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Constitutive equations for a simple shear flow of a disk shaped granular mixture
innr.J. Engng Sri. Vol. 22, No. 7, pp. 829-843. Printed in the U.S.A.
om3722sp34 $3.00 + .oo Pergamon Press Ltd.
1984
CONSTITUTIVE EQUATIONS FOR A SIMPLE SHEAR FLOW OF A DISK SHAPED GRANULAR MIXTURE HAYLEY H. SHENt and NORBERT L. ACKERMANN Department of Civil and Environmental Engineering, Clarkson College, Potsdam, NY 13676. U.S.A. (Communicated
by A. J. M. SPENCER)
Ahatraet-Constitutive equations are derived for a granular flow of disk shaped solids in a 2-dimensional, simple shear field. Binary collisions are assumed to be the major mechanism for momentum transfer. Stresses are computed as the average rate of momentum transfer across a surface due to the interparticle collisions. Stresses are formulated explicitly in terms of the shearing rate, concentration of solids, and the physical properties of the solid constituent. The anisotropic collision distribution on the circumference of the disks due to the mean flow gradient is explicitly quantified. The frictional impact during collisions is treated in sufficient detail so that the singularity which existed in previous constitutive equations [l, 21 is removed. This detailed analysis of collision geometry and its effect upon frictional dissipation includes important considerations that have not been included in the numerical simulation models of Campbell and Brennen [3] where particles, after impact, are considered to not have any relative tangential velocity. This assumption by Campbell and Brennen will most likely be the technique by which frictional effects will be commonly handled in the future. Hence the present study will provide an important reference for assessing the effects of using simplifying assumptions when determining frictional effects during particle collisions. The constitutive relationships developed in this investigation are compared to data obtained from computer simulated experiments [3]. The result of the 2dimensional analysis has practical applications in the study of ice flows on a river surface as reported in [4] and in the conveyance of bulk solids in chutes. Except for geometric complications, the procedure developed in the analysis can be extended to 3-dimensional flows of spherical particles.
1. INTRODUCTION
IN A RAPIDLY sheared granular flow, interparticle
collisions inevitably occur due to the relative velocity between adjacent particles. These collisions transfer momentum throughout the flow region and contribute to the generation of stresses within the granular mixture. In addition to this collisional transfer of momentum, stresses are also produced by the continuous rubbing of solids arising from prolonged intergranular contacts. As demonstrated experimentally, the collisional transfer of momentum becomes the dominating mechanism for stress generation as the solid concentration and the shearing rate increases [5,6]. When internal stresses are generated predominantly by particle collisions, the granular flow is said to be in the “grain inertia” regime as defined by Bagnold [5]. In that analysis, Bagnold formulated the stresses in the following manner zii=pi+
AMj
(1-l)
where zii is the stress on the surface normal to xi acting in the xi direction, pi is the number of particles per unit area normal to the xi direction, 3is the average collision frequency per particle and Aaj is the average momentum transfer per collision in the xi direction. The number of particles per unit area was obtained as a function of the packing pattern and the linear gap between adjacent particles. The momentum transfer per collision was obtained by assuming a perfectly elastic and frictionless collision between two particles. The relative velocity between the two particles was quantified by their mean motions. The collision frequency, however, was left undetermined. The results of Bagnold’s analysis were qualitatively verified by experiments in which neutrally buoyant particles were sheared between concentric cylinders rotating at different angular velocities. Savage and Jeffrey [7] have recently expanded upon the analysis by Bagnold [5]. In their tOn leave from Clarkson College from July 1983 to June 1984. Present address: Snow and Ice Branch, U.S. Army Cold Regions and Engineering Laboratory, Hanover, NH 03755, U.S.A. 829
830
H. H. SHEN and N. L. ACKERMANN
analysis, an approach similar to that employed in gas kinetic theory was adopted. The relative velocity between colliding particles and the distribution of a pair of particles’ relative location were both allowed to be random. They quantified (1.1) through a statistical average. The constitutive laws were formulated in terms of the undetermined root-meansquare of the fluctuation velocity. Jenkins and Savage [8] further improved upon the theory of Savage and Jeffrey by utilizing an energy balance law which quantified the root-meansquare of the fluctuation velocity. The stresses were obtained in terms of an unknown constant which measured the degree of anisotropy of collision contacts. In the above analyses, particles were assumed to be inelastic and frictionless. At approximately the same time other analyses were proposed by Kanatani [9] and Ogawa et al. [I] which abandoned Bagnold’s approach as embodied in (1.1). They derived constitutive laws from a pure energy consideration and a term matching procedure. Kanatani considered perfectly elastic but frictional particles in which the mean rotation of particles was incorporated. The fluctuation of rotation was neglected however. Ogawa et al. considered inelastic and frictional particle collisions. Neglecting the rotational motion, assuming an isotropic and constant fluctuation velocity and isotropic contact distribution, they obtained stresses that contained no undetermined constants. However, the calculated stresses were two orders of magnitude smaller than those obtained in laboratory studies [2]. Moreover, the constitutive laws exhibit a singular behavior as will be discussed later. Ogawa et al. [I] noted that in a shear flow, the contact distribution on the surface of a particle should not be isotropic. However, it is not clear how the anisotropy can be quantified with their model. Another approach emerged during the same time by the present authors which incorporated at a crude level a complete description of the dynamics of particle motion in a granular flow [2, lo]. The particles were allowed to fluctuate isotropically with a constant fluctuation speed. The collisions were inelastic and frictional. The effects of particle rotation were neglected as was done in [l]. The results of this approach gave explicit constitutive laws free of any undetermined parameters. In this approach, the distribution of particle fluctuation speed was not considered. The fluctuation speed was assumed a constant as in Kanatani [9] and Ogawa et al. [ 11.This simplification greatly reduced the amount of computational work and hence enabled the model to incorporate detailed mechanisms involved at a microscopic level. The results of this approach, however, still underestimated the stress generated in laboratory studies, and still exhibited singular behavior. In the present paper a planar flow of disk shaped particles will be considered because of their geometric simplicity. The degree of anisotropy of contact points on the surface of a particle due to the mean shear motion will be explicitly quantified. A mechanism associated with frictional forces between colliding particles will also be studied. This mechanism was completely ignored in all the previous studies that considered frictional collisions [ 1,2], and has caused the singular behavior of the stresses. The anisotropic effects and the frictional mechanism will be shown to increase the theoretically determined stresses and remove the existing singularity in previous models.
2. GENERAL
MODELING
An assembly of identical disk shaped particles of diameter D, thickness d, and density ps are considered. The rotational motion of the individual particles is neglected. The volume concentration of these disks is C. The disks are assumed to have a restitution coefficient 6 and friction coefficient p. The interstices of these disks is assumed to contain a vacuum. By considering frictional collisions while neglecting rotation motion, some inconsistency is created. However, including the rotational motion will greatly increase the complexity of the analysis and hence is withheld until further study. Consider a steady linear simple shear flow of this granular continuum as shown in Fig. 1. It is assumed that the granular solids are not subjected to any body force and that the average gap between adjacent particles is represented by s. A linear concentration 1, as defined by Bagnold [S],
Constitutive equations for a simple shear flow
831
Fig. 1. Definition sketch for the model of a simple shear flow of disks.
is expressed as I =
D/s.
As shown in Appendix A the volume concentration,
(2.1) C, is related to 1 as (2.2)
where C,, is the volume concentration corresponding to the densest packing arrangement. For the flow described in Fig. 1, the mean shear creates and maintains a fluctuating motion among the disks. Therefore, the instantaneous local velocity can be described as V=ui+v’
(2.3)
where v’ is the fluctuation component of the velocity. In reality v’ is a random variable which might have a wide spectrum of magnitudes and directional distributions. However, it will be assumed here that v’ has a constant magnitude and is isotropic in direction. This simplification enables the convenient investigation of frictional forces and contact distribution in the flow. 2.2 Energy balance in a granularjlow Consider the material volume within the square shown in Fig. 1, which is moving in the x, coordinate direction with a steady simple shear flow. The figure is not to scale in that the dimension of the disks are to be small compared to the length of the side of the square. Because the flow is steady, unidirectional and uniform in the x1 direction, the work done by the surface stresses on this volume do not increase the kinetic energy of the disks within the volume. This work can, therefore, only be used to change the internal energy within the material volume. This internal energy takes both thermal and mechanical forms. The mechanical form of internal energy is defined as the energy contained in the fluctuation mode of the particles [l 1, 121. The thermal energy is defined in terms of the temperature of the granular constituents. The passage of energy going from mechanical form to thermal
832
H. H. SHEN and N. L. ACKERMANN
form is made possible by the dissipative integranular collisions. The balance law for the mechanical internal energy can be formulated as [I] (2.4) where p is the bulk density of the granular continuum, Z = $u’~is the specific mechanical internal energy, Z?, is the symmetric strain-rate tensor, qj,i is the divergence of the flux of Z and y is the rate of dissipation of Z. The dissipation of mechanical energy y, as described in [I], serves as a source term in the balance equation for the thermal energy of the granular continuum. In the steady simple shear flow described in Fig. 1, the stresses, the solid concentration, and the velocity gradient are everywhere constant and hence Z must be uniform throughout the flow field. Thus, al/ax, and aZ,@x, both vanish and the flux of Z, q,, q2, and their gradients, 8q@x,, aq~lax~ are, therefore, all equal to zero. The energy balance law for the steady simple shear flow described in Fig. 1 is thus further reduced to
The fundamental important of dissipation in a granular flow can be seen in (2.5). If the granular material being sheared is not dissipative, there is no passage by which mechanical internal energy can be transformed to thermal energy. Therefore, y would be zero. In such a situation, however, steady state conditions could not occur. A qualitative description of this unsteady flow condition follows. In (2.5), under the condition that y is zero, the work done by the surface stresses is balanced by an increase in 1. As Z increases, the fluctuations become more intense. This results in an increase of the collisional activities. According to (1 . 1), these increased collisions produce continually increasing stresses so that steady state is never reached. One, therefore, concludes that in granular flows, if steady state can be obtained, the material must be dissipative. The reverse is also true in that a dissipative material can generate steady state stresses when subjected to a steady shearing motion. The final form of (2.4) for the steady flow described in Fig. 1 is reduced to du 721&
=
Y*
(2.6)
This same equation was adopted in Shen and Ackermann [2] where spherical particles were considered. 2.3 Stress formulation The stresses which are generated within the granular mixture are determined from (1.1) and act on planes perpendicular to the surface formed by the xi, x2 coordinate axis. Such a plane within the granular mixture would have a depth d equal to the thickness of the disks. The average number of particles, p, on a unit area of that plane can be obtained by assuming a random but isotropic distribution of disks. As is shown in Appendix A
p,=p2=p2-
ndD'
(2.7)
The dete~ination of the average momentum transfer in the x, and x2 direction, d2\;d and ,cU@~, is obtained from an analysis of the distribution of collisions over the surface of the disks. The values of da1 and An;i, are influenced by the relative motion of colliding particles and the coefficients of restitution, 6, and friction, Z.LTherefore, one may conclude that
Constitutive equations for a simple shear flow
833
To quantify the collision frequency, one must be able to evaluate the average speed of the disks, r, relative to their neighboring disks, and the average gap, s, between two adjacent disks. Since s can be quantified as a function of packing pattern and con~ntration as shown in (2.1) and (2.2), the ratio p/s gives the collision frequency once the average speed of disks P is determined. This average speed P is a function of the mean relative velocity of a disk to its neighboring particles and the fluctuation speed of the disk, or
9=
* (u’,(D+s)-$
v
(2.9)
2>
In the previous analyses [l, 21, it was assumed that P = Y’. The collisions that contribute to stresses on the surface of the material volume in Fig. 1 are generated only by those disks that are located exterior to the volume. A is the only exterior disk shown in Fig. 1. Hence, if an isotropic dist~bution of solids is assumed, only one half of the collisions experienced by a disk on the control surface shown in Fig. 1 are stress generating and the value of f-for (1.1) becomes (2.10) Substituting (2.7) and (2.10) into (1.1) one obtains (2.11) in which, if the functional forms defined in (2.8) and (2.9) are determined through a statistical averaging, the only unknown quantity is tr’. To quantify u’, the energy balance eqn (2.6), can be used by expressing the rate of dissipation y as
where N is the number of disks per unit volume in which the volume has a depth equal to the thickness d of the individual disks, and E is the average dissipation of energy per particle per collision. Since the volume concentration is C and is equal to N ELYd/4, it can be shown that
(2.13) where p is defined in (2.7). The dissipation .l? depends on the relative velocity of colliding disks and their mechanical properties as characterized by 6 and p. Hence
.
v',(D+s$+
(2.14)
2
Substituting (2.8) and (2.11)-(2.14) into (2.6) yields (2.15) If the functional forms of L&I?, and J? are obtained through a statistical averaging of all possible collisions, (2.15) can be used to compute ZJ’since it is the only unknown in that expression. After quantifying tt’, F, da, and Ah&, (2.11) will yield the desired stresses.
834
H. H. SHEN and N. L. ACKERMANN 3. DETAILED
FORMULATION
OF STRESSES
3.1General description of Aa,O,2 and I? As previously discussed, the problem of formulating the stresses defined in (1.1) reduce to determining the functional forms of A&?, oI 2 and ,??.Consider two disks A and B about to collide as shown in Fig. 2, where N and P are orthogonal unit vectors. Since the energy dissipation and momentum transfer in a collision process are independent of the inertial reference frame chosen, a reference frame is selected which moves with the instantaneous velocity, V,,particle B has acquired just prior to the collision. In this reference frame, the precollisional velocities of A and B are V,,and 0, where V,,is the relative velocity between disks A and B, or v,, = u_,,- uB+ v;B
(3.1)
where uAand us are the mean velocities of A and B and vLBis the relative velocity between A and B due to their fluctuations. A reference frame could also have been selected which moved with the mean velocity us obtained from the description of the simple shear field and the identical result would have been obtained. Since the centers of A and B were originally separated by a distance D + s at the angle 8 du uA- ug = (D + s)~ sin Oi.
(3.2)
2
Both A and B have a fluctuating component of their velocity with a magnitude V’ about their respective means. These fluctuations are isotropically distributed in all directions as assumed earlier. Hence, the root-mean-square relative velocity due to these fluctuations is 1 !I;,=
2R
2n
( 47r2Ll ss
l/2
~v’cos~Ai+v’sin~aj-v’cos~Bi-v’sin~~~2d~Ad~B
_
0
=Jzvf
(3.3)
Fig. 2. Two colliding disks.
835
Constitutive equations for a simple shear flow
where PA and fie are angles shown on disks A and B of Fig. 1. Therefore VAB= (D + s)$
sin 8i + ,/%
(3.4)
2
where v’ has no preferred direction. However, in order to ensure that A is approaching B rather than leaving B when a contact is considered to take place at 8, the angle a shown in Fig. 2 must be between --~/2 and 7c/2. This puts a constraint on the distribution of the direction of v’ as indicated by the angle a’. In Fig. 2 positive angles are measured counterclockwise from the line joining the centers of particles A and B. For a given contact location 8 and a given relative velocity VAB, the post collision velocities V,* and Vg can be computed using the conservation laws of particle dynamics [13]. The reason that particle dynamics can be appropriately utilized is because the rotational motion of these disks is neglected. These conservation laws are
V,*+%=V,,,
CV-WN= AL3* v
_~
(3.5a, b)
N
and either of the following equivalent expressions (3.6a,b) where Fp is the impulsive frictional force along the tangential direction of contact. When V,* and V2; are determined using (3Sa,b) and (3.6a,b), AM,,, and E for particle A can be computed for this particular collision as
AM,,
=
Ps-“D,2d(V: - VAB) . i,
n:2d(V: AMZA = Ps-
- VAB).j
(3.7a,b)
and
EA
=
n;2d;(v:, - vj2),
Ps-
(3.8)
These quantities associated with particle B are computed similarly. A@, or2and l? can then be obtained by integrating with respect to 0 and a’ throughout the range of each variable as determined from the following analysis identifying all possible collision geometries. 3.2 Determination of ranges of random variables Consider the disk B as shown in Fig. 3. As mentioned earlier, collisions only occur when a, as shown in Fig. 2, is between -n/2 and 7112.However, within that angle K the corresponding range for a’ as shown in Fig. 2, is not the same for different values of 8. For example consider the collision occurring at 0, as shown in Fig. 3. If v’ Q U, - U, the value of a: can vary over the full range of values 27r and collisions between particles A and B would still always occur. However at 8,, in Fig. 3, there are no values of a: for which collisions would occur if v’ * u, - us. This result for 8, stems from the fact that the velocity VA, would never have a normal component V,*, . N directed towards particle B. It can, therefore, be seen that this anisotropy is caused by the mean gradient of the flow field. A more detailed analysis will now be conducted using the contact points Or and 0, on B as in Fig. 3. If 8, is the angle of the contact point when A approaches B, the relative velocity V,*,, shown with the solid arrow, must lie within 90” of the normal direction of contact. The conditions shown in Fig. 3 represent a limiting case for a for collisions to occur. Because of the existence of mean shear, the fluctuation component of the relative velocity, $!v, lies in the range described by a:. Whenever v’ lies in a:, V& lies within 90” of the normal direction. Similarly, when 0, is the angle of the contact point, fiv’ lies
836
H. H. SHEN and N. L. ACKERMANN
Fig. 3. Anisotropic distribution of collisions.
in the range described by tl:. The probability density function of the collision distribution is quantified by the magnitude of the values of a* at the circumference of B as follows
P@)=
a*(e)
s zn
(3.9)
.
a*(e) dtI
0
By using the law of sines to the triangles defined by fiv, can be shown that (”
uA- ug and V,, in Fig. 3, it
+y$fT2e)
a*(O) = 7r - 2 sin-’
(3.10)
while the range of the direction for v’ as defined by a’ in Fig. 2 is
(3.11)
Using R = (D + s)(du/dx,)/2 &D’ as a parameter, the anisotropy of the collision distribution is shown in Fig. 4. (When R is greater than 1, an entire range of 0 about 45” on B will no longer be subjected to collisions.) This anisotropic behavior was described by Savage and Jeffrey [7] and Jenkins and Savage [8] and was shown to occur in a computer simulation of rapidly sheared disks by Campbell and Brennen [3]. However, in those works, an expression of the distribution function was not explicitly formulated as a function of the kinematic property of the flow and the material composition. In the present study u’ is quantified as a function of the mean flow gradient and the material properties of the granular constituents, and hence enable R to be determined explicitly.
Constitutive equations for a simple shear flow
0
90”
45”
837
135”
Fig. 4. Probability of collision distribution: R = (du&I(D
180”
+ s)/2,/%‘.
3.3 Post cdlision velocities To determine A@, or2and E for a given collision, the post collision velocities, V,* and Vg, of colliding disks A and B have to be determined as discussed in Section 3.1. (3.5a,b) and (36a,b) will be used to compute V,t and Vt. However, before (3.6a,b) can be utilized, the impulsive frictional force Fp has to be quantified. In the works by Ogawa et al. [l] and Shen and Ackermann [2], (3&b) was adopted. By using that equation it was assumed that
n;2d(V:, - V;) *N.
Fp= a~,---
(3.12)
The above equation states that the impulsive frictional force l$ is proportional to the impulsive normal force with the proportionality being p, the friction coefficient. There are limitations however to the applicability of (3.12). If V,*, comes “head-on” towards B (i.e. Q = 0 in Fig. 2) (3.12) indicates that some tangential component V$ *P will be generated for B after this collision. A contradiction is, therefore, created. As shown in Appendix B, (3.12) applies only when collisions occur “far away” from the “head-on” direction. This criteria can be described quantitatively as permitting the use of (3.12) when V,,*P-~(1
f&)V,,.N&O.
(3.13)
+t.)V,,*N
(3.14)
Equation (3.12) does not apply when VAB.P-/J(~
and the post collisional velocities along P should then be determined as v;.P=Vf-P
(3.15)
which says that the two disks A and B separate at no relative motion in the tangential direction. The region about the “head-on” direction, or the direction of N as described by (3.14), will be termed the “friction wedge” in this study. This wedge, as determined from solving (3.14), makes an angle n = tan-’ ,u( 1 + 6) about N. The range of the direction of v’ corresponding to this friction wedge can be obtained by geometric considerations. In
H. H. SHEN and N. L. ACKERMANN
838
Fig. 5, the outer edges of the friction wedge are formed by the sum of velocity vectors represented by dashed lines. By applying the law of sines to the two triangles composed of fiv’, uA- II, and V,,, the direction of v’ corresponding to the friction wedge can be described by its angle a‘ with respect to N as follows,
il ?l
=
(D + 7~
-sin-’
tlz
i
s)-$
IIzvl
* sinOsin(f3 TV)
.
(3.16)
1
3.4 Determination of AR,O,2 and I? Consider the situation when VABlies outside of the friction wedge as defined in Section 3.3. The post collision velocities of A and B can then be determined using (3.5) and (3.12). These velocities will be denoted by V,*,, V$,. If V,, lies inside of the friction wedge, the post collision velocities can be determined by using (3.5) and (3.15). These velocities will be denoted by V,*, V&. The average values of E and Ait? are then computed as
and
(3.18)
Fig. 5. Determination
of friction wedge: 9 = tan-’ ~(1 + c),
839
Constitutive equations for a simple shear flow
where m, = p,nD%/4 and a;, a;, q,, qz are defined in (3.11) and (3.16). Integrating (3.17) yields (see Appendix C),
E=m, [(1-E)2-$(1+&)2]
(3.19)
where
P(1+ e) - 6 cos’ q + 2 sin2 q
sin 2~ - ---+3 +cos2q(l
$(l + E)2 [n - sin 2?7(1 + cos2 7j)] -I‘2
+2cos2rt)l
(3.20)
and -4/~(1 +r)(l
-cos2q)+4,~~(1
+Q2 ( q + q?).
(3.21)
Integrating (3.18) yields (see Appendix C)
Afi=AdM,i-f-An;j;j=m
+s~i--mS(l
+$)+FM,l(D
+c)[-j&hv’
(3-22)
i
where 1 +ccdq)+$
-sin?
1
-cos2q)sinq
1.
and
FM, =
-cosq)+$l
1
(3.23)
(3.24)
3.5 Constitutive relationships l%e fluctuation speed v’ can now be computed after substituting (3.19) and (3.22) into (2.15), which then yields
&7’ =z
+z X
-- FE, 81r
j-62 4+----g---
’ ~(1
i-c)
~“‘(1+E)~
4
FE2 +8n
1
l/2
du dx,
1 (3.25)
840
H. H. SHEN and N. L. ACKERMANN
The collision frequency as defined in (2.10) can be obtained as
(3.26) where s and fiA are defined in Fig. 1 and il is defined in (2.1). All terms in the constitutive relatinons (1.1) are, therefore, specified. Substituting (2.7), (3.22) and (3.26) into (l.l), the stresses are obtained as
‘j2 ----+----_-~(1 +E) R 4
,u’(l +E)’ FE, 4 +81t
1 (3.27)
and
=22 = TC +
4~ + 37c2FM,
X
(3.28) where FE,, FE, and FM, are defined in (3.20), (3.21) and (3.23). 4. RESULTS
AND
CONCLUSIONS
The constitutive equations developed in this study demonstrate many of the characteristics that were observed by Bagnold [5], Savage [6], Savage and Sayed [14] and Sayed [15] in their experimental investigations using sphere shaped particles in a rapidly sheared granular flow. Common features to both theory and experiments include the fact that the stresses were proportional to the square of both the particle diameter D and the mean flow velocity gradient (du/dx,). Previous experimental results did not report values of the physical properties of the materials such as L and p. It is evident from the results of this investigation that such information is essential in order to be able to compare experimental results with theoretically predicted values. The significant influence of concentration upon the stress level is also demonstrated in Fig. 6 where conditions are compared when L = 2 and 3, = 5 or C = 0.4 and 0.63. The conversion from I to C is through adopting (2.2), in which C, = 0.906 is used which corresponds to a hexagonal packing. The increase in C from 0.4 to 0.63 produced an almost threefold increase in stress. Data produced by the
Constitutive equations for a simple shear flow
841
low 1
10-l
E = 0.7
.I 0
0.50
0.25
0.75
Fig. 6. Concentration
1.00
1.25
1.50
1.75
effect on shear stress.
computer simulated simple shear flow of disks also demonstrate this behavior, Campbell and Brennen [3]. The comparison between the theoretical results and the computer simulated data is presented in Fig. 7. It should be pointed out here that in the computer simulated model particle rotation was included and the frictional collisions were assumed to be such that the relative velocity at the contact point of a pair of disks was destroyed after the collisions. This assumption is equivalent to assuming all collisions are “inside” the friction wedge as defined in the present analysis. The main contribution of this paper is to quantify the anisotropic collision effect and treat the friction force correctly. If these two effects were neglected, the same outlined procedure will produce the following simplified stresses
al+6 r‘2,= c&D2 - 2 211
Tz2
r
l+c 1 311Gz
l-E2
, /iL(l +c)
112 -du 2 ~~(1 +c)”
10
---0 _
dxz
(4.1)
(4.2)
It
1
4
I
x
4
-I
A singular point exists for 721and 722. For any L, as
the stresses 7,, and zu both become infinite. As discussed in Section 2.2, when a dissipative material is used, a finite steady state stress must exist under the flow conditions studied here. This singular behavior also exists in Ogawa et al. [l] and an earlier work by Shen and Ackermann [2]. The comparison of calculated stresses using (3.27) and (4.1) are given in Fig. 8. Not only is the singular behavior of the stresses shown to be removed in the present analysis, but it also can be seen that when (3.27) is used the stress levels are
842
H. H. SHEN and N. L. ACKERMANN
0
E = 0.8
A E = 0.6
L
I
0.2
I
I
0.3
0.4
I
0.5
I
0!6
0.7
0
C
Fig. 7. Comparison between theoretical results and computer simulated data. Solid curve shows the theoretical result (3.7), data points are from [3].
increased for values of 6 and p which do not correspond to values in the immediate vicinity of the singular point. The value of R defined in Fig. 4 which measures the anisotropy of the collision distribution can be explicitly quantified. Since R = (D + s) (du/dx,)/2&‘, R can be determined by substituting o’ from (3.25) to obtain R=
*((($)(l+~)+FM,)&@$)+lii(:;f)2-$)
I
1 -fz2 4+-----
Al +c) 71
-“2 (43) ~‘(1 +c)~ ( FJ’% 4 87r
1
.
where FE,,FE, and FM, are defined in (3.20), (3.21) and (3.23). This shows that the anisotropy of collision contacts in a shear field is explicitly quantified when the concentration and material properties are specified. However, this anisotropy, as shown in Fig. 4, is exactly a sinusoidal function of the contact location. The extreme sensitivity to concentration and nonsinusoidal behavior in contact distribution observed in the computer simulation by Campbell and Brennen [3] are not predicted in the present analysis. The reason for this discrepancy is believed as a result of the assumption made in the present model. Since in this analysis the random velocity v’ and the contact point on the peripheral of a disk are both assumed to be completely random, the only anisotropy introduced is due to the mean shear. This kind of anisotropy may be called “dynamic anisotropy”. However, as solid concentration increases, another kind of anisotropy which may be called the “kinematic anisotropy” will dominate. This kind of anisotropy is due to the preferred orientation or “fabric” which takes place in a densely sheared material as observed in [3]. For a set of parameters used in Fig. 8 1 = 5, 6 = 0.7, and ~1= 0.2, R is computed as 0.43. The principal assumption used in the derivation of the constitutive relationships was that U’ > (D + S)(dz+/dxJ/2& This enables all points on a disk to be collisional points. If u’ is allowed to be smaller than (D + s)(du/dx,)/2&, R would be greater than 1 and some part on the circumference of a disk may not be allowed to experience collisions. This will then put a limitation on 13in the averaging processes defined in (3.17) and (3.18).
Constitutive equations for a simple shear flow
10
843
-1 0
I .25
I .50
I
I
.75
1.0
I 1.25
I 1.50
1. 75
Fig. 8. Anisotropic and friction effect on shear stress. Solid curve (3.27), dashed curve (4.1).
Whether or not the inequality u’ > (D + s)(du/dx,)/2,/ is satisfied depends upon the physical properties 6 and p, and the concentration of solids within the granular flow. Computations for v’/(D + S)(dUJdx& were performed using (3.25). It was found that u’(D + s)(du,/dx,) > l/2$2 was satisfied for the ranges 0 I /J < 2, 0 5 L I 1 and 1 2 2. The analysis of anisotropic collisions was not conducted for the case when (D + S)(dnJdxJ 9 u’. This situation could be realized when the granular solids were highly dissipative and interparticle distances became large.
Acknowledgement-This Foundation.
study was partially supported by a grant number CEE-821665 of the National Science
REFERENCES [I] S. OGAWA, A. UMAMURA and N. OSHIMA, ZAMP 31, 482 (1980). [2] M. SHEN and N. L. ACKERMANN, AXE Engng Mech. Diu. 108, EM5, 748 (1982). [3] C. S. CAMPBELL and C. E. BRENNEN, Proc. U.S.-Japan Seminar on New Models and Constitutive Relations in the Mechanics of Granular Materials, Cornell University, New York (1982). [4] N. L. ACKERMANN, H. T. SHEN and R. W. RUGGLES, LAHR Znt. Symp. Ice, p. 251. Quebec (1981). [5] R. A. BAGNOLD, Proc. Roy. Sac. London A225, 49 (1954). [6] S. B. SAVAGE, Proc. U.S.-Japan Seminar on Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials, p. 241. Sendai, Japan (1978). [7] S. B. SAVAGE and D. J. JEFFREY. J. Fluid Mech. 110. 2.55 (1981). 181J. T. JENKINS and S. B. SAVAGE, J. Fluid Mech. 130, 187 (1983). [91 K. I. KANATANI, Int. J. Ertgng Sci. 17, 419 (1979). DOI N. L. ACKERMANN and H. SHEN, ASCE Engng Mech. Div. 108, EMl, 95 (1982). [‘ll A. BLINOWSKI, Arch. Mech. 30, 27 (1978). WI S. OGAWA, Proc. U.S.-Japan Seminar on Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials, p. 208. Sendai, Japan (1978). u31 D. T. GREENWOOD, Principles of Dynamics, Chap. 4. Prentice-Hall, New York (1965). 1141S. B. SAVAGE and M. SAYED, EURO MECH 133, Proc. Statics and Dynamics of Granular Materials, Oxford (1980). [I51 M. SAYED, Ph.D. Thesis, McGill University, Montreal, Canada (1981).
(Received 20 August 1983)
om3722sp34 $3.00 + .oo Pergamon Press Ltd.
1984
CONSTITUTIVE EQUATIONS FOR A SIMPLE SHEAR FLOW OF A DISK SHAPED GRANULAR MIXTURE HAYLEY H. SHENt and NORBERT L. ACKERMANN Department of Civil and Environmental Engineering, Clarkson College, Potsdam, NY 13676. U.S.A. (Communicated
by A. J. M. SPENCER)
Ahatraet-Constitutive equations are derived for a granular flow of disk shaped solids in a 2-dimensional, simple shear field. Binary collisions are assumed to be the major mechanism for momentum transfer. Stresses are computed as the average rate of momentum transfer across a surface due to the interparticle collisions. Stresses are formulated explicitly in terms of the shearing rate, concentration of solids, and the physical properties of the solid constituent. The anisotropic collision distribution on the circumference of the disks due to the mean flow gradient is explicitly quantified. The frictional impact during collisions is treated in sufficient detail so that the singularity which existed in previous constitutive equations [l, 21 is removed. This detailed analysis of collision geometry and its effect upon frictional dissipation includes important considerations that have not been included in the numerical simulation models of Campbell and Brennen [3] where particles, after impact, are considered to not have any relative tangential velocity. This assumption by Campbell and Brennen will most likely be the technique by which frictional effects will be commonly handled in the future. Hence the present study will provide an important reference for assessing the effects of using simplifying assumptions when determining frictional effects during particle collisions. The constitutive relationships developed in this investigation are compared to data obtained from computer simulated experiments [3]. The result of the 2dimensional analysis has practical applications in the study of ice flows on a river surface as reported in [4] and in the conveyance of bulk solids in chutes. Except for geometric complications, the procedure developed in the analysis can be extended to 3-dimensional flows of spherical particles.
1. INTRODUCTION
IN A RAPIDLY sheared granular flow, interparticle
collisions inevitably occur due to the relative velocity between adjacent particles. These collisions transfer momentum throughout the flow region and contribute to the generation of stresses within the granular mixture. In addition to this collisional transfer of momentum, stresses are also produced by the continuous rubbing of solids arising from prolonged intergranular contacts. As demonstrated experimentally, the collisional transfer of momentum becomes the dominating mechanism for stress generation as the solid concentration and the shearing rate increases [5,6]. When internal stresses are generated predominantly by particle collisions, the granular flow is said to be in the “grain inertia” regime as defined by Bagnold [5]. In that analysis, Bagnold formulated the stresses in the following manner zii=pi+
AMj
(1-l)
where zii is the stress on the surface normal to xi acting in the xi direction, pi is the number of particles per unit area normal to the xi direction, 3is the average collision frequency per particle and Aaj is the average momentum transfer per collision in the xi direction. The number of particles per unit area was obtained as a function of the packing pattern and the linear gap between adjacent particles. The momentum transfer per collision was obtained by assuming a perfectly elastic and frictionless collision between two particles. The relative velocity between the two particles was quantified by their mean motions. The collision frequency, however, was left undetermined. The results of Bagnold’s analysis were qualitatively verified by experiments in which neutrally buoyant particles were sheared between concentric cylinders rotating at different angular velocities. Savage and Jeffrey [7] have recently expanded upon the analysis by Bagnold [5]. In their tOn leave from Clarkson College from July 1983 to June 1984. Present address: Snow and Ice Branch, U.S. Army Cold Regions and Engineering Laboratory, Hanover, NH 03755, U.S.A. 829
830
H. H. SHEN and N. L. ACKERMANN
analysis, an approach similar to that employed in gas kinetic theory was adopted. The relative velocity between colliding particles and the distribution of a pair of particles’ relative location were both allowed to be random. They quantified (1.1) through a statistical average. The constitutive laws were formulated in terms of the undetermined root-meansquare of the fluctuation velocity. Jenkins and Savage [8] further improved upon the theory of Savage and Jeffrey by utilizing an energy balance law which quantified the root-meansquare of the fluctuation velocity. The stresses were obtained in terms of an unknown constant which measured the degree of anisotropy of collision contacts. In the above analyses, particles were assumed to be inelastic and frictionless. At approximately the same time other analyses were proposed by Kanatani [9] and Ogawa et al. [I] which abandoned Bagnold’s approach as embodied in (1.1). They derived constitutive laws from a pure energy consideration and a term matching procedure. Kanatani considered perfectly elastic but frictional particles in which the mean rotation of particles was incorporated. The fluctuation of rotation was neglected however. Ogawa et al. considered inelastic and frictional particle collisions. Neglecting the rotational motion, assuming an isotropic and constant fluctuation velocity and isotropic contact distribution, they obtained stresses that contained no undetermined constants. However, the calculated stresses were two orders of magnitude smaller than those obtained in laboratory studies [2]. Moreover, the constitutive laws exhibit a singular behavior as will be discussed later. Ogawa et al. [I] noted that in a shear flow, the contact distribution on the surface of a particle should not be isotropic. However, it is not clear how the anisotropy can be quantified with their model. Another approach emerged during the same time by the present authors which incorporated at a crude level a complete description of the dynamics of particle motion in a granular flow [2, lo]. The particles were allowed to fluctuate isotropically with a constant fluctuation speed. The collisions were inelastic and frictional. The effects of particle rotation were neglected as was done in [l]. The results of this approach gave explicit constitutive laws free of any undetermined parameters. In this approach, the distribution of particle fluctuation speed was not considered. The fluctuation speed was assumed a constant as in Kanatani [9] and Ogawa et al. [ 11.This simplification greatly reduced the amount of computational work and hence enabled the model to incorporate detailed mechanisms involved at a microscopic level. The results of this approach, however, still underestimated the stress generated in laboratory studies, and still exhibited singular behavior. In the present paper a planar flow of disk shaped particles will be considered because of their geometric simplicity. The degree of anisotropy of contact points on the surface of a particle due to the mean shear motion will be explicitly quantified. A mechanism associated with frictional forces between colliding particles will also be studied. This mechanism was completely ignored in all the previous studies that considered frictional collisions [ 1,2], and has caused the singular behavior of the stresses. The anisotropic effects and the frictional mechanism will be shown to increase the theoretically determined stresses and remove the existing singularity in previous models.
2. GENERAL
MODELING
An assembly of identical disk shaped particles of diameter D, thickness d, and density ps are considered. The rotational motion of the individual particles is neglected. The volume concentration of these disks is C. The disks are assumed to have a restitution coefficient 6 and friction coefficient p. The interstices of these disks is assumed to contain a vacuum. By considering frictional collisions while neglecting rotation motion, some inconsistency is created. However, including the rotational motion will greatly increase the complexity of the analysis and hence is withheld until further study. Consider a steady linear simple shear flow of this granular continuum as shown in Fig. 1. It is assumed that the granular solids are not subjected to any body force and that the average gap between adjacent particles is represented by s. A linear concentration 1, as defined by Bagnold [S],
Constitutive equations for a simple shear flow
831
Fig. 1. Definition sketch for the model of a simple shear flow of disks.
is expressed as I =
D/s.
As shown in Appendix A the volume concentration,
(2.1) C, is related to 1 as (2.2)
where C,, is the volume concentration corresponding to the densest packing arrangement. For the flow described in Fig. 1, the mean shear creates and maintains a fluctuating motion among the disks. Therefore, the instantaneous local velocity can be described as V=ui+v’
(2.3)
where v’ is the fluctuation component of the velocity. In reality v’ is a random variable which might have a wide spectrum of magnitudes and directional distributions. However, it will be assumed here that v’ has a constant magnitude and is isotropic in direction. This simplification enables the convenient investigation of frictional forces and contact distribution in the flow. 2.2 Energy balance in a granularjlow Consider the material volume within the square shown in Fig. 1, which is moving in the x, coordinate direction with a steady simple shear flow. The figure is not to scale in that the dimension of the disks are to be small compared to the length of the side of the square. Because the flow is steady, unidirectional and uniform in the x1 direction, the work done by the surface stresses on this volume do not increase the kinetic energy of the disks within the volume. This work can, therefore, only be used to change the internal energy within the material volume. This internal energy takes both thermal and mechanical forms. The mechanical form of internal energy is defined as the energy contained in the fluctuation mode of the particles [l 1, 121. The thermal energy is defined in terms of the temperature of the granular constituents. The passage of energy going from mechanical form to thermal
832
H. H. SHEN and N. L. ACKERMANN
form is made possible by the dissipative integranular collisions. The balance law for the mechanical internal energy can be formulated as [I] (2.4) where p is the bulk density of the granular continuum, Z = $u’~is the specific mechanical internal energy, Z?, is the symmetric strain-rate tensor, qj,i is the divergence of the flux of Z and y is the rate of dissipation of Z. The dissipation of mechanical energy y, as described in [I], serves as a source term in the balance equation for the thermal energy of the granular continuum. In the steady simple shear flow described in Fig. 1, the stresses, the solid concentration, and the velocity gradient are everywhere constant and hence Z must be uniform throughout the flow field. Thus, al/ax, and aZ,@x, both vanish and the flux of Z, q,, q2, and their gradients, 8q@x,, aq~lax~ are, therefore, all equal to zero. The energy balance law for the steady simple shear flow described in Fig. 1 is thus further reduced to
The fundamental important of dissipation in a granular flow can be seen in (2.5). If the granular material being sheared is not dissipative, there is no passage by which mechanical internal energy can be transformed to thermal energy. Therefore, y would be zero. In such a situation, however, steady state conditions could not occur. A qualitative description of this unsteady flow condition follows. In (2.5), under the condition that y is zero, the work done by the surface stresses is balanced by an increase in 1. As Z increases, the fluctuations become more intense. This results in an increase of the collisional activities. According to (1 . 1), these increased collisions produce continually increasing stresses so that steady state is never reached. One, therefore, concludes that in granular flows, if steady state can be obtained, the material must be dissipative. The reverse is also true in that a dissipative material can generate steady state stresses when subjected to a steady shearing motion. The final form of (2.4) for the steady flow described in Fig. 1 is reduced to du 721&
=
Y*
(2.6)
This same equation was adopted in Shen and Ackermann [2] where spherical particles were considered. 2.3 Stress formulation The stresses which are generated within the granular mixture are determined from (1.1) and act on planes perpendicular to the surface formed by the xi, x2 coordinate axis. Such a plane within the granular mixture would have a depth d equal to the thickness of the disks. The average number of particles, p, on a unit area of that plane can be obtained by assuming a random but isotropic distribution of disks. As is shown in Appendix A
p,=p2=p2-
ndD'
(2.7)
The dete~ination of the average momentum transfer in the x, and x2 direction, d2\;d and ,cU@~, is obtained from an analysis of the distribution of collisions over the surface of the disks. The values of da1 and An;i, are influenced by the relative motion of colliding particles and the coefficients of restitution, 6, and friction, Z.LTherefore, one may conclude that
Constitutive equations for a simple shear flow
833
To quantify the collision frequency, one must be able to evaluate the average speed of the disks, r, relative to their neighboring disks, and the average gap, s, between two adjacent disks. Since s can be quantified as a function of packing pattern and con~ntration as shown in (2.1) and (2.2), the ratio p/s gives the collision frequency once the average speed of disks P is determined. This average speed P is a function of the mean relative velocity of a disk to its neighboring particles and the fluctuation speed of the disk, or
9=
* (u’,(D+s)-$
v
(2.9)
2>
In the previous analyses [l, 21, it was assumed that P = Y’. The collisions that contribute to stresses on the surface of the material volume in Fig. 1 are generated only by those disks that are located exterior to the volume. A is the only exterior disk shown in Fig. 1. Hence, if an isotropic dist~bution of solids is assumed, only one half of the collisions experienced by a disk on the control surface shown in Fig. 1 are stress generating and the value of f-for (1.1) becomes (2.10) Substituting (2.7) and (2.10) into (1.1) one obtains (2.11) in which, if the functional forms defined in (2.8) and (2.9) are determined through a statistical averaging, the only unknown quantity is tr’. To quantify u’, the energy balance eqn (2.6), can be used by expressing the rate of dissipation y as
where N is the number of disks per unit volume in which the volume has a depth equal to the thickness d of the individual disks, and E is the average dissipation of energy per particle per collision. Since the volume concentration is C and is equal to N ELYd/4, it can be shown that
(2.13) where p is defined in (2.7). The dissipation .l? depends on the relative velocity of colliding disks and their mechanical properties as characterized by 6 and p. Hence
.
v',(D+s$+
(2.14)
2
Substituting (2.8) and (2.11)-(2.14) into (2.6) yields (2.15) If the functional forms of L&I?, and J? are obtained through a statistical averaging of all possible collisions, (2.15) can be used to compute ZJ’since it is the only unknown in that expression. After quantifying tt’, F, da, and Ah&, (2.11) will yield the desired stresses.
834
H. H. SHEN and N. L. ACKERMANN 3. DETAILED
FORMULATION
OF STRESSES
3.1General description of Aa,O,2 and I? As previously discussed, the problem of formulating the stresses defined in (1.1) reduce to determining the functional forms of A&?, oI 2 and ,??.Consider two disks A and B about to collide as shown in Fig. 2, where N and P are orthogonal unit vectors. Since the energy dissipation and momentum transfer in a collision process are independent of the inertial reference frame chosen, a reference frame is selected which moves with the instantaneous velocity, V,,particle B has acquired just prior to the collision. In this reference frame, the precollisional velocities of A and B are V,,and 0, where V,,is the relative velocity between disks A and B, or v,, = u_,,- uB+ v;B
(3.1)
where uAand us are the mean velocities of A and B and vLBis the relative velocity between A and B due to their fluctuations. A reference frame could also have been selected which moved with the mean velocity us obtained from the description of the simple shear field and the identical result would have been obtained. Since the centers of A and B were originally separated by a distance D + s at the angle 8 du uA- ug = (D + s)~ sin Oi.
(3.2)
2
Both A and B have a fluctuating component of their velocity with a magnitude V’ about their respective means. These fluctuations are isotropically distributed in all directions as assumed earlier. Hence, the root-mean-square relative velocity due to these fluctuations is 1 !I;,=
2R
2n
( 47r2Ll ss
l/2
~v’cos~Ai+v’sin~aj-v’cos~Bi-v’sin~~~2d~Ad~B
_
0
=Jzvf
(3.3)
Fig. 2. Two colliding disks.
835
Constitutive equations for a simple shear flow
where PA and fie are angles shown on disks A and B of Fig. 1. Therefore VAB= (D + s)$
sin 8i + ,/%
(3.4)
2
where v’ has no preferred direction. However, in order to ensure that A is approaching B rather than leaving B when a contact is considered to take place at 8, the angle a shown in Fig. 2 must be between --~/2 and 7c/2. This puts a constraint on the distribution of the direction of v’ as indicated by the angle a’. In Fig. 2 positive angles are measured counterclockwise from the line joining the centers of particles A and B. For a given contact location 8 and a given relative velocity VAB, the post collision velocities V,* and Vg can be computed using the conservation laws of particle dynamics [13]. The reason that particle dynamics can be appropriately utilized is because the rotational motion of these disks is neglected. These conservation laws are
V,*+%=V,,,
CV-WN= AL3* v
_~
(3.5a, b)
N
and either of the following equivalent expressions (3.6a,b) where Fp is the impulsive frictional force along the tangential direction of contact. When V,* and V2; are determined using (3Sa,b) and (3.6a,b), AM,,, and E for particle A can be computed for this particular collision as
AM,,
=
Ps-“D,2d(V: - VAB) . i,
n:2d(V: AMZA = Ps-
- VAB).j
(3.7a,b)
and
EA
=
n;2d;(v:, - vj2),
Ps-
(3.8)
These quantities associated with particle B are computed similarly. A@, or2and l? can then be obtained by integrating with respect to 0 and a’ throughout the range of each variable as determined from the following analysis identifying all possible collision geometries. 3.2 Determination of ranges of random variables Consider the disk B as shown in Fig. 3. As mentioned earlier, collisions only occur when a, as shown in Fig. 2, is between -n/2 and 7112.However, within that angle K the corresponding range for a’ as shown in Fig. 2, is not the same for different values of 8. For example consider the collision occurring at 0, as shown in Fig. 3. If v’ Q U, - U, the value of a: can vary over the full range of values 27r and collisions between particles A and B would still always occur. However at 8,, in Fig. 3, there are no values of a: for which collisions would occur if v’ * u, - us. This result for 8, stems from the fact that the velocity VA, would never have a normal component V,*, . N directed towards particle B. It can, therefore, be seen that this anisotropy is caused by the mean gradient of the flow field. A more detailed analysis will now be conducted using the contact points Or and 0, on B as in Fig. 3. If 8, is the angle of the contact point when A approaches B, the relative velocity V,*,, shown with the solid arrow, must lie within 90” of the normal direction of contact. The conditions shown in Fig. 3 represent a limiting case for a for collisions to occur. Because of the existence of mean shear, the fluctuation component of the relative velocity, $!v, lies in the range described by a:. Whenever v’ lies in a:, V& lies within 90” of the normal direction. Similarly, when 0, is the angle of the contact point, fiv’ lies
836
H. H. SHEN and N. L. ACKERMANN
Fig. 3. Anisotropic distribution of collisions.
in the range described by tl:. The probability density function of the collision distribution is quantified by the magnitude of the values of a* at the circumference of B as follows
P@)=
a*(e)
s zn
(3.9)
.
a*(e) dtI
0
By using the law of sines to the triangles defined by fiv, can be shown that (”
uA- ug and V,, in Fig. 3, it
+y$fT2e)
a*(O) = 7r - 2 sin-’
(3.10)
while the range of the direction for v’ as defined by a’ in Fig. 2 is
(3.11)
Using R = (D + s)(du/dx,)/2 &D’ as a parameter, the anisotropy of the collision distribution is shown in Fig. 4. (When R is greater than 1, an entire range of 0 about 45” on B will no longer be subjected to collisions.) This anisotropic behavior was described by Savage and Jeffrey [7] and Jenkins and Savage [8] and was shown to occur in a computer simulation of rapidly sheared disks by Campbell and Brennen [3]. However, in those works, an expression of the distribution function was not explicitly formulated as a function of the kinematic property of the flow and the material composition. In the present study u’ is quantified as a function of the mean flow gradient and the material properties of the granular constituents, and hence enable R to be determined explicitly.
Constitutive equations for a simple shear flow
0
90”
45”
837
135”
Fig. 4. Probability of collision distribution: R = (du&I(D
180”
+ s)/2,/%‘.
3.3 Post cdlision velocities To determine A@, or2and E for a given collision, the post collision velocities, V,* and Vg, of colliding disks A and B have to be determined as discussed in Section 3.1. (3.5a,b) and (36a,b) will be used to compute V,t and Vt. However, before (3.6a,b) can be utilized, the impulsive frictional force Fp has to be quantified. In the works by Ogawa et al. [l] and Shen and Ackermann [2], (3&b) was adopted. By using that equation it was assumed that
n;2d(V:, - V;) *N.
Fp= a~,---
(3.12)
The above equation states that the impulsive frictional force l$ is proportional to the impulsive normal force with the proportionality being p, the friction coefficient. There are limitations however to the applicability of (3.12). If V,*, comes “head-on” towards B (i.e. Q = 0 in Fig. 2) (3.12) indicates that some tangential component V$ *P will be generated for B after this collision. A contradiction is, therefore, created. As shown in Appendix B, (3.12) applies only when collisions occur “far away” from the “head-on” direction. This criteria can be described quantitatively as permitting the use of (3.12) when V,,*P-~(1
f&)V,,.N&O.
(3.13)
+t.)V,,*N
(3.14)
Equation (3.12) does not apply when VAB.P-/J(~
and the post collisional velocities along P should then be determined as v;.P=Vf-P
(3.15)
which says that the two disks A and B separate at no relative motion in the tangential direction. The region about the “head-on” direction, or the direction of N as described by (3.14), will be termed the “friction wedge” in this study. This wedge, as determined from solving (3.14), makes an angle n = tan-’ ,u( 1 + 6) about N. The range of the direction of v’ corresponding to this friction wedge can be obtained by geometric considerations. In
H. H. SHEN and N. L. ACKERMANN
838
Fig. 5, the outer edges of the friction wedge are formed by the sum of velocity vectors represented by dashed lines. By applying the law of sines to the two triangles composed of fiv’, uA- II, and V,,, the direction of v’ corresponding to the friction wedge can be described by its angle a‘ with respect to N as follows,
il ?l
=
(D + 7~
-sin-’
tlz
i
s)-$
IIzvl
* sinOsin(f3 TV)
.
(3.16)
1
3.4 Determination of AR,O,2 and I? Consider the situation when VABlies outside of the friction wedge as defined in Section 3.3. The post collision velocities of A and B can then be determined using (3.5) and (3.12). These velocities will be denoted by V,*,, V$,. If V,, lies inside of the friction wedge, the post collision velocities can be determined by using (3.5) and (3.15). These velocities will be denoted by V,*, V&. The average values of E and Ait? are then computed as
and
(3.18)
Fig. 5. Determination
of friction wedge: 9 = tan-’ ~(1 + c),
839
Constitutive equations for a simple shear flow
where m, = p,nD%/4 and a;, a;, q,, qz are defined in (3.11) and (3.16). Integrating (3.17) yields (see Appendix C),
E=m, [(1-E)2-$(1+&)2]
(3.19)
where
P(1+ e) - 6 cos’ q + 2 sin2 q
sin 2~ - ---+3 +cos2q(l
$(l + E)2 [n - sin 2?7(1 + cos2 7j)] -I‘2
+2cos2rt)l
(3.20)
and -4/~(1 +r)(l
-cos2q)+4,~~(1
+Q2 ( q + q?).
(3.21)
Integrating (3.18) yields (see Appendix C)
Afi=AdM,i-f-An;j;j=m
+s~i--mS(l
+$)+FM,l(D
+c)[-j&hv’
(3-22)
i
where 1 +ccdq)+$
-sin?
1
-cos2q)sinq
1.
and
FM, =
-cosq)+$l
1
(3.23)
(3.24)
3.5 Constitutive relationships l%e fluctuation speed v’ can now be computed after substituting (3.19) and (3.22) into (2.15), which then yields
&7’ =z
+z X
-- FE, 81r
j-62 4+----g---
’ ~(1
i-c)
~“‘(1+E)~
4
FE2 +8n
1
l/2
du dx,
1 (3.25)
840
H. H. SHEN and N. L. ACKERMANN
The collision frequency as defined in (2.10) can be obtained as
(3.26) where s and fiA are defined in Fig. 1 and il is defined in (2.1). All terms in the constitutive relatinons (1.1) are, therefore, specified. Substituting (2.7), (3.22) and (3.26) into (l.l), the stresses are obtained as
‘j2 ----+----_-~(1 +E) R 4
,u’(l +E)’ FE, 4 +81t
1 (3.27)
and
=22 = TC +
4~ + 37c2FM,
X
(3.28) where FE,, FE, and FM, are defined in (3.20), (3.21) and (3.23). 4. RESULTS
AND
CONCLUSIONS
The constitutive equations developed in this study demonstrate many of the characteristics that were observed by Bagnold [5], Savage [6], Savage and Sayed [14] and Sayed [15] in their experimental investigations using sphere shaped particles in a rapidly sheared granular flow. Common features to both theory and experiments include the fact that the stresses were proportional to the square of both the particle diameter D and the mean flow velocity gradient (du/dx,). Previous experimental results did not report values of the physical properties of the materials such as L and p. It is evident from the results of this investigation that such information is essential in order to be able to compare experimental results with theoretically predicted values. The significant influence of concentration upon the stress level is also demonstrated in Fig. 6 where conditions are compared when L = 2 and 3, = 5 or C = 0.4 and 0.63. The conversion from I to C is through adopting (2.2), in which C, = 0.906 is used which corresponds to a hexagonal packing. The increase in C from 0.4 to 0.63 produced an almost threefold increase in stress. Data produced by the
Constitutive equations for a simple shear flow
841
low 1
10-l
E = 0.7
.I 0
0.50
0.25
0.75
Fig. 6. Concentration
1.00
1.25
1.50
1.75
effect on shear stress.
computer simulated simple shear flow of disks also demonstrate this behavior, Campbell and Brennen [3]. The comparison between the theoretical results and the computer simulated data is presented in Fig. 7. It should be pointed out here that in the computer simulated model particle rotation was included and the frictional collisions were assumed to be such that the relative velocity at the contact point of a pair of disks was destroyed after the collisions. This assumption is equivalent to assuming all collisions are “inside” the friction wedge as defined in the present analysis. The main contribution of this paper is to quantify the anisotropic collision effect and treat the friction force correctly. If these two effects were neglected, the same outlined procedure will produce the following simplified stresses
al+6 r‘2,= c&D2 - 2 211
Tz2
r
l+c 1 311Gz
l-E2
, /iL(l +c)
112 -du 2 ~~(1 +c)”
10
---0 _
dxz
(4.1)
(4.2)
It
1
4
I
x
4
-I
A singular point exists for 721and 722. For any L, as
the stresses 7,, and zu both become infinite. As discussed in Section 2.2, when a dissipative material is used, a finite steady state stress must exist under the flow conditions studied here. This singular behavior also exists in Ogawa et al. [l] and an earlier work by Shen and Ackermann [2]. The comparison of calculated stresses using (3.27) and (4.1) are given in Fig. 8. Not only is the singular behavior of the stresses shown to be removed in the present analysis, but it also can be seen that when (3.27) is used the stress levels are
842
H. H. SHEN and N. L. ACKERMANN
0
E = 0.8
A E = 0.6
L
I
0.2
I
I
0.3
0.4
I
0.5
I
0!6
0.7
0
C
Fig. 7. Comparison between theoretical results and computer simulated data. Solid curve shows the theoretical result (3.7), data points are from [3].
increased for values of 6 and p which do not correspond to values in the immediate vicinity of the singular point. The value of R defined in Fig. 4 which measures the anisotropy of the collision distribution can be explicitly quantified. Since R = (D + s) (du/dx,)/2&‘, R can be determined by substituting o’ from (3.25) to obtain R=
*((($)(l+~)+FM,)&@$)+lii(:;f)2-$)
I
1 -fz2 4+-----
Al +c) 71
-“2 (43) ~‘(1 +c)~ ( FJ’% 4 87r
1
.
where FE,,FE, and FM, are defined in (3.20), (3.21) and (3.23). This shows that the anisotropy of collision contacts in a shear field is explicitly quantified when the concentration and material properties are specified. However, this anisotropy, as shown in Fig. 4, is exactly a sinusoidal function of the contact location. The extreme sensitivity to concentration and nonsinusoidal behavior in contact distribution observed in the computer simulation by Campbell and Brennen [3] are not predicted in the present analysis. The reason for this discrepancy is believed as a result of the assumption made in the present model. Since in this analysis the random velocity v’ and the contact point on the peripheral of a disk are both assumed to be completely random, the only anisotropy introduced is due to the mean shear. This kind of anisotropy may be called “dynamic anisotropy”. However, as solid concentration increases, another kind of anisotropy which may be called the “kinematic anisotropy” will dominate. This kind of anisotropy is due to the preferred orientation or “fabric” which takes place in a densely sheared material as observed in [3]. For a set of parameters used in Fig. 8 1 = 5, 6 = 0.7, and ~1= 0.2, R is computed as 0.43. The principal assumption used in the derivation of the constitutive relationships was that U’ > (D + S)(dz+/dxJ/2& This enables all points on a disk to be collisional points. If u’ is allowed to be smaller than (D + s)(du/dx,)/2&, R would be greater than 1 and some part on the circumference of a disk may not be allowed to experience collisions. This will then put a limitation on 13in the averaging processes defined in (3.17) and (3.18).
Constitutive equations for a simple shear flow
10
843
-1 0
I .25
I .50
I
I
.75
1.0
I 1.25
I 1.50
1. 75
Fig. 8. Anisotropic and friction effect on shear stress. Solid curve (3.27), dashed curve (4.1).
Whether or not the inequality u’ > (D + s)(du/dx,)/2,/ is satisfied depends upon the physical properties 6 and p, and the concentration of solids within the granular flow. Computations for v’/(D + S)(dUJdx& were performed using (3.25). It was found that u’(D + s)(du,/dx,) > l/2$2 was satisfied for the ranges 0 I /J < 2, 0 5 L I 1 and 1 2 2. The analysis of anisotropic collisions was not conducted for the case when (D + S)(dnJdxJ 9 u’. This situation could be realized when the granular solids were highly dissipative and interparticle distances became large.
Acknowledgement-This Foundation.
study was partially supported by a grant number CEE-821665 of the National Science
REFERENCES [I] S. OGAWA, A. UMAMURA and N. OSHIMA, ZAMP 31, 482 (1980). [2] M. SHEN and N. L. ACKERMANN, AXE Engng Mech. Diu. 108, EM5, 748 (1982). [3] C. S. CAMPBELL and C. E. BRENNEN, Proc. U.S.-Japan Seminar on New Models and Constitutive Relations in the Mechanics of Granular Materials, Cornell University, New York (1982). [4] N. L. ACKERMANN, H. T. SHEN and R. W. RUGGLES, LAHR Znt. Symp. Ice, p. 251. Quebec (1981). [5] R. A. BAGNOLD, Proc. Roy. Sac. London A225, 49 (1954). [6] S. B. SAVAGE, Proc. U.S.-Japan Seminar on Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials, p. 241. Sendai, Japan (1978). [7] S. B. SAVAGE and D. J. JEFFREY. J. Fluid Mech. 110. 2.55 (1981). 181J. T. JENKINS and S. B. SAVAGE, J. Fluid Mech. 130, 187 (1983). [91 K. I. KANATANI, Int. J. Ertgng Sci. 17, 419 (1979). DOI N. L. ACKERMANN and H. SHEN, ASCE Engng Mech. Div. 108, EMl, 95 (1982). [‘ll A. BLINOWSKI, Arch. Mech. 30, 27 (1978). WI S. OGAWA, Proc. U.S.-Japan Seminar on Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials, p. 208. Sendai, Japan (1978). u31 D. T. GREENWOOD, Principles of Dynamics, Chap. 4. Prentice-Hall, New York (1965). 1141S. B. SAVAGE and M. SAYED, EURO MECH 133, Proc. Statics and Dynamics of Granular Materials, Oxford (1980). [I51 M. SAYED, Ph.D. Thesis, McGill University, Montreal, Canada (1981).
(Received 20 August 1983)