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In Section 2.2, it was shown that the equilibrium equations and the force (stress) boundary
conditions are equivalent to [Eq. (2.44)]
V
δ
u
T
u
T
D
T
σ
(A)
∗∗
S
p
δ
(
p
−
p
)
dS
=
(
+
p
V
)
dV
The
u
i
are the variations of the displacement field. Similarly, the strain-displacement equa-
tions and the kinematic boundary conditions follow from [Eq. (2.70)]
δ
V
δ
σ
T
p
T
(B)
∗∗
(
Du
−
)
dV
=
S
u
δ
(
u
−
u
)
dS
where
p
i
are the variations of the stresses and forces. Use of Gauss' integral
theorem, along with the kinematic admissibility conditions
δσ
ij
and
δ
δ
=
D
δ
u
in
V
and
δ
u
=
0
on
S
u
,
converts (A) into the principle of virtual work [Eq. (2.54)]
V
δ
T
σ
dV
u
T
p
V
dV
u
T
p
dS
(C)
∗∗
−
V
δ
−
S
p
δ
=
0
In a similar fashion, Gauss' integral theorem and the static admissibility conditions of
δσ
=
0in
V
and
δ
p
i
=
0on
S
p
convert (B) into the principle of complementary virtual
ij, j
work [Eq. (2.78)]
σ
T
dV
p
T
u
dS
−
V
δ
+
S
u
δ
=
(D)
∗∗
0
It is possible to relax the underlying assumptions that were required for each of these
principles. For example, in establishing (C), it was assumed that
δ
u
=
0on
S
u
. If this
assumption is abandoned, then the term
S
u
δ
u
T
p
dS
must be included in (C) giving
V
δ
T
σ
dV
u
T
p
V
dS
u
T
p
dS
u
T
p
dS
V
δ(
Du
)
−
−
S
p
δ
−
S
u
δ
=
0
(C)
where the new term is underlined, and the kinematic condition
=
Du
has been inserted.
Equation (C) follows directly from (A) if the underlined integral is not set equal to zero
when applying Gauss' theorem. Refer to Section 2.2.1, where the underlined integral is set
equal to zero in establishing Eq. (2.51).
Similarly, if it is
not
assumed that
δσ
=
0in
V
and
δ
p
i
=
0on
S
p
, then expression (D)
ij, j
must be appended as [see Eq. (2.73) versus Eq. (2.41)]
S
u
δ
V
δ
σ
T
dV
p
T
u
dS
D
T
σ
)
p
T
u
dS
−
+
−
V
δ(
u
dV
+
S
p
δ
=
0
(D)
These modified equations are summarized in Table 2.2. Recall that cases C and D can be
returned to the forms of cases A and B, respectively, by applying Gauss' integral theorem.
∗∗
These four relationships are so important to the fundamental theme of this work that they are given the special
labels A, B, C, and D, rather than equations numbers.
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