TY - JOUR

T1 - Domain Formation in Magnetic Polymer Composites

T2 - An Approach Via Stochastic Homogenization

AU - Alicandro, Roberto

AU - Cicalese, Marco

AU - Ruf, Matthias

N1 - Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

PY - 2015/11/4

Y1 - 2015/11/4

N2 - 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 ɛ and the magnets as classical $${\pm 1}$$±1 spin variables interacting via an Ising type energy. Under surface scaling of the energy we prove, in terms of Γ-convergence, that, up to subsequences, the (continuum) Γ-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 Γ-limit exists and that the integrand is given by an asymptotic homogenization formula which becomes deterministic if the lattice is ergodic.

AB - 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 ɛ and the magnets as classical $${\pm 1}$$±1 spin variables interacting via an Ising type energy. Under surface scaling of the energy we prove, in terms of Γ-convergence, that, up to subsequences, the (continuum) Γ-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 Γ-limit exists and that the integrand is given by an asymptotic homogenization formula which becomes deterministic if the lattice is ergodic.

UR - http://www.scopus.com/inward/record.url?scp=84938557098&partnerID=8YFLogxK

U2 - 10.1007/s00205-015-0873-y

DO - 10.1007/s00205-015-0873-y

M3 - Article

AN - SCOPUS:84938557098

SN - 0003-9527

VL - 218

SP - 945

EP - 984

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

IS - 2

ER -