Ground states in complex bodies

TL;DR

This paper develops a comprehensive mathematical framework to analyze the existence of ground states in complex elastic bodies, incorporating advanced tools like Sobolev mappings and Cartesian currents, with applications to quasicrystals.

Contribution

It introduces a unified approach using semicontinuity, Sobolev maps, and Cartesian currents to study ground states in complex bodies, including irregular minimizers and configurational balance.

Findings

Existence of ground states in complex elastic bodies established.

Application to thermodynamically stable quasicrystals demonstrated.

Balance equations for irregular minimizers derived.

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 mappinngs 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.