Campbell Stress-Tensor 1989
Campbell Stress-Tensor 1989
Campbell Stress-Tensor 1989
&473
Printed in Great Britain
449
The complete stress tensor has been measured using a computer simulation of an
assemblage of rough, inelastic spheres in an imposed simple shear flow. Only five
components of the stress tensor were found to be significantly different from zero.
These represent the disperssive normal stresses 7,,, 7yy and 7,, and the in-the-shearplane shear stresses 7,y and 7yz; furthermore, the two off-diagonal stresses, r X yand
rYz, were found to be equal so that the resultant stress tensor is symmetric. Two
modes of microscopic momentum transport produce the final macroscopic stress
tensor: the streaming or kinetic mode by which particles carry the momentum of
their motion as they move through the bulk material, and the collisional mode by
which momentum is transported by interparticle collisions. The contribution of each
to the final result is examined separately. The friction coefficient, the ratio of shear
to normal force, is shown to decrease at dense packings for both the collisional and
streaming modes. Also observed were normal stress differences, both in and out of the
shear plane, reflecting anisotropies in the granular temperature.
1. Introduction
Compared to most other branches of fluid mechanics, the flow of granular
materials is still quite a mystery. I n part, problems arise because a granular material
is, of course, solid and only adopts fluid behaviour under special circumstances.
When an appropriate state of stress is applied to a static granular material, it will
yield along stress characteristics, much like an ideal plastic material. If the
deformation is slow enough, the motion will continue in this fashion ; i.e. the material
will flow as large blocks, each consisting of many granules, moving relative to one
another along thin slip lines. This is the quasi-static regime of granular flow.
However, if the deformation occurs rapidly enough, the impact between particles
along the slip lines will be sufficient to dislodge the particles from their parent blocks
of granules, continually enlarging the intervening slip region until the entire mass of
material is moving as independent grains, each in relative motion with even their
nearest neighbours. To the eye, individual particles will appear to move in a random
manner about the average motion of the bulk material. This latter case is the rapid
flow or grain inertia regime. Within this regime, any contact between particles is
momentary, as the relative motion which drives the particles together will soon draw
them apart. A complete description of the flow field must then include both the
average velocity of the bulk material and some description of the individual random
particle velocities.
This concept of particles moving individually in a random manner within the
context of a bulk material moving as a mass under the influence of applied forces,
450
C. 8. Campbell
strongly evokes the image of the thermal motion of molecules in the kinetic theory
picture of gases. The analogy is so strong that the mean-squared average of the
random velocities has been dubbed the granular temperature and there has
recently been a great deal of success in adapting hard-sphere molecular models to
rapid granular flows, by using the granular temperature as a replacement for the
thermodynamic temperature. Indeed, the granular and thermodynamic temperatures share many of the same macroscopic effects : both generate pressures, both
are related to the local density and pressure through an equation of state, both
control the transport rates which result in the apparent viscosity and thermal
conductivity of the material, and both conduct thermal energy along their
gradients. (See Campbell & Brennen 1985b.) I n the parlance of granular materials,
the pressures associated with the granular temperature are referred to as dispersive
stresses as they act to force the particle centres apart (i.e. disperse the particles).
From a macroscopic point of view, the dispersive stresses keep the local solid
concentration small enough to maintain the bulk material in a fluidized state. Unlike
molecules, however, the interactions between particles are inelastic and thus,
breaking the analogy with the thermodynamic temperature, the granular temperature cannot be self-sustaining. Instead, to maintain the granular temperature,
energy must be continually pumped down into it, from the energy of the mean flow,
by the mechanism of shear work (i.e. the work done by stresses against the velocity
gradient). Thus there is a three-tiered energy flow path within rapid granular flows :
(i) work performed on the granular system by stresses applied a t boundaries and/or
by a body force such a gravity which collectively drive the bulk motion of the
material ; (ii) shear work generates granular temperature wherever there are velocity
gradients in the mean flow ; (iii) collisions between particles dissipate the granular
temperature into thermodynamic heat. Steady motion of a granular material implies
that the energy remains nearly constant so that whatever work is performed by
external forces on the granular systems must eventually be dissipated away as heat
by interparticle collisions. Understanding this energy path, is key to an understanding of the mechanical behaviour of rapid granular flows.
Bagnold (1954) performed the earliest detailed investigation into rapid granular
flow. He studied wax spheres suspended in a glycerine-water-alcohol mixture,
and sheared in a Couette shear cell. The results showed that even a t moderate
concentrations and shear rates, the composite ceases to behave like a Newtonian fluid
with a corrected viscosity and adopts the behaviour :
where rii is the stress tensor, pp is the density of the solid material, fii is a tensorvalued function of the solid fraction v , (v = pbulk/ppis the fraction of a unit volume
that is occupied by solid), R is the particle radius, and duldy is the local velocity
gradient. This rule has been confirmed for dry granular materials by Savage & Sayed
(1984), Hanes (1983), Hanes & Inman (1985), and by the fluid free computer
simulations of Campbell & Brennen (1985a), Campbell & Gong (1986), Walton &
Braun (1986a, b ) , Hopkins (1985) and Hopkins & Shen (1987).I n fact, as long as the
only timescale in the problem arises from the velocity gradient duldy, this behaviour
may be anticipated from a simple dimensional analysis. As such, it is not surprising
that all theoretical analyses, starting with the heuristic arguments of Bagnold (1954)
and continuing through the progressively more sophisticated work of McTigue
45 1
(1978), Kanatani (1979u, b, 1980), Ackermann & Shen (1979), Ogawa & Oshima
(1977),Oshima (1978, 1980), and Haff (1983),predict exactly the same behaviour for
simple shear flows. The most comprehensive studies along these lines, described in
Savage & Jeffrey (1981), Jenkins & Savage (1983), Lun et al. (1984), Lun & Savage
(1987),Jenkins & Richmond (1985a, b, 1986) and Nakagawa (1987),are derived from
Enskogs dense-gas model (see Chapman & Cowling 1970). The essential differences
between the predictions of all of the theories for simple shear flows lies in the nature
of the tensor-valued catchall function f i i ( v ) .
Equation ( 1 . 1 ) indicates that the apparent viscosity of a rapid granular flow varies
linearly proportionally to the shear rate. This is particularly interesting as Campbell
& Brennen ( 1 9 8 5 ~showed,
)
in their simple shear flow simulations, that thc granular
temperature is almost uniformly distributed across the gap and varies as the square
of the shear rate. In this light, (1.1) indicates that the apparent viscosity of a
granular flow varies as the square root of the granular temperature, much as simple
kinetic theory arguments dictate that the viscosity of a gas should vary as the square
root of the thermodynamic temperature. This is particularly intriguing as a recent
experimental study, Campbell & Wang (1986), indicates that the effective thermal
conductivity of a granular material in air also varies directly proportional to the
shear rate (and thus with the square root of the granular temperature), just as would
be expected from the kinetic theory of gases. (This may not be the case for more
complicated flows. Campbell & Brennen (1985b) show that this is not the case for flow
down an inclined chute, where there are large gradients in the granular temperature,
and give evidence of a conduction of granular temperature much like the
conduction of heat in a solid. Similar phenomena are predicted in many of the
theoretical models mentioned above.)
Recently the techniques of molecular dynamics computer simulations have been
adapted to the study of macroscopic particle flows. Based on well-defined models of
particle interactions - surface friction, collisions, elastic deformations, etc. - a
mechanical system of granules is set up on a computer. Body forces are applied or
the boundaries of the system are set in motion to induce flow within the particle
assembly. Experiments are then performed on the system by taking statistical
averages of the system properties. As the instantaneous positions and velocities of
the particles are known (which collectively describe the entire state of the system),
literally everything about the system can be found in this manner, including many
things that probably can never be found by direct experiment. This type of
investigation is especially valuable in granular flows where the large particlr
concentrations make laboratory measurements extremely difficult. The first work
along these lines was due to Cundall (1974),but while the utility of his simulation was
evident by the modelling of some rapid granular flows such as the emptying of a
hopper, the only quantitative measurements were of extremely slow flows which
involved only small deformations of a granular assembly. Campbell & Brennen
(1985u, b ) were the first to apply this type of simulation to rapid granular systems,
studying the flow down an inclined chute and in a Couette shear cell. This last was
extended to make detailed stress tensor measurements by Campbell & Gong (1986)
and studies of the effects of system boundaries by Campbell & Gong (1987) and
Campbell (1987). Two-dimensional stress tensor measurements were independently
performed by Walton & Braun ( 1 9 8 6 ~ )using a slightly different simulation
technique. The general type of simulation has also been used by Werner & Haff
(1985, 1986) and Haff & Werner (1986). Recently Hopkins (1985) and Hopkins &
Shen (1987) have adapted the Monte Carlo method t o granular flows, which shows
452
C. S. Campbell
remarkably good comparison with the more exact models described above. A review
of the various simulation methods can be found in Campbell (19863).
Until the smooth particle studies of Walton & Braun (19863), all of the above
simulation efforts were performed on two-dimensional flows of discs or cylinders.
This eased the extensive computational demands and allowed some excellent movies
to be produced, but somewhat complicated the interpretation of t h e results. I n
particular, the geometrical differences between the two- and three-dimensional cases
led to different interpretations of the space-filling effects of the solid fraction.
Furthermore, there could be no measurement of out-of-shear-plane normal forces
such as those reported in Savage (1979). The purpose of the current investigation is
to extend the studies of Campbell & Gong (1986) to make a detailed study of the
granular stress tensor in a simple shear flow of rough spheres.
2. Computer simulation
Other than its three-dimensional nature, this simulation is not markedly different
from those used previously by Campbell (1982), Campbell & Brennen (1985a,b ) and
Campbell & Gong (1986). Throughout the simulation, spherical particles (of mass m
and radius R ) are confined within a control volume such as that shown schematically
in figure 1. All of the sides of the control volume are bounded by periodic
boundaries ; as a particle passes through one periodic boundary it re-enters the other
with exactly the same position and relative velocity with which it left. This type of
boundary gets its name because it simulates a situation in which the control volume
and its particles are periodically repeated, infinitely many times upstream and
downstream of the central control volume. For these simulations, similar boundaries
are also used to close the top and bottom. This set-up greatly enhances the
computational efficiency of the simulation by limiting the number of particles to
those initially placed in the control volume. It has the drawback that it is only
applicable to flows with no gradients in the flow direction (i.e. steady, unidirectional
flows).
The major difference between this simulation and that used by Campbell & Brennen
(1985a,b ) and Campbell & Gong (1986) is that there are no solid boundaries
enclosing the control volume. I n the previous simulations, the shear flow was driven
by two solid walls, separated by a distance H in the y-direction, and set in relative
motion in the x-direction with velocity U , to impose a shear rate UIH. (Here and in
the following discussion, the x-direction will refer to the direction of mean motion.
The boundaries of the system generate a mean field velocity gradient in the ydirection. The out-of-the-shear-plane coordinate will be referred to as the zdirection.) In these most recent simulations, shown schematically in figure 1, the
solid walls are eliminated and the control volume is closed in the y-direction by
periodic boundaries separated by a distance H . To similarly impose a shear rate U I H ,
the periodic images that bound the top and bottom of the control volume are set in
motion with velocities gU and -;U, respectively, in the x-direction. That is, when a
particle exits the bottom of the central control volume, it re-enters the top with its
x-direction velocity increased by U and a displaced x-coordinate that reflects the
displacement of the origin of the moving periodic image. The opposite path is
followed by particles that exit through the top of the control volume. This type of
boundary is similar to that used by Walton & Braun (1986a,3) but can be attributed
originally to Lees & Edwards (1972).The major benefit is that non-uniformities, such
_j
Stationary
periodic
image
Central
Stationary
image
f-n
453
tA
as those revealed in Campbell (1987) and Campbell & Gong (1987),are not imposed
on the system by solid boundaries.
Most of the current work was performed on control volumes of 240 particles which
were originally arranged in a 6 x 10 x 4 (referring ot the x-,y- and z-directions)
cubical array. The sole exception were the simulations performed at the largest solid
fractions which were started from an initial 6 x 5 x 4 array. This was done to limit the
freedom of motion of the particles by introducing some relatively short-range order
into the system and thus to prevent bridges (an effectively solid percolation of
particles) which would transmit large stresses across the extent of the control
volume. In actual Couette flow experiments such bridges form across the shear gap,
producing large stress fluctuations such as those noted by Savage & Sayed (1984).
But to break such a bridge, and allow the continuance of the flow, the walls of their
apparatus had to be allowed to momentarily expand, as if vaulted on a pole of
particles, causing a momentary decrease in the local density. Exactly the same
behaviour could be seen in the wall-bounded simulations of Campbell & Brennen
(1985a) and Campbell & Gong (1986) which were equipped with moveable walls in
an almost exact reproduction of the Savage & Sayed (1984) apparatus. As the
dimensions of the control volume used in the current studies are fixed, in order to
maintain a uniform value of the instantaneous solid fraction, such bridges, if allowed
to form, would never clear and thus had to be prevented.
The particles interact by colliding with one another. Each collision is assumed to
occur instantaneously once the particle surfaces come into contact (this is essentially
the hard-sphere approximation often used in the kinetic theory of gases) and the
collision result is computed from a standard centre-of-mass collision solution.
Because the particles rotate as well as translate, two conditions are required to close
the system of equations : one for the relative particle velocities normal to and one for
454
G. S . Campbell
FIQURE
2. Diagram for the collision analysis.
the velocities tangential to the particle surfaces a t contact. The normal velocity
condition assumes that the particles are nearly elastic in the sense that energy is
dissipated as a result of the collision but the particles retain their spherical shape.
This is realized in the simulation through a coefficient of restitution F(F < l ) , which
is the ratio of the approach to recoil velocities, and is specified as an input parameter
to the program. For the tangential condition, the particle surfaces are assumed to be
fully rough in the sense that surface friction will always be large enough to stop any
relative motion of the particle surfaces tangential t o the point of contact. The
impulse, J , exerted by the collision is then (referring to figure 2 )
J = +(1 + E ) ( 4 . k )k + mp
2(1 + A
( q - ( q * k ) k + R ( w , + o ,x) k ) ,
(2.1)
where m is the mass of the particle, q = zi, -u, is the relative velocity of the particles
just before collision, w , and w , are the particle angular rotation rates, ~3is the ratio
of the square of the particle radius of gyration to the square of the particle radius,
and k = (x, - xl)/llx, - x111 is the unit vector pointing along the line connecting the
particle centres a t the instant of collisions. (Here, x, and x, are vectors pointing from
the origin to the centres of particles 1 and 2 respectively.)
After the initial configuration and velocities of the particles and boundaries are
chosen, the simulation is allowed to proceed as it will, with no outside intervention,
until it converges to a steady state. (For these simulations, a converged state was
assumed to occur when the total system kinetic energy achieves nearly constant
values. However, like all small thermodynamic systems, the kinetic energy will
fluctuate slightly with time, making the determination of convergence somewhat
difficult.) Starting from the initial state, convergence was achieved after as little as
500 collisions per particle for most of these simulations.
455
(uv)
(ulw)
(UV)
(Vf2)
(VW)
(UW)
(vwl)
(w2)
(3.1)
where the primed quantities indicate the instantaneous deviation from the mean
velocity. The symbol ( ) represents the average of the appropriate system properties,
sampled a t regular intervals, over a long period of system time about 2500
collisions per particle. (For more details about the averaging process, the reader is
referred to Campbell 1982.) Each term of the stress tensor is determined by the
formula
~
in exactly the same way as the Reynolds stresses in a turbulent fluid are computed
from hot-wire traces. The primed quantities represent the random mot,ion of the
particles and it is therefore appropriate to define the granular temperature, T, as
T = (u)>+R~(w~).
(3.31
(3.4)
where [Jk] is found by summing the dyadic product Jk for every collision and, a t
the conclusion of sampling, dividing the result by the system volume and the length
of the averaging period. (Physically the [ 1 average should be interpreted as the ( )
average multiplied by the collision rate.)
As in Campbell & Gong (1986), the complete stress tensor is determined by
summing the collisional and streaming contributions. The results are shown in
C. S. Cumpbell
456
200
100
50
30
20
h 10
s
3
[ 5
-3
c"
:::I
1
0.5
0.1
0.1
0.2
0.3
0.4
0.5
0.2
0.6
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.1
0.2
0.3
0.4
0.5
0.2
0.3
0.1 I
0.1
0.6
I
,
0
0.1
0.2
0.3
0.4
0.5
0.6
FIGURE
3. The complete dimensionless stress tensor as a function of the solid fraction v : ( a ) T ~
( 6 ) T , ~ ;(c) T ~ ( d ~) T ;~The
~ .lines are derived from Lun et al. (1984) and the solid symbols are from
the smooth-particle simulations of Walton & Braun (19860).
457
which had been found as a better fit to the data from molecular Monte Carlo
simulations. Here urn is the maximum shearable solid fraction which was taken to be
urn = 0.60. No theoretical line is plotted for B = 1.0 as, with no particle surface
friction, there would then be no energy dissipation mechanism within the Lun et al.
system to damp the granular temperature ; but granular temperature would still be
generated by shear work so that, in such a case, its magnitude would continually
increase and never reach steady conditions. Also, Lun et at. used only a first-order
correction to a Maxwellian velocity distribution function in their analysis, and as
such their results should only be applicable to flows that are not far from equilibrium,
458
C. X.Campbell
i.e. small velocity gradients and small energy dissipation. The curves plotted for
tz = 0.8 may just barely fall inside their range of validity while the curves for B = 0.4
and E = 0.6 probably do not.
The other item that cannot be accounted for in the Lun et al. theory is the possible
development of a shear-induced microstructure a t large solid concentrations within
the bulk material. Such a microstructure was observed in the two-dimensional
simulations of Campbell & Brennen ( 1 9 8 5 ~ )They
.
showed that, in order to maintain
a shear flow a t large density, the two-dimensional particles align themselves into
layers oriented in the direction of mean flow: this organization allows almost
unrestricted motion between the layers and thus permits a shear flow a t
concentrations that, without the layer formation, would probably exhibit sohd
behaviour. The layers affect the collisional stresses indirectly by inducing strong
anisotropies in the collision angle distribution (i.e. the probability that a collision will
occur a t a given unit vector k, connecting the particles centres). As the collisional
stress tensor is formed by the average of the dyadic product [Jk],favoured values of
k can strongly affect both the absolute and relative magnitude of the stress tensor
components. (This is doubly true as, from (2.1), the collision impulse Jitself depends
on k . ) No observations of an equivalent microstructure development have been
reported for assemblies of rigid spheres. However, molecular dynamics studies of
Leonard-Jones molecules performed by Heyes (1986) indicate that shearing forces
the molecules to align themselves into linear strings of molecules pointing roughly
in the direction of flow (corresponding to the x-direction in the current simulations).
The strings organize themselves in a triangular packing in what here would be the
(y, 2)-plane. A shear motion can be maintained a t high density within such a packing
by relative motion between the strings in much the same way as a two-dimensional
shear motion was maintained by relative motion between the layers. It seems
reasonable to expect that a similar microstructure forms a t high density in granular
shear flows, especially as it appears to be the least restrictive organization that would
kinematically permit a shear flow. One might guess, however, that owing to the
additional degree of freedom, the three-dimensional microstructure is much less
restrictive than its two-dimensional counterpart. At present, there is no theoretical
model for the evolution of the microstructure and thus the microstructure effects
could not be accounted for in the predictions of Lun et al. (1984).
The plots for 7,y and 7yy (figures 3b and 3e) also include data from the smoothparticle simulations of Walton & Braun (1986b). Unfortunately, most of their
simulations were performed for larger coefficients of restitution and the only data
that could be directly compared were for e = 0.8 and a single point a t E = 0.6. The
error bars on the points reflect the spread in their data with shear rate. The spread
exists because Walton & Braun (19863) use a soft-particle model, in which the
collision time is not instantaneous. (The rigid-particle model used here assumes that
collisions occur instantaneously. A discussion of the various simulation methods can
be found in Campbell 1986 b.) Now the only timescale in the rigid-sphcrc model is the
inverse shear rate, which makes equation ( 1 . 1 ) a dimensional certainty. The softparticle model introduces the collision time as a new timescale into the problem and
opens up the possibility that the ratio of collision time to the inverse shear rate may
have an effect on the functionf,(v) in (1.1). It is interesting to note that the largest
spread in the Walton & Braun data occurs a t the largest density when the collision
time becomes of the same order or longer than the time between collisions.
Similar to the results obtained from the two-dimensional simulation of Campbell &
459
Gong (1986),all of the rii curves shown in figure 3, show a characteristic U-shape with
asymptotes to infinity a t v = 0 and at the shearable limit, v +
0.6. (The points
shown a t v = 0 were actually computed a t a solid fraction of v z 0.01.) Exactly the
same behaviour is observed in the Walton & Braun data and is predicted by Lun
et al. (1984).As might be expected, there is very good agreement between the smoothparticle theory and the smooth-particle simulation, yet both predict significantly
larger stress levels than are found in the rough-particle simulations. (Some of the
apparent agreement between the rough- and smooth-particle data and the theory is
artificial because the E = 0.8 curve corresponds closest to the E = 1.O data points and
the E = 0.6 curve corresponds best with the E = 0.8 data etc. It would be wrong to
assume that the difference is due to the different simulation methods as comparisons
in Campbell (19863) show that both methods yield very similar results for rough
particle simulations.) The large degree of disagreement between the rough- and
smooth-particle data is somewhat surprising as wall friction appears to make only a
small contribution to the stresses. (The two-dimensional simulations of Campbell &
Gong 1986 show that the friction contributes about 10% of the total stress.) Thus
there must be another mechanism that accounts for the large differences between the
rough- and smooth-particle results. It may be reasonably speculated that the
difference is largely due to the role that particle wall friction plays in dissipating
away the granular temperature. Hence one would expect lower temperature levels
with rough particles and with the lower temperatures, smaller collision rates
(implying smaller collisional contributions to the stress tensor) and smaller streaming
contributions to the stress tensor (as it is apparent from (3.1)that the streaming
contribution is very closely related to the granular temperature).
The nature of the two asymptotes in figure 3 can be better understood by
comparing the individual contributions of the collisional and streaming modes which
are shown in figures 4 and 5 respectively. Each clearly accounts for one leg of the U
shape of the complete stress tensor and thus the low-density asymptote can be
attributed to the streaming contribution, and the high-density asymptote to the
collisional contribution. The physical underpinning of the high-density asymptote is
easy to understand. It occurs as the solid fraction approaches the shearable limit
(v+- 0.6) as beyond this limit, infinite stresses would be needed to initiate or
maintain a shear flow. The explanation for the asymptote as v + 0 is more elusive and
is best understood by digressing for a moment to consider the energy flow in a
granular material. Because energy is always dissipated in a collision, the energy
associated with the granular temperature must be continually supplied by the shear
work performed on the system, or else it would quickly dissipate away to nothing.
Thus, the magnitude of the granular temperature depends on a tradeoff between the
rate of shear work and the dissipation. The physical cause of the low-density
asymptote may be understood by remembering that the dissipation rate is
proportional to the collision rate while the stresses transmitted in the streaming
mode are independent of the collision rate. Hence as v + O , the collision rate and, with
it, the dissipation rate, go to zero. But, a t the same time, there is still shear work
performed as a product of the streaming stresses and the velocity gradient. Thus
temperature is being produced a t low density and, t o maintain a steady flow, it must
be dissipated by the few collisions that do occur. This implies that more energy must
be dissipated per collision, which, as all the dissipation mechanisms are proportional
to the impact velocity, implies large relative particle velocities and consequently a
large granular temperature. Thus in the limit as v i - 0 , the granular temperature
C. S . Campbell
460
E =
0.4 0
0.6 o
(4
E =
200
0.8 A
1.0 0
0.4 0
0.8 A
1.0 0
50
0.2
0.1
0.3
V
100
0.4
= 0.4
0.5
0.6
0.01
0.005
0.1
0.2
0.3
V
500
0.1-0
= 0.4
A ,
0.4
0.6
0.5
0.6
(4
0
0.8 A
0.8 A
1.0 0
50
0.2-0
A
O
A
0
A
:
A $ :
0.2-0
0.1-0
0.05 -
0.02
A g e o
0.01 -
0.005
FIGURE
4.The streaming contribution t o the stress tensor as a function of the solid fraction v :
( a ) T,,,; (21) 7,,y; ( c ) T ~ ( d )~7szz.
~ The
; lines are derived from Lun et al. (1984).
46 1
200
100
50
20
10
5
9
2
E
2
1
.3 0.5
0.2
0.1
0.05
0.02
0.01
0.001
t
I
= 0.4 CI
0.6
0.6 0
0.8 A
1.0 0
0.005
0.002
I
0.1
0.2
0.3
0.4
0.5
0.6
= 0.4 0
0.1
0.2
0.3
0.4
0.5
0.6
1.0 0
1.0 0
v:
C . S. Campbell
462
10
2
1
0.5
0.2
0.1
0.05
0.02
0.01
0.005
0.002
0.001
0.1
0.2
0.3
0.4
0.5
0.6
FIGURE
6. The dimensionless parameter S (equation (3.1))as a function of the solid fraction
The lines are derived from Lun et al. (1984).
u.
463
FIGURE
7. The ratio, T J T ~of streaming t o collisional contributions to the stress tensor as a function
of the solid fraction Y . The lines are derived from Lun et al. (1984).
stresses begin to dominate, i.e. just as the stresses begin to rise along the right leg of
the U-shaped complete stress pattern shown in figure 3.)
Notice, in figure 3, that the points of minimum scaled stress are shifted far to the
left of the minimum points predicted by Lun et al. (1984).A t low densities, streaming
stresses are dominant and the shifting of the minimum to the left in that region
reflects a reduction in their importance. Furthermore, the shift becomes larger as the
coefficient of restitution is reduced. This should be anticipated as the streaming
stresses are closely related to the granular temperature and a smaller 6 implies more
energy dissipation and consequently a smaller granular temperature. When the
streaming stresses are considered separately, as in figure 4, the reduction can be seen
throughout the range of solid fractions. But this only becomes apparent for the
complete stresses shown in figure 3 a t low densities where the streaming stresses
dominate. Notice that the minimum point for the E = 0.8 rough-particle simulation
is shifted just slightly to the left of its smooth-particle counterpart, a fact that can
similarly be attributed to the additional energy dissipated by the particle surface
friction.
The relative importance of the collisional and streaming contributions, T,/T,, is
shown in figure 7 along with the predictions of Lun el al. (1984). Note that the range
of v where the streaming stresses are important corresponds, as was anticipated, to
the region where S is small. The Lun et al. study predicts that there should be
minor variations in T,/T, with both density and stress tensor component. This
variation is reflected somewhat in the simulation data but the scatter of the data is
C. 8. Campbell
464
much larger than any variation predicted by the theory. However, these results do
show that the streaming and collisional contributions to the stress tensor have about
the same magnitude a t around u = 0.15, and that the streaming contribution
~ 0 . 1 ~ a~t )about u % 0.4. The streaming stress tensor thus
becomes negligible ( T x
seems to be significantly less important in these three-dimensional simulations than
for the two-dimensional flows examined by Campbell & Gong (1984). As it stands, it
seems possible to ignore the streaming contributions to the stress tensor (as was
popular in most of the early theoretical work) for much of the densities common in
granular flows, as long as one keeps in mind that it may still be very important in
some regions such as the low-density region observed near the chute bottoms by
Campbell & Rrennen (19856 ) .
By considering the nature of the collision impulse in equation (Z.l),it should be
surprising that the stress tensor ends up being symmetric. This is because the
component of the impulse which is conveyed through friction between the particle
surfaces, J,
k),
13.8)
J = * ( q - ( q * k ) k + R ( w , + o , ) x
2(1 +P)
is perpendicular to k and thus makes asymmetrical contributions to the collisional
stress tensor when formed, as in (3.4),into dyadic products with the unit vector k .
However, J is composed of two parts, that due to the relative motion of the particle
centres tangential t o the point of contact between the particles, ( q - ( q . k )k ) , (which
in an averaged sense is related to the velocity gradient U / H ) and that due to the
x k ) (which will be related to the mean rotation rate,
particle rotation (R(o,02)
(o),of the particles). Now, asymmetric stress are possible in a granular flow and
may be interpreted as torques on the particles. The macroscopic manifestation of the
torques will be either angular acceleration of the particles or spatial gradients in the
mean rotation rates; however, as in steady flow there can be no time change in the
mean rotation rate of the particles and, as the control volume configuration was
chosen to prohibit any spatial gradients, the stress tensor must be symmetric.
Campbell & Gong (1986) have shown in their two-dimensional simulation that,
considered separately, the two contributions to S do indeed make asymmetrical
contributions to the stress tensor but, when considered together, the asymmetries
cancel out. The granular flow accomplishes this naturally by fixing the mean rotation
rate (o)relative to the velocity gradient U / H . They showed that over most of the
density range, (o)H / U x -$, but it decreases sharply as the shearable limit is
approached. (Campbell 1986a has shown that the drop observed in the twodimensional simulations is due to the microstructure development within the
material.) The ratio -(a) H / U is plotted in figure 8 as a function of the solid
fraction v. Over the entire range, (o)has a value of about -$H/U and is oriented
perpendicular to the shear plane. This is more or less the same behaviour experienced
by a particle in a fluid shear flow. The lack of the precipitous drop observed by
Campbell & Gong (1986) at the larger solid fractions is one indication that the
microstructure development is not as restrictive in three-dimensional flows of
spheres as it is in two-dimensional disc flows.
As a side note, Campbell & Gong (1987) and Campbell (1987) have shown that
asymmetric stresses may be found near boundaries as the boundary can itself impose
a rotational state in the particles that collide with i t that would be different from
what the particle would naturally assume in a uniform shear flow. I n that case, the
torques induced in particles by the asymmetric stresses are balanced in steady flow
by gradients in the corresponding couple stress tensor.
'
465
00.6
0.3
0.2
0.1
E =
0.8 A
'
0.4 0
0.6 0
1.0 0
I
0.1
I
0.2
I
0.3
I
0.4
I
0.5
I
0.6
FIGURE
8. The scaled average rotation rate, -(o)H I U , as a function of the solid fraction v.
C. S . Campbell
466
1.6
0.6 0
0.8 a
1.0 0
1.4
1.2
c.
.-
1.0
.-
sa 0.8
t-a
---s
c"
0.6
0.4
0.2
0.1
0.2
0.3
0.4
0.5
0.6
1.6
1.4 .
0.1
0.2
0.3
U
0.4
0.5
0.6
(4
= 0.4 0
0.6 o
0.8 A
1.0 0
= 0.4
A//
1.2
h
I
2 1.0
8
:: 0.8
-=
c"
0.6
0.4
0.2
0.1
0.2
0.3
U
0.4
0.5
0.6
FIGURE
9. The friction coefficient T~./T,, as a function of the solid fraction v. ( a ) Collisional
contribution, (6) streaming contribution, and (c) complete stresses. The lines are derived from Lun
et al. (1984),and the dashed lines in (c) from Lun & Savage (1987) (hybrid).
467
(which neglected any streaming stresses and thus, by itself, compares poorly with the
simulation data); this is plotted in figure 9(c) and, once again, the altered theory
agrees better with the simulation results. In either case, the theories predict that
T , ~ / Twill
~ ~decrease for small values of v but will then increase towards the shearable
limit.
To improve the physical understanding, the results are divided up to show the
individual contributions of the collisional and streaming stress tensors. The streaming
contribution to ~ , ~ is/ a7decreasing
~ ~
function of v and contributes most to the
dramatic decline in the friction coefficient. Exactly the same trend in the streaming
stresses is predicted by Lun et al. (1984). At the same time, the collisional
contribution to the friction coefficient actually increases for small values of the solid
fraction and only decreases as the shearable limit is approached. This latter decrease
is not anticipated in the Lun et al. (1984) calculations, which predict that the friction
coefficient should rise as the shearable limit is approached. Campbell ( 1 9 8 6 ~has
)
shown that for two-dimensional flows the decrease in the collisional contribution to
7xy/7yy, near the shearable limit, can be explained by the formation of the internal
.
that the
microstructure observed by Campbell & Brennen ( 1 9 8 5 ~ ) Remember
microstructure induces preferred choices of the collision vector k which will strongly
affect the absolute and relative magnitudes of the components of the collisional stress
tensor T~ = [ J k ] .In two-dimensions the layer formation restricts a particle to collide
with particles in its own layers and those in its two nearest neighbouring layers. Now,
the particles from neighbouring layers restrict a particles motion to a narrow band
about the centreplane of the layer. Thus the collisions between particles within the
same layer are restricted to a narrow range of angles about the midplane of the laycr
and the collisions with particles from neighbouring layers are restricted to a similarly
narrow range of angles but are roughly perpendicular t o those that occur between
particles within the same layer. The similarity between the current results and their
two-dimensional counterparts indicates that the behaviour of the collisional friction
coefficient can be accounted for by an equivalent three-dimensional microstructure
development such as the organization of particles into stringsas observed by Heyes
(1986). Such a microstructure would affect on the three-dimensional collision angles
in much the same way as the layered microstructure does in two dimensions. That
is, collisions between particles within the same string occur about the poles of a
particle while collisions between a particle and those in neighbouring strings would
be more or less evenly distributed about the equator. This could explain the
reduction in the collisional friction coefficient as the density is increased (and thus
the microstructure becomes increasingly more confining). However, no clear
connection can be made between the microstructure and the reduction in the
streaming friction Coefficient. As mentioned previously, the microstructure development could not be accounted for the Lun et al. theory, which explains why their
predictions do not fall near the shearable limit.
Figures 10 and 11 show the variation of the normal stress ratios r,,/ryy and T ~ ~ / T , ,
as functions of the solid fraction v. No theoretical lines are plotted because Lun et al.
(1984) predict that there should be no normal stress differences. However, an earlier,
though less complete, paper in that series, Savage & Jeffrey (1981),does predict that
a t large values of the parameter S, T,, is smaller than r,, and rYy(the latter two of
which they predict to be equal). The data show that none of the normal stresses are
equal and that for all values of u , T,, is by far the largest, taking up to six times the
value of the smallest, rZz.
Figure 10 shows a plot of the ratio T , , / T ~ ~as a function of the solid fraction v.
C. S . Campbell
468
(b)
7-
6C
= 0.4
0.6 o
0.8 A
1.0 0
'2e
4-
2.
3
3
a a
2
0
0.1
0.2
0.3
0.4
A
0
1-
0.5
A 8
0 8 b o
I
0.6
FIQURE
10. The ratio of in the shear plane normal stresses T , , / T ~ ~as a function of the solid fraction
v. ( a ) Collisional contribution, ( b ) streaming contribution, and (c) complete stresses.
469
C. X.Campbell
470
1.4
O
1.2
2 0.8 c
o.6
0.2 I
0
= 0.4
0.6 o
0.8 A
1.0 0
0.1
0.2
0.3
0.5
0.4
1.4
0.4
1.2 -
A
0
1.0 0
0.1
0.2
0.3
0.4
0.5
11
0.6
o o o o
= 0.4
0.6 0
0.8 A
0.2 1
0
t
1
0.6
*
0
a 1.0-
0.6
A A S
0 0
c 0.8 -
--5
c
0.6
0.4
E = 0.4
1
-
0.21
0
0.6 o
0.8 A
1.0 0
0.1
0.2
0.3
0.4
0.5
0.6
FIGURE
11. The ratio of normal stresses T ~ ~ / T as
, , a function of the iolid fraction v. ( a )Collisional
contribution, ( b ) streaming contribution, and ( c ) complete stresses.
5. Conclusions
This paper has presented the results of a detailed study of the stress tensor that
is generated in an imposed simple shear flow of inelastic rough spheres. The study
was performed using a computer simulation which allows access to all the details of
the flow. The stress tensor was found to be symmetric under all the conditions
studied. Also, only four of the nine components, rxx,rxy,ryyand rzz,were reported
47 1
C. S . Campbell
472
CAMPBELL,
C. S. 1982 Shear flows of granular materials. Ph.D. thesis and Rep. E-200.7. Division of
473