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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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