Stochastic Plasticity Theory:

The Derivation and Analysis of a Stochastic Flow Rule

in 2-D Granular Plasticity

Ken Kamrin

Department of Mathematics, Massachusetts Institute of Technology

1 Introduction

Granular materials are surprisingly complicated. Despite the numerous achievements of

modern science over the last 100 years, basic phenomena like particle drainage and ramp

flow remain poorly understood. The simple microscopic makeup of granular materials and

their fundamental nature in everyday life adds special appeal to the granular enigma.

This paper is an effort to shed new light on the problem of quasi-2-D dense granular

flow. It details a novel way to consider how stresses within a granular material translate to

material flow. This new flow rule, called the Stochastic Flow Rule, reconfigures the field of

granular plasticity into a form which we call “Stochastic Plasticity”.

We first describe two current methods for obtaining flow: the Kinematic Model (and

its microscopic generators the Void and Spot Models) and Mohr-Coulomb Plasticity. The

Stochastic Flow Rule is then introduced and compared to these models. A rigorous mechan-

ical derivation ensues to assure us that the Stochastic Flow Rule, while intuitive, is indeed

supported by physics. We then delve into the means by which we may extend Stochastic

Plasticity to a general form, able to be used in any quasi-2-D boundary conditions.

2 Kinematic Model

The Kinematic Model (Nedderman, Tuzun, 1979) [?] is a phenomenological method by which

to understand granular flow. Its constitutive equations can be arrived at several different

ways, but we will begin by considering the principle of Void motion.

This principle, known itself as the Void Model (Litwiniszyn 1963, Mullins, 1972), claims

that dense drainage is governed by the motion of non-interacting voids which enter at the

1

orifice and propagate up the silo while randomly walking horizontally along a lattice of

packed particles (see Figure 1) similar to the flow of vacancies in a crystal. [?]

Figure 1: The Void Model’s depicition of granular drainage.

For a fully developed 2-D silo flow, the distribution of voids, ρ = ρ(x, y, t = ∞), must obey

the steady-state Fokker-Planck equation with constant upward drift and constant horizontal

variance in the steps:

0 = − (0, ∆y/τ)︸ ︷︷ ︸D1

·∇ρ +1

2τ

((∆x)2 0

0 (∆y)2

)

︸ ︷︷ ︸D2

∂2 ρ.

We now apply the approximation (∆x)2 ∼ ∆y ¿ 1. The equation now takes the form of a

diffusion equation but with time replaced by the vertical height y,

∂ρ

∂y= b

∂2ρ

∂x2,

where the Kinematic diffusion length b, is defined by (∆x)2

2∆y.

To obtain a velocity field from the void density, we utilize the notion that void concen-

tration should be proportional to the downward velocity. This is sensible since the motion

of a void always corresponds to the same vertical position change of a particle and thus high

void concentration means proportionally high downward flux of particles. Thus we write

∂v

∂y= b

∂2v

∂x2

for v the downward velocity component. To obtain the horizontal component, we assume

(approximate) incompressibility of the material:

∇ · ~v = ∇ · (u,−v) =∂u

∂x− ∂v

∂y= 0

=⇒ ∂u

∂x= b

∂2v

∂x2

2

by the diffusion equation for v. Thus

u = b∂v

∂x+ K(y)

for some function K. We choose K = 0 for physical reasons (i.e. to ensure 0 horizontal

particle drift) leaving us with our two constitutive equations for the Kinematic Model:

∂v

∂y= b

∂2v

∂x2, u = b

∂v

∂x.

The Kinematic Model has been quite successful in determining mean field velocity profiles

for drainage, especially when not far from the orifice. However, the parameter b must be

defined empirically. Attempts to use b values corresponding to typical 2-D lattice structures

(e.g. hexagonal, FCC) have failed to match experimentally obtained values. Experiments

consistently find b greater than the particle diameter d whereas b < d according to the

standard lattices [?].

The microscopic properties of flow predicted by the Void Model differ enormously from

experiment. To wit, there is too much mixing in the Void Model. To correct this, Bazant

constructed the Spot Model in 2000 which follows a similar argument but claims that ex-

tended “spots” of slightly lower density are the true random walkers, not fully empty voids

(Figure 2). A spot can be wider than one particle and when it moves, the particles through

which it passes likewise exhibit highly correlated motion. These localized correlations have

been seen experimentally and in simulations, suggesting that the principle of spot motion

may indeed be fundamental to the nature of dense granular flow. [?]

Figure 2: A “spot” propagating upward, randomly choosing to go rightward.

The steady state continuum limit of the Spot Model still yields the same Kinematic

constitutive equations for mean velocity. Even so, the fact remains that both the Spot and

Void Models are phenomenological, not physical, and likewise provide us no incite as to the

origins of the elusive b parameter and how it could depend on properties like friction.

3

3 Mohr-Coulomb Plasticity

With foundations dating back to the 19th century, the Mohr-Coulomb plastic analysis of

granular materials and soils is possibly the oldest methodology for determining granular flow.

Formulated mathematically by Sokolovskii in the 1950’s [?], it is based on the assumption

that granular material can be treated as an Ideal Coulomb Material (ICM), i.e. a rigid-plastic

continuous media which yields according to a Coulomb yield criterion

|τ/σ| = µ ≡ arctan φ

akin to a standard friction law (with no cohesion). Calculating the various stresses on such

a material in equilibrium is greatly simplified in 2-D with the aid of a tool known as Mohr’s

Circle. A solid mechanics “slide-rule” of sorts, Mohr’s Circle enables a quick determination

of the normal and shear stresses along any plane in a static material given the stresses

σxx, σyy, and τyx in the chosen Cartesian frame of reference (τyx = −τxy to balance torques,

so τxy is not a free variable). Figure 3 illustrate how Mohr’s Circle is used to determine the

stresses (σθ, τθ) within a 2-D material element. The lines τ = ±µσ have been drawn into the

Mohr’s Circle diagram and are called the “Internal Yield Locus”, or IYL. A material element

whose Mohr’s Circle is tangent to the IYL (as drawn) is in a state of incipient yield or yield

criticality. When this occurs, the two points of tangency between the IYL and Mohr’s Circle

represent the two lines along which the material element could fail. [?]

With only two equations for force balance (in 2-D), yield criticality is an appealing notion

since it puts a constraint on the 3 stress variables thereby closing the equations. Thus to the

ICM assumption, we add that granular materials are presumed to be everywhere critical.

θ

σ σ

τ

τ

σ

σ

xxxx

yy

yy

yxτ

yx

xy

τxyτ

θσθ

� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �

� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �

(σ , τ )xx xy

τ

σ(p,0)

yy xy(σ ,−τ )

(σ , τ )θ θ

2ψ2θφ

1σ 3σ

τ = −µσ

τ = µσ

Figure 3: (Left) Illustration of the stresses on a 2-D material element. (Right) The corresponding Mohr’sCircle for determination of (σθ, τθ).

It is apparent from Mohr’s Circle that there are always two planes along which the

material shear stress is zero. These planes are known as the major and minor principle

4

planes respectively and their associated stresses denoted σ1 and σ3 are called the principle

stresses. The parameters p and ψ are known as the stress parameters and fully define any

stress state for a material element in incipient yield, i.e.

σxx = p(1 + sin φ cos 2ψ)

σyy = p(1− sin φ cos 2ψ)

τyx = −τxy = p sin φ sin 2ψ.

As indicated on the Mohr’s Circle diagram, the parameter ψ is the angle anti-clockwise from

vertical along which the major principle plane lies.

Force balancing now leads to two PDE’s for the stress state of a static granular material

on the verge of yield:

(1 + sin φ cos 2ψ)px − 2p sin φ sin 2ψ ψx + sin φ sin 2ψpy + 2p sin φ cos 2ψ ψy = 0 (1)

sin φ sin 2ψ px + 2p sin φ cos 2ψ ψx + (1− sin φ cos 2ψ)py + 2p sin φ sin 2ψ ψy = γ (2)

where γ is the material’s weight density (weight/area). The system is hyperbolic and there-

fore may be solved using the method of characteristics. The characteristic equations are:

dp− 2p tan φ dψ = γ(dy − tan φ dx) alongdy

dx= tan(ψ − ε)

dp + 2p tan φ dψ = γ(dy + tan φ dx) alongdy

dx= tan(ψ + ε)

where ε = π/4 − φ/2. A couple manipulations along Mohr’s Circle shows us that the

characteristic lines through a point are always directed along the two slip lines through that

point.

Now that the stresses have been described, a flow rule must be inferred based on those

stresses. The continuous nature of the ICM assumption suggests that symmetry should be

kept with respect to the principle stress planes. Thus co-axiality (AKA Levy Flow Rule,

isotropy) is adopted wherein material must contract along the major principle stress plane

and expand along the minor [?][?].

Together with mass continuity, the following two velocity equations are hence formed:

uy + vx = (ux − vy) tan 2ψ (3)

ux + vy = 0. (4)

Given ψ(x, y) from equations 1 and 2, the above is another hyperbolic PDE system. One

might ask, why, if velocities were going to be inferred from the stresses, did we not include

convective terms in the stress equations? Solving a fully coupled 4-dependent-variable hy-

perbolic system is excruciating. Since the inertial terms are in general small compared to the

stress derivatives, we justify neglecting them, vastly simplifying our work by splitting the

5

PDE system into two smaller ones. But, in doing so, we should clarify as an assumption that

the flow we solve for cannot be fast and in fact must be quasi-static. With the assumptions

now all stated, we declare equations 1, 2, 3, and 4 as the constitutive equations for flow

according to Mohr-Coulomb Plasticity Theory.

In real life, we cannot actually create 2-D flows, so we must settle for quasi-2-D in the

sense that the third dimension of the flow is quite small compared to the other two. Once

in 3-D, we must consider the existence of a third principle stress pointing in a direction

orthogonal to the other two. Luckily, the equations for flow within quasi-2-D boundaries

reduce to the regular planar flow equations thanks to a property of the Coulomb yield

criterion which states that the intermediate principle stress does not effect yield. Judging

from the strain-rate of material in a typical quasi-2-D flow, we infer that the major and

minor principle stress directions are parallel to the plane of the flow. Any stress in the third

dimension must therefore create only an intermediate principle stress on the material and

consequently will not affect the constitutive equations. [?]

The theoretical shortcomings of Mohr-Coulomb Plasticity are mostly in its assumptions.

For one, a realistic granular material is not continuous; the microscopic grains composing it

are usually visible to the naked eye. Intuitively, packing geometry and particle size should

have a significant effect on the flow profile. Also, the model makes no modifications for

changes in the friction coefficient as material goes from static to dynamic. The quasi-static

assumption seems questionable as well and will be checked in depth later.

The predictive shortcoming of Mohr-Coulomb plasticity have been shown experimentally.

The fully hyperbolic nature of the constitutive PDE’s frequently entail discontinuous solu-

tions in setups where experiment indicates smooth, unbroken flow. For example, see Figure

4 [?].

Figure 4: Numerical solution to the plasticity equations in a conical hopper. Radial velocity componentdisplayed.

6

4 Stochastic Flow Rule

In an effort to patch some of these issues we develop a new flow rule to substitute co-axiality.

For simplicity, let us first consider how the rule works in one particular set of boundaries.

Later in this paper we will study how to generalize the rule to arbitrary setups.

Suppose we have a flat-bottomed, quasi-2D silo. In this special case, it just so happens

that ψ(x, y) is identically 0 and the stress characteristics are likewise perfectly straight lines

angled at ±ε from the horizontal axis as indicated in Figure 5. Let us add that the silo must

be ‘wide’ in that the slip-lines eminating from the orifice meet the surface of the material,

not the walls (i.e. height ≤ width· tan ε).

Silo bottom

Surface

Slip lines

Exit

����������

Figure 5: Slip lines in a wide, quasi-2-D silo.

Co-axiality, the standard flow rule in plasticity theory, claims the flow is governed by the

deformation process pictured in Figure 3. Note that the the orientation of the slip lines is

completely ignored. The material deforms based solely on principle plane alignment.

��

��

Figure 6: The Co-axial Flow Rule.

Now, in contrast, consider the sequence of events in Figure 7. This process illustrates

the Stochastic Flow Rule, a new manner by which to view granular flow. It claims the

flow is dictated by the partial-fluidization, or mobilization, of individual cells. A mobilized

cell essentially “moves” along established material slip-lines by inciting slip on neighboring

material. Each step, the mobilization transfers to a new cell by jumping ±a cos ε horizontally

and a sin ε upward. One may liken this to the Spot Model but with a lattice structure

determined theoretically, not empirically, by the mechanics of Mohr-Coulomb Plasticity.

The cell length D, however, is still required a priori and as yet has not been theoretically

derived. It is assumed to be approximately constant.

7

Partially fluidized

D

Slip

Slip

or

Figure 7: The Stochastic Flow Rule.

This flow methodology, wherein we execute the Stochastic Flow Rule on a lattice con-

structed from the slip-lines of Mohr-Coulomb Plasticity, is called Stochastic Plasticity Theory

(SPT). One apparent bonus of this perspective on granular flow is that it minimizes the de-

pendence of b on unknown parameters. The Kinematic Model defines b in terms of the lattice

spacings ∆x and ∆y, both theoretically undetermined, whereas SPT gives b solely in terms

of D, i.e.

b =D cos2 ε

2 sin ε.

The angle ε is not an undetermined parameter as it is directly given once the internal

friction angle φ has been measured. Moreover, a natural correlation exists between what

SPT refers to as D and what the Spot Model refers to as the spot diameter. Typical spot

diameters are from 3-5 particle diameters, and internal friction for glass beads is almost

always between 20o and 25o. Using these ranges and the b formula above, SPT predicts b in

the range

1.75d < b < 3.31d .

This compares quite well with the experimentally determined b range of 1.3d < b < 3.4d. It

seems as though SPT could be a promising and surprisingly simple way to describe granular

flow.

The theory is based on the same stress mechanics as Mohr-Coulomb Plasticity, yet the

invokation of the Stochastic Flow Rule enables us to account for several granular properties

beyond co-axiality’s realm:

1. Frictional Hysteresis: When a static material slips internally, the friction coefficient

on the mobilized side of the rupture will decrease to a dynamic value causing immediate

internal failure some distance from the rupture. In SPT, the cell length D can be

thought of as this hysteresis length.

2. Randomness: Granular media is inherently random in its packing. It thus makes

sense for a model of granular flow to have a random component accounting for the

8

small, unpredictable fluccuations in the stress profile due to non-uniform packing.

3. Discreteness: The discreteness of individual particles and the observation of spots of

correlated motion in dense flow make a fully continuum-mechanical model less appeas-

ing. The Stochastic Flow Rule is, at its most fundamental level, a treatment of flow

via the dynamics of discrete cells of material.

While the quasi-static assumption is still utilized by SPT, there is significant evidence

that this may not be of much concern. It has been experimentally verified by Choi et al. [?]

that in a dense silo flow, the Peclet number

Pe = UD/b

is strongly independent of flow rate. Thus a faster flow is very much like a slow flow in

fast forward. Consequently, a velocity profile obtained in the quasi-static limit should still

represent a developed flow up to a multiplicative constant factor.

5 Physical Formalization

The previous section has illustrated the basic mechanism of the Stochastic Flow Rule. In this

section we seek to formalize its use physically by answering the questions “what is exactly

meant by ‘partial-fluidization’?” and “why does the fluidized region cause slip excitation as

pictured in the flow rule?”.

Partial fluidization, a term defined by Aranson and Tsimiring in 2002 [?], is a state of

phase for granular material wherein the shear stresses depend partially on shear strain (as in

a fluid) and partially on applied shear stresses (as in a solid) depending on the value of the

order parameter ρ which varies continuously from 0 (fully liquid-like) to 1 (fully solid-like).

Suppose we choose the x direction to align with one of the slip-lines in a material element.

Now, suppose the material begins to fail along the x direction. The partially fluid tensorial

relationship is then

(σxx τxy

τyx σyy

)= −η

(2∂u

∂x∂u∂y

+ ∂v∂x

−(∂u∂y

+ ∂v∂x

) 2∂v∂y

)

︸ ︷︷ ︸fluid tensor

+

(σ0

xx ∓ρ(x, y)µσ0xx

±ρ(x, y)µσ0xx σ0

yy

)

︸ ︷︷ ︸static tensor

(5)

where η is a viscosity and

σ0 =

(σ0

xx ∓µσ0xx

±µσ0xx σ0

yy

)

is the tensor of applied stresses on the material element. The static tensor is thus the stress

tensor corresponding to what a material element in the same geometric configuration would

9

feel were it ‘frozen’ in place and held at incipient yield with µ → µρ. Thus, one can think of

ρ as the fractional change in µ brought on by material mobilization. In essence, the tensorial

relationship claims that the stresses on a material element can be obtained by multiplying

its strain-rate by η (as in a fluid) but then adjusting for the degree to which the flow operates

like a rupturing solid, able to support shear stress as a non-linear function of shear strain-rate

along the rupture.

For our purposes, whenever we refer to a fluidized cell, we assume the cell still has a large

solid component (as the flow is slow and dense) and likewise has ρ ≈ 1. Thus, the presence

of a fluidized cell negligibly affects all phenomenon caused by the static stress tensor such

as slip line orientation and non-fluid related stress interactions between adjacent cells.

Observe the situation in Figure 8. It shows a partially fluidized cell filling an un-pictured

cell to its the lower right. Let us now analyze the effect this has on cells 1 and 2.

D

1 2

Velocity profile in partially fluidized cell.

Figure 8: A partially fluidized cell flowing.

To start with, cells 1 and 2 are both in a state of incipient yield along their boundaries.

Specifically, they are in a state called passive incipient yield meaning that each cell is being

squeezed horizontally moreso than vertically as we might expect for a silo flow in which

material converges horizontally as it falls.

σ1 σ2

σ4

σ3

µσ2

µσ3µσ4

µσ1

Figure 9: The stresses on a solidified cell in passive incipient yield. Cells 1 and 2 begin with these appliedstresses.

A static analysis of cell 1 is shown in Figure 10. We draw a control volume which

extends slightly beyond the cell. The incipient yield stresses on the cell’s boundary have

been omitted from the diagram for ease of viewing. Note that the particle flux from the

10

1

Momentum Flux

1

σnet /2

σnet /2

Figure 10: Analysis of cell 1.

fluidized cell beneath has created a net momentum flux through the lower right face of the

volume. This is statically equivalent to placing a net stress of σnet = 〈ρ~v · ∇~v〉 · n on the

volume (we refer to σnet as the convective perturbation (CP) on cell 1), which can be statically

redistributed to the lower left and upper right edges of the cell. The stresses on the cell’s

boundaries must cancel these additional stresses if the cell is to remain stable. Revisiting

Figure 9, we see that this would imply the lower left boundary becomes relieved of its critical

state while the upper right becomes supercritical. We have thus shown that the fluidized

cell beneath translates to a yield excitation along cell 1’s upper right edge.

Now we consider cell 2. A somewhat different looking control volume diagram is helpful

here (see Figure 11). The particle flow near the lower left edge of cell 2 shears significantly

as in boundary layer flow. Since the fluidized region has a viscosity, τ1 + τ2∼= 2τ1 must be

approximately ηL cos 2ε n⊥ · (n · ∇~v). This precipitates into a reaction shear stress τ1 on

the lower left edge of cell 2 as indicated (we call τ1 the viscous perturbation (VP) on cell 2).

Statically, this is equivalent to placing a net moment and force on the cell which in turn can

be distributed as shown in the figure. As before, we determine if yield is excited anywhere

by analyzing how the applied stresses on these boundaries would have to change in order

to maintain stability. Both the lower right and upper left edges would require additional

shear stress. The judgement then follows from the fact that the upper left edge would have

a decrease in normal stress whereas the lower right would have an increase. We can assert,

then, that ratio τ/σ only goes up for the upper left edge, and likewise, the upper left edge

receives a yield excitation.

At this point we may close the analysis by specifying the scaling relationship on which

our argument hangs:

|σ0 − σstatic| ¿ CP ≈ V P ¿ 1.

We have now physically deduced that the presence of a fluidized cell excites slip-lines on

its neighbors in a way consistent with the initial presentation of the Stochastic Flow Rule in

Figure 7.

11

2τ1

2

velocity

τ1

τ2

2τ1 /2

τ1 /2

L

Figure 11: Analysis of cell 2.

6 Generalizing the Stochastic Flow Rule

So far, our entire analysis of SPT has been taken from the simple case of a wide flat-bottom

silo. We are now in a position to generalize its usage to other boundary conditions and in

doing so, state the Stochastic Flow Rule in its full form.

In any set of boundaries, we can apply equations 1 and 2 to obtain ψ(x, y). At every

point in the material, the slip-lines are angled at ψ ± ε. In general, the ψ field will not

be constant as it was in our simple example and likewise the slip-lines will probably have

curvature.

In order to preserve as much of an analogy with the flat wide silo, let us now enumerate

two ways to view what happens to a cell of mobilization:

1. Each step, the mobilization travels a constant distance D along one slip direction.

2. The mobilization travels along along a lattice of slip-lines, each step choosing between

two neighboring lattice points an equal distance away.

For non-constant ψ(x, y) these two perspectives are no longer synonymous. Therefore,

the generalized Stochastic Flow Rule has two forms, one in accordance to perspective 1 above

and one in accordance with 2.

We should also clarify that to use either of these interpretations requires we have some

general knowledge of the overall flow direction. Without this knowledge, we wouldn’t know

which adjacent pair of cells gets excited when a neighboring cell is fluidized. For typical

drainage boundaries, the answer is usually obvious, but should one wish to apply the theory

under obscure boundary conditions, one should first apply co-axiality to obtain a general flow

direction, then propagate the fluidization such that the drift makes a negative dot product

with the co-axiality-predicted particle flow.

12

7 Mean Field Stochastic Flow Rule: Interpretation 1

Under this interpretation, mobilized cells perform a random walk in which the step PDF is

determined by the cell’s current location (x, y):

p(x′, y′|x, y) =1

2δ ((x′ − x, y′ − y)−D(cos(ψ(x, y) + ε), sin(ψ(x, y) + ε)))

+1

2δ ((x′ − x, y′ − y) + D(cos(ψ(x, y)− ε), sin(ψ(x, y)− ε)))

We solve for the mean field approximation to the steady-state mobilized-cell density ρ using

the corresponding steady-state Fokker-Planck equation with non-constant Dn, i.e.

0 = −∇ · (D1(x, y) ρ) + ∂2(D2(x, y) ρ).

For a constant step time length τ , the drift coefficient is

D1(x, y) =D sin ε

τn⊥ψ

where n⊥ψ ≡ (− sin ψ(x, y), cos ψ(x, y)). Let us also define nψ ≡ (cos ψ(x, y), sin ψ(x, y)).

Observe that n⊥ψ is the drift direction (refer to Figure 5).

The coefficient D2 is most easily obtained by noting that in the nψ, n⊥ψ coordinate system,

it is a diagonal matrix. So the rotation sequence gives,

D2(x, y) =1

2τ

(cos ψ − sin ψ

sin ψ cos ψ

)(D2 cos2 ε 0

0 D2 sin2 ε

)(cos ψ sin ψ

− sin ψ cos ψ

)

As before, we implement the scaling approximation that ∆y = D sin ε ∼ (∆x)2 = D2 cos2 ε ¿1. This enables us to reduce the expression to

∇ · (ρn⊥ψ ) =D cos2 ε

2 sin ε

[∂2

∂x2(ρ cos2 ψ) + 2

∂2

∂x∂y(ρ sin ψ cos ψ) +

∂2

∂y2(ρ sin2 ψ)

].

Just as in the derivation of the Kinematic constitutive equations, we now claim that the

component of velocity in the −n⊥ψ direction is proportional to ρ. In fact, since our above

equation for ρ is homogeneous, we can just solve for

~v = f(x, y)nψ − ρ(x, y)n⊥ψ

where f(x, y) is yet to be determined. We obtain it by asserting incompressibility:

∇ · ~v =∂

∂x(f cos ψ + ρ sin ψ) +

∂

∂y(f sin ψ − ρ cos ψ) = 0.

13

This implies:

∂

∂x(f cos ψ) +

∂

∂y(f sin ψ) = ∇ · (ρn⊥ψ )

=D cos2 ε

2 sin ε

[∂2

∂x2(ρ cos2 ψ) + 2

∂2

∂x∂y(ρ sin ψ cos ψ) +

∂2

∂y2(ρ sin2 ψ)

]

=D cos2 ε

2 sin ε

(∂

∂x

[∂

∂x(ρ cos2 ψ) +

∂

∂y(ρ sin ψ cos ψ)

]+

∂

∂y

[∂

∂y(ρ sin2 ψ) +

∂

∂x(ρ sin ψ cos ψ)

]).

Therefore,

f(x, y)nψ

=D cos2 ε

2 sin ε

(∂

∂x(ρ cos2 ψ) +

∂

∂y(ρ sin ψ cos ψ),

∂

∂y(ρ sin2 ψ) +

∂

∂x(ρ sin ψ cos ψ)

)+ curl ~F (x, y)

for any arbitrary, smooth vector field ~F . We choose ~F = ~0 to prevent anomalous particle

drifting. Since f and −ρ are the nψ and n⊥ψ components of ~v, we write the final form of the

mean field Stochastic Flow Rule (Interpretation 1) as:

∇ · ((~v · n⊥ψ )n⊥ψ)

= D cos2 ε2 sin ε

[∂2

∂x2 (~v · n⊥ψ cos2 ψ) + 2 ∂2

∂x∂y(~v · n⊥ψ sin ψ cos ψ) + ∂2

∂y2 (~v · n⊥ψ sin2 ψ)]

~v · nψ = −D cos2 ε2 sin ε

sec ψ(

∂∂x

(~v · n⊥ψ cos2 ψ) + ∂∂y

(~v · n⊥ψ sin ψ cos ψ))

.

The full set of constitutive equations would then consist of equations 1 and 2 (to obtain

ψ(x, y)) together with these two equations. To check that our flow rule was properly derived,

note that when ψ is uniformly 0, the rule turns into the Kinematic equations just as we would

expect.

8 Mean Field Stochastic Flow Rule: Interpretation 2

Under this interpretation, the step length is not necessarily constant. It is allowed to vary

in order to preserve some sort of unbiased lattice structure wherein a mobilized cell at one

lattice point must choose between two equally distant lattice points, though that distance

may not be the same for other cells. We cannot assume, at the outset, that any arbitrary

smooth ψ permits such a lattice to be formed. We must determine conditions on ψ which

permit the existence of a lattice and in those cases determine how to build such a lattice.

We seek a transformation mapping horizontal grid lines to trajectories of nψ and vertical

lines to trajectories of n⊥ψ . The transformation must stretch uniformly in all directions to

14

α

β

x

y

ε

Slip line

ε

ensure an equal distance from one lattice point to all its neighbors (in the differential limit). A

byproduct of this condition is that the map is necessarily conformal, ensuring us that nψ and

n⊥ψ remain perpendicular and that slip-lines are always angled ±ε off nψ. A conformal map on

R2 can be written as a holomorphic function on C wherever its derivative is non-vanishing.

We may thus formulate our task as follows: We seek conditions on ψ(x, y) such that there

exists a holomorphic function f(α, β) fulfilling f ′(α, β) = g(x(α, β), y(α, β))eiψ(x(α,β),y(α,β))

for some function g, the linear stretching factor.

The condition on f ′ can be stated as f ′(f−1(x, y)) = g(x, y)eiψ(x,y). Going further we may

write, h(x, y) = log(f ′(f−1(x, y))) = log g(x, y) + iψ(x, y). f ′ is holomorphic and f−1 and

log are holomorphic on their (branched) ranges, so h is likewise holomorphic on the range of

f . Thus the imaginary part of h must be a harmonic function, i.e.

∇2ψ = 0.

Suppose ψ is harmonic. Let u(x, y) = log g(x, y). By the fundamental theorem of line

integrals,∫

C∇u d~r = u(x, y) − u(x0, y0) for any curve C connecting (x0, y0) and (x, y). By

the Cauchy-Riemann equations, this is equivalent to writing

u(x, y)− u(x0, y0) =

∫

C

(ψy dx− ψx dy).

This implies

g(x, y) = exp(u(x, y)) = g(x0, y0) exp

(∫

C

(ψy dx− ψx dy)

).

We must now set a reference step length. Say the step size at (x0, y0) is D. We then set

g(x0, y0) to 1 and deduce that the step size at any other location is D g(x, y).

We may now reformulate the Fokker-Planck equation for this situation. The PDF of the

steps is

p(x′, y′|x, y) =1

2δ ((x′ − x, y′ − y)− g(x, y)D(cos(ψ(x, y) + ε), sin(ψ(x, y) + ε)))

+1

2δ ((x′ − x, y′ − y) + g(x, y)D(cos(ψ(x, y)− ε), sin(ψ(x, y)− ε))) .

15

Thus the Fokker-Plank equation gives

∇ · (gρn⊥ψ ) =D cos2 ε

2 sin ε

[∂2

∂x2(g2ρ cos2 ψ) + 2

∂2

∂x∂y(g2ρ sin ψ cos ψ) +

∂2

∂y2(g2ρ sin2 ψ)

].

We are assuming a constant step time τ , so the particle velocity in the −n⊥ψ direction is

now proportional to gρ since larger g corresponds to longer steps. Proceeding as in the first

interpretation, we obtain the Stochastic Flow Rule (interpretation 2) constitutive equations

∇ · ((~v · n⊥ψ )n⊥ψ)

= D cos2 ε2 sin ε

[∂2

∂x2 (g~v · n⊥ψ cos2 ψ) + 2 ∂2

∂x∂y(g~v · n⊥ψ sin ψ cos ψ) + ∂2

∂y2 (g~v · n⊥ψ sin2 ψ)]

~v · nψ = −D cos2 ε2 sin ε

sec ψ(

∂∂x

(g~v · n⊥ψ cos2 ψ) + ∂∂y

(g~v · n⊥ψ sin ψ cos ψ))

.

As before, we may check this relationship in the case of a wide flat silo. In such boundaries,

ψ = 0 as before and g = 1 since the lattice spacing is uniform throughout. This indeed

reduces to the Kinematic equations.

It is important to emphasize at this time, however, that the likelihood of being able to

use this interpretation is actually quite low. Recall that ψ is generated from a system of

two nonlinear PDE’s. The probability that such an output also fulfills Laplace’s Equation is

very low. Be that as it may, in the few cases where we can actually compute a continuous

solution for ψ, it does appear that ψ can be approximately harmonic given certain boundary

conditions.

The Jenike radial solution to the Mohr-Coulomb Plasticity equations calculates the ψ

field for wedge-shaped, 2-D hoppers using a similarity ODE [?] . The similarity requires a

non-zero radial over-pressure on top of the flowing material, but this is typically the case

toward the bottom of the flow. ψ comes out as a function only of θ, measured vertically

from the hopper orifice anti-clockwise. Figure 12 displays a plot of ψ(θ) for a wedge hopper

with half-apex 45o, material internal friction angle 24o, and wall friction angle 20o. Notice

that the curve is approximately straight. In fact, the Laplacian of ψ turns out numerically

to be bounded by ≈ 5/r2 which may be argued in some sense as being “good enough”. But

these cases are, of course, diamonds in the rough. Other Jenike solutions under different

boundaries can yield ψ fields with high maximal values of the Laplace operator.

9 Open Questions and Future Work

There still remains much experimental/simulational verification of SPT. In the upcoming

months we shall build a detailed comparison of how well SPT predicts the b factor in a

flat-bottomed silo flow with material of varying internal friction. We also intend to show

how the first interpretation holds up in other simple geometries.

16

−50 −40 −30 −20 −10 0 10 20 30 40 50−100

−80

−60

−40

−20

0

20

40

60

80

100ψ(θ)

ψ (

degr

ees)

θ (degrees)

Figure 12: Numerically generated ψ(θ).

On the theoretical side, one of the biggest challenges which remains is finding a method

for determining regions of failure in 2-D flow. Recall that the Mohr-Coulomb stress equations

are only valid in regions of incipient failure. Likewise, SPT should only hold in regions of

incipient failure. Frequently, flows have stagnant or plug-like regions where material is well

below the Coulomb yield criterion. We can sidestep this problem by restricting our window

of view to the direct vicinity of an orifice, where the material is presumably failing. But to

truly apply SPT in a global flow, we would need to know exactly where the un-failing parts

of a flow are, and then exclude these regions from the domain on which we compute the

ψ field, essentially ensuring that we “dodge” any regions below yield. Future work on this

issue may utilize elastic/plastic energy minimization methods to determine the locations of

elastically governed regions in equilibrium.

Lattice-based interpretations of the Stochastic Flow Rule also need more theoretical

development. It very well could be that all flows do invoke a lattice of cell mobilization

where the cell dimensions are not necessarily equal (unlike the second interpretation) but

rather are determined by the excitation forces CP and VP.

Furthermore, work still needs to be done on the issue of determining the cell length D

for the simple flat-bottomed silo case. The original order parameter work by Aranson and

Tsimiring gives a non-linear diffusion equation for the time evolution of ρ based on free

energy dynamics [?]. This may be a good starting point, though it very well could lead to a

scaling argument for cell size, not a specific length. Of course, there is no reason to presume

that D must remain constant in a flow.

Also on the table is the issue of how to extend SPT to full, 3-dimensional flows. This is

limited only by our lack of ability to extend the Mohr-Coulomb stress equations into 3-D.

When we impose incipient failure on a 3-D system, we equivalently claim that the ratio of

major to minor principle stress is at a fixed value throughout. The indeterminacy arises

17

because the intermediate principle stress is unconstrained and not forced to reside along

one direction as in the quasi-2-D case. Many hypotheses have been devised to describe the

behavior of the intermediate principle stress such as the Haar Von Karman Hypothesis for

axi-symmetric 3-D flows. Even so, a fully rigorous extension of the Mohr-Coulomb stress

equations would be needed to apply SPT in 3-D with generality. [?][?]

10 Conclusion

Stochastic Plasticity is a cute and simple new perspective on granular flow. It combines

the empirical successes of the Kinematic Model with the physical rigor of Mohr-Coulomb

Plasticity in one intuitive flow rule. It enables a theoretical determination of the Kinematic

b factor and is general enough to be applied under arbitrary boundary conditions (i.e. be-

yond simple drainage). While there is still much work to do, SPT indeed appears to be a

promising direction in the theoretical development of granular material flow.

References

[1] Aranson, I.S. and Tsimiring, L.S. (2002), Phys. Rev. Lett., 65, 20.

[2] Bazant, M.Z. (2004) arXiv:cond-mat/0307379

[3] Caram, H. and Hong, D.C. (1991) Phys. Rev. Lett., 67, 828.

[4] Choi, J., Kudrolli, A., Rosales, R. R., Bazant, M. Z. (2004) Phys. Rev. Lett., 92, 174301.

[5] Drescher, A. (1991) Analytical Methods in Bin-Load Analysis, Elsevier.

[6] Gremaud, P. http://www4.ncsu.edu/eos/users/g/gremaud/WWW/trr.png

[7] Jenike, A.W. (1964) Journ. Appl. Mech., 31, 499.

[8] Mullins, W.W.J. (1972) Appl. Phy., 43, 665.

[9] Nedderman, R.M. (1992) Statics and Kinematics of Granular Materials, Cambridge

Univ. Press.

[10] Nedderman, R.M and Tuzun, U. (1979) Powder Technol., 22 243.

[11] Sokolovskii, V.V. (1965) Statics of Granular Materials, Pergamon/Oxford.

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