Local anisotropy in globally isotropic granular packings

We study local stresses and elastic moduli defined at various coarse-graining scales, $R$, and volume fractions, $\phi$, in a two dimensional (2D) mixture of frictionless granular particle packings. We measure the average deviatoric stress normalized by pressure, $\tau/p$, and normalized anisotropic component of the shear modulus, $\delta\mu/\mu$, as a function of $R$. As the packings are prepared isotropically, both $\tau/p$ and $\delta\mu/\mu$ vanish at large $R$. However, in local regions, where single force chains dominate, the response can be quite anisotropic. We show that $\tau/p$ exhibits two power-law regimes in $R$ with a cross-over that is only weakly dependent on $\phi$. In contrast, $\delta\mu/\mu$, behaves like a pure power law up to $R\sim640D$ (where $D$ is the characteristic particle diameter) at all $\phi$.