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linear problems the extremum of
is an absolute minimum. Thus, this principle, in the
case of a solid for which Hooke's law is applicable, is often referred to as the principle of
minimum potential energy . It is also known as Dirichlet's 4
principle.
2.2.3 Principle of Complementary Virtual Work
For kinematically admissible displacements, the principle of virtual work and its associ-
ated theorems hold for a system in equilibrium. A “dual” to the principle of virtual work
is the principle of complementary virtual work. For stresses (forces) that satisfy the condi-
tions of equilibrium and the static boundary conditions—so-called statically admissible
stresses—the principle of complementary virtual work and its corollary theorems hold for
a kinematically compatible system. The kinematic (strain-displacement) conditions plus
the displacement boundary conditions and this complementary principle are equivalent in
the sense that the principle of complementary virtual work is a global (integral) form of the
kinematic equation and the kinematic boundary conditions. In Section 2.2.1, variations in
the displacements for fixed external forces were prescribed; now the stresses and forces for
fixed displacements will be varied. Thus, the stresses and forces are the fundamental unknowns
for the principle of complementary virtual work.
To establish the principle of complementary virtual work, proceed in a fashion similar
to that used for the principle of virtual work. However, rather than dealing with a solid in
equilibrium, begin with a body for which the local kinematic conditions [Chapter 1, Eqs.
(1.19) or (1.21) and (1.61)] are satisfied, i.e.,
1
2 (
ij =
u i, j +
u j,i )
or
=
Du
in
V
(2.69a)
=
=
u i
u i
or
u
u
on
S u
(2.69b)
where u i or u are prescribed displacements on surface area S u .
Multiply Eq. (2.69a) by a virtual stress field
δσ
ij , and integrate over the volume. Multiply
Eq. (2.69b) by the virtual force
δ
p i , and integrate over the surface. The sum of the two
expressions gives
V (
u i,j
)δσ
ij dV
S u (
u i
u i
p i dS
=
0
(2.70)
ij
or
V δ σ T
S u δ
p T
(
Du
)
dV
(
u
u
)
dS
=
0
According to the fundamental lemma of the calculus of variations, Eq. (2.70) is equivalent
to the kinematic conditions of Eq. (2.69).
To alter Eq. (2.70) so that the integrals can be identified as work expressions, use Gauss'
integral theorem of Eq. (2.41) along with the condition of statically admissible stresses. As
a “dual” to
u i , introduce the concept of statically admissible stresses
σ
= σ
+ δσ
ij which,
ij
ij
like the actual stresses
σ
ij , satisfy the equilibrium conditions [Chapter 1, Eqs. (1.52) or (1.54)]
4 Lejeune Dirichlet (1805-1859) studied mathematics at Jesuit college in Bonn, Germany, where one of his teachers
was Ohm. He illuminated the works of his friend and father-in-law Jacobi and of Gauss whom he replaced
in G ottingen. His own original efforts in mathematics, especially his celebrated Fourier theorem, were very
significant. During his studies of mathematical physics, he treated a boundary value problem, now referred to as
Dirichlet's problem , in which a solution is sought to Laplace's equation having prescribed values on a given surface.
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