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December  2009, 4(4): 667-708. doi: 10.3934/nhm.2009.4.667

Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers

Marco Cicalese 1, , Antonio DeSimone 2, and  Caterina Ida Zeppieri 2,

1. 

Dipartimento di Matematica e Applicazioni "R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126 Napoli
2. 

SISSA-International School for Advanced Studies, Via Beirut 2-4, 34014 Trieste, Italy

Received  January 2009 Revised  June 2009 Published  October 2009

In the framework of linear elasticity, we study the limit of a class of discrete free energies modeling strain-alignment-coupled systems by a rigorous coarse-graining procedure, as the number of molecules diverges. We focus on three paradigmatic examples: magnetostrictive solids, ferroelectric crystals and nematic elastomers, obtaining in the limit three continuum models consistent with those commonly employed in the current literature. We also derive the correspondent macroscopic energies in the presence of displacement boundary conditions and of various kinds of applied external fields.
Keywords: nematic elastomers., liquid crystals, discrete systems, ferroelectric crystals, $\Gamma$-convergence, magnetostrictive solids, linear elasticity.
Mathematics Subject Classification: Primary: 49J45, 49M25; Secondary: 74B05, 76A15, 82D45, 82D4.
Citation: Marco Cicalese, Antonio DeSimone, Caterina Ida Zeppieri. Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers. Networks & Heterogeneous Media, 2009, 4 (4) : 667-708. doi: 10.3934/nhm.2009.4.667
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