$\eta$ - modulus of rigidity

We know that

$y=3 k(1-2 \sigma)$

$\sigma=\frac{1}{2}\left(1-\frac{ y }{3 k }\right)$     $.........(i)$

$y =2 \eta(1+\sigma)$

$\sigma=\frac{y}{2 \eta}-1$       $........(ii)$

From Eq.$(i)$ and Eq. $(ii)$

$\frac{1}{2}\left(1-\frac{ Y }{3 k }\right)=\frac{ y }{2 \eta}-1$

$1-\frac{y}{3 k}=\frac{y}{\eta}-2$

$\frac{y}{3 k}=3-\frac{y}{\eta}$

$\frac{y}{3 k}=\frac{3 \eta-y}{\eta}$

$\frac{\eta y }{3 k }=3 \eta- y$

$k =\frac{\eta y }{9 \eta-3 y }$