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, Tuz¨ un, 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
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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.