Criticality of the "critical state" of granular media: Dilatancy angle in the tetris model

TL;DR

This paper investigates the critical behavior of the dilatancy angle in granular media using the tetris spin model, revealing a phase transition at a critical density and its relation to the medium's texture.

Contribution

It introduces a simple spin model to analyze dilatancy in granular media and establishes a critical density threshold with a mapping to directed percolation.

Findings

Existence of a critical density $ ho_c$ for dilatancy behavior.

Dilatancy angle exhibits strong anisotropy in random packings.

Dilatancy angle effectively characterizes the medium's texture.

Abstract

The dilatancy angle describes the propensity of a granular medium to dilate under an applied shear. Using a simple spin model (the ``tetris'' model) which accounts for geometrical ``frustration'' effects, we study such a dilatancy angle as a function of density. An exact mapping can be drawn with a directed percolation process which proves that there exists a critical density $\rho_c$ above which the system expands and below which it contracts under shear. When applied to packings constructed by a random deposition under gravity, the dilatancy angle is shown to be strongly anisotropic, and it constitutes an efficient tool to characterize the texture of the medium.