GATE 2024 AE – Question 56
Consider the plane strain field given by
$\varepsilon_{xx} = 10xy^2$, $\varepsilon_{yy} = -5x^2y$ and $\gamma_{xy} = Axy(2x - y)$
where, A is a constant and $\gamma_{xy}$ is the engineering shear strain. The value of the constant A for the strain field to be compatible is _____________ (rounded off to 1 decimal place).
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Correct answer: 4.97 to 5.03
Explanation
The compatibility condition is $\frac{\partial^2\varepsilon_{xx}}{\partial y^2} + \frac{\partial^2\varepsilon_{yy}}{\partial x^2} = \frac{\partial^2\gamma_{xy}}{\partial x\partial y}$. The left side is $20x - 10y$. For $\gamma_{xy} = A(2x^2y - xy^2)$ the mixed derivative is $A(4x - 2y)$. Equating, $4A = 20$ and $2A = 10$, so $A = 5$.