Calculus of Variations and Geometric Measure Theory
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R. Alicandro - M. Cicalese - M. Ruf

Domain formation in magnetic polymer composites: an approach via stochastic homogenization

created by cicalese on 06 Oct 2014
modified on 11 Mar 2016

[BibTeX]

Published Paper

Inserted: 6 oct 2014
Last Updated: 11 mar 2016

Journal: Arch. Ration. Mech. Anal.
Volume: 218
Number: 2
Pages: 945–984
Year: 2015

Abstract:

We study the magnetic energy of magnetic polymer composite materials as the average distance between magnetic particles vanishes. We model the position of these particles in the polymeric matrix as a stochastic lattice scaled by a small parameter $\varepsilon$ and the magnets as classical $\pm 1$ spin variables interacting via an Ising type energy. Under surface scaling of the energy we prove, in terms of $\Gamma$-convergence that, up to subsequences, the (continuum) $\Gamma$-limit of these energies is finite on the set of Caccioppoli partitions representing the magnetic Weiss domains where it has a local integral structure. Assuming stationarity of the stochastic lattice, we can make use of ergodic theory to further show that the $\Gamma$-limit exists and that the integrand is given by an asymptotic homogenization formula which becomes deterministic if the lattice is ergodic.


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