 Bending strain of a rod

For planar director kinematics of a rod measured in reference arc length $S$, define the signed bending strain by $d{\mathbf{d}}_3/dS=u{\mathbf{d}}_1$. If $s$ is current arc length and $\alpha =ds/dS>0$ is the stretch ratio, the signed spatial curvature is $u/\alpha$: the chain rule gives $d{\mathbf{d}}_3/ds=(u/\alpha ){\mathbf{d}}_1$. Thus bending strain and spatial curvature coincide for an inextensible rod, but differ in general. An intrinsic value $\hat{u}$ is the preferred reference-length bending strain in an energy density $B(u-\hat{u}{)}^2/2$ per unit reference length.