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For a differentiable material frame of a rod, there is a unique vector $\mathbf{u}$ such that ${\mathbf{d}}_i^{\prime }=\mathbf{u}\times {\mathbf{d}}_i$. To see this, form the orthogonal matrix $R=({\mathbf{d}}_1,{\mathbf{d}}_2,{\mathbf{d}}_3)$. Differentiation of $R{R}^T=I$ shows that $R^{\prime }{R}^T$ is a skew-symmetric matrix; every three-dimensional skew-symmetric matrix has the unique form $\mathbf{v}\mapsto \mathbf{u}\times \mathbf{v}$. Primes may denote differentiation with respect to reference or current arc length, but that choice must be stated. In a planar bend about ${\mathbf{d}}_2$, reference differentiation gives ${\mathbf{d}}_3^{\prime }=u{\mathbf{d}}_1$ and ${\mathbf{d}}_1^{\prime }=-u{\mathbf{d}}_3$.

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.