Derivation of effective theories for thin 3D nonlinearly elastic rods with voids

Abstract

We derive a dimension-reduction limit for a three-dimensional rod with material voids by means of $\mathrm{\Gamma }$-convergence. Hereby, we generalize the results of the purely elastic setting [M. G. Mora and S. Müller, Derivation of the nonlinear bending-torsion theory for inextensible rods by $\mathrm{\Gamma }$-convergence, Calc. Var. Partial Differential Equations18 (2003) 287–305] to a framework of free discontinuity problems. The effective one-dimensional model features a classical elastic bending–torsion energy, but also accounts for the possibility that the limiting rod can be broken apart into several pieces or folded. The latter phenomenon can occur because of the persistence of voids in the limit, or due to their collapsing into a discontinuity of the limiting deformation or its derivative. The main ingredient in the proof is a novel rigidity estimate in varying domains under vanishing curvature regularization, obtained in [M. Friedrich, L. Kreutz and K. Zemas, Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces, preprint (2021), arXiv:2107.10808].