9.4 Principle of Complementary Virtual Work
541
where, δ X , δY and δ Z are the virtual changes of the surface forces corresponding to
the virtual stresses.
Using the Gauss formula and the integration by parts, the following relations hold
˚
V
∂u
∂ x
δσ x dV =
˚
V
∂
∂ x
(uδσ x )dV −
˚
V
∂δσ x
∂ x
udV =
¨
S
uδσ x ldS −
˚
V
∂δσ x
∂ x
udV (9.4.9)
˚
V
∂v
∂ x
δτ xy dV =
˚
V
∂
∂ x
(vδτ xy )dV −
˚
V
∂δτ xy
∂ x
vdV =
¨
S
vδτ xy ldS −
˚
V
∂δτ xy
∂ x
vdV
(9.4.10)
Similarly, the two relations similar to Eq. (9.4.9) and the five relations similar to
Eq. (9.4.10) can be written out, substituting these relations into Eq. (9.4.8), we give
˚
V
(ε x δσ x + ε y δσ y + · · · + γ xy δτ xy )dV +
˚
V
∂δσ x
∂ x
+
∂δτ xy
∂ y
+
∂δτ zx
∂z
u
+
∂δτ xy
∂ x
+
∂δσ y
∂ y
+
∂δτ yz
∂z
v +
∂δτ zx
∂ x
+
∂δτ yz
∂ y
+
∂δσ z
∂z
w
dV
−
¨
S
[u(δσ x l + δτ xy m + δτ zx n) + v(δτ xy l + δσ y m + δτ yz n)
+ w(δτ zx l + δτ yz m + δσ z n)]d S
+
¨
S u
[(u − ¯
u)δ X + (v − ¯
v)δY + (w − ¯
w)δ Z ]d S = 0
(9.4.11)
Note that Eqs. (9.4.3) and (9.4.4), and using Eqs. (9.4.5) and (9.4.11) can be
transformed into
˚
V
(ε x δσ x + ε y δσ y + · · · + γ xy δτ xy )dV −
¨
S
( ¯
uδ X + ¯
vδY + ¯
wδ Z )dS = 0
(9.4.12)
Note that the surface integral in the Eq. (9.4.12) originally shall only be carried out
on the displacement boundary S u , but because on the stress boundary S σ Eqs. (9.4.4)
hold, the surface integral in Eq. (9.4.12) can be extended to the entire surface. Equation (9.4.12) is called the principle of complementary virtual work of an elastic
body. The principle of complementary virtual work is also called the principle of
virtual stress.
According to Eqs. (9.1.2), (9.1.5), and (9.4.12) can be written as
˚
V
ε i j δσ i j dV =
¨
S
¯
u i δ X i dS
(9.4.13)
The left side of Eq. (9.4.13) expresses the total complementary virtual energy of
an elastic body, but the right side expresses the work done by the variation of surface
force on the real displacement. The principles of virtual displacement expressed by
Eqs. (9.4.13) and (9.2.29) become the complementary form.