Jammed granular packing
Once the volume fraction exceeds the critical density, granular materials are jammed (Fig. 1(c)), where irreversible rearrangements of granular particles determine their mechanical responses to deformations.
Approach: To understand the anomalous mechanical responses of granular materials, we have proposed a stochastic method from a point of view of statistical mechanics [10]. In a microscopic scale, the mechanical response is determined by a random force-chain network (Fig. 1(d)), where any macroscopic quantities (e.g. stress tensor) can be deduced from the probability distribution function (PDF) of inter-particle forces. During quasi-static deformations, the force-chains are randomly reconstructed by the rearrangements, where some bonds in the force-chains are broken or generated. Therefore, we generalize the force-chain network by using the Delaunay triangulations (Fig. 1(d)) to include not only the particles in contacts, but also the nearest neighbors without contacts. Then, we introduce a Master equation for the PDFs, where transition rates in the Master equation capture all kinds of reconstruction of force-chains including the breakage and generation of contacts. We simulate isotropic (de)compressions of jammed granular materials by MD simulations to measure the transition rates from scatter plots of inter-particle forces (Fig. 3(a)). We find that the transition rates are symmetric around mean forces and self-similar against a dimensionless parameter, defined as the ratio between an applied strain and a distance from the critical density (Fig. 3(b)). We also find that the deviation from affine deformation and the width of transition rates are linearly scaled by the dimensionless parameter, which implies that the strength of non-affine deformations and spatial correlations of inter-particle forces increase as the system approaches the jamming transition. The evolution of the PDF is well described by the Master equation (Fig. 3(c)), where the irreversible mechanical responses of jammed granular materials are also well reproduced by the Master equation (Fig. 3(d)). A part of our results, e.g. transition rates in the Master equation, have already been validated by experiments of wooden cylinders (this work is the collaboration with Joseph Fourier University and is still in progress).
Method overview: We used MD simulations of bidisperse frictionless granular particles and polydisperse frictional granular particles, where the different size ratio is used to avoid crystallization. The applicant made all the numerical codes and drew the 2D-graphics (including force-chains) by using the PGPLOT libraries.