= Solution
Apply equal compressive <pressure> $p$ in all directions; here $\sigma$ denotes <Poisson's ratio>. The three principal normal <stresses> are $-p$. For an isotropic linearly elastic material, <Hooke's law> gives
$$
\epsilon_x=\frac{-p-\sigma(-p-p)}{E}
=-\frac{p(1-2\sigma)}E,
$$
and similarly for $y,z$. At small <strain>, the fractional volume change is the sum of these normal <strains>:
$$
\frac{\Delta V}{V}=\epsilon_x+\epsilon_y+\epsilon_z
=-\frac{3p(1-2\sigma)}E.
$$
The <bulk modulus> is $B=-p/(\Delta V/V)$, giving the <isotropic elastic modulus relation>,
$$
\boxed{B=\frac{E}{3(1-2\sigma)}.}
$$
The result assumes <linear elasticity> and fixed isotropic moduli. In the incompressible limit $\sigma\to1/2$ with finite <Young's modulus>, the bulk modulus tends to infinity.
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