  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 et al. on isotropic and hexagonal packings of
photoelastic grains can be fit 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.

Soft Condensed Matter