Strongly non-linear continuous model for wave propagation in 2D granular media: derivation and hyperbolicity analysis
Abstract
In this work, we develop a nonlinear continuum model for wave propagation in two-dimensional granular materials, building on key ideas from Nesterenko's work. Starting from the Hertzian contact law and neglecting contact friction, we derive an effective continuum description through homogenization, assuming spherical, monodisperse grains, local periodicity, and the Cauchy--Born hypothesis. The resulting strain-gradient model depends on both the first and second gradients of the displacement field. We analyse the hyperbolicity of the leading-order system through its eigenvalues and eigenvectors, establishing conditions for hyperbolicity within a restricted deformation domain. The model is solved using an explicit finite-volume method to investigate gravitational settling and wave propagation. We derive analytical expressions for the longitudinal and transverse wave speeds and highlight their dependence on depth. Finally, the predictions of the continuum model are compared with existing discrete-element simulations. This formulation allows several extensions, including more complex packing geometries, contact laws, and large-deformation regimes in which the contact network can evolve.
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BibTeX
@misc{cominelli2026strongly,
title = {{Strongly non-linear continuous model for wave propagation in 2D granular media: derivation and hyperbolicity analysis}},
author = {Sebastiano Cominelli and Régis Cottereau and Sergey Gavrilyuk and Bruno Lombard and Fabien Petitpas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31738},
eprint = {2610.31738},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31738},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Strongly non-linear continuous model for wave propagation in 2D granular media: derivation and hyperbolicity analysis
AU - Sebastiano Cominelli
AU - Régis Cottereau
AU - Sergey Gavrilyuk
AU - Bruno Lombard
AU - Fabien Petitpas
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.31738
UR - https://arxiv.org/abs/2610.31738
AB - In this work, we develop a nonlinear continuum model for wave propagation in two-dimensional granular materials, building on key ideas from Nesterenko's work. Starting from the Hertzian contact law and neglecting contact friction, we derive an effective continuum description through homogenization, assuming spherical, monodisperse grains, local periodicity, and the Cauchy--Born hypothesis. The resulting strain-gradient model depends on both the first and second gradients of the displacement field. We analyse the hyperbolicity of the leading-order system through its eigenvalues and eigenvectors, establishing conditions for hyperbolicity within a restricted deformation domain. The model is solved using an explicit finite-volume method to investigate gravitational settling and wave propagation. We derive analytical expressions for the longitudinal and transverse wave speeds and highlight their dependence on depth. Finally, the predictions of the continuum model are compared with existing discrete-element simulations. This formulation allows several extensions, including more complex packing geometries, contact laws, and large-deformation regimes in which the contact network can evolve.
ER -