15.8 Lagrange Equilibrium Variation Equation
209
δA =
V
δW dV =
V
∂W
∂ε x
δε x + . . .
dV =
=
V
ρ(Xδu + Y δv + Zδw)dV +
S
(X ν δu + Y ν δv + Z ν δw)dS,
(15.43)
where X ν , Y ν , Z ν are still projections onto the axes of the Cartesian coordinate
system Oxyz of the external surface loading acting in the point with the normal line
ν; ρX, ρY, ρZ are the projections of mass forces on the same axes, and S is the
surface confining the area V .
Let us substitute strains in formula (15.43) with their expressions through
displacements under the formulas
ε x =
∂u
∂x
, . . . , γ zx =
∂u
∂z
+
∂w
∂x
.
Furthermore, we will substitute derivatives and variations in the second integral of
these formulas with the following expressions:
∂W
∂ε x
= σ x , . . . , ; δε x = δ
∂u
∂x
=
∂
∂x
δu, . . . .
By substituting the last expressions into formula (15.43) and grouping addends for
the variations δu, δv, and δw, we obtain as follows
δA =
V
σ x
∂
∂x
+ τ xy
∂
∂y
+ τ xz
∂
∂z
δu
+
τ yx
∂
∂x
+ σ y
∂
∂y
+ τ yz
∂
∂z
δv
+
τ zx
∂
∂x
+ τ zy
∂
∂y
+ σ z
∂
∂z
δw
dV .
(15.44)
For further transformation of expression (15.44), let us use the Ostrogradsky–Green
formula
V
∂P (x, y, z)
∂x
dV =
S
P (x, y, z)lds; [l = cos( ˆ
ν, x)],
(15.45)
as well as equations
∂
∂x
(σ x δu) =
∂σ x
∂x
δu + σ x
∂
∂x
δu;
σ x
∂
∂x
δu =
∂
∂x
(σ x δu) −
σ x
∂x
δu.
(15.46)