Metric Description of Defects in Amorphous Materials

TL;DR

This paper demonstrates that classical elasticity can describe all defect types in amorphous materials using only metric tensor fields, unifying defect modeling without extra geometric structures.

Contribution

It shows that classical elasticity frameworks can encompass all defects via metric tensor fields, avoiding the need for additional structures like torsion or non-metricity.

Findings

Classical elasticity can model defects using only metric structures.

Defects correspond to monodromy in affine manifolds.

Two-dimensional defects with trivial monodromy are locally Euclidean.

Abstract

Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with a configuration of a body in classical elasticity is the sum of local contributions that arise from a discrepancy between the actual metric and the reference metric. In contrast, the modeling of defects in solids has traditionally involved extra structure on the material manifold, notably torsion to quantify the density of dislocations and non-metricity to represent the density of point defects. We show that all the classical defects can be described within the framework of classical elasticity using tensor fields that only assume a metric structure. Specifically, bodies with singular defects can be viewed as affine manifolds; both disclinations and dislocations are captured by the monodromy that maps curves that surround the loci of the defects into affine transformations. Finally, we show that two dimensional defects with trivial monodromy are purely local in the sense that if we remove from the manifold a compact set that contains the locus of the defect, the punctured manifold can be isometrically embedded in Euclidean space.