Stable disarrangement phases arising from expansion/contraction or from simple shearing of a model granular medium

A principal challenge in modeling granular media is to connect the macroscopic deformation of the aggregate of grains with the average deformation of a small number of individual grains. We used in previous research the two-scale geometry of structured deformations (g,G)(g,G) and the theory of elastic bodies undergoing disarrangements (non-smooth submacroscopic geometrical changes) and dissipation to obtain, in terms of the free-energy response Ψ of the body, an algebraic tensorial consistency relation between the macroscopic deformation F=∇gF=∇g and the grain deformation G  , as well as an accommodation inequality detF⩾detG>0detF⩾detG>0 that guarantees that the aggregate provides enough room at each point for the deformation of the grains. These two relations determine all of the disarrangement phases G corresponding to a given F. We use the term stable disarrangement phase to denote a grain deformation G   that minimizes the stored energy density for the aggregate Ψ(G′)Ψ(G′) among all the disarrangement phases G′G′ corresponding to F  . Stability in this sense is determined solely by the constitutive response function ΨΨ and, therefore, may be described as a notion of material stability with respect to changes in microstructure. In this article we determine for a model aggregate and for two familiar families of macroscopic deformation – simple shears and uniform expansions or contractions – all of the stable as well as all of the unstable disarrangement phases of the model aggregate. For the stable disarrangement phases, we determine the connections between aggregate deformation and grain deformation. We showed in an earlier article that each stable disarrangement phase of this model aggregate cannot support tensile tractions, and our present results confirm that no-tension property of stable disarrangement phases for the model granular medium. Consequently, the appearance of tensile tractions in the present model granular medium would entail the loss of the material stability that we consider. This loss of stability is expected to be the rule, rather than the exception, because only special boundary conditions turn out to be compatible with one or more of the stable disarrangement phases at the disposal of the material.