Problem $8\mathrm{.}2.$ The stress$-$strain relationships for a cubic crystal are

${}_{11}=c_{11}{e}_{11}+c_{12}(e_{22}+e_{33}),$

${}_{22}=c_{11}{e}_{22}+c_{12}(e_{33}+e_{11}),$

${}_{33}=c_{11}{e}_{33}+c_{12}(e_{11}+e_{22}),$

and

${}_{12}=2c_{44}{e}_{12},{}_{23}=2c_{44}{e}_{23},{}_{31}=2c_{44}{e}_{31}$

$($a$)$ Derive the inverse relationships expressing the strain components $e_{ij}$ in terms of stress components ${}_{ij}$ and the elastic constants $c_{ij}.$

$($b$)$ Derive the relationships between $c_{11},c_{12},c_{44}$ and $E,v,G($Young$\text{'}$s modulus, Poisson's ratio, shear modulus$).$