Harvard University, SEAS
Room 312, Pierce Hall
29 Oxford St.
Cambridge, MA
02139
Jump to Stochastic Flow Rule
Jump to Nonlinear Elasto-Plasticity
Understanding The Stochastic Flow Rule (SFR)
This work provides a conceptual basis for uniting the Spot Model for dense granular flow with the Slip-Line Theory (also known as limit-state Mohr-Coulomb plasticity) for granular stresses. In doing so, the SFR becomes a highly general flow rule used for predicting velocity profiles in arbitrary quasi-2D granular flow apparati. The description below attempts to describe the major concepts of this model with minimal mathematics. For more details please see the paper Stochastic Flow Rule for Granular Materials.
I. The Spot Model
The Spot Model originated as a method for simulating full silo flows. Its main argument is that a meso-scale object called a "spot" is responsible for the motion of granular matter. A spot is a localized region of collective motion of roughly 3-5 particle diameters in width. Intuitively speaking, the spot principle resonates with the basic notion that an individual grain cannot flow independent of its nearby surroundings, but rather must always drags along some of its neighbors. This is verified by experiments and simulations which show strong velocity correlations in this range.
 |
||
A spot performing a displacement to the upper right, causing a block-like particle displacement to the lower left. |
The spot is viewed as a persistent entity which travels through the material as a random-walker. Every step of its random walk is accompanied by a collective particle displacement in the opposite direction. Spots are viewed as mutually non-interactive in the sense that they pass through one another without any repulsion or attraction. The full flow profile of the particles can then be obtained by superposing the effects of many spot trajectories on top of one another.
The simplest environment in which to apply the Spot Model is the flat bottomed silo where flow is initiated by opening an orifice at the bottom. This environment is simple because the random walk parameters for a spot follow from intuition alone. To wit, a spot should always move upward so that particles sift downward under gravity's pull. The side-to-side motion of the spot is left as the random contribution to the spot trajectory. Thus a spot step anywhere in the material consists of a step upward plus a random horizontal displacement. Ultimately, this stochastic component to the spot step results in flow predictions that share the "diffusive" looking quality previously observed in silo flow experiments.
 |
||
Spots which are injected at the orifice perform a random walk (with upward drift) through the silo. |
The most essential parameter here is the effective diffusion length b, that is, the variance in a spot displacement's sideways motion divided by twice its deterministic upward climb. Once b is measured empirically, the continuum limit for the spot distribution at steady-state is obtained via the Fokker-Planck (drift-diffusion) equation.
For a point-source of spots at the orifice, the result is the well-known Green function solution to the diffusion equation except with time replaced by vertical distance from the silo bottom. The particle velocity field is then found as the reverse flux of all the spots, giving us the time-tested result that the downward component of the velocity field spreads horizontally from the orifice as the square-root of the height and that the particle trajectories are parabolas which converge on the orifice.
 |
||
Generalizing the Spot Model requires a theory for how spot motion depends on the local material state. |
Beyond the continuum limit, the Spot Model may also be applied as a multi-scale algorithm for simulating full particle flows; spots are inserted one-by-one into a computer-generated silo packing and travel through the packing causing particles to flow accordingly.
But how can the Spot Model be applied beyond the silo drainage environment? This deep question is certainly non-trivial, since the components of a spot displacement are not intuitive beyond the simple silo geometry. To answer this question, we must have a broader understanding of granular stresses.
II. Slip-Line Theory
The dynamics of spot motion should depend on the material stress state, so first we must bring in a model for constructing granular stress profiles. Since our interest is in slow, dense, 2D flows, we choose the Slip-Line Theory of plasticity which has been used in hopper and silo design for years, but is general in its nature and applicable in any 2D flow environment.
 |
||
A 2D material element and its contact stresses. |
All quasi-static continuum stress theories must include torque and force balance at the level of a small granular element. Balance of torque implies
and inserting this result into the force balance relationship gives
These two equations invoke three unknowns and thus are not a closed system. The third and final relationship must be selected to reflect properties of granular material. In the case of Slip-Line Theory, the concept of yield is invoked for the final equation.
The theory claims that 2D bulk granular material fails and begins to plastically flow whenever the shear stress and normal stress acting along some line through the material attain a certain ratio &mu called the internal friction coefficient. This is akin to the rudimentary law of frictional sliding. The next assumption, which is responsible for the name of the theory is that everywhere within a slow flow, the material possesses a slip-line, i.e. a direction along which the shear and normal stresses are precisely at yield. This is justified by the fact that when granular materials are set into motion, even the "stagnant" regions are actually flowing at a very slow rate. Also called the "incipient failure" (IF) hypothesis, the assumption is bold and still debated in the granular community, but for a first-order understanding of stresses, it should suffice. Combining the IF hypothesis with the force balance equations gives a closed, hyperbolic system of PDE's for stresses which can be solved a number of ways.
 |
||
A material element at yield has two slip-lines intersecting through it angled &epsilon up and down from the direction angled &psi anti-clockwise from the horizontal. |
A material element at yield has some distinctive properties regarding its full stress state. The existence of a slip-line actually implies there must be exactly two slip-lines intersecting through a yielding material element, and the angle between the slip-lines is always the constant value 2&epsilon where &epsilon = &pi/4 - arctan(&mu)/2. The stress state is parameterized by two variables, the average normal stress p (i.e. pressure), and the angle from the horizontal to the major principal stress direction &psi (i.e. stress angle). The two slip-lines are symmetric about the stress angle.
Globally, the slip-lines compose two intersecting families of curves that extend throughout the granular packing forming a mesh. Indeed, the slip-lines are only "lines" at the level of a differential granular element; the continued path of a slip-line is typically curved.
III. The SFR
Defining a spot's random walk boils down to determining two spatial fields: the spot drift vector field D1 and the spot diffusion tensor field D2. The drift vector is proportional to the average displacement of a spot and represents the bias in a spot's step. For example, in the silo, the drift is upward. The diffusion tensor indicates the spatial variance in a spot displacement. With these coefficients defined, a single spot step can be broken down into two pieces: one part proportional to the drift vector plus a completely random part obtained by sampling a distribution with spatial variance D2. We immediately simplify the problem by asserting that the diffusion is isotropic and thus representable by a diffusion scalar D2. The steady-state distribution of spots can be approximated in the continuum limit using the Fokker-Planck equation
for &rhos the spot concentration.
We now seek a mechanically relevant way of defining the drift and diffusion fields. The first step is a restriction on the magnitude of these fields. Namely, if all motion stems from dynamics at the scale of a spot, the coefficients governing spot motion should also be set at the scale of a spot. We assert this hypothesis by requiring that all lengths referred to in the definitions of the drift and diffusion be set to the spot width Ls
where n is the as yet undefined drift direction. To determine n, which indicates the direction of bias on a spot step, we observe what happens to a static cluster of particles at incipient failure when a spot enters. The spot mobilizes the cluster causing the packing stresses to drop from the static friction value &mu to the kinetic value &muk. It is easiest to assess this process if we let the cluster occupy a space of width Ls between two pairs of intersecting slip-lines:
where &phi = arctan(&mu). This drop in contact friction brings the static granular cluster to an unbalanced state. The sum of the force contributions on the mobile collection can be approximated using a corollary of the divergence theorem giving
This net force pulls on the material, thus the negative of this force can be viewed as pulling on the spot causing a bias in the next spot displacement. The material is in a state of yield along the slip-lines however, and thus the cluster is far more likely to slide along a slip-line. Thus to determine the direction of spot drift n, we project the negative net force on the two slip directions and take the average. In simpler words, we claim that spots randomly walk through the slip-line mesh and that the negative net force is what nudges the spot along on its path.
The primary source of diffusion comes from the spot having to "choose" between two slip-lines each step of its trajectory. We also must accept though, that continuous slip-lines are an idealism and that the discreteness of the granular packing, especially at the meso-scale, will perturb slip-line orientations. To account for this additional source of noise, we continue with our isotropic diffusion presumption and incorporate the slip-line orientation only in the drift.
Now that n is defined, both the drift and diffusion of the spot trajectory are determined. The Fokker-Planck equation above can now be solved for steady-state spot distributions. With the spot distribution known, the particle velocity is obtained by superposing the affect of all the spots on all the particles, that is, by convolving the reverse of the spot flux -D1&rhos + D2&nabla&rhos with the spot shape.
IV. Summary
The model is now complete. It requires two input parameters only, neither of which are fitted. The first is the internal friction coefficient &mu which is measurable in a shear cell among other means. Friction values typically range from 0.25 to 0.55. The second is the spot width Ls which is always between 3 and 5 particle diameters and can be measured from the velocity correlations in a sample flow. The progression of steps to attaining a steady granular flow solution proceed as follows:
1) Solve for the stress profile in the desired geometry using Slip-Line Theory.
2) Compute the spot drift direction from the stress profile.
3) Solve the Fokker-Planck equation for the steady spot distribution.
4) Reverse the spot flux and convolve with the spot shape to attain the velocity field.
V. Results
We apply this model to a variety of setups to test its generality. First, we check that this new approach has not spoiled the Spot Model's applicability to the silo geometry.
 |
||
Left: The flat-bottomed silo geometry. The slip-line field computed from Slip-Line Theory is the crisscrossing black mesh and the red arrows represent the spot drift vector field. |
As can be seen from the above comparison to a full particle-by-particle simulation, the SFR does a quantitatively accurate job predicting silo flow. It correctly orients the spot drift upward and the spot diffusivity appears to capture the right amount of spreading in the velocity profile.
The true test, however, is whether the SFR can describe flow in other environments. One of the most common environments for studying dense flow is the annular Couette cell where grains fill the gap between two concentric rough cylinders and flow is initiated by rotating the inner wall holding the outer wall fixed.
 |
||
Left: The annular Couette geometry. The inner wall is rotated anti-clockwise to cause flow. The crisscrossing black mesh is the slip-line field and the red vector field is the spot drift. |
From the above plot, we see the range of possible SFR predictions appears to envelope the range of results obtained from a vast assortment of experiments and simulations of this flow geometry. The exponentially decaying character of the known results is excellently matched by the SFR prediction. This is the same model, with no new parameters, that predicted the previous silo flow results. And yet there are striking differences between the two environments, most notably the annular cell has curved boundaries and is driven by surface tractions, not gravity.
Continuing in this fashion, we move on to another common granular flow environment, that of a bed of grains being sheared from above by a rough plate. In this geometry, gravity is present but the flow is driven by the top boundary condition. This should serve as a strong test of the SFR's capability to reconcile boundary and body force effects on the flow structure.
|
||
Left: The plate dragging geometry (gravity is downward). The plate is dragged leftward along the top surface. The crisscrossing black lines are the slip-line mesh and the red vector field is the spot drift. |
Though the experimental shear band width is fatter than the SFR predicted range, we do note yet again that the exponentially decaying character of the experimental data is matched by the SFR prediction. It is indeed noteworthy that the SFR correctly localizes the flow in the annular Couette and plate-dragging environments even though the reason for localization is different for the two cases; in the annular environment geometry causes flow localization, whereas in the plate-dragging environment gravity causes flow localization. The reason for the shear band width mismatch may be due to an effect on the slip-line field known as slip-line admissibility, which measures the onset of rate-dependence and is discussed in depth in our long paper.
We have also tested the SFR in flow down an inclined plane. The trouble with this geometry is that different experimental studies have found significantly differing results. It has been determined that the shape of the flow profile down the incline depends quite sensitively on the precise angle of the incline, going from continuous avalanche-type flow near the repose angle, to linear flow at slightly steeper angles, and eventually to "Bagnold flow" where the velocity profile is characterized by a 3/2 power law.
Since the SFR is a slow, dense flow rule, it is of little surprise that the solution it gives for inclined plane flow is of avalanche type, the flow field known to occur right at the onset of yielding. The half-width of the avalanches is well-matched by the findings of Lemieux and Durian, Phys. Rev. Lett., 2000.
VI. Conclusions
The SFR provides a general extension of the Spot Model and is capable of predicting flows accurately in many environments. We do not claim to have proven the validity of either the Spot Model or Slip-Line Theory individually, but rather, we have shown that the SFR, which springboards off these theories, manages well as a flow rule in its own right. The success of the SFR is highlighted by the fact that it utilizes no fitting, and requires only two measured parameters which both exist in a narrow range.
Under certain geometric conditions, as previously mentioned, we believe the SFR is trumped by rate-dependent effects and we have already begun splicing the SFR with Bagnold scaling, the most common rate-sensitive law. In doing so, the joint theory has been shown to give the correct variety of flows for the inclined plane geometry as well as provide a means of smoothly connecting the banded flow of plate-dragging under gravity to the linear flow of gravity-free simple shearing.
Understanding Nonlinear Granular Elasto-Plasticity
(Under construction...)