Nonlinear elastic stress response in granular packings.

We study the nonlinear elastic response of a two-dimensional material to a localized boundary force, with the particular goal of understanding the differences observed between isotropic granular materials and those with hexagonal anisotropy. Corrections to the classical Boussinesq result for the stresses in an infinite half space of a linear, isotropic material are developed in a power series in inverse distance from the point of application of the force. The breakdown of continuum theory on scales of order of the grain size is modeled with phenomenological parameters characterizing the strengths of induced multipoles near the point of application of the external force. We find that the data of Geng [Phys. Rev. Lett. 87, 035506 (2001)] on isotropic and hexagonal packings of photoelastic grains can be fitted within this framework. Fitting the hexagonal packings requires a choice of elastic coefficients with hexagonal anisotropy stronger than that of a simple ball-and-spring model. For both the isotropic and hexagonal cases, induced dipole and quadrupole terms produce propagation of stresses away from the vertical direction over short distances. The scale over which such propagation occurs is significantly enhanced by the nonlinearities that generate hexagonal anisotropy.