Question 3 of 8: Strain Compatibility and Recovery of Displacements

Question 3: Strain Compatibility and Recovery of Displacements (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A plane strain field $\varepsilon_x=c(-3x^2+7y^2)$, $\varepsilon_y=c(x^2-5y^2)$, $\gamma_{xy}=bxy$, with $u=v=0$ at the origin.

Find. (a) the constraint linking $b$ and $c$; (b) the displacements $u,v$ (as multiples of $c$) at $(4,6)$.

Approach. Apply the single 2-D compatibility equation to fix $b$; then integrate the strain–displacement relations, using the origin condition to kill the integration constants.

1. Compatibility. The plane condition is $\dfrac{\partial^2\varepsilon_x}{\partial y^2}+\dfrac{\partial^2\varepsilon_y}{\partial x^2}=\dfrac{\partial^2\gamma_{xy}}{\partial x\,\partial y}$. Here $\dfrac{\partial^2\varepsilon_x}{\partial y^2}=14c$, $\dfrac{\partial^2\varepsilon_y}{\partial x^2}=2c$, $\dfrac{\partial^2\gamma_{xy}}{\partial x\,\partial y}=b$, so $$\boxed{b=16c}$$
2. Integrate $\varepsilon_x=\partial u/\partial x$. $u=c(-x^3+7xy^2)+f(y)$.
3. Integrate $\varepsilon_y=\partial v/\partial y$. $v=c\!\left(x^2y-\tfrac{5}{3}y^3\right)+g(x)$.
4. Impose the shear relation. $\gamma_{xy}=\dfrac{\partial u}{\partial y}+\dfrac{\partial v}{\partial x}=14cxy+f'(y)+2cxy+g'(x)=16cxy+f'(y)+g'(x)$. Matching $\gamma_{xy}=16cxy$ forces $f'(y)+g'(x)=0$, i.e. both are constants representing rigid-body motion; with no rotation and $u=v=0$ at the origin, $f=g=0$.
5. Evaluate at $(4,6)$. $u=c(-4^3+7\cdot4\cdot6^2)=c(-64+1008)$, $v=c(4^2\cdot6-\tfrac53\cdot6^3)=c(96-360)$, giving $$\boxed{u(4,6)=944\,c,\qquad v(4,6)=-264\,c}$$