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

Marco Cicalese; Antonio DeSimone; Caterina Ida Zeppieri; Marco Cicalese; Antonio DeSimone; Caterina Ida Zeppieri

doi:10.3934/nhm.2009.4.667

Networks and Heterogeneous Media

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

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

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

Received: 01 January 2009 Revised: 01 June 2009
Primary: 49J45, 49M25; Secondary: 74B05, 76A15, 82D45, 82D40.

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.
- $\Gamma$-convergence,
- discrete systems,
- linear elasticity,
- liquid crystals,
- magnetostrictive solids,
- ferroelectric crystals,
- nematic elastomers
Citation: Marco Cicalese, Antonio DeSimone, Caterina Ida Zeppieri. Discrete-to-continuum limits for strain-alignment-coupled systems:Magnetostrictive solids, ferroelectric crystals and nematicelastomers[J]. Networks and Heterogeneous Media, 2009, 4(4): 667-708. doi: 10.3934/nhm.2009.4.667

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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.
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