Two-dimensional defects in amorphous materials

TL;DR

This paper introduces a Riemannian-based definition of defects in amorphous materials, enabling a unified elastic description and analysis of localized 2D defects through multipole expansions, including dislocations and higher-order deformations.

Contribution

It proposes a novel Riemannian framework for defining and analyzing defects in amorphous materials using a reference metric field, with analytical, experimental, and numerical insights into multipole defect representations.

Findings

Analytical and numerical study of edge dislocation as a dipole defect.

Development of a multipole expansion for 2D localized defects.

Identification of higher multipoles as potential fundamental entities in plasticity.

Abstract

We present a new definition of defects which is based on a Riemannian formulation of incompatible elasticity. Defects are viewed as local deviations of the material's reference metric field, $\bar{\mathfrak{g}}$, from a Euclidian metric. This definition allows the description of defects in amorphous materials and the formulation of the elastic problem, using a single field, $\bar{\mathfrak{g}}$. We provide a multipole expansion of reference metrics that represent a large family of two-dimensional (2D) localized defects. The case of a dipole, which corresponds to an edge dislocation is studied analytically, experimentally and numerically. The quadrupole term, which is studied analytically, as well as higher multipoles of curvature carry local deformations. These multipoles are good candidates for fundamental strain carrying entities in plasticity theories of amorphous materials and for a continuous modeling of recently developed meta-materials.