3.17 Numerical Examples
91
Example 3.10 Under what conditions are the following expressions for the
components of strain at a point compatible?
ε x = 2ax y
2
+ by
2
+ 2cx y
ε y = ax
2
+ bx
γ xy = αx
2 y + βx y + ax
2
+ ηy
Solution: For compatibility, the strain components must satisfy the compatibility
equation.
i.e.,
∂
2
ε x
∂ y 2 +
∂
2
ε y
∂ x 2 =
∂
2
γ xy
∂ x∂ y
(i)
or
∂
2
ε x
∂ y 2 +
∂
2
ε y
∂ x 2 −
∂
2
γ xy
∂ x∂ y
= 0
( i i )
Now, ε x = 2ax y
2
+ by
2
+ 2cx y
∴
∂ε x
∂ y
= 4ax y + 2by + 2cx
∂
2
ε x
∂ y 2 = 4ax + 2b
ε y = ax
2
+ bx
∂ε y
∂ x
= 2ax + b
∂
2
ε y
∂ x 2 = 2a
γ xy = αx
2 y + βx y + ax
2
+ ηy
∂γ xy
∂ x
= 2αx y + βy + 2ax
∂
2
γ xy
∂ x∂ y
= 2αx + β
∴ (i) becomes
4ax + 2b + 2a = 2αx + β