Elastic energy ($E_{\mathrm{elastic}}$)

Elastic energy is the recoverable energy stored by deforming an elastic material. In the small-deflection model of an elastic filament, the bending contribution is a positive quadratic functional of the curvatures, such as $\frac{1}{2}\int (A{y}^{\prime \prime 2}+B{z}^{\prime \prime 2})dx$. Work by applied loads must also be included when minimizing the total potential energy.

The part of elastic energy associated with a change of material length. A simple quadratic constitutive equation assigns energy density $C(\alpha -1{)}^2/2$ per reference arc length, where $C>0$ is stretching stiffness and $\alpha$ is the stretch ratio. Such a small-strain model need not remain realistic as $\alpha \to 0$.

The part of an elastic rod's elastic energy that penalizes departure of its bending strain of a rod $u$ from the intrinsic value $\hat{u}$. Here $B$ is the bending stiffness and $S$ is reference arc length. For an inextensible rod, the same formula can be written using spatial curvature and current arc length.

Let a convex strain energy density have derivative equal to the actual stress tensor. Under mixed displacement and traction data, any equilibrated stress ${\sigma }_0$ with the prescribed body force and boundary tractions represents the external work. For any compatible trial strain ${\epsilon }^{\prime }$, the convex supporting-plane inequality gives $W({\epsilon }^{\prime })-W(\epsilon )\ge \sigma :({\epsilon }^{\prime }-\epsilon )$. The integral of $(\sigma -{\sigma }_0):({\epsilon }^{\prime }-\epsilon )$ vanishes by integration by parts. Thus the actual strain minimizes $\int (W-{\sigma }_0:\epsilon )$.