26
2 The First Basic Problem of Elasticity Theory
u ≈ ε x x + yγ xy + zγ xz + zω y − yω z ,
v ≈ ε y y + xγ yx + zγ yz + xω z − zω x ,
w ≈ ε z z + xγ zx + yγ zy + yω x − xω y .
(2.7)
2.4 Saint-Venant Identities
By directly checking, we can make sure that the following identity is true for a
combination of certain derivatives:
∂
∂x
∂v
∂z
+
∂w
∂y
+
∂
∂y
∂w
∂x
+
∂u
∂z
−
∂
∂z
∂u
∂y
+
∂v
∂x
≡ 2
∂ 2 w
∂x∂y
.
The right part of the identity is obtained after reduction of similar terms in its left
parts. By differentiating using z, we will find that
2
∂ 3 w
∂x∂y∂z
≡
∂
∂z
∂
∂x
∂v
∂z
+
∂w
∂y
+
∂
∂y
∂w
∂x
+
∂u
∂z
−
∂
∂z
∂u
∂y
+
∂v
∂x
,
or, taking into account formulas (2.2) and (2.3),
2
∂ 2 ε z
∂x∂y
≡
∂
∂z
∂γ yz
∂x
+
∂γ xz
∂y
−
∂γ xy
∂z
.
(2.8)
By differentiating formulas (2.2) and (2.3), we can easily obtain that
∂ 2 γ xy
∂x∂y
≡
∂ 2 ε x
∂y 2 +
∂ 2 ε y
∂x 2 .
(2.9)
From formulas (2.8) and (2.9) by circular permutation of indexes, we can obtain
four more similar ratios. These six ratios are called [2] Saint-Venant identities.
2.5 Compatibility Conditions
Substituting formulas (1.11) and (1.12) into identities (2.8) and (2.9) gives
2
E
∂ 2
∂x∂y
σ z − ν(σ y + σ x )
=
1
G
∂
∂z
∂τ yz
∂x
+
∂τ xz
∂y
−
∂τ xy
∂z
,
(2.10)