Experimental Demonstration of Snell’s Law for Shear Zone Refraction in Granular Materials

We present experiments on slow shear flow in granular materials. Under appropriate conditions shear localizes in narrow shear zones. We demonstrate that when the shear zone crosses a material boundary, it refracts in accordance with Snell’s law in optics—an effect first found in simulations [Phys. Rev. Lett. 98, 018301 (2007)]. The shear zone is the one that minimizes the dissipation rate upon shearing, i.e., a manifestation of the principle of least dissipation. We have prepared the materials as to form a granular lens. Shearing through the lens is shown to give a very broad shear zone, which corresponds to fulfilling Snell’s law for a continuous range of paths through the cell.

Figure 1

Sketch of the experimental setup. To the left: the geometry of the shear cell. Upper right: top view of a cut through the cell. Lower right: the filling of material makes color layers.

Figure 2

Refraction at the material boundary. Its orientation $\varphi$ is (a) 22°, (b) 36°, (c) 45°, (d) 54°, and (e) 67°. The images are from layer 7 in (b)–(d) and layer 5 in (a) and (e). The shear zone clearly deviates from the shortest path in all cases.

Figure 3

The angles of incidence (defined in Fig. 1) are shown as a function of depth for the $\varphi =45°$ experiment. Here, about seven layers show roughly the same behavior. There is a crossover towards the bottom to meet with the imposed boundary conditions. The error bars reflect the variation within each layer (no error bar means that only one image was obtained in that layer).

Figure 4

For each experiment the material index ratio is estimated. For the $\varphi =22°$ experiment there was problems with slip at the material boundary and the value is based on one layer only. Otherwise, the error bars reflect the fluctuations of the data.

Figure 5

Refraction at two material boundaries. (a) A wedge of glass beads in sand and (b) a wedge of sand between glass beads. For both wedges the shear zone was sharply defined. (c)–(f) A “granular lens” of sand showed a very wide shear zone. The pictures are all from within layer ten at the positions (measured from the top of the layer) : (c) 5, (d) 9, (e) 13, and (f) 18 mm. Since the position of the color transition is dependent of depth, surface height variations may explain curved lines between yellow and black.

Figure 6

The shearing of one color layer in the lens experiment. The system is translation invariant and all motion is parallel to the axis. Here, the shearing is not localized to a narrow zone, but takes place on a set of equivalent “rays” between the two cuts in the cylinder wall. The next layer above is sheared in the same manner, and has the colors grey and black. Thus, when looking at a horizontal cross section through the cell after shearing, the location of the color transition will depend on the exact vertical position of the cut within the layer. Two cross sections are illustrated on the right.