Quasi-Static Deformation of a Granular System with a Regular Arrangement of Particles

In order to study the growth of heterogeneity in granular systems, we investigate the quasi-static deformation of monodisperse particle systems with regular initial arrangements. We construct a linearized model by limiting our consideration to infinitesimal deformations. Assuming Coulomb friction, we introduce new variables to express the slippage distances of particles in contact. The results of our numerical simulations show that heterogeneities appear through a fingering-like instability and develop into several microscopic shear zones. The equations in our model can be reduced to simpler equations in the limit of small tangential interactions using a perturbative analysis. In the long wavelength limit, the equations are analogous to models of Laplacian growth, such as the dielectric breakdown problem. We investigate the instability of critical stress states and the competitive growth of microscopic shear zones.