Calculus of Variations and Geometric Measure Theory

M. Friedrich - L. Kreutz - K. Zemas

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

created by kreutz on 12 Apr 2023
modified by zemas on 13 Jan 2024


Accepted Paper

Inserted: 12 apr 2023
Last Updated: 13 jan 2024

Journal: Math. Models Methods Appl. Sci. (M3AS)
Year: 2023

ArXiv: 2304.05289 PDF


We derive a dimension-reduction limit for a three-dimensional rod with material voids by means of $\Gamma$-convergence. Hereby, we generalize the results of the purely elastic setting 57 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 32.