Calculus of Variations and Geometric Measure Theory

A. Gloria - M. Ruf

Loss of strong ellipticity through homogenization in 2D linear elasticity: A phase diagram

created by ruf on 16 May 2018
modified on 01 Aug 2018


Accepted Paper

Inserted: 16 may 2018
Last Updated: 1 aug 2018

Journal: Arch. Ration. Mech. Anal.
Year: 2018
Doi: 10.1007/s00205-018-1290-9

ArXiv: 1805.05150 PDF


Since the seminal contribution of Geymonat, Müller, and Triantafyllidis, it is known that strong ellipticity is not necessarily conserved through periodic homogenization in linear elasticity. This phenomenon is related to microscopic buckling of composite materials. Consider a mixture of two isotropic phases which leads to loss of strong ellipticity when arranged in a laminate manner, as considered by Gutiérrez and by Briane and Francfort. In this contribution we prove that the laminate structure is essentially the only microstructure which leads to such a loss of strong ellipticity. We perform a more general analysis in the stationary, ergodic setting.