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TABLE 2.2
Global Forms of the Fundamental Equations and the Classical Variational Principles
A
B
The force (static) boundary
The kinematical equations and
conditions and the conditions of
displacement (kinematical)
equilibrium (Eq. 2.44)
boundary conditions (Eq. 2.70)
S
p
δ
u
T
V
δ
σ
T
(
p
−
p
)
dS
(
Du
−
)
dV
u
T
D
T
σ
+
p
T
=
V
δ
(
p
V
)
dV
=
S
u
δ
(
u
−
u
)
dS
Gauss' Integral (Divergence) Theorem
Gauss' Integral (Divergence) Theorem
C
D
↓
↓
The principle of virtual work
The principle of complementary
(Eq. 2.54)
virtual work (Eq. 2.78)
V
δ
u
T
p
V
dV
V
δ
σ
T
dV
T
σ
dV
S
u
δ
p
T
u
dS
V
δ(
Du
)
−
−
+
V
δ(
S
p
δ
u
T
p
dS
u
T
p
dS
D
T
σ
)
p
T
u
dS
−
S
p
δ
−
S
u
δ
−
u
dV
+
=
0
=
0
C and D are usually written without the underlined terms and then require admissible
displacements and stresses, respectively.
Observe that a complete description in variational form of the solid is provided by sum-
ming, for example, Eqs. (A) and (B)
u
T
D
T
σ
σ
T
−
V
δ
(
+
p
V
)
dV
+
V
δ
(
Du
−
)
dV
(
A
+
B
=
AB
)
u
T
p
T
+
S
p
δ
(
p
−
p
)
dS
−
S
u
δ
(
u
−
u
)
dS
=
0
(2.86)
That is, this expression is equivalent to
D
T
σ
+
p
V
=
0
in
V
(2.87a)
=
Du
in
V
(2.87b)
with boundary conditions for forces:
A
T
σ
=
p
=
p
on
S
p
(2.87c)
for displacements:
=
u
u
on
S
u
(2.87d)
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