$4$ Stress Function

In class, we considered a possible stress function for a thin rectangular plate, below.

The plate has surface tractions$/$stresses on its boundaries, with ${}_x=8,{}_y=-4,$ and ${}_{xy}=5.$ Consider a

stress function of the form

$=A{x}^2-Bxy+C{y}^2$

We defined ${}_x=\frac{de{l}^2}{del{y}^2},{}_y=\frac{de{l}^2}{del{x}^2},$ and ${}_{xy}=-\frac{de{l}^2}{\text{d}\text{e}\text{l}\text{x}\text{d}\text{e}\text{l}\text{y}}.$ Complete the following:

find numerical values for $A,B,$ and $C($after some derivatives, this should be trivial$)$

determine the normal and shear strains $lo{n}_x,lo{n}_y,$ and ${}_{xy},$ in terms of Young's modulus $E,$ Poisson's ratio

$u,$ and shear modulus $G$

determine whether the below displacement profile would be compatible with this strain state, and

explain why $/$ why not

$u=1+\frac{4}{E}(2+$

$u)x+\frac{5}{G}y$

$v=-1+\frac{3}{G}x-\frac{4}{E}(1+2$

$u)y$