Connecting Atomistic and Continuous Models of Elastodynamics

TL;DR

This paper establishes the long-time existence and convergence of atomistic elastodynamics models to continuum models, providing rigorous links between discrete atomic interactions and continuum elasticity.

Contribution

It proves the convergence of atomistic models to continuum nonlinear elastodynamics and introduces a new atomistic Gårding inequality for stability analysis.

Findings

Proved long-time existence of atomistic elastodynamics solutions.

Established convergence to continuum nonlinear elastodynamics.

Developed a new atomistic Gårding inequality for stability.

Abstract

We prove long-time existence of solutions for the equations of atomistic elastodynamics on a bounded domain with time-dependent boundary values as well as their convergence to a solution of continuum nonlinear elastodynamics as the interatomic distances tend to zero. Here, the continuum energy density is given by the Cauchy-Born rule. The models considered allow for general finite range interactions. To control the stability of large deformations we also prove a new atomistic G{\aa}rding inequality.