Why Effective Medium Theory Fails in Granular Materials

Experimentally it is known that the bulk modulus $K$ and shear modulus $\mu$ of a granular assembly of elastic spheres increase with pressure $p$ faster than the $p^{1/3}$ law predicted by effective medium theory (EMT) based on Hertz-Mindlin contact forces. To understand the origin of these discrepancies, we perform numerical simulations of granular aggregates under compression. We show that EMT can describe the moduli pressure dependence if one includes the increasing number of grain-grain contacts with $p$. Most important, the affine assumption (which underlies EMT), is found to be valid for $K\left(p\right)$ but breaks down seriously for $\mu \left(p\right)$.