d
θ
d
φ
0
2π
a
2
π
0
=
F
3
sinθ
d
θ
6π
Ca
2
π
0
cos
θ
a
3
cos
2
3
(
a
cos
θ
−
ξ
3
)
a
=
θ
+
cosθ
(1.500)
3λ
+
a
4
sinθ
d
θ
2
3
2μ
λ
+
μ
+
ξ
3
−
3λ
+
cos
2
cosθ
π
0
a
cos
3
6
π
C
a
1
1
3
2μ
λ
+
μ
ξ
3
4
cos
4
=
−
θ
+
θ
+
θ
−
=
0.
For Kelvin's problem at the image point, the stresses are found from expres-
sions (1.321) and (1.322) for Kelvin's problem at the source point, by replacing
Search WWH ::
Custom Search