Convergence of equilibria for bending-torsion models of rods with inhomogeneities

TL;DR

This paper proves that inhomogeneous elastic rods' equilibrium configurations converge to those predicted by a limit theory as their thickness diminishes, extending previous homogeneous results to more complex material compositions.

Contribution

It establishes convergence of stationary points for inhomogeneous rods to the variational limit, generalizing prior homogeneous material results.

Findings

Stationary points of energy functionals converge as thickness approaches zero.

The convergence holds for non-periodic inhomogeneities.

Results extend to inhomogeneous materials with appropriate energy scaling.

Abstract

We prove that, in the limit of vanishing thickness, equilibrium configurations of inhomogeneous, three-dimensional non-linearly elastic rods converge to equilibrium configurations of the variational limit theory. More precisely, we show that, as $h \to 0$, stationary points of the energy $E^h$, for a rod $\Omega_h \subset \mathbb R^3$ with cross-sectional diameter $h$, subconverge to stationary points of the $\Gamma$-limit of $E^h$, provided that the bending energy of the sequence scales appropriately. This generalizes earlier results for homogeneous materials to the case of materials with (not necessarily periodic) inhomogeneities.