A consistency study of coarse-grained dynamical chains through a nonlinear wave equation of mixed type

TL;DR

This paper investigates the consistency of coarse-grained dynamical chains with a nonlinear wave equation model, analyzing how well simplified atomistic models approximate macroscopic material behavior and addressing stability issues.

Contribution

It provides a rigorous analysis of the consistency between coarse-grained atomistic models and continuum equations, including methods to improve approximation accuracy.

Findings

Coarse-grained models can approximate macroscopic behavior effectively.

Adding viscous terms or space averaging improves model consistency.

Instability and defect formation are linked to large deformations.

Abstract

A dynamical atomistic chain to simulate mechanical properties of a one-dimensional material with zero temperature may be modelled by the molecular dynamics (MD) model. Because the number of particles (atoms) is huge for a MD model, in practice one often takes a much smaller number of particles to formulate a coarse-grained approximation. We shall mainly consider the consistency of the coarse-grained model with respect to the grain (mesh) size to provide a justification to the goodness of such an approximation. In order to reduce the characteristic oscillations with very different frequencies in such a model, we either add a viscous term to the coarse-grained MD model or apply a space average to the coarse-grained MD solutions for the consistency study. The coarse-grained solution is also compared with the solution of the (macroscopic) continuum model (a nonlinear wave equation of mixed type) to show how well the coarse-grained model can approximate the macroscopic behavior of the material. We also briefly study the instability of the dynamical atomistic chain and the solution of the Riemann problem of the continuum model which may be related to the defect of the atomistic chain under a large deformation in certain locations.