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

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.