9.4 Principle of Complementary Virtual Work
543
It needs to specially point out that no matter what the stress-strain relationship
of material namely the constitutive relation, the principle of complementary virtual
work holds.
9.5 Principle of the Minimum Complementary Energy
The principle of minimum complementary energy is also called the principle of
total complementary energy. According to the general theory of elastic mechanics,
in the case of small deformation, the complementary function U
c
0 is a positive definite single-valued function, dU
c
0 must be total differential, therefore there are the
following relationships
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
ε x =
∂U
c
0
∂σ x
, ε y =
∂U
c
0
∂σ y
, ε z =
∂U
c
0
∂σ z
γ yz =
∂U
c
0
∂τ yz
, γ zx =
∂U
c
0
∂τ zx
, γ xy =
∂U
c
0
∂τ xy
(9.5.1)
Thus, the principle of complementary virtual work (9.4.13) becomes
˚
V
∂U c
0
∂σ x
δσ x +
∂U c
0
∂σ y
δσ y + · · · +
∂U c
0
∂τ xy
δτ xy
dV −
¨
S
( ¯
uδ X + ¯
vδY + ¯
wδ Z )dS = 0
(9.5.2)
or
˚
V
δU
c
0 dV −
¨
S u
¯
u i δ X i dS = 0
(9.5.3)
Note that on the stress boundary S σ , δ X i = δ X i = 0, on the displacement
boundary S u , ¯
u, ¯
v and ¯
w are constants, thus Eq. (9.5.3) can be written as
δ
˚
V
U
c
0 dV −
¨
S
¯
u i X i dS
= 0
(9.5.4)
or
δΠ
c
= δ(U
c
− W
c
) = δ(U
c
+ V
c
) = 0
(9.5.5)
where
U
c
=
˚
V
U
c
0 dV
(9.5.6)