Effects of friction and disorder on the quasistatic response of granular solids to a localized force

The response to a localized force provides a sensitive test for models of stress transmission in granular solids. Elasto-plastic models, traditionally used by engineers, have been challenged by theories and experiments that suggest a wavelike (hyperbolic) propagation of the stress, as opposed to the elliptic equations of static elasticity. Simulations of two-dimensional granular systems subject to a localized external force have been employed to examine the nature of stress transmission in these systems as a function of the magnitude of this force, the frictional parameters, and degree of disorder. The results indicate that in large systems (as considered by engineers) the response is close to that predicted by isotropic elasticity, whereas for small systems (or strongly forced ones) it is strongly anisotropic. In the latter case the applied force induces changes in the contact network accompanied by frictional sliding and gives rise to hyperboliclike stress propagation. The larger the static friction, the more extended the range of forces for which the response is elastic, and the smaller the anisotropy. Increase in the degree of polydispersity (in the studied range, up to 25%) decreases the range of elastic response. This paper is an extension of a previously published Letter [C. Goldenberg and I. Goldhirsch, Nature (London) 435, 188 (2005)].