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Networks and Heterogeneous Media (NHM)
 

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

Pages: 667 - 708, Volume 4, Issue 4, December 2009      doi:10.3934/nhm.2009.4.667

 
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Marco Cicalese - Dipartimento di Matematica e Applicazioni "R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126 Napoli, Italy (email)
Antonio DeSimone - SISSA-International School for Advanced Studies, Via Beirut 2-4, 34014 Trieste, Italy (email)
Caterina Ida Zeppieri - SISSA-International School for Advanced Studies, Via Beirut 2-4, 34014 Trieste, Italy (email)

Abstract: 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:  $\Gamma$-convergence, discrete systems, linear elasticity, liquid crystals, magnetostrictive solids, ferroelectric crystals, nematic elastomers.
Mathematics Subject Classification:  Primary: 49J45, 49M25; Secondary: 74B05, 76A15, 82D45, 82D40.

Received: January 2009;      Revised: June 2009;      Available Online: October 2009.