Keywords: bipotential, contact dynamics, discrete element method.

In this paper an improved 3D Discrete Element Method is proposed. It is based on the bipotential method developed by de Saxcé [1] and applied to granular systems by Fortin in 2D [2]. A collection of rigid particles is considered during the motion of which contacts can occur or break. The energy dissipated during the collisions is taken into account by means of restitution coefficients [3]. The dry friction is modelled by Coulomb's law which is typically non-associated: during the contact, the sliding vector is not normal to the friction cone. The non-associativity of the constitutive law is responsible for numerical difficulties.

The main feature of the proposed algorithm is to overcome this kind of difficulties by means of the bipotential theory [1]. There are two goals: it leads to a fast and easily implemented predictor-corrector scheme involving just an orthogonal projection onto the friction cone [1] and it enables the use of a convergence criterium based on an error in the constitutive law [4]. The application shows the convergence and the robustness of the algorithm. In Figure 1 an example of the flow of 500 spheres in a hopper is shown.

1

de Saxcé G. and Feng Z.-Q., "The bipotentiel method: a constructive approach to design the complete contact law with friction and improved numerical algorithms", Mathl. Comput. Modelling, vol. 28, 4-8, pp. 225-245, 1998. doi:10.1016/S0895-7177(98)00119-8

2

Fortin J., "Simulation numérique de la dynamique des systèmes multi-corps appliquée aux milieux granulaires", thèse, Université de Lille I, janvier 2000.

3

Moreau J.-J, "Some numerical methods in multibody dynamics : application to granular materials", Eur. J. Mech, A/Solids, vol. 13, 4 suppl., pp.93-114, 1994.

4

Ladevèze P., "Mécanique non linéaire des structures : nouvelle approche et méthodes de calcul non incrémentales", Hermès, Paris, 1996.

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