## Coulombian plastic friction: a world of bifurcations

### Félix Darve

Grenoble Institut Polytechnique

Like turbulence in fluid mechanics is giving rise to various kinds of bifurcations, it seems that coulombian friction is inducing also different classes of instabilities, leading to various failure modes. The most known example of media involving plastic friction of Coulomb's type is probably the one of granular materials, whose mechanical behaviour is essentially governed by the intergranular friction. In plasticity theory, these materials are characterised by a "non-associate" behaviour, what makes the elasto-plastic tensor non-symmetric. From a mathematical point of view, it is well recognised that the singularities of a non-symmetrical matrix are a lot more varied than for symmetrical matrices (related to an "associate" behaviour, corresponding to the usual metal plasticity).

It will be shown that, indeed, some instabilities can appear strictly inside the plastic limit surface for some kinematically constrained loading paths. The most famous example is certainly the so called "undrained" (isochoric) triaxial loading path on loose sands, leading to liquefaction phenomenon. A proper general criterion for all divergence instabilities will be proposed: "the second order work criterion", related to the loss of positive definitiveness of the elasto-plastic matrix. The link between this criterion and bifurcation theory will be shown through a regime transition from a quasi-static evolution to a sudden dynamic one. The conditions for such a transition will be examined.

Some phenomenological analyses by elasto-plastic relations and some discrete element simulations with the code "YADE", developed in Grenoble, will be presented to show the existence of a bifurcation stress domain and of some "instability cones", corresponding to the "isotropic cones" of the quadratic form associated to the elasto-plastic matrix. The influence of perturbations on the bifurcation states is emphasized.