Geology Reference
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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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