Understanding Strain Energy Density

Question

The strain energy density is: $$\frac{1}{2}\sigma_{ij}\epsilon_{ij}$$ Where $\sigma$ is the Cauchy stress tensor ($\sigma_{ij}=T_j(\mathbf{e}_i))$ and $\epsilon^e$ is the infinitesimal strain tensor ($\epsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right))$ and summation notation was used.

Is there a way to derive this relationship? Intuitively, the stress tensor represents the force per unit area and the strain tensor represents the displacement per unit length, so their product represents the "work" done by the internal stresses per unit volume, but I can't find a way to state that in a more rigorous derivation.

Any advice would be appreciated

Answer 1

Infinitesimal stress–strain work is $\sigma V\,d\varepsilon=C\varepsilon V\,d\varepsilon$, where $C$ is the stiffness tensor. Integrate from initial strain 0 to final strain $\varepsilon$, replace an $\varepsilon$ with an equivalent $\sigma/C$, and normalize by volume to get the strain energy density: $\frac{1}{2}\sigma \varepsilon$.