530
9 Variational Principles in Mechanics and Their Applications
and for the differential equations of equilibrium (9.2.3) and the stress boundary
conditions (9.2.10) on S σ , according to the principle of virtual displacement, there
is
−
˚
V
∂σ x
∂ x
+
∂τ xy
∂ y
+
∂τ zx
∂z
+ F x
δu +
∂τ xy
∂ x
+
∂σ y
∂ y
+
∂τ yz
∂z
+ F y
δv
+
∂τ zx
∂ x
+
∂τ yz
∂ y
+
∂σ z
∂z
+ F z
δw
dV
+
¨
S
[(X − X )δu + (Y − Y )δv + (Z − Z )δw]dS = 0
(9.2.23)
where, dV = dxdydz and dS denote the volume element of the elastic body and area
element on the surface of the elastic body respectively. Note that the surface integral
should originally be conducted on the stress boundary S σ , but the expression (9.2.8)
holds on the displacement boundary S u , so in the expression (9.2.23) the surface
integral can be extended to the entire surface.
According to the relations (9.1.1) and the property that the sequence of variation
and derivation can be exchanged, there are
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
δε x = δ
∂u
∂ x
=
∂δu
∂ x
, δε y = δ
∂v
∂ y
=
∂δv
∂ y
, δε z = δ
∂w
∂z
=
∂δw
∂z
δγ xy = 2δε xy = δ
∂u
∂ y
+ δ
∂v
∂ x
=
∂δu
∂ y
+
∂δv
∂ x
δγ yz = 2δε yz = δ
∂v
∂z
+ δ
∂w
∂ y
=
∂δv
∂z
+
∂δw
∂ y
δγ zx = 2δε zx = δ
∂w
∂ x
+ δ
∂u
∂z
=
∂δw
∂ x
+
∂δu
∂z
(9.2.24)
Equations (9.2.24) are called the relational expressions of virtual strains and
virtual displacements. Equations (9.2.24) can be written as
δε i j =
1
2
(δu i, j + δu j,i )
(9.2.25)
Taking the first term in the volume integral of the expression (9.2.23), using the
Gauss formula and integration by parts, and taking note that the first expression in
Eqs. (9.2.24), there is
˚
V
∂σ x
∂ x
δudV =
˚
V
∂
∂ x
(σ x δu) − σ x
∂δu
∂ x
dV
=
¨
S
l(σ x δu)dS −
˚
V
σ x
∂(δu)
∂ x
dV =
¨
S
σ x lδudS −
˚
V
σ x δε x dV (9.2.26)
Similarly, performing the same treatment to the terms containing σ y and σ z in
volume of the expression (9.2.23), the two relations similar to the expression (9.2.26)
can be obtained.