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potential energy, the internal and external forces must be derivable from a potential. For
the principle of virtual work, there is no such limitation.
The principle of stationary potential energy will be quite useful. For example, one way of
determining the relative merits of approximate solutions is to compare their total potential
energies. The approximate solution for which the potential energy has an extreme value is,
in some sense, the best approximation.
The principle can also be written as
δ
=
0or
→
extremum
(2.65)
where
are assumed to be satisfied.
=
Du
in
V
u
=
u
on
S
u
Information on the type of extremum in Eq. (2.65) can be found by studying the sec-
ond variation of
(
u
i
+
δ
u
i
)
−
(
u
i
)
=
,
the
difference between the potential energies corresponding to the displacement state
u
i
and
the neighboring state
u
i
+
δ
.From
of Eq. (2.64), we can form
u
i
,
as
=
(
u
i
+
δ
u
i
)
−
(
u
i
)
=
[
U
0
(
ij
+
δ
ij
)
−
U
0
(
ij
)
−
p
Vi
δ
−
p
i
δ
]
dV
u
i
dV
u
i
dS
V
V
S
p
Expand
U
0
(
ij
+
δ
ij
)
in a Taylor series, giving
U
0
(
ij
)
+
∂
U
0
∂
ij
∂
ij
+
1
2
2
U
0
∂
kl
∂
ij
δ
kl
δ
ij
+···
∂
U
0
(
ij
+
δ
ij
)
=
Using Eq. (2.60), we get
∂
∂
kl
σ
ij
1
2
U
0
(
ij
+
δ
ij
)
−
U
0
(
ij
)
=
δ
U
0
+
δ
kl
δ
ij
+···
1
2
δσ
ij
δ
ij
+···
=
δ
U
0
+
1
2
E
ijkl
δ
kl
δ
ij
+···
=
δ
U
0
+
1
2
δ
T
E
=
δ
U
0
+
δ
+···
Substitution of this expression and the definition of
δ
in
leads to
=
δ
+
δ
2
+···
terms of higher order
(2.66)
2
with
δ
, the
second variation
of
,defined as
1
2
2
V
δ
T
E
δ
(
u
i
)
=
δ
dV
(2.67)
δ
=
According to the principle of stationary potential energy,
0 for a system in equilib-
δ
T
E
δ
is a positive definite quadratic function,
rium. Since
V
δ
T
E
1
2
2
(
u
i
)
=
δ
dV
≥
=
δ
0
(2.68)
Since for a linearly elastic solid the change in potential energy away from the state of
equilibrium is always positive, i.e.,
≥
0or
(
u
i
+
δ
u
i
)
≥
(
u
i
)
,
we conclude that for
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