For a general isotropic medium, the stress tensor ${}_{ij}$ and strain tensor $e_{ij}$ are related by

${}_{ij}=\frac{vE}{(1+v)(1-2v)}{e}_{kk}{}_{ij}+\frac{E}{(1+v)}{e}_{ij}$

where $E=$ Young's modulus and $v=$ Poisson's ratio. By considering an isotropic body subjected to a uniform hydrostatic pressure, $p($no shearing stresses$),$ show that the bulk modulus, $K$ is related by the ratio of the pressure to the fractional decrease in volume, i$.$e$.K=\frac{E}{3(1-2v)}=\frac{p}{e_{ii}}$