Incremental non linearity in phenomenological and multi-scale constitutive relations

Throughout this chapter we have attempted to discuss one outstanding feature of the constitutive behavior of granular materials, namely their incrementally nonlinear character. For this purpose, two types of models were considered: the first one is the incrementally nonlinear model, which is essentially a phenomenological model; the second one is the micro-directional model, built from a multi-scale approach. These two models have clearly pointed out the incrementally nonlinear character of granular materials. It is quite remarkable that these two models, which are basically different in their approaches, highlight both this outstanding aspect of granular masses. In addition, the micro-directional model, which is micro-mechanically based, gives rise to an interpretation of the origin of this property; it appears that this feature (as far as the rolling motion of particles can be omitted) is related to the dependence of the mechanical state of each contact (in the elastic or plastic regime) with respect to the direction of local loading. We should emphasize that the rolling of particles, responsible to a large extent for the so-called buckling effect [Oda, 1982], could constitute another micro-structural source of incremental non linearity. If this property can be finally regarded as a basic ingredient of the incrementally nonlinear model, it stands rather as a consequence of the description of contacts in the case of the micro-directional model.