Compaction of elastoplastic granular media
Two states of matter have been extensively studied by the physics community: classical (Newtonian or not) fluids, whose viscosity induces short-time memory effects, and dry dense granular media composed of elastic grains, which can reorganize when submitted to external forces but always keep the memory of the initial shape of the grains -- i.e. present an infinitely long memory at the smallest scale -- and thus belong to what is commonly referred as "complex fluids". The presence of this small-scale memory is responsible for long-range correlations in an elastic granular medium: any heterogeneity in the granulometry or contact properties of the grains results in the emergence of a mesoscopic scale above the one of the grain, due to the heterogeneity of the contact forces between grains: the force exerted on the entire system is essentially carried by a loose skeleton of high stress carrying bonds, with long-range spatial correlations. The existence of these correlations is the principal source of difficulty for the so far unsuccesful attempts of describing a granular medium as a classical fluid with a given constitutive relationship relating time derivatives of average strain and stress at a given mesoscale, corresponding to the "elementary representative volume" for the engineering community. They are responsible for major qualitative differences in both the dynamics and quasi-static properties between classical fluids and dense granular media: whereas a fluid will flow regularly, the flow of a granular medium is dominated by quick bursts or avalanches. The difference in the quasistatic properties is well illustrated by the existence of the Janssen effect in granular assembly piled in a tube or a hopper: the long-range selforganization of the stress-carrying network induces arching phenomena, and the load of the pile is essentially carried by the side walls instead of being exerted on the bottom of a container. The average pressure exerted on the bottom boundary is then independent of the height of the pile, contrarily to the case of a classical fluid where it is proportional to this height. Both of those two regimes have been extensively studied.
There is however an intermediate regime between those two which emerges when the granular medium is composed of elements which can afford a certain plastic deformation above a given yield stress and/or after a certain characteristic time (viscous case). There are no other studies of such soft plastic granular media at the moment to our knowledge. Experiments are currenttly carried at the University of Oslo, on the dynamics of granular packings with grains composed of silicon oil. This regime could be relevant to describe the deformation of sediments in the lower-crust: different mechanisms of plastic deformation for such sediments interplay to transform a rather loose sand packing to a denser sandstone for example, or to pack together a clay material formed by debris encountered in turbidites. These mechanisms include "pure" plasticity of the composing grains, and dissolution/deposition mechanisms induced by stress corrosion.
In order to investigate analytically and numerically this intermediate regime, we have built up a simple 2D Tetris model where grains interact elastically, and are furthermore allowed to change there reference stress-free shape by modifying their aspect ratio while conserving their total surface (incompressibly), when the differential stress exerted on them exceeds a certain yield threshold. These studies are yet on a preliminary stage. The forces exerted on those model Tetris grains include gravity, frictionless elastic interaction forces between particles, and elastic interactions with a Coulomb friction criterion along the boundary walls. The present model fixes their orientations, and we intend to release this constraint in a further advanced version of this model. The first results obtained on this model show that the arching phenomena are indeed partially destroyed when this plasticity of the grains is taken into account: in Fig. 1, 300 particles are piled in a vertical tube. The local pressure on each grain is mapped, and the color scale indicator on the side gives as well the size of the gravitational constant: it corresponds to the pressure map in a column of classical fluid at rest, of the same density and on the same spatial scale. The grains are purely elastic in the first case, and in the second case they can yield plastically when the local differential stress exceeds the pressure corresponding to a pile of five grains.
|purely elastic grains||viscoplastic grains|
By comparing the two cases in presence or absence of plasticity of the grains, it is clear that the arches present in the purely elastic case are destroyed when plasticity is allowed, which gives rise to an overall vertical pressure gradient in the entire sample, only visible on the top of it in the elastic case. This pressure gradient is nonetheless lower than in the case of a classical Newtonian fluid at rest (color range on the left), which shows that tangential forces along the boundaries are still present even if the arches are smoothened: this example shows clearly that this plastic granular regime is intermediate between the two extreme classical cases of Newtonian fluid and elastic granular medium.
Those first results show that this intermediate regime of matter is indeed a rich and wide open field for research. We intend to investigate further the geometrical properties of the major stress-carrying skeleton in such systems.