Shear Band Formation in Granular Media as a Variational Problem

Strain in sheared dense granular material is often localized in a narrow region called the shear band. Recent experiments in a modified Couette cell provided localized shear flow in the bulk away from the confining walls. The nontrivial shape of the shear band was measured as the function of the cell geometry. First, we present a geometric argument for narrow shear bands that connects the function of their surface position with the shape in the bulk. Assuming a simple dissipation mechanism, we show that the principle of minimum dissipation of energy provides a good description of the shape function. Furthermore, we discuss the possibility and behavior of shear bands that are detached from the free surface and are entirely covered in the bulk.

Figure 1

Setup geometry. The rotating (white) and the stationary (gray) parts induce shear flow in the granular material held by the container.

Figure 2

The symbols are experimental data showing the shear zone radius $r$ measured in the bulk at height $h$, taken from 10. +, ×, and ○ correspond filling heights 25, 37, and 49 mm, respectively; $R_s$ is 95 mm. The rectangle in the middle shows the estimated errors in both directions for all data [12] (plotted only for one data point). The solid line is the experimentally found fit curve for the surface positions. The dashed lines are the calculated bulk positions based on the solid curve.

Figure 3

Results obtained from the variational principle. Symbols show bulk profiles; from top to bottom $H/R_s=0.15$, 0.25, 0.35, 0.45, and 0.55, respectively. The two lines denote the surface positions as the function of the total height. The dashed line comes from our model; the solid line is the experimental fit function.

Figure 4

Open and closed shear bands with cylindrical symmetry. (a) upper radius, (b) bulk profiles for several filling heights ($0<H/R_s<7$), and (c),(d) height of the shear band $h_{\mathrm{top}}$.

Figure 5

Hysteresis emerging in the variational problem. Upper radius (a) and height (b) of the shear band in units of $R_s$. The dashed line denotes the transition of the global minimum.