We report on two-dimensional computer simulations of frictionless granular packings at various area fractions $\varphi$ above the jamming point ${\varphi }_c$. We measure the anisotropy in coarse-grained stress ${\epsilon }_s$ and shear modulus ${\epsilon }_m$ as functions of coarse-graining scale, $R$. ${\epsilon }_s$ can be collapsed onto a master curve after rescaling $R$ by a characteristic length scale $\xi$ and ${\epsilon }_s$ by an anisotropy magnitude $A$. Both $A$ and $\xi$ accelerate as $\varphi \to {\varphi }_c$ from above, consistent with a divergence at ${\varphi }_c$. ${\epsilon }_m$ shows no characteristic length scale and has a nontrivial power-law form, ${\epsilon }_m\sim R^{-0.62}$, over almost the entire range of $R$ at all $\varphi$. These results suggest that the force chains present in the spatial structure of the quenched stress may be governed by different physics than the anomalous elastic response near jamming.

• Received 3 May 2011

DOI:https://doi.org/10.1103/PhysRevLett.107.268001