$2\mathrm{.}2$ Analysis of plane strain $({}_z={}_{vz}={}_{xz}=0)$ shows that non$-$zero strains can be represented by:

${}_x=a+b(x^2+y^2)+x^4+y^4,{}_v=c+d(x^2+y^2)+x^4+y^4,{}_{xv}=e+fxy(x^2+y^2-g^2)$

where $a,b,c,d,e,f$ and $g$ are constants. Show that these are only possible if $f=4$ and $b+d+2g^2=$

$0.$ Hence derive the displacements from the strains.

Answer: $u=Ax+b(\frac{x^3}{3}+y^{2}x)+\frac{x^5}{5}+y^{4}x+C_{3}y+C_1,v=cy+d(x^{2}y+\frac{y^3}{3})+x^{4}y+\frac{y^5}{5}+C_{4}x+C_2$