Justification of Leading Order Quasicontinuum Approximations of Strongly Nonlinear Lattices

TL;DR

This paper justifies leading order quasicontinuum approximations for strongly nonlinear granular lattices, demonstrating their accuracy through error bounds, and explores shock wave development and connections to NLS and KdV models.

Contribution

It provides a rigorous justification of the quasicontinuum limit for precompressed granular media and links it to classical nonlinear wave equations.

Findings

Error bounds established for the approximation

Shock waves predicted and numerically confirmed

NLS and KdV approximations applicable in weakly nonlinear regime

Abstract

We consider the leading order quasicontinuum limits of a one-dimensional granular medium governed by the Hertz contact law under precompression. The approximate model which is derived in this limit is justified by establishing asymptotic bounds for the error with the help of energy estimates. The continuum model predicts the development of shock waves, which are also studied in the full system with the aid of numerical simulations. We also show that existing results concerning the Nonlinear Schrodinger (NLS) and Korteweg de-Vries (KdV) approximation of FPU models apply directly to a precompressed granular medium in the weakly nonlinear regime.