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3
Related Variational and Energy Principles
The two fundamental variational theorems of solid mechanics are the principles of virtual
work and complementary virtual work. Important corollaries to these theorems, the prin-
ciples of stationary potential and complementary energies, respectively, were derived in
Chapter 2 based on the existence of appropriate potentials for the forces. Further useful
corollaries will be presented in this chapter. These additional variational and energy prin-
ciples are usually considered to be the classical energy techniques for practical problem
solving in solid mechanics.
3.1
The Principle of Virtual Work Related Theorems
The principle of virtual work contends that a solid is in equilibrium if the sum of the external
and internal virtual work is zero for kinematically admissible virtual displacements. This
principle is expressed by the equations
−
δ
−
δ
=
W
i
W
e
0
V
σ
δ
ij
dV
−
p
V
i
δ
u
i
dV
−
p
i
δ
u
i
dS
=
0
ij
V
S
p
or
V
δ
T
σ
dV
u
T
p
V
dV
u
T
p
dS
−
V
δ
−
S
p
δ
=
0
(3.1)
with kinematically admissible
δ
u
i
,
i.e.,
δ
u
i
that satisfy the kinematical equations
=
Du
in
V
u
=
u
on
S
u
If a potential exists for both the internal and external forces, the principle of virtual
work can be specialized to the principle of stationary potential energy. This principle states
that among the kinematically admissible deformations, the actual deformations which cor-
respond to forces in equilibrium are the ones for which the total potential energy
is
stationary. This is expressed by the equation
δ
=
=
+
−→
0or
U
i
U
e
Stationary
(3.2)
135
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