Interpenetration of matter in plate theories obtained as Gamma-limits
Abstract
We consider a potential pathology in the derivation of plate theories as Gamma-limits of 3-dimensional nonlinear elasticity by Friesecke James and Muller (Comm. Pure Appl. Math., 55:1461-1506 and Arch. Ration. Mech. Anal., 180:183-236), which consists in recovery sequences of invertible maps that converge to a non-invertible limit. These pathologies have been noted by Muller and Spector (Arch. Ration. Mech. Anal. 131:1-66) in a different context. Using a combination of degree theory, the approximation of Sobolev functions by Lipschitz functions and geometric rigidity we show that the potentially pathological situation in the derivation of plate theories provides sufficient conditions for the self-intersection of the graph of the recovery sequence element, and thus the pathology is excluded.
Keywords
Cite
@article{arxiv.1310.2865,
title = {Interpenetration of matter in plate theories obtained as Gamma-limits},
author = {Heiner Olbermann and Eris Runa},
journal= {arXiv preprint arXiv:1310.2865},
year = {2015}
}
Comments
20 pages, 4 figures. Proof of main proposition corrected (now Lemma 2 and Proposition 1), minor corrections elsewhere. Section about von-Karman plates removed, remarks added