Dissipation in granular materials

Fo r two simple sct-up:.;, we discuss how dissipative grain-grain interactions give rise to unexpected properties on large scales. ln the first example, a spherical particlc rolling along a rough surface experiences an effective viscous friction on large lime scales. It is due to temporal correlations among collisions with incomplete normal restitution. The second cxample is a sheared granular packing:. There, spatial correlations among non-sliding and sliding contacts with Coulomb frictio n suggest that the devialoric stress, although responsible for the dissip<nion, is Jocalized on bonds that are non-sliding and bence non dissipative

Granular media (such as sand) are classical many-particle systems with dissipative inleractions (for a recent review see Jager et al. ( 1996)).In the absence of external driving, the kinetic energy of the grains decreascs in each collision owing to the irreversible transfer of energy into the internai degrees of freedom of the grains!.This is called collisional cooling §.Oftcn the grains fonn lasting contacts within a finite time, which is called inelastic collapse (McNamara and Young 1994).
It happens if the relative velocity of the grains (i.e. the granular tempera ture) drops to zero.white the local pressure stays finite.An example is a heap o f sand.The weight of the grains provides a finite pressure in the pile, while the co llisional cooling eliminates ali relative motion.That this happens within a finite time can easily be illustrated with a single steel bail bouncing back from a p lane and losing a finite fraction of its kinetic energy with each co llision.ln the absence of a gravitational field the inelastic collapse may still happen in a finite region of the granular material if the outer regions provide enough pressure to compactify the inner part.ln order to keep granular media in a steady o r periodic motion, the translationa) and rotational degrees of freedom of the grains have to be agitated externally.One can regard it as the second characteristic pro pert y, besides the dissipa tive interactions, distinguishing the c1ass of granular materials from other many partic1e systems such as liquids or solids.tha t the typical agitation energy per degree of freedom has to be much larger than the thermal cnergy k 8 T /2.A sa nd pile is frozen into a mctastab\e configuration, for instance.In order to agitate any gra in, a minimal t e-mail:d.wolf(aluni-duisburg.de  :::The partial regain of translational energy from internai degrees of freedom has bcen discussed by Giesc and Zippelius ( 1996) and leads to a stochastic res titution coefficient.This will not be co nsidered further in this paper.§The term 'cooling' herc refl!rs to the so-called granular temperature which is the mean squa re deviation of the velocities from their average.energy of the order of mgr is needed, where 111 and r denote tht! mass and radius respcctively of a grain and g is the gravitational acceleration.For r > l J.tm this becomes larger than the thermal encrgy of room temperature.Therefore.for cxample Brownian motion is unimportant for granular media.In so-called dry granular media, cohesion and hydrodynamic interactions can also be neglccted.This lS the case consrdered in this papcr.
Two types of irreversible interaction mode) are important in granular ma terials.Therc is on the one hand the incomplete normal restitution in head-on collisions.The corresponding phenomenological material parameter is the normal restitution coefficient which is the ratio of the relative velocities after and bef ore the collision.lt is smaller than unity, because a fraction 1 -e~ of the kinetic energy is irreversibly lost to internai degrees of freedom of the grains.
On the other hand, therc is cnergy dissi pation during a sliding contact due to Co ulomb friction.at a rate F , t\.The friction force F 1 is proportional to the normal force Fn pressing the grains together.The dynamic friction coefficient is the ratio of th e two forces: Of course, th ese d issi pa tive grai n-grain interact ions characterized by equatio ns ( 1) and ( 2) are idealized (sec fo r example Schafer et al. (l996) and Baumbergcr and Rerthoud (1997) for refinements), but they arc Jegitimat.estarting points for eluciciating the dissipation phenomena on large scales, wh ich is the aim of this paper.
In general the d issipation in a granular matcrial is d ominated by o nly one of thesc two irreversible grain-grain interactions.For cxample, Coulomb friction is unimportant as a so urce of dissipation in a granular gas, where the dynamics are mostly due to binary collisions.On the other hand, the plastic deformation of granular packing involves ahnost exclusively sliding of particles with respect to cach other, so that the incomplete normal resti tution in the few collisions contributes only very little to the ovcra ll dissi pation.
Accordingly we present the following two examplcs: the fi rst in which the incomplete normal restitu tion is the microscopie source of dissipation, and the second in which the Coulomb frict ion is the microscopie source of dissipation.Jn both cxamples the dissipation on large scalcs shows qua litatively new fcatures which a re not obvious from the microscopie laws.The incomplete normal restitu tion givcs rise to a velocity-independent ('viscous') friction for a sphere rolling along a rough surface, and the Coulomb friction leads to a localization of dissipation on a small fraction (about 8°/tl) of contacts, when a granular packing is plastically deformed.assume thal the spheres of the incline are densely packed.More general situations have been investigated and do not lead to any important differences (Dippel et al. 1996).

Veloâty-:fàrce diagrams
Let us first consider three simpler cases.A sphere rolling down a plane accclerates.It never reachcs a steady state.By contrast, a solid block shding down the tilted plane may reach a steady stale, albeit a trivial one.If the inclination is small enough, the block simply stops slid1ng.If the driving force F = mg sin () exceeds the Coulomb friction force, however, the block will accelerate for cver.Figure 2 shows the Coulomb graph.which allows one to read off the points (v, F) for which a steady statc with velocity v and driving force F (which of course is compensated hy the friction force in a steady state) cxists.Thcse points lie on the bold lines.Ali other points (v, F) do not correspond to a stcady state but evolve in time along the broken flow lines.For example, for v= 0 any driving force smaller than F_, will be compensated by static Coulomb friction.However, for a driving force F > Fs the block will start to slide and be accelerated with the force F -Fd.The Coulomb graph should be contrasted with the corresponding diagram for viscous friction (figure 3).A sphcre falling in a viscous medium experiences a frictional force wh ich is linear for laminar and quadratic fo r turbulent How.Here any driving force leads to a steady state.The veloci ty adjusts itself such that the viscous friction compensates the driving force .
Having a rugged instcad of a flat incline, little as this change may scern, leads to a surprisingly rich velocity-force diagram (figure 4).There are at least three force intervals, scparated by with any velocity v will bccomc trapped, that is it stops rolling after passing a number of substrate spheres.If FAs is exceedcd and the initial velocity is large enough to go over the first little bump on the incline, the rolling spherc reaches a steady state.lf the driving force is larger than F 6 c, howcver, no stable state exists any longer, as indicated by the flow lines.The sphere starts to make larger and Jarger bounces.In the following we discuss the effective friction which guarantees a steadystate motion in the intcrmediate regime between FAR and Fsc• 2.2.The ej(ective ji-iction in the steady-state regime ln a steady state the driving force F can be identified (apart from the sign) wit h an effective frictional force which descrihes the dissipation of the energy input.
Molecular dynamics simulations (Wolf 1996) and experiments (Ristow et al. 1994) show that this effective frictional force depends on the velocity approximately like (3) The offset vA B is a function of the ratio of the radius R of the rolling sphere to the radius r of the surface sphcres and approaches zero for increasing R/ r (figure 5).
Equation ( 3) means that the rolling particle e.ffcctively feels a viscous friction.
There is a seçond even more remarkable observation.The data for ditTerent restitution coefficients e 11 (see figure 5) and also for different friction coefficients /-id are indistinguishable.The cffecti ve friction depends very little on the mate rial coefficients charactcrizing the dissipation on the scale of one grain.
These two key observations will now be explained qualitatively.A quantitative analysis will be given in the next section.For the explanation of equation ( 3 ln this limiting case the motion becomes particularly simple.Because P•d ___, oo the sphere has no slip relative to the bumps on the incline.It must roll without dissipation.The only dissipation happens when the rolling spherc first hits a new bump.Recause en = 0, the kinetic encrgy stored in the motion perpendicular to the new bump surface is dissipated at once.The moving sphcre does not bounce back.For simplicity we consider only the case when it stays always in contact with the inclined surface (no detachment due to centrifugai force).
The kinetic encrgy at any point on bump k + 1 is related to that at the corresponding point on the preoeding bump by The last term is the energy dissipated in one collision, and is the difference in potential energy.Obviously, for a steady state, one must have This explains why the effective friction has a quadratic vclocity dependcnce. (6) Simple geometry (figure 6) shows that the normal component of the velocity 1.' just before the collision with a new bump is where 'Ymax = sin -1 [r /( R + r) ].This explains why the steady-state velocity increases with incrcasing ratio R / r > 1 of the radii of the rolling sphere and those forming the incline (sce figure 5).The la rger the rolling sphere, the smaller is the normal component of its velocity, when it hits the new bump, and hence the Jess eflicient is the dissipa ti on .
With equations (7) and (6) and some elementary trigonometrie transformations the steady-state velocity v just before the collision with a new bump is ., ln the ncxt section we shall see that vis representative for the average velocity ii.
The surprising rcsult that the effective friction force is nearly independent of the value of e 11 can be understood in the following way.We found by computer simulation thal a steady state requires essenlially that the moving sphere undergoes an inelastic collapse on each substrate particle.One can assume that it has formed a lasting contact when it reaches the next substrate partide, with almost the same tangentia1 ve1ocity as in the case en = O.Whether or not the inelastic collapsc can be completed bcforc the next surface bump is hit depends on the coefficient of restitution.Therefore Fac is a function of <'w lncreasing en tends to destabihze the steady state.The results are independent of the friction coefficient /l'ct' as long as the moving sphcre roUs whcn it is in contact with the surface.

Anal_vtic calculation <~lthe veloàty-:fàrce diagram
In order to show the stability of the steady state and to cvaluatc the average velocity, one necds to know the kinctic energy Ekin in equation ( 4).As the sphere is roll ing, it rotates about its centre of mass with angular frequency w = vf R. The kinetic energy has a translational and a rotational contribution: (9) with an effective mass mcff = m( l + J / mR 1 ) , where J denotes the moment of inertia.
Hence equation ( 4) becomes As E ctisJ Ekin is smaller than unity, b.vJ.: converges to zero exponentially.Now we use these rcsuhs to determine the average velocity in the steady state.Knowing the vclocity v at 1ma. \ (see figure 6) any previous velocity u(! ) can he obtained from energy conservation for -'l'max < r < l'max: Solving this for 'V 2 ('y ).one obtains ( 15 ) The average velocity v is given by the arc lcngth 21'max (R + r) divided by the duration T of the contact with one bump: lnserting equation ( 13) into ( 17) the average veloci ty ( 16) is determined by The integral in equation ( U~) is an elliptic integral of the first ki nd and cannot be solved in closed form.The theoretical curves in figure 5 are obtained by numerical evaluation of this formula and are in excellent agreement with the data.
Howevcr, one can gct additional insight by expanding ii for small r/ R (or small <.:. (equation ( 8))), keeping the driving force F (i.e.th e inclination 8) fixed.This is done in the appendix.The result is This specifies the proportionality constant in equation ( 3) and shows tbat the offsets vanish in the limit r / R ~O.
Finally.we determine the value FAs of the driving force, bclow which v= 0 is the only steady state.The integral ( 18) is only finite if b > l.It diverges for h ---t 1 like lln (b-l )1, that is v vanishes like 1 /f ln (b -1) 1• We conclude that b = 1 implicitly determines F.I\B• Th is can also be seen from equation (13).While rolling over a substrate spherc, the highest point (~f = --fJ) is rcachcd with zero velocity, if h = !.
lnscrting equations ( 8) and ( 14) into equation ( 15), one finds that with the ab breviation Suhstituting b = l and solving for BAB= sin-1 (FA 8 /mg) , one obtains (see appendix) Figure 7 shows a comparison of equation ( 22) with simulation results.BAB vanishes for large R/r like cj2a, that is  ) is the analytical result for e 11 = O. § 3. GRANULAR PACKINGS: DISSIPATION BY FRICTION As in the collisional regime, the quasistatic deformation of granular materials involves both gain (collision) and loss of contacts between particles, which result in the evolution of the contact network and the appearance of an induced anisotropy.However, the average lifetime of contacts is compara ble with the macroscopic time scalc associated with the global deformation of the system.Owing to the non-smooth character of the Coulomb friction law, at those 'lasting' contacts, the particles may roll upon eac h othcr.This is a non-dissipative microscopie mechanism of deformation.If ail particlcs could roll upon one anothcr, then a granular assembly would defonn without dissipa tion.This is, h owever, forbidden by the frustration of particle rotations (Radjai and Roux 1995).This means that it is impossible for ali particles to move and rota te without sliding at sorne contacts.We see that a granular medium in slow deformation is heterogeneous not on1y because of the heterogeneous distribution of contact forces~ but also owing to the sliding (dissipative) and non-sliding states of contacts.l t is our aim here to show that this heterogeneity due to dissipation is correlated with that of the force network.

The lt'eak and the strong network of/orees
The numerical study of the statistical distribution of forces in static pac kings using the contact dynamics approach (Radjai et al. 1996) shows tha t the probability distributions PF of normal or (absolute values of) tangential forces have two distinct parts separated by the average force.The forces lower than the average ('weak' forces) are power law distributed, whereas the higher forces ('strong' forces) have an exponentia Il y decreasing probability: We observed the samc behaviour also in a simulated granular system under quasistatic; biaxial c;ompression (Radjai et al. 1997).The absence of a characteristic force in the weak part of the distrihution indicates that the weak subnetwork (supporting v.:eak forces) does not dircctly feel the influence of the deviatoric load.In contrast.the strong forces have a characteristic force set by the external Joad and the average coordination number.

The stress tem•ors
What are the respective contributions of the weak and the strong subnetwork to the transmission of forces through the system?The answer to this question needs the investigation of the stress tensor.In a granular system.the stress tensor involves both the contact forces and the contact orientations.It is obtained from (Christoffersen er al. 1981) where the summation is over ali contacts and Vis the volume of the system.Ft is the i component of the total force (normal plus tangential) at the contact c, and dj' is the j component of the vector connecting the centres of the contacting particles.
ln ordcr to separate the contributions of weak and strong forces to the stress tensor, th!! summïJtion is to be rcstricted to contacts in the weak or the strong subnetwork respectively.Figure 8 shows the principal axes and the corresponding eigenvalues of the weak stress tensor aw and the strong stress tensor <Jsat the shear peak for the wholc sample.The surprising phenomenon observed here is that the wcak subnetwork provides a negligibly small contribution to the dcviatoric part of the total slress tensor <J = aw + cl, while it represents 28% of the mean pressure.The whole deviatoric load is thus sustained by the strong subnetwork.
The absence of a deviatoric stress is the charactcristic property of fluids in static equilibriurn.The solids, by contrast.can bear a finite deviatoric stress.In this respect, the weak subnetwork in our system can be viewed as a liquid, whcreas the

.
Figure l.A sphere rolling along a rugged surface experiences an effective viscous friction.

Figure 2 .
Figure 2. T he points (v, F) on the bold li nes correspond tc steady stmes of a sol id block subject to Coulomb friction.For tixed driving force the velocity evolves along the broken lines.

FvFigure 3 .
Figure3.Frictional force F of a sphere moving with velocity •1.' in a viscous medium.For fixed drivi ng force the velocity evolves along the broken linc.
Fig ure 4. Schematic vclocity-force diagram of a sphere rolling along a ruggcd surface.Velocity v <.lveraged over dUI•arion of the contact with one surface sphere.The bold lines correspond to s teady states.Below a driving force FAn the sphere stops rolling.Betwee n FAB and Fsc there exists a stcady state wi th finite velocit y.Abovc FBc computer simul ations indicate a force intcrval, for which a steady-state vclocity can be reached from below.but not fr om ahove.For even larger driving force, no steady state is rcached.

Figure 5 .
Figure 5. Simula tion results of f i as a function of the driving force F "-mg sin e.The ratio of the radius R of the rolling sphere to the radius r or surface spheres is R/ r = 1.75 for the lowest curves ( e ), 2.25 in the middle (•) and 3 for the uppcrmost curves (.A.).The data for en = 0.7 (-----) and 0.5 (----•• ) arc indistinguishable from those ror e 0 = 0.1 (e , •• .A.).Also shown in the analytical prediction for e 11 = 0 ( --).

Figure 6 .
Figure 6.At angle ; = 'Ym:~x the rolling sphere hits a new suhstrate sphere and looses the norma l component t'n of ils velocity.The kinetic energy of the tangential motion is redistrihuted between the rotational and translational degrees of frcedom.
0)The stability of the steady-statc solution is now easily chccked.Let D.v~ = v 2 -11~ denote the distance from the steady-state value.Then the iteration has the simple form 13) with the characteristic veloci ty Vo = ( 2m g(R + r)) l/2 111.;(( and the dimensionlcss constant b = (• u )

Figure 8 .
Figure 8. Principal axes and eigen values of the stress tensor.The contribu tion of weak contact fon:es to the stress tensor is isotropie.The deviatoric stress is transmitted by the strong forces.