d
Y- Younge modulus, \(K\) - Bulk modulus,

\(\eta\) - 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. \((ii)\)

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

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

\(\frac{y}{3 k}=3-\frac{y}{\eta}\)

\(\frac{y}{3 k}=\frac{3 \eta-y}{\eta}\)

\(\frac{\eta y }{3 k }=3 \eta- y\)

\(k =\frac{\eta y }{9 \eta-3 y }\)