Two Scale Model (FEM-DEM) For Granular Media
The macroscopic behavior of granular materials, as a consequence of the interactions of individual grains at the micro scale, is studied in this paper. A two scale numerical homogenization approach is developed. At the small-scale level, a granular structure is considered. The Representative Elementary Volume (REV) consists of a set of N polydisperse rigid discs (2D), with random radii. This system is simulated using the Discrete Element Method (DEM) – molecular dynamics with a third-order predictor-corrector scheme. Grain interactions are modeled by normal and tangential contact laws with friction (Coulomb’s criterion). At the macroscopic level, a numerical solution obtained with the Finite Element Method (FEM) is considered. For a given history of the deformation gradient, the global stress response of the REV is obtained. The macroscopic stress results from the Love (Cauchy-Poisson) average formula including contact forces and branch vectors joining the mass centers of two grains in contact. The upscaling technique consists of using the DEM model at each Gauss point of the FEM mesh to derive numerically the constitutive response. In this process, a tangent operator is generated together with the stress increment corresponding to the given strain increment at the Gauss point. In order to get more insight into the consistency of the two-scale scheme, the determinant of the acoustic tensor associated with the tangent operator is computed. This quantity is known to be an indicator of a possible loss of uniqueness locally, at the macro scale, by strain localization in a shear band. The results of different numerical studies are presented in the paper. Influence of number of grains in the REV cell, numerical parameters are studied. Finally, the two-scale (FEM-DEM) computations for simple samples are presented.
KeywordsGranular materials Computational homogenization Discrete element method Finite element method 2D deformations
- M.P. Allen, D.J. Tildesley, Computer Simulations of Liquids (Clarendon Press, Oxford, 1989)Google Scholar
- G. Bilbie, C. Dascalu, R. Chambon, D. Caillerie, Micro-fracture instabilities in granular solids, in Bifurcations, Instabilities, Degradation in Geomechanics, ed. by G. Exadaktylos, I. Vardoulakis (Springer, Berlin/Heidelberg, 2007), pp. 231–242Google Scholar
- J.R. Rice, The localization of plastic deformation, in Theoretical and Applied Mechanics, ed. by W.T. Koiter (North-Holland Publishing Company, Amsterdam, 1976), pp. 207–220Google Scholar