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Discrete Element Methods for the Study of Unconsolidated Granular Materials through Computational ModelingA numerical code based on Discrete Element Modeling (DEM) has been developed for computing the response of particulate materials such as unconsolidated granular materials, concentrated emulsions or fractured rocks. The systems under study in this project may be considered as ``unconsolidated'' aggregates of particulate materials which acquire stiffness as a result of the applied stress. The constitutive properties of such materials, which also include cohesionless soils, industrial powders, droplets, foams and colloidal suspensions, are determined mainly by interparticle contact properties such as friction and viscoelastic response. In DEM the dynamical evolution of the particles is obtained by solving Newton's equation for an assembly of spherical particles constrained by a given external pressure. In the case of granular materials, the particles are modeled as viscoelastic spheres of millimeter size with different coefficients of friction. Interparticle forces are computed using the principles of contact mechanics. Full details are given in previous papers of the PI. The normal force has the typical 3/2 power law dependence on the overlap between two spheres in contact (Hertz force), while the transverse force depends linearly on the shear displacement between the spheres, as well as on the value of the normal displacement (Mindlin tangential elastic force). An extra sliding condition is also added, according to Coulomb friction. Viscous dissipative forces are included in the force acting between viscoelastic particles, proportional to the relative normal and tangential velocity of the particles. This model is the starting point for our simulations. More complicated interactions, such as rolling friction or other forms of tangential elasticity, will be incorporated at a later stage.
ResultsA numerical code has been developed and tested for the two-dimensional case and, preliminary, for three dimensions. The code is written in Fortran 90 and is ready to be ported to a parallel architecture.
Two dimensional caseWe tested the code by performing DEM simulations of systems of the order of $10000$ particles in serial computers. Our calculations begin with a numerical protocol designed to mimic the experimental procedure used to prepare dense packed granular materials. The simulations begin with a gas of spherical particles located at random positions in a periodically repeated cubic cell of side $L$. At the outset, a series of strain-controlled isotropic compressions and expansions are applied until a volume fraction slightly below the critical density is reached. The system is then compressed and extended slowly until a specified value of the stress and volume fraction is achieved at static equilibrium. The structure of packings depends in detail on the forces acting between the grains. In Figure 1 we show a typical micromechanical structure of a packing of 10000 spherical disks. We plot the forces between the particles as lines joining the center of the particle. The width of the lines represent the absolute value of the interparticle forces. We show the case of a packing of identical disks of 1 mm diameter (Fig. 1a) and a mixture composed of 5000 disk of 1 mm and 5000 disks of 1.5 mm (Fig. 1b). The degree of crystallization and the difference in the packing structure is obvious from the pictures. We have further tested the code by conducting a simple numerical experiment which measures the response of the granular packing to a infinitesimal perturbation consisting of moving a grain in the center of the packing. We monitor the stress relaxation as a function of time. Figure 2 show snapshots of our simulations where we plot the stress as a function of time. The color blue corresponds to particle under compression (negative stress) and purple corresponds to particles under extension (positive stress). A movie of a simulation of the propagation os stress is available here for frictional spheres and frictionless spheres .
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Three-dimensional resultsWe have also tested a preliminary version of the three-dimensional code by developing a thermodynamic formulation for slow granular matter with computer simulations. The preliminary results of our first year of research show very promising results. Using the code developed under this grant we have performed a preliminary numerical study of a diffusion-mobility test for granular materials. This work was published in Nature and has received an enthusiastic News & Views Editorial commentary by B. Behringer.
CollaboratorsJorge Kurchan, ESPCI, Paris, Nicolas Gland, ENS, Paris, Eugene Mananga, Won-Geun Kim. |
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