540
9 Variational Principles in Mechanics and Their Applications
where, the volume forces F x , F y and F z are all the given external forces, they do
not change.
Subtracting Eqs. (9.4.2) and differential equations of equilibrium (9.2.3), we get
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂δσ x
∂ x
+
∂δτ xy
∂ y
+
∂δτ zx
∂z
= 0
∂δτ xy
∂ x
+
∂δσ y
∂ y
+
∂δτ yz
∂z
= 0
∂δτ zx
∂ x
+
∂δτ yz
∂ y
+
∂δσ z
∂z
= 0
(9.4.3)
On the stress boundary S σ , there are
⎧
⎨
⎩
δ X = δσ x l + δτ xy m + δτ zx n = 0
δY = δτ xy l + δσ y m + δτ yz n = 0
δ Z = δτ zx l + δτ yz m + δσ z n = 0
(9.4.4)
It is that the on the stress boundary S σ the surface forces are given, therefore the
virtual changes of the surface forces are all zero. On the displacement boundary S u
the virtual changes of the surface forces
⎧
⎨
⎩
δ X = δσ x l + δτ xy m + δτ zx n
δY = δτ xy l + δσ y m + δτ yz n
δ Z = δτ zx l + δτ yz m + δσ z n
(9.4.5)
Because it has assumed that the strain and the corresponding stress are real, the
elastic body has the following relationships
ε x −
∂u
∂ x
= 0, ε y −
∂v
∂ y
= 0, . . . , γ zx −
∂u
∂z
−
∂w
∂ x
= 0
(9.4.6)
On the boundary S u given the displacements there are
u − ¯
u = 0, v − ¯
v = 0, w − ¯
w = 0
(9.4.7)
where, ¯
u, ¯
v and ¯
w are given displacements on the displacement boundary S u .
Assume that the elastic body in the state of equilibrium carries a set of arbitrary,
infinitesimal virtual stresses δσ x , δσ y , δσ z , δτ yz , δτ zx , δτ xy , then there is
˚
V
ε x −
∂u
∂ x
δσ x +
ε y −
∂v
∂ y
δσ y +
ε z −
∂w
∂z
δσ z
+
γ xy −
∂u
∂ y
−
∂v
∂ x
δτ xy +
γ yz −
∂w
∂ y
−
∂v
∂z
δτ yz +
γ zx −
∂u
∂z
−
∂w
∂ x
δτ zx
dV
+
¨
S u
[(u − ¯
u)δ X + (v − ¯
v)δY + (w − ¯
w)δ Z ]dS = 0
(9.4.8)