

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.

In a realistic threedimensional setup, we simulate the slow
deformation of idealized granular media composed of spheres undergoing
an axisymmetric triaxial shear test. We follow the selforganization
of the spontaneous strain localization process leading to a shear band
and demonstrate the existence of a critical packing density inside
this failure zone. The asymptotic criticality arising from the dynamic
equilibrium of dilation and compaction is found to be restricted to
the shear band, while the density outside of it keeps the memory of
the initial packing. The critical density of the shear band depends on
friction (and grain geometry) and in the limit of infinite friction it
defines a specific packing state, namely the dynamic random loose
packing.

When granular materials (like sand) deform, strain is often localized to
sliding planes called shear zones.
We found a new effect for shear zones that are created in layered granular
materials. When two materials with different frictional properties are
layered on top of each other, shear zones are refracted at the
interface[1,2]. The phenomenon is in complete analogy with the
refraction
of light. The angle of refraction follows Snell's law from geometric
optics.
The effect of refraction is tested by discrete element
simulations based on the algorithm of Contact Dynamics. We analyzed
slow shear flow of 100000 spherical grains
confined in a cylindrical drum. The cylinder is cut in two along the
axis and each half slides along the axis in opposite
directions which leads to the formation of a shear zone. The
simulations confirmed the phenomenon and also the law of
refraction[2].
A recent model of shear zones can account for the effect
refraction. According to this model, shear zones are optimal in the
sense
that their shapes correspond to the least possible rate of energy
dissipation[3]. This approach was first applied
for a modified Couette
geometry to describe the open and closed shapes of shear zones. Within
the framework of this model the effect of refraction can be understood as
follows: the selection principle of minimum dissipation, when applied
at material interfaces, has exactly the form of Fermat's principle
of
optics[2]. Only, in case of shear zones, the
effective friction coefficient
plays the role of the index of refraction. Based on the same selection
principles the same laws can be derived for the angle of refraction. Thus
Snell's law turns out to be valid also for shear zones in granular media.
[1]
A little light reading;
Nature Physics, 3, 76 (2007)
[2]
Refraction of shear zones in granular materials,
pdf
Phys. Rev. Letters, 98 , 018301 (2007),
[3]
Shear band formation in granular media as a variational problem,
pdf
Phys. Rev. Lett., 92, 214301 (2004).

Using threedimensional Distinct Element Method with spherical particles we simulated shear band formation of granular materials in axisymmetric triaxial shear test. The calculated threedimensinoal shear band morphologies are in good agreement with those found experimentally. We observed spontaneous symmetry braking strain localization provided it was allowed by the boundaries. If the symmetry was enforced, we found strain hardening. We discuss the formation mechanism of shear bands in the light of our observations and compare our results with high resolution NMR experiments.

We performed computer simulations based on a twodimensional
Distinct Element Method to study granular systems of magnetized
spherical particles. We measured the angle of repose and the
surface roughness of particle piles, and we studied the effect
of magnetization on avalanching.
The movies which can be downloaded from this page
are recordings from our simulations.



f = 0,
[click here for larger image]

f = 3,
[click here for larger image]

f = 5,
[click here for larger image]

100_b_visu.mpg,
38MB, higher quality

103_b_visu.mpg,
47MB, higher quality

105_b_visu.mpg,
54MB, higher quality

100_b_visu_s.avi,
14MB, lower quality

103_b_visu_s.avi,
15MB, lower quality

105_b_visu_s.avi,
13MB, lower quality

  



f = 7,
[click here for larger image]

f = 16,
[click here for larger image]

f = 24,
[click here for larger image]

107_b_visu.mpg,
61MB, higher quality

116_b_visu.mpg,
74MB, higher quality

124_b_visu.mpg,
81MB, higher quality

107_b_visu_s.avi,
13MB, lower quality

116_b_visu_s.avi,
12MB, lower quality

124_b_visu_s.avi,
11MB, lower quality

The particles were magnetized by a vertical external field.
The particle magnetization was modeled by magnetic dipoles.
There was no coupling between the particle rotation states and the
dipole orientations, i.e. the particles could rotate freely while
their magnetic dipoles remained vertical at any time.
The particleparticle and the
particlewall interactions were calculated, and the system
was integrated based on the Newton equations.
The Hertz contact model with appropriate
dumping and Coulomb sliding friction was used.
The particles touching the base wall stuck to the wall.
The particles were placed gently on the pile (i.e. with zero
impact velocity).
We found a difference in
avalanche formation at small and at large
interparticle force ratios f, defined as the magnetic dipole
interaction at contact divided by the gravitational force.
For f < 7 small vertical chains follow each other
at short times (granular regime), while for
f > 7 the avalanches are typically formed
by one single large particle cluster (correlated regime).
The transition is not sharp.
The movies show also the normal contact forces with lines connecting
the centers of neighboring particles. The thickness of these lines
is proportional to the corresponding contact forces.
The AVI files are created with the DivX 5.0.5 codec.


