Bending energy (${\mathcal{E}}_b=\frac{1}{2}\int B(u-\hat{u}{)}^{2}dS$)

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.