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as
σ
T
]
0
E
−
1
u
p
V
0
dV
u
D
T
u
T
u
T
p
dS
[
δ
δ
−
−
S
p
δ
D
u
−
σ
V
δ
σ
T
]
0A
T
A0
u
σ
0
Au
dS
u
T
−
[
δ
−
=
0
(2.90)
S
u
If a state vector
z
is defined as
u
σ
z
=
(2.91)
then expression
CB
of Eq. (2.90) appears as
V
δ
z
T
0
E
−
1
z
p
V
0
dV
u
D
T
u
T
p
dS
−
−
S
p
δ
−
D
u
z
T
0A
T
A0
z
0
Au
dS
−
S
u
δ
−
=
0
(2.92)
The other generalized principles can be converted to matrix notation in a similar fashion.
Definitions similar to those in Eq. (2.89) and identities of the form
u
T
D
T
σ
u
T
D
T
σ
σ
T
D
σ
T
δ
u
i
σ
=
δ
(
)
=
δ
σ
=
(
)δ
u
=
D
δ
u
(2.93a)
ij, j
σ
1
2
(
σ
T
σ
T
D
u
u
T
u
T
u
D
T
δσ
u
i, j
+
u
j,i
)
=
δ
(
Du
)
=
δ
=
(
Du
)
δ
σ
=
δ
σ
(2.93b)
ij
u
T
p
u
T
A
T
σ
p
T
σ
T
A
δ
u
i
p
i
=
δ
=
δ
=
δ
u
=
δ
u
(2.93c)
must be used. The resulting generalized principles are shown in Table 2.4. In these expres-
sions, the material law of Chapter 1, Eq. (1.43), which incorporates the effects of initial
strains, has been employed, i.e.,
E
T
σ
0
=
+
(2.94)
It is of interest to observe in the variational forms of Table 2.4 that the matrices appearing
in functions
CB
and
AD
are symmetric, whereas this is not the case, in general, for
AB
and
CD
. Whether or not this symmetry exists in particular instances can play an important role
in the development of numerical solution techniques.
2.3.2 Related Forms
A number of useful expressions can be derived from the generalized principles. For exam-
ple, if it is assumed that the applied loads can be expressed in terms of a potential, then
form
CB
can be written as
U
0
(σ )
δ
−
dV
+
V
σ
ij
u
i, j
dV
−
u
i
p
Vi
dV
V
V
dS
−
−
(
−
)
=
u
i
p
i
dS
p
i
u
i
u
i
0
(2.95)
S
p
S
u
which can be obtained from case
CB
of Table 2.3 and can be considered as the variation of a
functional which will be denoted by
R
. Then Eq. (2.95) is equivalent to setting
δ
R
equal
to zero. The functional
U
0
(σ )
R
=−
dV
+
V
σ
ij
u
i, j
dV
−
u
i
p
Vi
dV
V
V
−
−
(
−
)
u
i
p
i
dS
p
i
u
i
u
i
dS
(2.96)
S
p
S
u
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