30
3 The Second Primary Problem of Elasticity Theory
τ xy = G
∂u
∂y
+
∂v
∂x
, τ yz = G
∂v
∂z
+
∂w
∂y
,
(3.4)
τ zx = G
∂w
∂x
+
∂u
∂z
.
3.2 Equations of Elastic Body Strain
Let us substitute formulas (3.2) and (3.4) into equilibrium equations (2.1). We will
obtain a system of three differentiated equations of relative movement u, v, w of
elastic body points. The first of these equations will look as follows:
2G
∂ 2 u
∂x 2 + λ
∂∂
∂x
+ G
∂
∂y
∂u
∂y
+
∂v
∂x
+ G
∂
∂z
∂u
∂z
+
∂w
∂z
= 0
or
G
∂ 2 u
∂x 2 +
∂ 2 u
∂y 2 +
∂ 2 u
∂z 2
+ G
∂
∂x
∂u
∂x
+
∂v
∂y
+
∂w
∂z
+ λ
∂∂
∂x
= 0.
By applying formula (1.13), we will obtain
G
∂ 2 u
∂x 2 +
∂ 2 u
∂y 2 +
∂ 2 u
∂z 2
+ (λ + G)
∂∂
∂x
= 0.
Similar equations can be obtained from two last equations of equilibrium (2.1).
By using the common designation for Laplacian [3]
=
∂ 2
∂x 2 +
∂ 2
∂y 2 +
∂ 2
∂z 2 ,
(3.5)
let us represent these equations as follows:
u +
1
1 − 2ν
∂∂
∂x
= 0,
v +
1
1 − 2ν
∂∂
∂y
= 0,
w +
1
1 − 2ν
∂∂
∂z
= 0.
(3.6)
In this manner, the second primary problem of elasticity theory is as follows.