Critical packing in granular shear bands

In a realistic three-dimensional setup, we simulate the slow deformation of idealized granular media composed of spheres undergoing an axisymmetric triaxial shear test. We follow the self-organization of the spontaneous strain localization process leading to a shear band and demonstrate the existence of a critical packing density inside this failure zone. The asymptotic criticality arising from the dynamic equilibrium of dilation and compaction is found to be restricted to the shear band, while the density outside of it keeps the memory of the initial packing. The critical density of the shear band depends on friction (and grain geometry) and in the limit of infinite friction it defines a specific packing state, namely the dynamic random loose packing.

Figure 1

(Color online) (a) Grains placed between two horizontal platens and surrounded by an elastic membrane were subjected to a vertical load and a lateral confining pressure. (b) The membrane was modeled with overlapping spheres [20].

Figure 2

(Color online) Example of shear intensity and volume fraction histogram maps (${\eta }_0=0.641$, $\mu =0.5$). The shear intensity is measured in arbitrary units. In (a) and (b) the occurrences (scaled to [0,1]) are encoded with the color scale shown at the top. In (a) the white curve marks the shear intensity threshold $S_{\mathit{TH}}$. In (b) it marks the shear band volume fraction ${\eta }_{SB}$. The dotted and dashed vertical lines in (a) and (b) at 4.4% and 16% axial strain mark the position of the histograms (c), (d) and (e), (f), respectively.

Figure 3

(Color online) Volume fraction measured in high shear intensity regions (a) and globally (b) as a function of the axial strain. The different lines correspond to different initial packing densities ${\eta }_0$ (see Table ) decreasing from top to bottom. The two panels use the same notations.

Figure 4

(Color online) Critical packing of shear bands as a function of friction measured at 20% axial strain. The fitted curve shows ${\eta }_c\left(\mu \right)={\eta }_c^{\infty }-({\eta }_c^{\infty }-{\eta }_c^0)\exp (-\mu ∕{\mu }_c^0)$, where ${\eta }_c^0=0.637$, ${\eta }_c^{\infty }=0.578$, and ${\mu }_c^0=0.23$.