Shock and Rarefaction Waves in Generalized Hertzian Contact Models

TL;DR

This paper investigates shock and rarefaction wave formation in generalized Hertzian contact models for granular crystals, using analytical and numerical methods to predict wave profiles and formation times based on nonlinearity exponent p.

Contribution

It introduces a quasi-continuum approximation approach to analyze wave dynamics in generalized Hertzian models, capturing shock formation and oscillations beyond the approximation.

Findings

Both shock and rarefaction waves can occur depending on p.

The quasi-continuum model accurately predicts wave profiles near shock formation.

Discrete oscillations emerge after the shock formation time, not captured by the continuum approximation.

Abstract

In the present work motivated by generalized forms of the Hertzian dynamics associated with granular crystals, we consider the possibility of such models to give rise to both shock and rarefaction waves. Depending on the value $p$ of the nonlinearity exponent, we find that both of these possibilities are realizable. We use a quasi-continuum approximation of a generalized inviscid Burgers model in order to predict the solution profile up to times near the shock formation, as well as to estimate when it will occur. Beyond that time threshold, oscillations associated with the discrete nature of the underlying model emerge that cannot be captured by the quasi-continuum approximation. Our analytical characterization of the above features is complemented by systematic numerical computations.