6.2 Strain–Displacement Relations
165
Fig. 6.2 Deformed element
in three dimensions
The strain–displacement relations for the most general state of stress are given by
ε r =
∂u
∂r
, ε θ =
1
r
∂v
∂θ
+
u
r
, ε z =
∂w
∂z
γ r θ =
∂v
∂r
+
1
r
∂u
∂θ
−
v
r
γ θ z =
1
r
∂w
∂θ
+
∂v
∂z
γ zr =
∂u
∂z
+
∂w
∂r
(6.8)
6.3 Strain-Compatibility Equation
We have from the strain–displacement relations:
Radial strain, ε r =
∂u
∂r
(6.9a)
Tangential strain, ε θ =
1
r
∂v
∂θ
+
u
r
(6.9b)