In this article we consider a variant of the Simo-Reissner theory for a rod but restrict the study to two-dimensional motion where the rod undergoes flexure, shear and extension but not torsion. Linear elastic behaviour is assumed to formulate constitutive equations; the constitutive equations of the Timoshenko theory adapted for extension and large rotation. We call the model the Local Linear Timoshenko rod model. We show that this model serves as a framework for a class of simpler mathematical models for slender solids in various applications. The advantage is that the more general model can be used to evaluate and compare the simpler models.

The Local Linear Timoshenko (LLT) model for the planar motion of a rod that undergoes flexure, shear and extension, was recently derived in Van Rensburg et al. (2021). In this paper we present an algorithm developed for this model. The algorithm is based on the mixed finite element method, and projections into finite dimensional subspaces are used for dealing with nonlinear forces and moments. The algorithm is used for an investigation into elastic waves propagated in the LLT rod. Interesting properties of the LLT rod include the increased propagation speed of elastic waves when compared to the linear Timoshenko beam, and the appearance of buckled states or equilibrium solutions for compressed LLT beams. It is also shown that the LLT rod is applicable to large displacements and rotations for a wide range of slender elastic objects; from beams to highly slender flexible rods.