The Effect of Microstructure Development on the Collisional Stress Tensor in a Granular Flow. C. S. Campbell, Los Angeles, California.

Transcription

1 Aet~ i~eehanica 63, (I986) ACTA by Springer-Verlag 1986 The Effect of Microstructure Development on the Collisional Stress Tensor in a Granular Flow By C. S. Campbell, Los Angeles, California With 4 Figures (Received October 13, 1985; revised December 18, 1985) Summary Many oi the observed but yet unexplained phenomena associated with granular flow have been attributed to the development of a layered microstructure within the material. In particular the microstructure development has been called on to explain ma, ny of the experimental observations that cannot be accounted for in the presently avmlable theoreticm studies. While the attributions are based on sound physical arguments, the complexity of the processes involved has prohibited the inclusion of microstructure development in theoretical models. As a result no one has yet demonstrated how and to what extent the microstructure can account for the phenomena to which it has been ascribed. This paper addresses the effect of microstructure development for two dimensional flows by incorporating a heuristic model of the microstructure into a simple theoretical analysis. I. Introduction It is now apparent that flowing granular material represents an extremely complicated case of fluid mechanics. The closer the problem is surveyed, the more complex the physical processes involved become. At present the problem is being studied in two limiting flow regimes. Slow deformations fall into the quasistatic regime in which the materim flows in macroscopic (i.e. much larger than a particle) pieces of material moving relative to one another along slip lines. At the opposite extreme is the "grain inertia" or rapid flow regime. Here the granules move individually and behave much like the molecules in a gas. They exhibit random velocity fluctuations so reminiscent of the thermal motion of molecules that their associated energy has been dubbed the "granular temperatare", and interact only in the short range, i.e. by collision. (The major difference between rapid granular flows and hard sphere models of gases is that granular flows are dissipative and hence the continuance of the flow depends on a flux of

2 62 C.S. C~mpbell: energy from the mean flow down to the energy associated with the random velocities.) This paper is directed toward the kinematic constraints on the flow that is imposed by high density. The first macroscopic evidence of the microstrueture development was the observation [1], [2], [11] that the coefficient of friction, the ratio of shear to normal forces, in a simple shear granular flow, was a decreasing function of the density. This is in direct contrast to the myriad of soil strength experiments which show that the friction coefficient at yield is an increasing function of the packing. This phenomenon was explained when Campbell and Brennen [1] demonstrated in their two-dimensional computer simulations that a distinct layered microstructure appears at high density. In order to maintain a high density shear flow, the particles organize themselves into layers oriented perpendicular to the velocity gradient, each layer moving with the mean velocity corresponding to its position in the velocity field. This is a property of fully developed granular flows and is not reflected in the yield strengths of static granular constructions such as would be measured in soft mechanics testing. Campbell and Brennen [1] showed that the layers strongly influenced the angular distributions of collisions, which, by affecting the direction of momentum transport, alter the relative magnitude of the stress tensor components and produces the observed reduction in the friction coefficient. However, the detailed analysis of the stress tensor by Campbell and Gong [2] showed that the majority of the reduction in the friction coefficient occurred in the streaming or kinetic part of the stress tensor rather the collisional part and can therefore not be directly attributed to the microstructure development; they showed that the collisional friction coefficient actually increased with packing at small densities in accord with the theory of Lun, Savage, Jeffrey and Chepurniy [9], and only decreased, probably due to the mierostructure development, at large density. Another observation that was attributed by Campbell and Gong to the mierostructure development was that the ratio of the mean rotational rate of the particles, to the shear rate decreased sharply, from the value of one-half (which is found in most cases of shear flow which apply an off center force to particles or molecules), as the shearable limit is reached. Also, Campbell and Gong observed in-the-shear-plane normal stress differences, which are also possible byproducts of the microstructure as they had also not been evident in the previous analyses of granular flow. :For this paper, a heuristic model of the mierostructure is developed to examine the consequent effects on the collisional stress tensor and thus to help sort out which effects may or may not be attributed to the microstructure development. The paper builds itself around the relatively primitive analysis of Savage and Jeffrey [10], which by virtue of its simplicity easily allows the inclusion of the microstructure at the expense of the completeness of the analysis. The majority of the work described here was completed several years ago and was referred to in the discussions, and invaluably aided the understanding of the data, in Campbell and Brem:en [1] and Campbell and Gong [2].

3 The Effect of Microstructure Development Analysis It is common practice in the study of rapid granular flows to model the particle interactions as instantaneous collisions. The stress tensor in a rapid granular flow may then be determined by evaluating the integral : : fjkg. k/(:)(v, c. ~, rl 2Rk, c2, ej~) dk de1 dc~ d~ dc~ g,k>o where rl, rx + 2Rk, e~, e2, ol and r are the instantaneous positions, translational and rotational velocities of the particles at co]lision, _8 is the particle radius, ]t~) is the pair distribution function, (which is defined so that : f2)(r~, c~, ~ol, r2, c2, e~2) dr~ dr2 c[cl dc2 d~ d~2 is the probability of finding a particle in dr, de~ and dco~ neighborhoods about r~, c~ and r and another particle in dr2, de2 and dr neighborhoods about r2, c2 and r k = r2 -- rl, is the vector connecting the centers of the two particles, g ~ el -- c~ is the relative velocity, and J is the impulse applied by the collision. For perfectly rough (i.e. infinite surface friction coefficient) particles with an inelasticity realized through a coefficient of restitution, E, the impulse Jis given by: j = -~ ~ (g. k) k + 2(i + ~ -~ where fl is the ratio of the squared radius of gyration to the squared particle radius. The next step is to make the Enskog assumption as to the form of the pair distribution function, namely that the velocities and relative positions of the particles are uncoupled so that the pair distribution function/r may be written as the product of single particle velocity distribution functions/tl) which govern the velocities of the particles, and a distribution function 2(r. r2) relating their relative positions : ](2)(~'1, C1~I, r2, C2, r -- p(rl, v2)/(1!(cl, ~1 ; rl)/(1)(c2, ~o2 ; r2)- (A form of p(rl, r2) will be developed in the next section to permit the inclusion of a layered microstructure into the model. The parametric dependence of/(i) on r is included to allow for spatial variations in the mean values of e and co.) This assumption has been used by every author who has adapted the dense theory of gases to granular materials [6], [7], [8], [9], [10]. However, at the densities common to realistic flow problems it is unreasonable to expect that there should not be: some relationship between the instantaneous velocities of a particle and its near neighbors. (See the discussion at the beginning of Chapter 16 of Chapman and Cowling [4].) To compound the approximation, the single particle pair distribution

4 64 C.S. Campbell: functions are assumed to be Maxwellian: where u(r) and ~u(r) are the mean velocities and rotation rates of the particle and T is the "granular temperature" which in two-dimensions may be written: 2 T = y (<u'~} + <v"~} + flr2<o/~}), where <u '~} represents the mean fluctuating velocity and is found by taking the difference between the mean squared velocity and the mean velocity squared. The computer simulations of Campbell [3] and Hopkins [5] have shown that in appearanee, the distribution functions do not differ far from Maxwellians, however, Campbell and Gong [2] recorded a significant streaming contribution to the shear stresses along with anisotropies in the temperature distribution, both of which would be absent from if the distributions were truly Maxwellian. Still, determining the required perturbation to the distribution function, such as were computed by Lun et al. [9] and Jenkins and gichmond [7], [8], will be strongly influenced by the mierostructure development and seems to be outside the scope of current techniques. Finally, this paper shall examine simple shear flows, i.e. flows of constant velocity gradient. Particle one is used as a test particle whose center instantaneously corresponds to the origin of an appropriate reference frame (see Fig. 1) moving with the mean velocity corresponding to its location. Hence u(rl) = 0 and C/U ay {where el represents a unit vector in the/-direction). :Fig. 1. The coordinate system of the problem showing the layered mierosl~ructure about a l~est particle

5 The Effect of Microstructure Development 65 After making these assumptions the velocity integrations may be performed to yield for the two dimensional case: $2 0] "r = --4R'~m f p(o,2rk) {[--1/2 exp [= y cos~ sin2 cososino kk where, and + fi + Z-----) (~ -- sin 20) exp -- cos ~ 0 sin SsinOcosOerfc cososino lk } do k = cos 0ex + sin 0e u I = --sin Oex cos 0% is the vector perpendicular to k following the right hand rule. Here the parameters is the ratio of the shear rate to the square root of the granular temperature: du 2R-- dy T1/2 and is related to the production and dissipation of granular temperature. For this paper, S as a function of v and ~ will be taken from the data of Campbell and Gong [2]. (In the first iteration of this analysis, S was computed using an energy balance between the production and dissipation of temperature, similar to that performed by Jenkins and Savage [6] and in the first part of Lun et al. [9]. However, both studies predict that S should be a function only of the coefficient of restitution, s and not show the ]arge variation with density shown by Campbell and Gong and predicted by the latter part of Lun et al. Due to the density perturbation, p(r, Ir~ ), there was some density variation of S but not nearly that required to provide a reasonable fit to the data; this is probably due to the exclusion of the work done by the streaming stress tensor. The data of Campbell and Gong is used to close this analysis so that the results will show only the effects of the microstructural development without inaccuracies introduced by poor predictions of S.) The factor ~ is the ratio of the rotation rate to the shear rate. The terms proportional to the dyadic product lk are the contributions from the particle rotation and the relative tangential velocities of the particles. As/k is a dyadic formed from perpendicular vectors, it is necessarily antisymmetrie and produces 5 Acta Mech. 63/1--4

6 66 C.S. Campbell: asymmetries in the stress tensor. As the effect of asymmetry in a granular flow is to apply torques that alter the rotational state of the particles and as in steady flow there can be no net torque on the particles, the stress tensor must be symmetric. Thus x is chosen so as to cancel any asymmetric contribution. 3. The Spatial Distribution Function: p(r~, r2) Far from any solid boundaries there is nothing to influence the position of a single particle. Hence, the probability of finding a particle at a point r, is just the number of particles per unit volume, n. Once a particle has been located it influences the positions of the particles about it. In particular, for a high density shear flow, a particle must reside within a layer which fixes, to some extent, its location relative to the other particles within the layer and the location of adjacent layers. As a result p(rl, v2) can be written: p(rl, r2) = np(r~ ] r~) where P(r2 I rl) is read as the probability that a particle whose center is in a neighborhood df about r~ given that there is a particle center at r~. In particular, this analysis requires the values of P(r~ ~- 2Rk ] r,), the probability of finding the two particles in contact. On top of the spatial preferences due entirely to the layered configuration of the material, a spatial preference for collisions is also imposed by the velocity field. A velocity gradient forces particles with different mean velocities together. Hence, it is more likely to find two patricles in contact when their relative mean velocities, due to different spatial locations within the velocity gradient, drive them together and less likely when their relative mean velocities drive them apart. Campbell and Br.ennen [1] showed that the correction suggested by Savage and Jeffrey [10] fit their data extremely well at low densities. (At high densities, the angular distribution imposed by the layered microstrueture obscured the velocity perturbation.) Hence at low densities, the model generated here should reduce to : + 2Rk 1 erfc Rkk : Fu). In their two-dimensional simulation, Campbell and Brennen [1] measured the development of P(r2 ] rl) as a function of the solid fraction v. They divided P(r2 {r~) into two parts: the probability of finding two particles as a function of their separation distance in the direction parallel to the velocity gradient (y direction in Fig. 1) and the probability of finding two particles separated by a distance perpendicular to the velocity gradient (x direction in Fig. 1), the latter with the restriction that their y direction separation cannot exceed one particle diameter. The first shows the development of the layers and the second shows the development of structure within the layer. (The velocity

7 The Effect of Microstructure Development 67 gradient will wipe out any x-direction correlations between particles in different layers layers.) Their data shows that both measurements are similar: At low densities, both show a peak at a separation of 2R, which, as the density increases, grows while other peaks, each separated from its neighbors by 2R, appear, until, as the maximum shearable density is approached, both distributions are composed of large well defined peaks separated by a distance 2R. Working on the basis of the Campbell and Brennen [1] results, imagine that the space is filled by layers each separated by 2R. Each layer is formed by a Gaussian distribution about the center of the layer with a half width at half maximum equal to the distance, ~, between particle surfaces (~ ---- h -- 2R, where, in two dimensions, h = (u/v) 112 R is the mean spacing between particle centers). To find P(r~ I r~), one must consider a test particle located in one of the layers. For convenience, locate'the test Particle at the origin for which P(r~[r~) is represented as P(r2 ] 0). Within its own layer the particles sees its fellow particles within the layer as located within Gaussians likewise separated by 2R. In the latter eases, however the particles, are also confined by Gaussians in the y direction. There is no kinematic reason for any correlation for the x-direction ]positions between particles in different layers, meaning that the probability of finding a particle from a different layer at some relative distance in the x-direction from the test particle is uniform and equal to 1/h. As the test particle is itself corffined within Gaussians in the x and y directions, P(r2 ] 0) must be found by integrating the the re/ative positions of the particles in x and y. The result is that P(r~ t0) appears as a sum of Gaussians but with the half width at half maximum expanded by a factor of the square root of two. The result is: P(2Rk ] 0) = -~- erfc 1/-2" cos 0 sin 0 S ~ exp sin 20 exp -- (cos i) 2 )] "~ 2-~ where the erfc term is the aforementioned velocity perturbation, the first bracket term represents the other particles within the test particle's layer, and the second bracket term represents the contribution from other layers. The pair distribution is multiplied by a factor of 2R/h so that the solid fraction obtained by integrating across the probability distribution is consistent with the value for which the distribution was derived and also insures that the layered structure disappears at low densities. As the actual kinematic constraints that lead to the microstructure development are left out of this analysis, this model should at best be considered_ phenomenologically accurate.

8 68 C.S. Campbell: 4. Results The final values for the stress tensor components are found by substituting the assumed value of p(r2, rl) into the appropriate expression from section 2, and then numerically integrating on a computer. (For computational ease, the analysis was limited to five layers on either side of the test particle. The integrations are performed using Gaussian integration with special attention paid to the peaks in the distribution. Figs. 2, 3, and 4 show the results for the collisional component of the friction angle, the ratio of the normal stresses and the ratio between the mean rotation rate and the velocity gradient. Fig. 2 shows friction coefficient, vxy/rvv, as a function of the solid fraction, v. Also plotted are the same results that would be obtained in the absence of any structural development and the data of Campbell and Gong [2]. The most noticeable point is that both sets of curves lie about 30 per cent below the corresponding data points. At low density, before the layers become distinct, the two sets are identical. At high density the no-structure curves appear to asymptote to a constant while the curves from the analyses that include the microstructure decrease as the shearable limit is approached. Therefore, as has been often speculated [1], [2], [11], the drop in the friction coefficient with density can be attri- t.4 - i I I I 'I i I I I - - Present Results t.2- "... No Structure z~ ~= o ~=0.6 "~ t.0- [3, =0.8 ~ 0,=1.0 Zx A Z~ - 0,8 - Z~ to Ix _j 0 0 =0.4 o X 5,=0.6_ g~ L #~ - 0.?.~ 0 O.Z 0,4 0, SOLID FRACTION, "~ ]~ig. 2. The collisional friction coefficient, Vxy/Vvy, as a, function of solid fraction r, for both the structured and the unstructured cases. The data points are from Campbell and Gong [2]

9 The Effect of Microstructure Development 69 buted to the development of the microstructure. However, as the tlheoretical lines lie below the data, there must be deficiencies in the model. Most probably, this is due to the anisotropies in the granular temperature measured by Campbell and Brennen [1]. Furthermore the data for the larger coefficients of restitution do not show the large decrease with solid fraction that appear for the lower coefficients of restitution and are predicted by the present theory. Fig. 3, shows the ratio of the in-the-shear-plane normal stresses, ~/ru~ as a function of the solid fraction ~,. In the absence of any microstructure development, this, as well as most theoretical results [6], [7], [8], [9], [10], indicate that all the normal stresses should be of equal magnitude. The data of Campbell and Gong [2] show that at low densities, this ratio decreases along unique lines for each coefficient of restitution. This effect is not accounted for by the microstructure development as it occurs in a region where the structures are diffuse. However, at high densities, the model predicts a sharp increase in rz~/ruy just as was observed, and assumed to be anomalous behavior, by Campbell and Gong. This observation may be understood by noting that the development of the microstructure forces particles withir~ the same layer to remain in close proximity. Hence a larger number of collisions will occur between particles in the same layer than between those in different layers. For collisions between particles in the same layer, the vectors, /r connecting the particle centers at collision, will be oriented nearly parallel to the x-axis. Thus the majority of momentum will be transported in the x-direction and z~ should be significantly greater than ~. With these ideas in mind, the most surprising thing about this figure is how close the solid fraction must be to the shearable limit before this effect makes its appearance. Fig. 4 plots the ratio of the mean rotation rate, ~, to the shear rate, U/H, (Here, U is the moving plate velocity and H is the plate separation in the Couette shear cell modelled by Campbell and Gong [2].) In the absence of any structure. I~H/U] is a constant value of one-half, independent of density. In correspondence with the data, the present model predicts a sharp drop in the ratio, just as the shearable limit is approached. This is consistent with most other theoretical results. As speculated by Campbell and Gong [2] this behavior can be attributed to the microstructure development. All of the predicted points fall onto a single curve which indicates that this effect is a simple function of the solid fraction and independent of any other parameters of the problem. Much like the previous case, this effect can be attributed to the large number of collisions that occur within the same layer. With a positive velocity gradient, such as that shown in Fig. 1, collisions between particles in different layers will induce a clockwise rotation. However, collisions within the same layer between clockwise rotating particles will induce rotation in the counterclockwise direction. The net effect should be to randomize the particle rotation rate and send the mean value to zero. Once again, it is surprising that such a large solid fraction must be attained before this effect becomes apparent.

10 70 C.S. Campbell: 5,0, I I I I 1 I l i'. Present Results A r O ~ =0,6 [3 9 =0,8 cy 5,C 9 0 ~= 1.0 r Q/ '1 ~ 2.0~ J < 1.0, or" 0 Z yoaa -/". u~ A. Z I I 1 I 1 1 I I! I ,4 0,6 0,8 1,0 SOLID FRACTION, :Fig. 3. The ratio of the collisional ir~-the-shear-plane normal stresses, lr~z/tryy, as a function of solid fraction, v. The data points are from Campbell and Gong [2] 0,6 I '0,5 13 J P 0,4 < Z o_ 0,3 I-. o ~: 0,2 C~ W.J < O,l I l 1 i I 1 1 I I A A A O 0 \A Present Results A ~=0.4 O ~=0,6 r-i ~=0,8 O~=1,0 O <3( [3 0 I I I 1 I I I I I SOLID FRACTION, -v Fig. 4. The ratio, I~H/UI, as a function of the solid fraction, v. The data points are from Campbell and Gong [2]

11 The Effect of Hicrostructure Development Conclusions :Previous studies have shown that a distinct mierostructure develops in the material as the shearable limit is approached. In order to maintain a shear flow under such conditions, the particles align themselves into layers which are oriented perpendicular to the velocity gradient. The results published in this paper have investigated the effects of the microstructure development on the co]lisioual stress tensor in a two-dimensional granular shear flow in a format for which the effects of the microstructure can be uncoupled from other effects. The analysis was based on a heuristic model that, built to fit the best information available, phenomenologically modeled the microstrueture development. The analysis was used to examine three effects that have been attributed to the microstructure development: the decrease in friction coefficient at high density, the dropoff in the ratio of mean rotation rate to the shear rate at high density, and the in-the-shear-plane normal stress differences. The results show that some, but not all of the experimental observations can be attributed directly to the microstructure development and that the attributable effects occur only at very high densities. Other effects, such as anisotropies in the temperature distribution (which may be indirectly related to the microstructure) must account for some aspects that have previously been attributed to the microstructure development. Acknowledgments This work was supported under NSF grant MEA and additional funding by IBM Inc., TI~W Inc. and the Ralph Parsons Foundation to which the author is deeply indebted. Special thanks to Dr. George Lea, George Gelb, David Wang, Ailing Gong, Yi Zhang, Mike Fashano and the Hughes Aircraft Co. References [I] Campbell, C. S., Brennen, C.E.: Computer simulation of granular shear flows. J. Fluid Mech. 151, 172 (1985). [2] Campbell, C.S., Gong, A. : The stress tensor in a two dimensional granular shear flow. J. Fluid Mech. 164, 107 (1986). [3] Campbell, C. S. : Velocity distribution functions in two-dimensional granular shear flow. (In preparation.) [4] Chapman, S., Cowling, T. G.: The mathematical theory of non-uniform gases, third ed., Cambridge Univ. Press [5] Hopkins, hi. A.: Collisional stresses in a rapidly deforming granular flow. M. S. Thesis, Clarkson University, Potsdam NY, [6] Jenkins, J. T., Savage, S. B. : A theory for the rapid flow of identical, smooth, nearly elastic particles. J. Fluid Mech. 130, 187 (i983). [7] Jenkins, J.T., ~ichmond, M.W.: Grad's 13-moment system for a dense gas of identical, rough, inelastic, circular disks. Arch. Rat. Mech. and Anal. 87, 355 (1985).

12 72 C.S. Campbell: The Effect of Microstructure Development [8] Jenkins, J. T., M. W. Richmond: Kinetic theory for a dense gas of identical, rough, inelastic, circular disks. Physics of Fhfids (1986) (submitted). [9] Lun, C. K.K., Savage, S. ]3., Jeffrey, D.J., Chepurniy, in.: Kinetic theories for granular flow: Inelastic particles in Couette flow and slightly inelastic particles in a general flowfield. J. Fluid Mech. 140, 223 (1984). [10] Savage, S. B., Jeffrey, D. J. : The stress tensor in a granular flow ~t high shear rates. J. Fluid Mech. 110, 255 (1981). [11] Savage, S. ]3., Sayed, ~. : Stresses developed by dry cohesionless granular materials in an annular shear cell. J. Fhfid Mech. 142, 391 (1983). C. S. Campbell Department o[ Mechanical Engineering University o] Southern California Los Angeles, CA U.S.A.