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Thus,
−
1
p
ij
u
j
dS
lim
→
=
lim
→
u
i
[
(
1
−
2
ν)
+
3
,i
,i
] sin
ϕ
d
ϕ
d
θ
8
π(
1
−
ν)
0
0
S
S
u
i
−
1
3
2
3
1
2
u
i
=
(
1
−
2
ν)
2
π
+
=−
(9.87)
8
π(
1
−
ν)
Note that when
2,
but can assume other values (Problem 9.12). Actually, it will be seen later that this value
does not need to be determined explicitly.
The final integral equation on the boundary is
ξ
is at a location which is not smooth, the coefficient of
u
i
may not be
−
1
/
u
j
p
ij
dS
p
j
u
ij
dS
p
Vj
u
ij
dV
(ξ )
+
=
+
c
i
u
i
(9.88a)
S
S
V
or in matrix form
p
∗
u
dS
u
∗
p
dS
u
∗
p
V
dV
cu
(ξ )
+
=
+
(9.88b)
S
S
V
where
diag
(
c
1
c
2
c
3
)
for three-dimensional problems
c
=
diag
(
c
1
c
2
)
for two-dimensional problems
1
when
ξ
is inside the boundary
1
2
c
i
=
when
ξ
is on smooth boundary
i
=
1
,
2
,
3
Other value
when
ξ
is at a corner
[
u
1
u
3
]
T
u
2
for three-dimensional problems
u
=
u
2
]
T
[
u
1
for two-dimensional problems
[
p
1
p
3
]
T
p
2
for three-dimensional problems
p
=
p
2
]
T
[
p
1
for two-dimensional problems
are the displacement and traction vectors at the boundary,
[
p
V
1
p
V
3
]
T
p
V
2
for three-dimensional problems
p
V
=
p
V
2
]
T
[
p
V
1
for two-dimensional problems
is the body force vector and
p
11
p
12
p
13
u
11
u
12
u
13
p
∗
=
u
∗
=
p
21
p
22
p
23
u
21
u
22
u
23
for three-dimensional problems
p
31
p
32
p
33
u
31
u
32
u
33
p
11
u
11
p
12
u
12
p
∗
=
u
∗
=
for two-dimensional problems
p
21
p
22
u
21
u
22
are the fundamental solution coefficient matrices.
Equation (9.88) constitutes the basic integral equation for the boundary element formu-
lation.
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