# 3.13 Show that the elastic stress-strain relations for an isotropic material are \begin{array}{l} \varepsilon_{1}=\frac{\sigma_{1}}{E}-v \frac{\sigma_{2}}{E}-v \frac{\sigma_{3}}{E} \\ \varepsilon_{2}=\frac{\sigma_{2}}{E}-v \frac{\sigma_{1}}{E}-v \frac{\sigma_{3}}{E} \\ \varepsilon_{3}=\frac{\sigma_{3}}{E}-v \frac{\sigma_{1}}{E}-v \frac{\sigma_{2}}{E} \end{array} Hence derive an equation for

the dilatation in terms of Poisson's ratio,Young's modulus, and the sum of the principal stresses. For what value of v is the volume change zero?