Geometric Theory of Mechanical Screening in two-dimensional solids

TL;DR

This paper develops a geometric theory of stress screening in two-dimensional solids, revealing a hierarchy of screening modes and proposing a new mechanical definition for the hexatic phase, with implications for amorphous materials.

Contribution

It introduces a novel geometric framework for stress screening in 2D solids, connecting local stress relaxation mechanisms to electrostatic analogies and redefining the hexatic phase mechanically.

Findings

Hierarchy of screening modes characterized by internal length scales

Stress screening analogous to electrostatic polarization phenomena

Hexatic phase can be defined by mechanical properties in amorphous materials

Abstract

Holes in mechanical metamaterials, quasi-localized plastic events in amorphous solids, and bound dislocations in a hexatic matter are different mechanisms of generic stress relaxation in solids. Regardless of the specific mechanism, these and other local stress relaxation modes are quadrupolar in nature, forming the foundation for stress screening in solids, similar to polarization fields in electrostatic media. We propose a geometric theory for stress screening in generalized solids based on this observation. The theory includes a hierarchy of screening modes, each characterized by internal length scales, and is partially analogous to theories of electrostatic screening such as dielectrics and Debye-H{\"u}ckel theory. Additionally, our formalism suggests that the hexatic phase, traditionally defined by structural properties, can also be defined by mechanical properties and may exist in amorphous materials.