epl draft Stresses in smooth flows of dense granular media

The form of the stress tensor is investigated in smooth, dense granular flows which are generated in split-bottom shear geometries. We find that, within a fluctuation fluidized spatial region, the form of the stress tensor is directly dictated by the flow field: The stress and strain-rate tensors are co-linear. The effective friction, defined as the ratio between shear and normal stresses acting on a shearing plane, is found not to be constant but to vary throughout the flowing zone. This variation can not be explained by inertial effects, but appears to be set by the local geometry of the flow field. This is in agreement with a recent prediction, but in contrast with most models for slow grain flows, and points to there being a subtle mechanism that selects the flow profiles. Introduction. – Granular media are amorphous and athermal materials which can jam into stationary states, but which can also yield and flow under sufficiently strong external forcing [1, 2]. Slowly flowing granulates, for which momentum transfer by enduring contacts dominates over collisional transfer, are characterized by a yielding criterion and rate independence. The former expresses that granulates only start to flow when the applied shear stresses exceed a critical yielding threshold [1–3], while the latter signifies that a change in the driving rate leaves both the spatial structure of the flow and the stresses essentially unaltered [4–8]. Solid friction exhibits a similar combination of yielding and rate-independence: According to the Coulomb friction law, a block of material resting on an inclined plane starts to slide when its ratio of shear to normal forces exceeds the static friction coefficient. And, once the block slides, the same ratio is given by a lower dynamical friction coefficient, which is essentially rate independent. There is no unique manner in which these friction laws can be translated into a continuum theory, and there exists a plethora of approaches describing slow granular flows [3, 8–14]. To test these theories, one would like to determine the stresses and strain rates within the material. However, experiments can not easily access the flow in the bulk of the material, nor probe the stress tensor in sufficient detail. In addition, slow grain flows often exhibit sharp gradients, thus casting doubt on the validity of continuum theories [3–6,9]. Finally, granular flows are notoriously sensitive to subtle microscopic features [5], which often translates into a substantial number of tunable parameters in the models [10]. As far as we are aware, no direct comparison between the full stress and strain rate tensor has been undertaken for slow granular flows. In this Letter, we numerically study grain flows in splitbottom geometries as shown in fig. 1. Recently, these flows were shown to exhibit robust and continuum-like flow profiles that are numerically tractable and are governed by a number of universal, i.e. grain-independent, scaling relations, making them eminently suitable for our purpose. We relate the stress tensor to the strain-rate tensor in these flows, thus providing a benchmark for the testing and development of theoretical models for smooth and dense grain flows. Experiments and numerics so far have focussed on the flow in a cylindrical geometry (fig. 1c), where a wide shear zone is generated by rotating a centre bottom disc with respect to the cylindrical container [7, 14, 15]. We present some data for this cylindrical case, but focus on the linear version of this geometry (fig. 1a), where we find a wide shear zone to emanate from the relative motion of two bottom plates along their “fault line”. In this system, the physics behind the stresses is easier to disentangle because the stream lines are not curved. Our main finding is that, throughout the flowing zone, the stress and strain tensor are co-linear, meaning that their eigen-directions, or equivalently, their principle dip-1 ar X iv :c on dm at /0 61 10 80 v2 [ co nd -m at .s of t] 2 9 A pr 2 00 7 Martin Depken et al. Fig. 1: (a) Linear shear geometry where a split along the middle of the system generates a wide shear zone in a layer of grains. The curves indicate sheets of constant velocity. (b) Cuboid element of material showing the definition of the angle θ, the stresses in the SFS framework, and the labelling of the axis — the grey objects in a) and c) are examples of such elements. (c) Cylindrical split-bottom geometry, where the grains are driven by the rotation of a bottom disc. The two surfaces of rotation indicate sheets of constant angular velocity. Note that in the limit Rs→∞ one obtains the linear geometry (a). rections, coincide. Moreover, we find that the ratios of the non-zero stress components, such as the effective friction coefficient, which is the ratio between shear and normal stresses acting on a shearing plane, are not constants but vary throughout the flowing zone. This variation is crucial to understand the finite width of the shear zones, and is not due to the variation in the magnitude of the local strain rate. Both of these findings are in accord with the main features of theory developed in [8], and constitute an important step forward in establishing a general framework for the modelling of grains flows. SFS framework. – We formulate our results in the context of the theoretical framework recently developed by Depken et al. [8]. The central assumption of this socalled SFS theory is that, once the material is flowing, strong fluctuations in the contact forces enable otherwise jammed states to relax within a spatial region which we refer to as the fluctuation fluidized region. In this region there can not be a shear stress without a corresponding shear flow. This assumption can be interpreted as stating that the yielding threshold, which determines the onset of flow, is no longer relevant once part of the material flows, since this induces strong non-local fluctuations in the contact forces. Further one observes that the flows can be locally (and in the present cases also globally) seen as comprised of material sheets, with no internal average strain rate, sliding past each other (see fig. 1). Combining these two ingredients, it follows that both the shear strains and shear stresses in these material sheets are zero, and we refer to them as a Shear Free Sheet (SFS). It also follows that the stress and strain-rate tensors are co-linear. The major and minor principle directions of the strain-rate tensor are at an angle of 45◦ with respect to the SFSs, and in the more intuitive basis specified by these sheets (see fig. 1b) the stress tensor takes the form: σSFS =  P ′ 0 0 0 P τ 0 τ P  . (1) To test this prediction, we check whether the numerically obtained stresses are co-linear with the strain rate tensor and thus are of the form (1). Moreover, when no further assumptions are made, the three components P , P ′, and τ will be different, and in general vary throughout the sample. In fact, if the stress is of this form, a simple stress balance argument shows that μeff := τ/P has to vary throughout the shear zones [8]: A constant μeff would correspond to a shear zone of zero width, clearly inconsistent with the available data [7, 15]. To put these predictions in perspective, let us briefly consider the case of faster flows, where collisions play a role. The arguments for the form of the stress tensor can be extended to apply also for such systems, and Pouliquen and co-workers [13] have suggested that the stress is of the form eq. (1). However, they introduce the following restriction: P ′=P and τ = μeff(I)P , where the effective friction is a material dependent function of the so-called inertial number I = γ̇d/ √ P/ρ [16], and d and ρ are the particles diameter and density, respectively [12, 13]. For the slow flows under consideration here, we should consider the limit I → 0. If we only consider μeff to depend on I, μeff becomes a material constant, which is, as we explained above, incompatible with the finite width of the shear zones [8,14,17]. Our study will thus illuminate how subtle details of the form of the stress tensor have significant consequences for the grain flow. Method. – The simulations are carried out with a discrete element method (DEM) for 80− 100k monodisperse Hertzian spheres satisfying the Coulomb friction laws. The relevant parameters describing the material properties of the spheres are the normal stiffness kn = 2 × 10mg/d, the tangential stiffness kt = 2/7kn, the normal and the tangential viscous damping coefficients γn = 50 √ g/d , γt = 0, and the microscopic coefficient of friction μm = 0.5. Here d and m are the diameter and the mass of spheres, and g is the gravitational acceleration. The characteristic timescale t0 is given by √ d/g (e.g., t0 = 0.0101sec if d = 1mm). We have studied a range of driving rates varying from from ±0.05 to ±0.005 d/t0 and 0.015 to 0.005 rad/t0 for the linear and circular geometries, respectively. Stresses and velocities are averaged over the symmetry direction (along split) and are resolved with a resolution of 0.9d in the cross section. The stress tensor within this volume is the sum of contact and collisional stresses [18], where the latter is three orders of magnitude smaller than the former. The linear setup has dimensions 20d in the shearing direction (periodic boundary conditions), a width of 80d, and a height of 50d. The details of the specific implementation can be found elsewhere [18].