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It is of interest that this fundamental weighted-residual relationship can be derived in
a variety of ways. Probably the most common derivation is based on Betti's reciprocal
theorem of Chapter 3, Eq. (3.36), i.e.,
p
Vi
u
i
dV
p
i
u
i
dS
p
∗
Vi
u
i
dV
p
i
u
i
dS
+
=
+
(9.49)
V
S
V
S
Con
si
der the boundary
S
to be divided into the two parts
S
p
and
S
u
,
with
u
=
u
on
S
u
and
p
=
p
on
S
p
. Then write the surface integrals of Eq. (9.49) as
p
i
u
i
p
i
u
i
p
i
u
i
dS
=
dS
+
dS
S
S
p
S
u
p
i
u
i
dS
p
i
u
i
dS
p
i
u
i
dS
=
+
S
S
p
S
u
Introduce these relations and the equilibrium conditions
p
∗
Vi
=−
σ
ij, j
into Eq. (9.49) to
obtain
V
σ
ij, j
u
i
dV
p
Vi
u
i
p
i
u
i
p
i
u
i
p
i
u
i
dS
p
i
u
i
dS
+
dV
=−
dS
−
dS
+
+
V
S
p
S
u
S
p
S
u
(9.50)
Apply Gauss' integral theorem, in the form of Eq. (II.9) of Appendix II, twice to the first
integral of Eq. (9.50) to obtain
V
σ
ij
u
i, j
dV
V
σ
ij, j
u
i
dV
p
i
u
i
dS
=
−
S
p
i
u
i
dS
V
σ
ij
u
i, j
dV
=
−
S
p
i
u
i
dS
p
i
u
i
ij, j
u
i
=
−
dS
+
V
σ
dV
(9.51)
S
S
Here, use was made of Chapter 3, Eq. (3.31) and the relationship [Eqs. (3.29) and (3.33)]
σ
ij
u
i, j
=
σ
ij
ij
=
σ
ij
u
i, j
=
σ
ij
ij
Remember that
S
S
u
and insert Eq. (9.51) into Eq. (9.50). This leads directly to
Eq. (9.48), the extended Galerkin's formula.
=
S
p
+
9.3.2 Fundamental Solutions
The boundary element formulation of the linear elasticity problem starts with Eq. (9.50)
which is then converted into an integral equation with unknowns
u
i
,p
i
on the boundary.
To do so, the first step is to isolate
u
i
from the first term of Eq. (9.50). Let a concentrated
force
δ(ξ
,x
)
in an arbitrary direction be applied at point
ξ
,
so that the equilibrium equation
is
σ
ij, j
=−
δ(ξ
,x
)
a
i
(9.52)
where
and
x
stand for two points inside the body,
a
i
is the direction cosine between
the concentrated force and the
x
i
direction. In the derivations that follow, quantities with
superscript
ξ
∗
δ(ξ
)
are due to the force
,x
a
i
. Insertion of Eq. (9.52) in Eq. (9.50) and use
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