Refraction of Shear Zones in Granular Materials

We study strain localization in slow shear flow focusing on layered granular materials. A heretofore unknown effect is presented here. We show that shear zones are refracted at material interfaces in analogy with refraction of light beams in optics. This phenomenon can be obtained as a consequence of a recent variational model of shear zones. The predictions of the model are tested and confirmed by 3D discrete element simulations. We found that shear zones follow Snell’s law of light refraction.

Figure 1

(a) Schematic view showing the shear cell and the refraction of the shear zone. Dense granular material is sheared between two half-cylinders. The shear direction is indicated by the arrows. (b) A snapshot of the simulation. The upper light-gray (yellow) material and lower dark-gray (blue) material have different frictional properties (blue beads have stronger friction). If the velocity of any bead drops below half of the external shear velocity, it is overpainted by dark gray (red). Despite the fluctuations of single bead velocities, a dark (red) zone appears in the bulk that separates the two moving parts of the system. (c) The spatial distribution of the shear rate which shows the structure of the shear zone. Lighter colors represent stronger deformations (orange means the largest shear rate, followed by red and blue, while black means almost no deformation). The straight line indicates the material interface.

Figure 2

(a) ${\varphi }_1$, (b) ${\varphi }_2$, and (c) the relative index of refraction measured for shear zones in computer simulations. The horizontal axis show the tilt angle of the interface between the two materials. Data points denoted by triangles, squares, and circles stand for systems $A$, $B$ and $C$, respectively. The pairs of microscopic friction that are used in these systems are indicated in the figure. The straight lines show the theoretically predicted values (the upper line for systems $A$ and $B$, the lower one for system $C$).

Figure 3

Effective friction versus microscopic friction. The open circles correspond to the materials used in systems $A$, $B$, and $C$.