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with shear stresses
3
x
1
x
1
2
R
0
−
1
−
C
x
3
R
0
τ
13
=−
,
(1.413)
3
C
x
1
x
1
x
2
x
2
x
3
−
−
τ
23
=−
,
(1.414)
R
0
3
x
1
x
1
2
x
2
τ
12
=−
C
x
2
−
−
R
0
−
1
R
0
x
3
)
2
1
R
0
(
R
0
+
+
λ
+
2μ
λ
+
μ
−
x
1
−
x
1
2
3
R
0
+
x
3
.
(1.415)
R
0
(
R
0
+
x
3
)
From expressions (1.383) and (1.388), for the total normal stress to vanish on
the free surface
x
3
=
0, the remaining free constants must be related by
μ
λ
+
μ
A
=
C
.
(1.416)
The combined extra displacement fields produce a traction per unit area on the
hemisphere, centred on the point of application of the tangential force, with com-
ponents,
x
1
−
x
1
2
C
a
2
−
C
2
λ
+
μ
λ
+
μ
F
ν
1
=
a
4
⎣
1
⎦
,
x
1
−
x
1
2
2
x
1
−
x
1
2
x
3
a
2
x
3
−
C
λ
+
2μ
λ
+
μ
+
−
a
a
2
x
3
2
−
(1.417)
x
3
)
2
a
2
−
a
2
(
a
+
C
x
1
x
1
x
2
x
2
3
x
3
)
2
−
−
a
2
−
λ
+
2μ
λ
+
μ
2
F
ν
2
=−
,
(1.418)
a
2
(
a
+
3
C
x
1
−
x
1
x
3
a
4
F
ν
3
=−
.
(1.419)
The components of the total traction on the hemisphere are
a
2
2π
0
π/2
a
2
2π
0
1
F
ν
i
sinθ
d
θ
d
φ
=
F
ν
i
dud
φ,
T
i
=
(1.420)
0
0
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