Question $1(5\mathrm{.}2-5\mathrm{.}3)$

$($a$)$ Show that for small strains the fractional change in volume is the sum of the normal strain components associated with a set of three perpendicular axes.

$($b$)$ Using this result, prove that for an isotropic, linearly elastic material,

$\frac{V}{V}=\frac{({}_x+{}_y+{}_z)(1-2v)}{E}$

$($c$)$ The bulk modulus of a linearly elastic material is defined as the ratio of the hydrostatic pressure to the fractional decrease in volume.

$B=\frac{p}{-(\frac{V}{V})}$

Show that for an isotropic material the bulk modulus is given by

$B=\frac{E}{3(1-2v)}$