Geology Reference
In-Depth Information
The components of the traction per unit area on the hemisphere, coming from
the stresses, (1.383) through (1.388), generated directly by the application of the
tangential force, are
⎣
⎦
,
3
x
1
x
1
2
−
A
a
2
+
μ
λ
+
μ
F
ν
1
=−
(1.428)
a
2
3
A
x
1
x
1
x
2
x
2
−
−
F
ν
2
=−
,
(1.429)
a
4
3
A
x
1
x
1
x
3
a
4
−
F
ν
3
=−
.
(1.430)
Substituting these in (1.420), we find that they contribute the components of the
total traction on the hemisphere,
2
A
π
λ
+
2μ
λ
+
μ
,
T
2
T
1
=−
=
0,
T
3
=
0.
(1.431)
For equilibrium, the component
T
1
must balance the applied force, giving
T
1
=−
P
and
P
2π
λ
+
μ
λ
+
A
=
2μ
.
(1.432)
From (1.409) and (1.416), it follows that
P
4π
1
λ
+
μ
,
C
P
2π
μ
λ
+
B
=
=
2μ
.
(1.433)
Replacing the three constants in their respective displacement field expressions by
these values, we find the total displacement field for the Cerruti problem to be
⎣
⎝
⎠
⎦
,
x
1
−
x
1
2
x
1
−
x
1
2
(
R
0
+
P
4πμ
1
R
0
μ
λ
+
μ
R
0
u
1
=
1
+
R
0
+
x
3
−
(1.434)
R
0
+
x
3
)
2
⎣
⎦
,
x
1
x
1
x
2
x
2
−
−
P
4πμ
1
R
0
−
μ
λ
+
μ
1
=
u
2
(1.435)
R
0
x
3
)
2
(
R
0
+
⎣
⎦
.
x
1
x
1
−
P
4πμ
x
3
R
0
+
μ
λ
+
μ
1
R
0
+
x
3
u
3
=
(1.436)
R
0
1.4.11 The Mindlin problems
When the downward point force is below the surface of the elastic half-space, the
Boussinesq problem becomes Mindlin problem I. Similarly, when the horizontal
point force is below the surface, the Cerruti problem becomes Mindlin problem II.
The
Mindlin problems
were first solved by Mindlin (1936) after whom they are
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