Calculus of Variations and Geometric Measure Theory
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H. Olbermann - E. Runa

Interpenetration of matter in plate theories obtained as Gamma-limits

created by olbermann on 23 Jul 2019

[BibTeX]

preprint

Inserted: 23 jul 2019

Year: 2013

ArXiv: 1310.2865 PDF

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.

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