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where we have used (1.214) to express the surface traction
t
i
in terms of the stress
τ
ji
. Transforming the surface integral to a volume integral by Gauss's theorem, the
work done becomes
∂
+
F
i
u
i
dV
=
∂
x
j
u
i
+
F
i
u
i
dV
.
∂
x
j
τ
ji
u
i
τ
ji
∂
u
i
∂τ
ji
∂
x
j
+
(1.261)
V
V
From the equilibrium equation (1.221),
∂τ
ji
∂
x
j
=−
ρ
f
i
=−
and
∂τ
ji
∂
x
j
u
i
=−
F
i
u
i
.
F
i
(1.262)
The pair of terms,
∂τ
ji
∂
x
j
u
i
+
F
i
u
i
,
(1.263)
then combine to
F
i
u
i
+
F
i
u
i
=
−
0.
(1.264)
Substitution for the stress τ
ji
from Hooke's law (1.259), gives the work done as
∂
u
i
∂
x
j
+
∂
u
i
∂
x
j
dV
i
∂
u
i
∂
u
j
∂
x
i
j
λ
Θ
δ
∂
x
j
+
μ
V
dV
∂
x
j
∂
u
i
∂
x
i
∂
u
i
∂
x
j
+
μ
∂
u
j
λ
ΘΘ
+
μ
∂
u
i
=
∂
x
j
V
⎣
λ
ΘΘ
+
μ
∂
u
i
⎦
∂
u
j
∂
x
i
∂
x
j
∂
u
i
∂
x
j
+
μ
∂
u
i
=
dV
.
(1.265)
∂
x
j
V
On interchange of primed and unprimed quantities, this expression is unchanged,
and thus we have
Betti's reciprocal theorem
as stated,
t
i
u
i
dS
F
i
u
i
dV
t
i
u
i
dS
F
i
u
i
dV
.
+
=
+
(1.266)
S
V
S
V
Substituting for the stress, as given by Hooke's law (1.259), in the equation of
equilibrium (1.221), yields an equation for the displacement field,
∂
u
i
∂
x
j
+
∂
u
j
∂
x
i
∂
∂
x
j
i
j
λ
∇·
u
δ
+
μ
=−
F
i
.
(1.267)
In uniform media, λ and μ are not functions of position, and carrying out the dif-
ferentiation gives,
2
u
j
∂
x
i
∂
x
j
=−
2
u
i
∂
x
j
∂
x
j
+
μ
∂
λ
∂
∂
x
i
+
μ
∂
(
∇·
u
)
F
i
.
(1.268)
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