Dry granular flows – rheological measurements of the ( ) − I μ Rheology

Granular materials do not always flow homogeneously like fluids when submitted to external stress, but often form rigid regions that are separated by narrow shear bands where the material yields and flows. This shear localization impacts their apparent rheology, which makes it difficult to infer a constitutive behavior from conventional rheometric measurements. Moreover, they present a dilatant behavior, which makes their study in classical fixedvolume geometries difficult. These features led numerous groups to perform extensive studies with inclined plane flows, which were of crucial importance for the development and the validation of the ( )− I μ rheology. Our aim is to develop a method to characterize granular materials with rheometrical tools. Using rheometry measurements in an annular shear cell, dense granular flows of 0.5mm spherical and monodisperse beads are studied. A focus is placed on the comparison between the present results and the ( )− I μ rheology. INTRODUCTION Granular matter shows both solid and fluid behavior [1]. These materials are very sensitive to various parameters: geometry of the flow, wall roughness, flow rate, shape and size distribution of the grains, and coupling with the interstitial fluid [2]. In the dry case, the rheology is solely governed by momentum transfer and energy dissipation occurring in direct contacts between grains and with the walls. Despite the seeming simplicity of the system, the behavior of dry granular material is very rich and extends from solid to gaseous properties depending on the flow regime. In the absence of a unified framework, granular flows are generally divided into three different regimes. (i) At low shear, particles stay in contact and interact frictionally with their neighbours over long periods of time. This “quasistatic” regime of granular flow has been classically studied using modified plasticity models based on a Coulomb friction criterion [3]. The response in terms of velocity or solid fraction profiles is independent of the shear rate. (ii) Upon increasing the deformation rate, a viscouslike regime occurs and the material flows more as a liquid [5]. In this intermediate regime, the particles experience multicontact interactions. (iii) At very high velocity, a transition occurs toward a gaseous regime, in which the particles interact through binary collisions [4]. For the modeling of dense granular flows, the concept of inertial number has been widely used and investigated with regard to its relationship with dynamic parameters, such as velocity, stress, and friction coefficient, which leads to constitutive relations for granular flows. Thus, “dynamic dilatancy” law and “friction” law were deduced from discrete simulation of two dimensional Dry granular flows – rheological measurements of the ( )− I μ Rheology Abdoulaye Fall, Michel Badetti, Guillaume Ovarlez, François Chevoir, and Jean-Noël Roux 1 Laboratoire NAVIER, UMR 8205 CNRS-ENPC-IFSTTAR, Champs sur Marne, France 2 CNRS, LOF, UMR 5258, 33600 Pessac, France ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 25, 2017


INTRODUCTION
Granular matter shows both solid and fluid behavior [1]. These materials are very sensitive to various parameters: geometry of the flow, wall roughness, flow rate, shape and size distribution of the grains, and coupling with the interstitial fluid [2]. In the dry case, the rheology is solely governed by momentum transfer and energy dissipation occurring in direct contacts between grains and with the walls. Despite the seeming simplicity of the system, the behavior of dry granular material is very rich and extends from solid to gaseous properties depending on the flow regime. In the absence of a unified framework, granular flows are generally divided into three different regimes. (i) At low shear, particles stay in contact and interact frictionally with their neighbours over long periods of time. This "quasistatic" regime of granular flow has been classically studied using modified plasticity models based on a Coulomb friction criterion [3]. The response in terms of velocity or solid fraction profiles is independent of the shear rate. (ii) Upon increasing the deformation rate, a viscouslike regime occurs and the material flows more as a liquid [5]. In this intermediate regime, the particles experience multicontact interactions. (iii) At very high velocity, a transition occurs toward a gaseous regime, in which the particles interact through binary collisions [4]. For the modeling of dense granular flows, the concept of inertial number has been widely used and investigated with regard to its relationship with dynamic parameters, such as velocity, stress, and friction coefficient, which leads to constitutive relations for granular flows. Thus, "dynamic dilatancy" law and "friction" law were deduced from discrete simulation of two dimensional Dry granular flows -rheological measurements of the ( )− I µ Rheology Abdoulaye Fall 1 , Michel Badetti 1 , Guillaume Ovarlez 2 , François Chevoir 1 , and Jean-Noël Roux 1 simple shear of a granular material without gravity [5]. It was observed that both dimensionless quantities: the internal friction coefficient and the solid fraction φ are functions of I [5,6].
Following general results from simulations of planar shear [5] and successful applications to inclined plane flows [7], the experiments of Jop et al. [8] were carried out to quantify, for glass beads, the rheology from the quasistatic to the rapid flow regime, corresponding to moderate I as ( ) ( ) in which, s µ , 2 µ and 0 I are three fitting parameters dependent on material properties. However, the asymptotic value of at high I was not obtained by da Cruz et al. [5] who observed an approximately linear increase of the internal friction coefficient from the static internal friction value. In 3D-simulation studies, Hatano did not either observe the asymptotic value of at high I: from In the present work (see Fall et al. [10] more details), we show that it is not necessary to develop specific setups (such as the inclined plane) to study dense granular flows. Indeed, we show that a simple annular shear cell can be adapted to a standard rheometer to study the rheology of granular materials under controlled confining pressure. It allows us, in particular, to obtain the dilatancy law ( ) I φ and also to study very accurately the quasistatic limit. Thus, from the steady state measurements of the torque and the gap during an imposed shear flow under an applied normal confining stress N σ , we report two laws in which the internal friction coefficient and the solid fraction are functions of I. An effort is then made to compare the present results with the − ) (I µ rheology described in the literature.

MATERIALS AND METHODS
To investigate the steady flows of dry granular materials and determine the − ) (I µ rheology, three main features are required: (i) to avoid shear banding, (ii) to apply a confining stress in the velocity gradient direction, and (iii) to allow volume fraction variations. These requirements led us to develop a home-made annular shear cell, in which pressure-imposed measurements can be performed. Annular  In our rheometer, instead of setting the value of the gap size for a given experiment, as in previous studies and generally in rheometric measurements, we impose the normal force (i.e., the confining normal stress) and then, under shear, we let the gap size vary in order to maintain the desired value of the normal force. We measure then the torque T and the gap h as a function of strain (or time). In this case, the solid fraction ( ) t φ is not fixed but adjusts to the imposed shear. However, it remains important to notice that, in order to keep the imposed shear rate constant, the rheometer adjusts the rotation velocity Ω since the gap varies as will be discussed below. The system reaches a steady state after a certain amount of shear strain.

RESULTS AND ANALYSIS
We have measured the driving torque and the gap as a function of shear strain for various imposed normal force F N (between 1 and 5N) and applied shear rate γ! (between 0.01 and 77 s -1 ) for a given gap. In Figs. 1(a,b), we show those measurements for F N =3N and various γ! . At low shear rate, the driving torque increases slowly before reaching a steady plateau within strain of order of unity. Meanwhile, the gap fluctuates around its initial value. (We show the gap size rescaled by its initial value before shearing h/h 0 called the rescaled gap in the following.).
Upon increasing the imposed shear rate, an overshoot occurs: Its amplitude increases with increasing the shear rate. In steady state, a rate dependence of the torque is observed [ Fig. 1(a)]. Moreover, increasing the imposed shear rate causes an increase of the gap size [ Fig. 1(b)] allowing to quantify the dynamic dilatancy of the granular material. Notice that, with increasing the applied shear rate, large fluctuations of the driving torque and also of the gap size evolution occur in steady state flows. Similarly, in experiments in which different normal forces are imposed at a given shear rate [ Figs. 1(c,d)], the steady torque is observed to increase while the steady solid fraction (steady gap) decreases when the normal force is increased. Figure. 1: Evolution as a function of the strain at 3N imposed normal force under different applied shear rates of: (a) The driving torque and (b) the rescaled gap size (only two curves are shown for clarity). Evolution as a function of the strain at 77 s -1 imposed shear rate under different applied normal forces of: (c) The driving torque and (d) the gap size rescaled by its initial value before shearing.
Once the above described experiments are combined, we can obtain the constitutive laws of the dry granular material, i.e., the dependence of the steady solid fraction φ and the ratio between shear and normal stresses N σ τ µ / = variation on shear rate. Indeed, from macroscopic quantities T, Ω , F N , and h, the shear stress τ , the normal stress N σ , shear rate γ! , and the solid fraction φ can be computed [10] and one can plot the shear stress in the steady state as a function of the normal stress as shown in Fig. 2(a). The first observation is that a linear relationship between the shear and normal stresses is seen for all imposed shear rates, with a slope that increases from 0.265 to 0.6 with increasing the shear rate. If an internal friction coefficient µ is defined as the ratio between shear and normal stresses, we evidence here that µ is rate dependent: it increases with γ! . The second observation is that, for all imposed shear rates, the steady value of the solid fraction decreases when one decreases the normal stress [ Fig. 2(b)]. Indeed, since the grains cannot escape from the cell, one can measure unambiguously the solid fraction from the gap variation as ( )   At the same time, the solid fraction variation 0 /φ φ with the inertial number I is shown in Fig. 3(b). Once again, all the data collapse on a single curve. At low I, 0 /φ φ is quasiconstant: this is the quasistatic regime. When I increases, the inertia starts influencing the flow and the system becomes rate dependent: the ratio 0 /φ φ decreases; this regime corresponds to the dense flow in which the granular material dilates. Moreover, in order to check the robustness of our results, we have varied the initial size of the gap. Here, with our annular shear geometry, the same experiments discussed above are made with different gap sizes from 6d p to 22d p and different sets of imposed shear rate and normal force. It is shown in Fig. 3 that changing the gap does not significantly affect these results. This suggests a total absence of shear localization at high inertial number. However, at small inertial number, the resolution of our measurements is not sufficient to dismiss the possibility that shear localization arises.

CONCLUSION
From the steady state measurement of the torque and the gap, the internal friction coefficient, the solid fraction, and the inertial number I are measured. For low I, the flow goes to the quasistatic limit and the internal friction coefficient and the solid fraction profiles are independent of I. Upon increasing I, dilation occurs and the solid fraction decreases linearly when I increases while the friction coefficient increases. Comparing now these experimental results with existing models such as those of Jop et al. [15] and Hatano [9] on dry granular flows in terms of quasistatic and dense flow behaviors, our experimental data include points in range of I from 10 -7 to 0.1 which covers quasistatic and dense flow regimes. Our measurements then show that, in this range of I, both models can describe our data. As a consequence, we bring evidence that rheometric measurements can be relevant to describe dry granular flows. However, additional experimental work should be carried out in order to measure the dependence of the boundary layer constitutive law on the state of the bulk material, so as to be able to describe properly the rheology when approaching the quasistatic limit.