Reflections on the structural and constitutive approaches to the theory of defects

TL;DR

This paper explores the integration of structural and constitutive models in the theory of defects using G-structures, highlighting their geometric foundations and applications to liquid crystals.

Contribution

It demonstrates how G-structures unify microscopic and macroscopic defect theories, providing a common geometric framework for both approaches.

Findings

G-structures naturally arise from symmetry groups in defect theories

The approach unifies microscopic and macroscopic defect models

Application to smectic liquid crystals illustrates the framework

Abstract

An attempt is made to bring into harmony two of the paradigms commonly used in the theory of continuous distributions of defects. It is shown that the common differential geometric apparatus is provided neatly by the theory of G-structures. In the case of a structural model, based on putative experimental observations at the microscopic level, a G-structure can be shown to emerge from the group of linear transformations that preserve a tensorial quantity. For the phenomenological (macroscopic) constitutive model, the G-structure arises from the notion of material isomorphism and the underlying local symmetry group of the constitutive law. A comparative example is presented in the framework of certain smectic liquid crystals.