Shear zones in granular materials: Optimization in a self-organized random potential

We introduce a model to describe the wide shear zones observed in modified Couette cell experiments with granular material. The model is a generalization of the recently proposed approach based on a variational principle. The instantaneous shear band is identified with the surface that minimizes the dissipation in a random potential that is biased by the local velocity difference and pressure. The apparent shear zone is the ensemble average of the instantaneous shear bands. The numerical simulation of this model matches excellently with experiments and has measurable predictions.

Figure 1

(a) The experimental arrangement and calculated shear bands. The dotted circle is the bottom split; the definition of the most important quantities are noted in the figure. The gray scaling on all plots is proportional to the logarithm of the occurrence probability at the given point. $R_s=150$ for (a)–(c) and $H=115$, 100, and 150 for plots (a), (b), and (c), respectively.

Figure 2

The surface positions of the shear zones in normal scale (a) and in log-log scale (b). The solid line is from [5]; the dashed line is the experimental [3] curve. Symbols were obtained for systems with $R_s=90$ and $\alpha =0$, 0.43, 0.5, and 0.6 for +, ×, 엯, and ⊡, respectively.

Figure 3

The surface width of the shear zones. The solid line is ${(H∕R_s)}^{2∕3}$. The lower curves are for $\alpha =0.43$ with $R_s=90$, 150, 300, and 600 with symbols +, ×, ∗, and ⊡, respectively. The upper curves are for $\alpha =0$ with $R_s=150$ and 300 with symbols ◇ and ▵, respectively. The breakdown of scaling close to the transition is shown in the inset for $\alpha =0.43$.

Figure 4

The scaled center position [curves (a)] and width [curves (b)] of the closed shear zones. The solid line is the calculated position from [5]. Data points (+, ×, 엯, ⊡) connected with dashed lines have $R_s=90$, 150, 300, 600 and $\alpha =0.43$.

Figure 5

(Color online) The order parameter $m$ for $\alpha =0$. System size $R_s=15,35,75,90,150,300,600$ from right to left, respectively (+, ×, ∗, ⊡, ◇, ▵, +). Experimental data from [6] are shown with solid symbols of ☉, ▿, and ⊡ for $R_s=45$, 65, and $95\mathrm{cm}$, respectively. The inset shows the width of the tranisition versus $R_s$ for + experiments ☉. The dashed line has a slope of 0.5.