Material frame of a rod ($\{{\mathbf{d}}_1,{\mathbf{d}}_2,{\mathbf{d}}_3\}$)

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