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the co-ordinate ξ
3
by
−
ξ
3
. The Galerkin vector
G
4
=
e
3
DQ
then yields the normal
stresses
μ
λ
+
μ
−
−
ξ
1
)
2
Q
2
D
x
3
+
ξ
3
Q
3
3
(
x
1
τ
11
=
,
(1.501)
μ
λ
+
μ
−
−
ξ
2
)
2
Q
2
D
x
3
+
ξ
3
Q
3
3
(
x
2
τ
22
=
,
(1.502)
μ
λ
+
μ
+
+
ξ
3
)
2
Q
2
D
x
3
+
ξ
3
Q
3
3
(
x
3
τ
33
=−
,
(1.503)
and the shear stresses
μ
λ
+
μ
+
+
ξ
3
)
2
Q
2
D
x
1
−
ξ
1
Q
3
3
(
x
3
=−
τ
13
,
(1.504)
μ
λ
+
μ
+
3
(
x
3
+
ξ
3
)
2
Q
2
D
x
2
−
ξ
2
Q
3
τ
23
=−
,
(1.505)
3
D
(
x
1
−
ξ
1
)(
x
2
−
ξ
2
)(
x
3
+
ξ
3
)
Q
5
τ
12
=−
.
(1.506)
The stress field generated by the Galerkin vector
G
4
produces the traction per unit
area, on a small sphere of radius
a
surrounding the image point, with components
3
D
(
x
1
−
ξ
1
)(
x
3
+
ξ
3
)
a
4
F
1
=−
,
(1.507)
3
D
(
x
2
−
ξ
2
)(
x
3
+
ξ
3
)
F
2
=−
,
(1.508)
a
4
a
2
μ
+
ξ
3
)
2
a
2
D
1
3
(
x
3
F
3
=−
λ
+
μ
+
.
(1.509)
Thus,
T
1
=
T
2
=
0and
a
2
2π
0
π
a
2
μ
sinθ
d
θ
d
φ
D
3cos
2
T
3
=−
λ
+
μ
+
θ
0
2π
D
μ
λ
+
μ
π
0
=−
4π
D
λ
+
2μ
λ
+
μ
.
cos
3
=
cosθ
+
θ
(1.510)
The arbitrary constants
A
,
B
,
C
,
D
are to be determined by a balance of the point
force at the source point (ξ
1
,ξ
2
,ξ
3
) and traction at the image point (ξ
1
,ξ
2
,
−
ξ
3
), and
by the further condition that the surface
x
3
0).
Omitting the factor 2μ
HP
is equivalent to setting it to unity, or taking the mag-
nitude of the point force at the source point to be
=
0isstressfree(τ
13
=
τ
23
=
τ
33
=
4π
λ
+
2μ
λ
+
μ
.
P
=
(1.511)
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