Theory of rigidity and numerical analysis of density of states of two-dimensional amorphous solids with dispersed frictional grains in the linear response regime

TL;DR

This paper develops a theoretical framework using the Jacobian matrix to analyze the rigidity and density of states in two-dimensional amorphous solids with frictional grains, aligning well with molecular dynamics results.

Contribution

It provides a novel theoretical expression for rigidity and density of states in frictional amorphous solids, including the effects of rotational and translational modes.

Findings

Rigidity matches molecular dynamics simulations.

Two distinct modes in the density of states for small tangential to normal stiffness ratio.

Rotational modes shift with changing stiffness ratio and contribute minimally to rigidity.

Abstract

Using the Jacobian matrix, we obtain theoretical expression of rigidity and the density of states of two-dimensional amorphous solids consisting of frictional grains in the linear response to an infinitesimal strain, in which we ignore the dynamical friction caused by the slip processes of contact points. The theoretical rigidity agrees with that obtained by molecular dynamics simulations. We confirm that the rigidity is smoothly connected to the value in the frictionless limit. For the density of states, we find that there are two modes in the density of states for sufficiently small $k_{T}/k_{N}$, which is the ratio of the tangential to normal stiffness. Rotational modes exist at low frequencies or small eigenvalues, whereas translational modes exist at high frequencies or large eigenvalues. The location of the rotational band shifts to the high-frequency region with an increase in $k_{T}/k_{N}$ and becomes indistinguishable from the translational band for large $k_{T}/k_{N}$. The rigidity determined by the translational modes agrees with that obtained by the molecular dynamics simulations, whereas the contribution of the rotational modes is almost zero for small $k_{T}/k_{N}$.