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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 three-dimensional setup, we simulate the slow deformation of idealized granular media composed of spheres undergoing an axisymmetric triaxial shear test. We follow the self-organization 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 three-dimensional Distinct Element Method with spherical particles we simulated shear band formation of granular materials in axisymmetric triaxial shear test. The calculated three-dimensinoal 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 two-dimensional 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 f = 3 f = 5
  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 f = 16 f = 24
  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 particle-particle and the particle-wall 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.