Divergence of Viscosity in Jammed Granular Materials: A Theoretical Approach

A theory for jammed granular materials is developed with the aid of a nonequilibrium steady-state distribution function. The approximate nonequilibrium steady-state distribution function is explicitly given in the weak dissipation regime by means of the relaxation time. The theory quantitatively agrees with the results of the molecular dynamics simulation on the critical behavior of the viscosity below the jamming point without introducing any fitting parameter.

Figure 1

The density dependence of (a) the shear viscosity ${\tilde{\eta }}^{\prime }$ and (b) the granular temperature. The result of the theory is shown in the (blue) solid line, while that for the MD is shown in (red) triangles, (blue) diamonds, and (green) rectangles for ${\dot{\gamma }}^{*}={10}^{-3},{10}^{-4},{10}^{-5}$. (Inset) The log-log plots for the results near ${\phi }_J=0.639$.

Figure 2

The density dependence of the relaxation time, ${\tilde{\tau }}_{\text{rel}}=\epsilon {\tau }_{\text{rel}}^{*}$. The result for the shear rate ${\dot{\gamma }}^{*}={10}^{-4}$ is shown in the (blue) solid line for the theory and (blue) diamonds for the MD simulation.