Existence and Convergence of Solutions of the Boundary Value Problem in Atomistic and Continuum Nonlinear Elasticity Theory

TL;DR

This paper proves the existence of solutions in atomistic nonlinear elasticity and demonstrates their convergence to continuum solutions as interatomic distances decrease, under small data conditions.

Contribution

It establishes the existence and convergence of solutions in atomistic elasticity to continuum models, extending understanding of their relationship.

Findings

Solutions exist for small data near stable lattices.

Solutions converge to continuum elasticity solutions as interatomic distance approaches zero.

Results hold for general finite-range interaction potentials.

Abstract

We show existence of solutions for the equations of static atomistic nonlinear elasticity theory on a bounded domain with prescribed boundary values. We also show their convergence to the solutions of continuum nonlinear elasticity theory, with energy density given by the Cauchy-Born rule, as the interatomic distances tend to zero. These results hold for small data close to a stable lattice for general finite range interaction potentials. We also discuss the notion of stability in detail.