Ground states in complex bodies

¹ DICeA, University of Florence, via Santa Marta 3, 50139 Firenze, Italy; paolo.mariano@unifi.it
² Dipartimento di Matematica Applicata “G. Sansone”, University of Florence, via Santa Marta 3, 50139 Firenze, Italy; giuseppe.modica@unifi.it

Received: 8 November 2007
Revised: 17 January 2008

Abstract

A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappings and Cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and Cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance of configurational actions follows. After describing a list of possible applications of the general results collected here, a concrete discussion of the existence of ground states in thermodynamically stable quasicrystals is presented at the end.