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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.

An elastic rod is a slender elastic body represented by a centreline and an attached orthonormal director frame. Its elastic energy penalizes stretching, bending and twisting relative to a preferred configuration. The director kinematics of a rod describe changes of the frame along the centreline. An unshearable elastic rod has its third director parallel to the centreline tangent vector.

The right-handed material frame of a rod has ${\mathbf{d}}_2\times {\mathbf{d}}_1=-{\mathbf{d}}_3$ and ${\mathbf{d}}_2\times {\mathbf{d}}_3={\mathbf{d}}_1$. Hence the director kinematics of a rod reduce toSolving these constant-coefficient ordinary differential equations with the given initial frame givesThe inextensible rod has ${\mathbf{r}}^{\prime }={\mathbf{d}}_3$. An integration using $\mathbf{r}(0)=0$ yields, for $\hat{u}\ne 0$,Indeed,so the centreline is a circular arc of radius $1/|\hat{u}|$ in the plane perpendicular to ${\mathbf{e}}_2$. The continuous $\hat{u}\to 0$ limit is ${\mathbf{d}}_1={\mathbf{e}}_1$, ${\mathbf{d}}_3={\mathbf{e}}_3$, and $\mathbf{r}=S{\mathbf{e}}_3$.

The material frame of a rod obeys the director kinematics of a rodIn the stress-free reference configuration, the constitutive equation gives $u_1=0$, $u_2=\hat{u}$, and $u_3=0$, since all three stiffnesses are positive. Thus $\mathbf{u}=\hat{u}{\mathbf{d}}_2$. Its cross product with ${\mathbf{d}}_2$ vanishes, soConsequently this director is a fixed vector in space, even though the other two directors rotate along the elastic rod.

Because the two elastic rods each have thickness $a$, their centrelines are separated by $a$ along the common director ${\mathbf{d}}_1$. The bonded geometry isDifferentiate with respect to reference arc length. The first rod is an inextensible rod, and the printed director kinematics of a rod giveFor the physically admissible orientation-preserving configuration $1-au>0$, the rods have the same unit tangent ${\mathbf{d}}_3^{+}={\mathbf{d}}_3$ and the same planar material frame of a rod. ThereforeIn particular, the second rod is shortened on the inside of a bend with $u>0$.

The bending strain of a rod is defined here by differentiation with respect to reference arc length. Since the reference labels match,The distinction from spatial curvature matters: the second centreline hasThus $u/(1-au)$ is its signed spatial curvature, rather than the $u^{+}$ appearing in the given elastic energy.