Shear zone refraction and deflection in layered
Tamás Börzsönyi, Tamás Unger and Balázs Szabó
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
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
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
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
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.
Click on the image to see
the movie taken during shearing.
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
Shear zone refraction
Click on the image below to see a movie
of the subsequent layers from top to bottom during removing.
Click on the image below to
see a movie of the subsequent layers from top to bottom during removing.
The computer simulation is
performed based on the fluctuating narrow band model [Phys.
Rev. Lett., 92, 214301
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
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.
The experimental results were obtained using a straight
split bottom cell (see picture). The internal cross section had
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.
(click on the image to
Here we present further
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,
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