Shear zone refraction and deflection in layered granular materials
Tamás Börzsönyi, Tamás Unger and Balázs Szabó
[Phys. Rev. E  80, 060302(R) (2009)]   download pdf

When granular materials deform under external stress the deformation is often localized into narrow regions. These shear zones  act as internal slip surfaces between solid-like blocks of the bulk. The formation of shear zones is a crucial deformation mechanism in fine powders, sand and soil (landslides). Geological faults are themselves large scale examples of shear zones. Here we study experimentally a recent theoretical prediction that shear zones behave in striking analogy with geometric optics  [T. Unger, Phys. Rev. Lett. 98, 018301 (2007)], and compare our results with numerical simulations. We show that shear zones alter their orientations when crossing media boundaries similarly to light refraction, i.e. Snell's law is valid, but here the frictional properties of the materials take the role of the optical index of refraction. We find that the refraction phenomenon also exists in the presence of gravity, i.e. under natural pressure conditions. In certain configurations we observe another effect, namely that shear zones can be deflected by the material interface.
In the present experiments we use glass beads for low friction material (seen as yellow at the sketches below) and corundum for high friction material (seen as brown at the sketches below). For both experiments the interface of the two materials was tilted. Shearing was performed by translating one of the L shaped sliders (cell wall) according to the red arrows. For both materials we used two samples with different colors, therefore four regions appear at the top surface as it is seen on the grayscale images taken from the top surface of the material. Two of these regions correspond to glass beads while the other two correspond to corundum. This way not only the interface of the two materials is visualized (oriented vertically on the images below) but the displacement profile is also directly seen, as it is illustrated with blue arrows. The strain at the top surface was optically detected during translation and is shown as a function of y for both measurements below the images. The shear zone is refracted or deflected inside the material as it is illustrated by the red line on the sketch. In a homogeneous material the zone would be vertically aligned in the center of the cell as it is illustrated with the dashed blue lines on the sketches. Measurements on the geometry of the shear zone are presented in the next section.
Shear zone refraction
Click on the image to see the movie taken during shearing.


  Shear zone deflection
Click on the image to see the movie taken during shearing.

3D reconstruction
After each experiment the displacement profile in the bulk d(y,z) was reconstructed by removing the top surface of the material carefully layer by layer. These profiles are presented in the images below for both refraction and deflection, together with sample images taken during the excavation process. The gradient of the displacement provides the local shear strain inside the material in the y-z plane which is shown at the plots below the displacement profiles. In the first case the zone, starting from the bottom, takes a short path towards the interface and by reaching the low friction region it changes direction abruptly and heads straight to the top. In the second case the zone starts in the low friction material it deflects to avoid the high friction region even if it takes a much longer path. Further measurements to explore the internal deformation using MRI are on the way.
Shear zone refraction

Click on the image below to see a movie of the subsequent layers from top to bottom during removing.

Shear zone deflection

Click on the image below to see a movie of the subsequent layers from top to bottom during removing.

Numerical simulation results
The computer simulation is performed based on the fluctuating narrow band model [Phys. Rev. Lett.,  92, 214301 (2004)Phys. Rev. E 75, 011305 (2007)]. According to this model the deformation occurs along the weakest sliding surface through the random medium. An ensemble average over the random realizations of the granular medium provides the shape and width of the shear zone. A cross section of the numerical system is shown below for two simulations corresponding to the two experimental systems.
(click on the images to magnify)

Verifying Snell's law

Numerous experiments and simulations have been performed to test whether Snell's law is valid for this system. As described above it is expected that the relative index of refraction is replaced by the ratio of the effective frictions. We estimated the ratio of the effective frictions by measuting the angle of repose for both materials and the experimentally obtained value was 1.63. We compare this number with the raio of the sines of the angles of incidence (see inset of graph b below). The ageement is very good for both experiments and simulations.

Experimental details

The experimental results were obtained using a straight split bottom cell (see picture). The internal cross section had dimensions of 4.2 cm x 4.5 cm. The shear cell included two 60 cm long L shaped sliders, one of which was slowly translated in experiments with total displacement between 5 and 6 cm. The internal walls of the sliders were covered by sandpaper to prohibit slip at the walls. Here the cell is filled with corundum doped with poppy seeds (tracers). The cell is not filled until the end sothat during translation the material is not pushed by the end walls. Translation is obtained by the slow rotation of the screws.

  Picture of the experimental setup:
(click on the image to magnify)

Further experimental results
Here we present further four configurations as illustrated on the images below. In the first two configurations we present the shear localization in a homogeneous material. In all geometries two measurements were performed (i) first the right slider was translated, (ii) second the left slider was traslated. Surface distorsions are presented in the four graphs below for each measurement (click on the image to magnify).

As it is seen on the graph, the shear-zone is wider for glass than for corundum (see first two graphs) which is a consequence of the difference in grain size (0.655mm vs 0.36mm). In the first two experiment the middle of the zone is in the middle of the cell as it is expected. In the third experiment the shear zone is shifted towards the left hand side as friction is lower in glass beads. As seen above in such a configuration the shear zone is not refracted but it is simply deviated. In experiment No. 4 the shear zone is shifted towards the right hand side at the surface corresponding to shear zone refraction.
By clicking on the images below movies corresponding to the experimental data can be seen.


corundum (doped with poppy seeds)

glass - corundum

corundum - glass

We acknowledge discussions with János Kertész and Ralf Stannarius.