Question #42203

A cylindrical rod of a novel metamaterial has a constant bulk modulus $K = 1.0 \times 10^9 \text{ N/m}^2$. During a slow 1D longitudinal tensile test, its Poisson's ratio $\nu$ varies with the longitudinal strain $\epsilon$ as $\nu(\epsilon) = 0.25 - 10\epsilon$. The longitudinal stress $\sigma$ and strain $\epsilon$ are related by $\sigma = Y(\epsilon)\epsilon$, where $Y(\epsilon) = 3K(1 - 2\nu(\epsilon))$ is the effective Young's modulus. If the rod is stretched from an unstrained state ($\epsilon = 0$) to a final strain of $\epsilon = 0.05$, what is the work done per unit volume (strain energy density) on the rod?