# Journal Club Theme of November 2009: Steady granular flow

How does sand flow? A surprisingly difficult question.  This entry tells the story behind a model detailed in [6] for dry, dense, steady-flowing granular materials.

As an initial thought experiment, consider a long, 2D simple shear cell filled with viscoelastic disks.  The disks are slightly bidisperse (to avoid crystal packings), and have average diameter d and mass m.  The rough plates of the cell are compressed together with pressure P and sheared relative to each other at a rate dv/dy.  Supposing the disks are always composed of the same material, we ask the question:  How does the steady-state packing fraction φ and shear stress τ depend on the given parameters?  Dimensional arguments are helpful here. Aside from φ, there are two other dimensionless groups that can be constructed: I = (dv/dy) √(m/Pd) is the normalized shear-rate, and μ = τ/P is the effective friction.  The problem should have a unique steady behavior, so it follows that μ = g(I) and φ = h(I), since I is the only dimensionless group determined solely from the problem setup.  These dependences generalize "Bagnold scaling" [2];  Bagnold was the first to observe that at fixed packing fraction, the pressure exerted on the wall of a granular shear cell depends quadratically on the shear-rate.

Disk shear simulations of [3], revealed that the functions g and h always have a relatively simple form in the "inertial regime"  (1e-3 < I < 1e-1), characteristic of shearing in day-to-day flows like an hourglass.  Namely, φ = h(I) stays roughly constant at the random-close-packing value, and μ = g(I) ≈ μ_s + β I indicative of a rate-dependent flow stress with static yield criterion μ_s.  Here, rate-sensitivity arises from the dominating role of impact dissipation in the inertial regime; increasing the normalized shear-rate increases both the frequency of collisions and the energy loss per collision.

Next, the work of Jop et al. [5] extended these 2D results into a 3D rheology for monodisperse spheres by trying a straightforward re-interpretation:  Replace τ and dv/dy with the equivalent shear stress and equivalent shear-rate, let P be the hydrostatic pressure, presume incompressibility, and enforce codirectionality to relate the direction of the deviatoric stress tensor to the deformation-rate tensor (non-associative).  In essence, a Bingham fluid treatment of granular matter was proposed with Drucker-Prager yield criterion μ_s.  The model was heralded as a major step forward, as it turned out to be capable of predicting highly inhomogeneous 3D flow profiles with accuracy.  The applicability of the continuum rheology was bolstered in [7], where it was shown that the size-scale for RVE behavior in 3D inertial flow is generally a mere 5 particle diameters.

However, a key ingredient was missing.  Bingham models give no stress computation in static regions (below μ_s).  Solid-like zones are common in steady granular flow, and must be described in order for the law to be mechanically well-posed under admissible kinematic/traction boundary conditions.  Thus, the next step was to splice a granular elasticity law into this framework, converting the Bingham model to an elasto-plastic model.  The elasticity of a static granular material element is itself a complicated phenomenon, since even in the small-strain limit, the elastic response cannot be approximated as linear because grain assemblies do not support tension.  The work of Jiang and Liu [4] proposed a nonlinear granular elasticity model, which expanded on successful mean-field theories of Hertzian contacts.  With demonstrated experimental validation, both in terms of acoustic and static behavior, the Jiang-Liu elasticity law seemed to fit the bill.

My goal in [6] was to merge the Jiang-Liu elasticity model with the Jop flow law, to produce a unified granular constitutive law, which can be implemented in FEM, and used to predict steady stress and flow profiles throughout any 3D geometry. Following similar theories for thermodynamically compatible elasto-plasticity in [1], my approach was to presume a multiplicative Kröner-Lee decomposition of the deformation gradient.  Running the model to steady-state as a VUMAT in ABAQUS, several flows were computed in different 3D geometries (e.g. silo, inclined chute, annular shear).  The predicted flow and stress profiles compared favorably against the data of a number of experimental and discrete simulation studies.

While there is no shortage of remaining open questions in granular flow, it seems the most important to the study of dense, well-developed flow would be a general description of the "quasi-static" flow regime (I < 1e-3), which characterizes many geological applications.  Its rheology appears significantly more complex than the inertial regime, with rate-independent flow stresses and a non-local size dependence.

### question for Ken on non-local size dependence

Ken,

Thanks for the excellent and very interesting discussion.

As you probably know, Bob Behringer here at Duke has been studying granular flow problems for some time, and has focused attention on the "force chains" that develop in these systems.  Essentially one sees something that to me looks like non-local, long range interaction between grains. Is this what you were alluding to in your last sentence?

I assume that the approach with mean-field theories here in part is to average such effects out?  How successful do you feel they've been at that?

### force chains

Hi John,

Indeed, Bob Behringer has done some very nice work on this topic.  Your question is important, and is evaluated closely in the paper [7].  Bob has shown that for 2D disks, there can be long force chains that seem to preclude the use of a continuum approach on the size-scales we care about.  However, in [7], we have shown that the force chains shrink dramatically in length when one switches to 3D (i.e. flows of spheres rather than disks).  This can be understood mechanically as an artifact of there being more ways to attain equilibrium in 3D ---- in 3D flows, each grain has on average more neighbors than a disk in 2D, so it becomes much less likely that only two contacts are significant.

We have found (in [7]) that a 3D volume element having a width of roughly 5 particle diameters, contains a diffuse network of short force chains (usually <3 diameters long) so that the "chain" effect does indeed average out, giving a well-homogenized continuum law at that scale.

### Re: Force chains in 3D vs in 2D

Dear Ken,

The subject is new to me and I haven't had a look at the papers (or the topic in general), but going purely by my engineering sense, for whatever it is worth (and while acknowledging that such things as the aforementioned "sense" can rather easily be misleading), I still tend to think that, perhaps, more research would be necessary to settle the issue regarding force chains in 2D vs. 3D. And, indeed, what you've written seems to go more or less in this same direction.

After all, if the volume has a width of about 5 dias, and the force chains can go up to 3 dias, it already is 60% length-wise, and so, one already begins to think in terms of whether (i) all the experimental parameters and measurement methods were comparable or not, and (ii) if the measurements in 3D had sufficient resolution or not.

Just a thought in the passing, that's all... Don't take it very seriously... :) [Not my field!!]

--Ajit

### Validity of plasticity models

Ken,

Thanks for posing this nice topic.

I have two questions:

1- Are the dimensionless parameters history dependent as well?

2- Under which condition plasticity theories can be used
for granular matter i.e. continuum approach for discrete particles?

### history dependence

Hello Kazem,

Both good points. Yes, there is indeed a history dependence to all these parameters, though here we are looking only at the steady-state flow, which should (hypothetically) saturate the histroy effects. Definitely one of the next steps in a model like this is to account for the transients.  For instance, a granular assembly that begins very loosely packed has a much different stress/strain curve in simple shear than one that starts tightly packed.  Granular models such as Critical State Theory have had success in representing this kind of behavior so perhaps some variant of that approach could work.

As for your second comment, I encourage you to have a look at reference [7].  In that work, we looked at discrete particle simulations of various granular flows involving hundreds of thousands of spherical grains.  We observed that within the flow regime we care about (i.e. inertial flow), a volume element of size ~ 5 particle diameters is actually big enough for a continuum law to apply.  For example, coaxiality of the stress and strain-rate tensors is observed when space-averaged (instantaneously) at this scale, and ~ deterministic relationships relating I to μ and φ are also observed.  For more about the discrete vs. continuum debate on granular flow, you can also have a look at my response to John's comment.  Indeed the question of how much of a "continuum" a collection of grains can be, is an interesting one, and definitely not obvious.

### Does moisture matter?

I know the model in [6] is for dry granular flow. However, can this model with some minor modifications be applied to granular flow with small amount of moisture ? My daily work is related to coal dry feed with 1% to 10% moisture...

### Effects of moisture

I'd imagine that moisture in that range is likely to cause some measurable cohesion (depending on the surface properties of the grains).  If this is so, it seems that the most important adjustment would be to switch from a purely Drucker-Prager failure condition to a mixed condition that adds a cohesion term.  Also, if the grains cluster during flow because of the moisture, the value of d would have to be adjusted to reflect the "effective" grain size.  These are my intial thoughts.