Three dimensional multi scale bifurcation analysis of granular media

This paper deals with Hill's bifurcation criterion, which is very well suited to describe various failure modes in granular media. The first part of this paper is dedicated to the analytical and nu- merical investigation of this criterion by considering phenomenological constitutive relations: the incrementally piece-wise linear and non-linear relations proposed by Darve. The 3D bifurcation domain and 3D cones of unstable directions are given for these two relations. A similar analysis is performed with a micro mechanical model in the second part of the article. Finally a qualitative comparison of the results obtained with these two different approaches leads us to some key conclusions about this material instability criterion, which is studied for the first time for general 3D conditions, by considering these non conventional constitutive relations.