A thin-walled cylindrical pressure vessel has an inner radius \(r\), thickness \(t\), and length \(L\). If it is subjected to an internal pressure \(p\), show that the increase in its inner radius is \(d r=r \epsilon_{1}=p r^{2}\left(1-\frac{1}{2} u\right) / E t\) and the increase in its length is \(\Delta L=p L r\left(\frac{1}{2}-u\right) / E t\). Using these results, show that the change in internal volume becomes \(d V=\pi r^{2}\left(1+\epsilon_{1}\right)^{2}\left(1+\epsilon_{2}\right) L-\pi r^{2} L\). Since \(\epsilon_{1}\) and \(\epsilon_{2}\) are small quantities, show further that the change in volume per unit volume, called volumetric strain, can be written as \(d V / V=\operatorname{pr}(12.5-2 u) / E t\).