Bifurcation in granular materials: Micro-mechanical bases

It is now well established that for non-associated materials such as geomaterials, abroad domain exists, strictly within the plastic limit, where different failure modes can coexist. In particular, material instability as defined by Hill, related to the vanishing of the second-order work, can potentially occur. In this paper, the notion of loss of sustainability of a mechanical state in a granular assembly is investigated. The vanishing of the second-order work, defined on the macroscopic scale from tensorial variables, is shown to play a fundamental role in detecting the occurrence of this type of bifurcation. Then, we enlarge the debate by addressing this question from a micro-mechanical point of view. By considering that each contact between adjoining particles can be regarded as the fundamental constitutive unit of a grain assembly, the standard macroscopic second-order work defined from tensorial variables was established to coincide with the sum of the microscopic discrete second-order works defined on each contact with respect to a suitable frame. The microscopic second-order work can be computed at each contact as the scalar product of the incremental relative displacement with the incremental contact force between the two adjoining particles in contact. This equivalence of both formulations is of great interest because the microscopic formulation, implying local variables, can give rise to a micro-mechanical interpretation of the vanishing of the second-order work in granular materials.