Question 3 of 8: Strain compatibility and integration of a plane strain field

Question 3: Strain compatibility and integration of a plane strain field

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. The plane strain components $\varepsilon_x=c(-3x^2+7y^2)$, $\varepsilon_y=c(x^2-5y^2)$, $\gamma_{xy}=bxy$.

Find. (a) the compatibility relation between b and c; (b) the displacements at (1, 5) with the origin fixed and b = 5.

Approach. Enforce the 2-D compatibility equation, then integrate the strain–displacement relations, fixing the rigid-body constants at the origin.

1. Compatibility (part a). The single 2-D compatibility condition is $$\frac{\partial^2\varepsilon_x}{\partial y^2}+\frac{\partial^2\varepsilon_y}{\partial x^2}=\frac{\partial^2\gamma_{xy}}{\partial x\,\partial y}.$$ Here $\partial^2\varepsilon_x/\partial y^2=14c$, $\partial^2\varepsilon_y/\partial x^2=2c$, and $\partial^2\gamma_{xy}/\partial x\partial y=b$, so $$14c+2c=b\;\Rightarrow\;\boxed{b=16c}$$
2. Integrate the direct strains. From $\varepsilon_x=\partial u/\partial x$ and $\varepsilon_y=\partial v/\partial y$, $$u=c\!\left(-x^3+7y^2x\right)+f(y),\qquad v=c\!\left(x^2y-\tfrac{5}{3}y^3\right)+g(x).$$
3. Enforce the shear strain. Substituting into $\gamma_{xy}=\partial u/\partial y+\partial v/\partial x=16cxy+f'(y)+g'(x)$ and equating to $bxy=16cxy$ gives $f'(y)+g'(x)=0$. With no rigid-body rotation and $u(0,0)=v(0,0)=0$, all constants vanish, so $f=g=0$.
4. Evaluate at (1, 5) with b = 5. Then $c=b/16=0.3125$ and $$u(1,5)=c(-1+7\cdot25)=174c=\boxed{54.4},\qquad v(1,5)=c\!\left(5-\tfrac{5}{3}\cdot125\right)=-203.3c=\boxed{-63.5}$$ (in the same length units as the coordinates).