Korn type Inequalities for Objective Structures

TL;DR

This paper develops discrete Korn inequalities for objective structures, extending classical elasticity concepts to complex particle systems with symmetry, providing tools for stability analysis in atomistic models.

Contribution

It introduces new discrete Korn inequalities applicable to a broad class of objective structures, generalizing previous lattice-based results to more complex symmetries.

Findings

Established full discrete Korn inequalities for space filling configurations with space group symmetry.

Provided intrinsic rigidity estimates for systems with non-trivial codimension.

Enabled energy estimates and stability analysis for atomistic particle systems.

Abstract

We establish discrete Korn type inequalities for particle systems within the general class of objective structures that represents a far reaching generalization of crystal lattice structures. For space filling configurations whose symmetry group is a general space group we obtain a full discrete Korn inequality. For systems with non-trivial codimension our results provide an intrinsic rigidity estimate within the extended dimensions of the structure. As their continuum counterparts in elasticity theory, such estimates are at the core of energy estimates and, hence, a stability analysis for a wide class of atomistic particle systems.