Since the seminal contribution of Geymonat, Müller, and Triantafyllidis, it has been 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.
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We warmly thank Gilles Francfort for discussions on ellipticity conditions, and acknowledge the financial support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2014-2019 Grant Agreement QUANTHOM 335410).
Conflict of interest
The authors declare that there is no conflict of interest.
Communicated by S. Müller
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Gloria, A., Ruf, M. Loss of Strong Ellipticity Through Homogenization in 2D Linear Elasticity: A Phase Diagram. Arch Rational Mech Anal 231, 845–886 (2019). https://doi.org/10.1007/s00205-018-1290-9