Elastic rod

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.

An inextensible rod preserves material arc length, so its stretch ratio is one. In an unshearable elastic rod, $d\mathbf{r}/dS={\mathbf{d}}_3$ for reference arc length $S$.

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.

A right-handed orthonormal frame attached to the cross-sections of an elastic rod. For an unshearable elastic rod, ${\mathbf{d}}_3$ is the unit tangent vector to its centreline, while ${\mathbf{d}}_1$ and ${\mathbf{d}}_2$ follow the orientation of its material cross-section. Unlike a frame constructed only from the centreline curvature, the material frame records the twist of the material about the tangent.

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 ${\mathbf{d}}_1^{\prime }=-u{\mathbf{d}}_3$.