The incremental response of a stressed, anisotropic granular material: loading and unloading

In this paper, we investigate the incremental response of a transversely isotropic granular material through numerical simulations (Distinct Element Method) and a theoretical model. A granular material is idealized by a random aggregate made of elastic, identical, frictional particles. We consider an initial isotropic compression followed by a uni-axial deformation, at constant pressure. The regime of deformation of our interest is quite narrow and it encompasses shear strains small compared to the volume strain associated with the pressure. In this regime, the contact network is almost the same as in the initial, isotropic, state, and anisotropy is induced by the applied strain through the contacts. In numerical simulations, particles deform according to local force and moment equilibrium, given an applied strain. In the theory, we do something similar and we allow a pair of contacting particles to deform while satisfying force and moment equilibrium, approximately. An average expression of the first moment of the contact forces is employed to obtain the stiffness tensor AijklAijkl relating the increments in stress with the increments in total average strain. We determine the non-zero components of AijklAijkl in stressed, anisotropic, states. The results refer to two cases: (a) when the contact friction coefficient is the same as in the uni-axial compression; (b) when a relatively high-contact friction coefficient is introduced (e.g. elastic response with a full mobilization of contact network). In the latter case, we recover, within a reasonable approximation, the typical structure of a transversely isotropic stiffness tensor AijklAijkl, itself a function of five independent constants; in the former, in case of forward incremental loading, we find the lack of major symmetry of the stiffness tensor, Aijkl≠AklijAijkl≠Aklij. We show that this occurs because particle deformation is not affine and because anisotropy is present in the aggregate. Theory and numerical DEM simulations agree qualitatively.