Journal
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
Volume 218, Issue 2, Pages 945-984Publisher
SPRINGER
DOI: 10.1007/s00205-015-0873-y
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- [SFB/Transregio 109]
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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 E > and the magnets as classical spin variables interacting via an Ising type energy. Under surface scaling of the energy we prove, in terms of I-convergence, that, up to subsequences, the (continuum) I-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 I-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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