We have studied how it depends on the local parameters describing the interactions between grains. (restitution and friction), and on the global parameters of ...
30th Leeds-Lyon Symposium on Tribology, Lyon, septembre 2003 Macroscopic friction of dry granular materials a
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a
F. da Cruz , F. Chevoir , J-N. Roux and I. Iordanoff
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Laboratoire des Matériaux et des Structures du Génie Civil, UMR LCPC-ENPC-CNRS, Institut Navier, 2 allée Kepler, 77 420 Champs sur Marne, France b
Laboratoire de Mécanique des Contacts et des Solides, INSA de Lyon, 20 avenue Einstein, 69 621 Villeurbanne Cedex, France As a way to improve the understanding of dry contact behaviour, we have measured the macroscopic friction coefficient of a dense assembly of dry grains sheared between two rough walls, using discrete numerical simulations. We have studied how it depends on the local parameters describing the interactions between grains (restitution and friction), and on the global parameters of the shear mechanism (wall velocity and confining pressure). In a large range of parameters, the macroscopic friction depends essentially on a single dimensionless inertial number which gathers the global parameters, increasing from a static to a dynamic value when the velocity increases and/or the pressure decreases.
1. INTRODUCTION The third body concept introduced by Godet in the 70’s with its attached “tribological circuit”, added by Berthier in the 80’s, has improved the understanding of dry contact behaviour [1,2]. However, the link between the macroscopic friction coefficient, measured for given contact conditions, and the microscopic properties of the granular material trapped between the surfaces in contact, is still an open question. In order to understand the third body flow inside a contact, we have performed discrete numerical simulations of dense flows of dry grains in the plane shear geometry, with no gravity. In this simple geometry, the shear state is homogeneous, which provides detailed informations on the constitutive law for dense flows of dry grains [3-5]. We measure the macroscopic friction coefficient and study its dependence on the global parameters of the shearing mechanism and the local parameters of the material. We first describe the simulated systems and the numerical method. Then, taking advantage of the uniformity of the sheared system, we introduce a dimensionless number associated to the shear mechanism, and show that the macroscopic friction as well as volume fraction and velocity fluctuations depend essentially on this single parameter.
2. SIMULATED SYSTEM 2.1. Plane shear flow In the plane shear geometry (Fig. 1), the material is sheared between two parallel rough walls, distant of H; one is fixed and the other is moving at the velocity Vw. Experiments have been performed with gravity [6,7], or in annular shear cell [8-10]. In the case of dry contact, the pressure due to gravity is usually much smaller than the confining pressure, so that gravity can be neglected. This case with no gravity is much simpler, since then the stress distribution is homogeneous inside the sheared layer, from conservation of momentum. Controlling the shear force applied to the moving wall allows to study the flow threshold [11], but most of the results found in the litterature are for fixed velocity [12-21]. In the following, we present results of two discrete particles simulations for fixed velocity, which parameters are summarized in Tab.1. The main difference between the two studies is that in one case the volume is fixed and the pressure P is measured (fixed volume FV), whether in the other case the pressure is controlled and the volume fraction is measured (controlled pressure CP).
P
Vw
H
P
Vw
y x
Figure 1 : Plane shear geometry : picture of the force network for small I (left) and high I (right). The grains are disks or spheres, but both simulations are purely two-dimensional. The grains are slightly polydisperse, and we call d and m their mean diameter and mass. x is the flow direction and y is the normal direction. Periodic boundary conditions are applied along the flow direction, and the length of the simulated window is L. This assumption should be consistent with contacts for which the length is much larger than the width (H/L 0.1. For I < 0.1, the volume fraction is a linear function of I :
ν ( I ) = ν max − bI
(2)
with νmax = 0.84 and b = 0.38. Even if the flows remain dense (0.75 < ν < 0.84), the small dilatancy has strong consequences on the contact network (see Fig. 1). Very small I corresponds to the quasi-static regime, associated to a dense network of enduring contacts [23], and large I corresponds to the dynamic inertial regime, associated to binary collisions. 4.2. Macroscopic friction The macroscopic friction coefficient µ* in dense granular flows has been measured in confined shear geometries [8, 25], or down inclined planes [26, 5]. It is then the ratio of shear to normal force at the walls. It can also be defined as the ratio of the shear stress σxy to the pressure P inside the material (both components are constant in the sheared layer). Both definitions (the first being used in FV simulations and the second in CP simulations) give approximately the same results, but the fluctuations are stronger at the wall that inside the flowing layer. Fig. 5 shows that µ* starts from a finite value φ at small I, which should correspond to the internal Coulomb friction in the critical state of soil mechanics [24], and then increases with I. This
the velocity Vw increases or when the pressure P decreases. This variation is approximately linear :
µ * ( I ) = φ + aI
(3)
with φ = 0.11 and a = 1,62. We shall discuss later the regime of large I (greater than around 0.1). The quasi-static regime, for very small I (smaller than around 10-3), is shear rate independent. Then, for I larger than 10-3, the variations of µ* with the shear rate or the pressure become significant [13,14,19,21]. The transition between the quasi-static and the dynamic regime is progressive and we call intermediate the regime where Eq. (3) is significant, that is to say approximately 10-3< I < 0.1. The small quantitative differences between the two simulations, in Fig. 4 and 5, may come from the different shape of grains and polydispersity. 4.3. Fluctuations Dimensional analysis suggests that the relative velocity fluctuations δ v / γ (using the natural scale of velocity γ , in dimensionless unit) should depend on the dimensionless number I. Indeed, the following scaling law is observed on Fig. 6a :
δ v / γ ∼ I −β
(4)
with β ≈ 1/2 (0.47 in Fig. 6a). This means that the velocity fluctuations are more pronounced in the quasi-static regime, where they may lead to intermittency of the flow [27,4,5]. Eq.(4) can also be written :
δ v ∼ γ 1/ 2 P1/ 4
(5)
Consequently the dispersion of velocity fluctuations shown in Fig. 2c , for simulations at constant L and Vw, should be reduced when rescaled by P1/4. This
is confirmed in Fig. 6b, at least in the center of the shear layer, far from the influence of the walls.
0,85
0,80
ν 0,75
0,70
1E-3
0,01
0,1
I
Figure 4 : Mean volume fraction as a function of I : FV - e = 0.1 (black circles) and 0.8 (black squares), CP - e = 0.1(white circles) and 0.9 (black circles).
0,40 0,35 0,30 0,25
µ*
0,20 0,15 0,10 0,05 0,00
1E-3
0,01
I
0,1
Figure 5 : Macroscopic friction as a function of I measured at the wall (FV - e = 0.1 (black squares) and 0.9 (black circles) and inside the flow (CP - e = 0.1 (white squares) and 0.9 (white circles)).
(a) 10
δv/γ
.
1
1E-4
1E-3
0,01
I
0,1
1
5.2. Large I
(b) 16 14 12
y/d
10 8 6 4 2 0 0,10
0,12
0,14
0,16
δv/P
0,18 1/4
(contrarily to the velocity fluctuations). But there is some influence of friction µ [4,5]: the maximum volume fraction νmax decreases slightly with µ (from 0.84 for µ = 0 to 0.80 for µ = 0.8) and the internal friction φ jumps from 0.11 for µ = 0 to 0.22 for µ ≠ 0. The prefactor b in Eq.(2) slightly depends on the friction coefficient (0.38 for µ = 0 - 0.30 for µ = 0.4 and 0.37 for µ = 0.8). The prefactor a in Eq.(3) varies from 1.62 for µ = 0 to 1.0 for µ ≠ 0 . There is a strong difference between frictional and frictionless grains, but only a small influence on the intensity of the friction.
0,20
0,22
Figure 6 : (a) Relative velocity fluctuations as a function of I (CP - e between 0,1 and 0,9 - µ between 0 and 0.8); the slope of the guideline is 0.47; (b) Velocity fluctuations profiles rescaled by P1/4 (FV - e = 0.8 - I = 0.19 (diamond), 0.22 (down triangle) and 0.25 (circle)). 5. INFLUENCE OF THE MATERIAL We now discuss the influence of the microscopic parameters e and µ, that is to say the local dissipation mechanisms. 5.1. Small I We first discuss the quasi-static and intermediate regime (I < 0.1). Then, the volume fraction and velocity profiles, as well as the effective friction and the relative velocity fluctuations, do not depend on restitution e
We now discuss the regime of large I (> 0.1). Then the influence is essentially due to the dissipation coefficient α. When α ≈1 (slightly dissipative material), deviations from the linear dependencies of the dilatancy and the effective friction are apparent in Fig. 4 and 5. The dilatancy becomes much stronger, and the macroscopic friction µ* saturates, or even starts to decrease : contrarily to the quasi-static and intermediate regime, it now depends on the restitution coefficient e : it is constant for e < 0.6, and decreases when e increases from 0.6 to 1. An analysis of the average duration of contacts [27] concludes that this evolution with e and I corresponds to a transition towards a regime of binary collisions, where the assumptions of the kinetic theory become valid. Fig. 7 shows the range of binary collision regime as a function of volume fraction and restitution coefficient. 0,84
Multiple Collisions 0,80
ν 0,76
Binary Collisions 0,72 0,6
0,7
0,8
e
0,9
1,0
Figure 7 : Transition between binary and multiple
collisions regime as a function of restitution coefficient and volume fraction (FV simulations). In the dilute regime and for a slightly dissipative material (I > 0.1 and e ≈ 1), we now compare the simulated value of the macroscopic friction with the predictions of the kinetic theory, in the simple case of a two-dimensional homogeneous shear flow of identical frictionless disks in the dense limit [28,5]. Then µ∗ is given by an equilibrium between the work of the shear stress and the dissipation due to the slighltly dissipative collisions, and should not depend on I. Fig.8 shows that the agreement is fair for e > 0.8. The quantitative discrepancy may come from the influence of the walls.
Figure 8 : Prediction of the kinetic theory for the macroscopic friction in homogeneous shear flow (straight line). The FV simulated value are indicated by black circles. 6. CONCLUSION The flow of the third body inside a contact has been studied by discrete numerical simulations of the plane shear of an assembly of dry rigid grains between two rough walls without gravity. Two boundary conditions (fixed volume and controlled pressure) have been compared and shown to provide essentially the same results. We have measured the profiles of volume fraction, velocity and velocity fluctuations in steady uniform flow, and studied their dependencies on the local parameters characterizing the interactions
between grains (restitution and friction), and on the global parameters describing the shear mechanism (confining pressure and wall velocity). From dimensional analysis, we have defined a dimensionless inertial number I, which compares the inertial stress to the confining pressure. For small I (small velocity and/or large pressure), the flow is homogeneous (uniform shear rate, volume fraction and stress tensor), and the constitutive law only depends on this single inertial number. When I increases, the small dilatancy changes the interactions between grains from a dense network of enduring contacts to dilute binary collisions. The macroscopic friction, which represents the global energy dissipation in the system, increases from the static internal friction of soil mechanics to a higher dynamic value. There is no influence of the restitution coefficient, and a shift of the friction curve when going from frictionless to frictional grains. For large I (large velocity and/or small pressure), the flow enter the kinetic regime, where the influence of the restitution becomes significant. In the limit of a slightly inelastic material, we observe a saturation of the macroscopic friction, in quantitative agreement with the prediction of the kinetic theory. The interpretation of those behaviours may be found in the analysis of the transfer of momentum and energy between grains, taking into account the evolution of the contact network (orientation of the contacts, mobilization of friction [4,5]. We suspect that for very small I (quasi-static regime), free continuous flows are no more possible and a jamming transition occurs, leading to shear localization. This shear localization is also observed once gravity is added [4,5]. Our main conclusion is that in usual dense flows (intermediate value of I), and for usual dry granular materials (frictional dissipative grains), the macroscopic friction do not depend on the parameters of the material, but only on the shear mechanism through the single inertial number. This study carried out in a very simple case outlines the complexity of interaction between the first body (wall roughness), the third body (granular material), the mechanism (inertial number) and the response of the system (volume fraction and macroscopic friction).
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