Exercise $4$: Bulk modulus

The bulk modulus $K$ is a measure of the pressure $p$ needed to change a material's volume $V_0$ by $V$ :

${}_p=-k*\frac{V}{V_0}$

The minus sign accounts for the fact that positive pressure usually causes a reduction in the volume.

a$)$ Derive the relation between $K$ and the elastic constants $E$:$G,$ and

$u$ in an isotropic material at small deformations!

b$)$ Calculate the bulk modulus of a material with a Poisson's ratio of

$($