# Quantitative understanding of the onset of dense granular flows

Kasra Farain and Daniel Bonn  Van der Waals–Zeeman Institute, Institute of
Physics, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The
Netherlands
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

The question when and how dense granular materials start to flow under stress,
despite many industrial and geophysical applications, remains largely
unresolved. We develop and test a simple equation for the onset of quasi-
static flows of granular materials which is based on the frictional aging of
the granular packing. The result is a non-monotonic stress-strain relation
which – akin to classical friction – is independent of the shear rate. This
relation suffices to understand the below-threshold deformations of aging
granular media, and its solid-to-liquid transition. Our results also elucidate
the (flow) history dependence of the mechanical properties, and the
sensitivity to initial preparation of granular media.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

Under small stresses, the physical state of an accumulation of solid grains,
sand for example, can evolve or age slowly over long periods of time, in a
phenomenologically similar way as an ensemble of atoms or molecules in
electron glasses 1  and polymers 2 . Such slow relaxation and aging phenomena
in granular media, as non-equilibrium systems of discrete grains, have been of
considerable interest in the past years to understand the underlying
statistical principles of glassy and jammed dynamics generically 3 ; 4 ; 5 ; 6
. The evolution of granular systems, on the other hand, are critically
important in their own right due to many industrial and geophysical
applications. The problems of stability and flow of granular matter arise in
systems as diverse as cosmetics and foods, nuclear reactors 7 , soil mechanics
8 ; 9 , and earthquakes 10 ; 11 ; 12 . A well-known example is the settling of
soil; newly created Dutch polders, made by projecting sand onto an area, are
left to settle for several months or more before building can begin; otherwise
they are too soft.

Granular materials are often thought of as yield stress materials. They can
behave solid-like under small forces, e.g. forming granular piles with a
finite slope. But, they can also flow more or less like a liquid when
subjected to a shear stress above some threshold value, e.g. in an hourglass.
The solid-like phase resists deformation due to the presence of a force-
bearing network of grains that are in frictional contact with each other. The
strength of the system is, therefore, controlled by the stresses that these
frictional contacts can withstand. When enough of these contact points fail
under large stresses, the whole microstructure fails in an avalanche like
behavior and the material yields to flow 3 ; 13 . The rheological properties
of such ‘rapid’ granular flows, which are characterized by inertially-
dominated and hence rate-dependent dynamics, have been investigated
extensively in different settings; and constitutive equations, based on
dimensional analysis, have been proposed for the relation between stresses and
shear rates 13 ; 14 ; 15 ; 16 ; 17 . The quasi-static deformations of the
solid-like phase close to but below the threshold, and the solid to liquid
transition itself, however, despite much work 18 ; 19 ; 20 ; 21 ; 22 , still
lack a unified, empirically supported, theoretical framework.

The onset of granular flow is often described by a friction coefficient 15 ;
23 ; 24 , the ratio of shear to normal stresses at the failure point or,
equivalently, the tangent of the angle of repose of the material. Such zero-
order description, although it has had success in describing the aging and
humidity-dependence of the granular friction 23 ; 24 , suggests that there is
no flow before the complete failure of the whole network. Yet, granular media
can move slowly or creep even under stresses below the yield strength 18 ; 25
; 26 ; 27 . Such below-threshold deformations are evidently rate-independent,
contrasting with the inertial rapid-flow regime 18 ; 28 ; 29 .

Here, we exploit the similarities between granular materials and a larger
subclass of materials, thixotropic yield stress materials, with a time-
dependent structure that couples with the flow 30 , to develop a simple
analytical relation for the onset of quasi-static flows of granular materials.
The interactions between the grains in the slow flows considered here occur
solely through frictional sliding (not collision); and the inertia can be
safely ignored in such flows. The relation predicts a stress overshoot for the
onset of dry granular flows, qualitatively similar to those in the yield
stress fluids and dense suspensions 31  but with a shear-rate scaling that
originates from the dry friction between the grains. We subsequently test our
model by comparing it to slow shear experiments on a granular medium of hard
spherical particles. We find that the model quantitatively explains the stress
response of the granular matter subjected to small shear rates. The model also
accounts for the microscale below-threshold deformations and explains the
threshold behavior.

The model. — We begin our formulation of the onset of quasi-static flow by
specifying the basic assumptions that are considered in the model. First, it
is assumed that the granular system experiences a logarithmic aging under
stresses below the threshold (‘yield’) stress 23 ; 32 ; 33 . This may be
written as

$F_{v=0}=a~{}\ln(\frac{t}{\tau}),$ (1)

where $t$ is the time since the preparation of the system in a certain state,
$\tau$ is a reference time scale that makes $t$ dimensionless and $a$ is a
constant. By taking the derivative of this equation, the part of the friction
evolution that is imposed by the inherent aging is $dF/dt=a/t$. This means
that granular friction starts with a dramatic increase which slows down
quickly. The second assumption is that, just like the case of the thixotropic
yield stress fluids 30 , the effect of shear is a continuous ‘rejuvenation’ or
de-aging which is proportional to the sliding velocity, $v$. Finally, the
rejuvenation rate toward the steady-state flow is taken to be also
proportional to how far the current state of the granular system is from the
steady-state flow, i.e. to $F(t)-F_{d}$, where $F_{d}$ is the steady-state
frictional resistance of the system. Therefore, aging with a rate of $a/t$
strengthen the granular medium, while shear rejuvenates it with $\alpha
v(F-F_{d})$, where $\alpha$ is a shear constant of the granular material with
dimension of $m^{-1}$. This leads to a differential equation for the friction
force:

$\frac{dF}{dt}=\frac{a}{t}-\alpha v(F-F_{d}),$ (2)

that is similar (but not identical) to the one proposed for thixotropic yield
stress fluids 30 . In our granular case, solutions are of the form

$F(t)=ae^{-\alpha vt}\int_{t_{0}}^{t}\frac{e^{\alpha
vt^{\prime}}}{t^{\prime}}~{}dt^{\prime}+F_{d}(1-e^{-\alpha vt}),$ (3)

where $t_{0}$ is a constant that determines the initial state of the granular
system. This constant can be considered as equivalent to the initial state
variable in the rate-and-state equations for rock friction 34 . It should be
noted that $t_{0}\geq\tau$, because the assumption of Eq. (1) that we began
with is not correct for $t<\tau$ (frictional strength is finite and positive).
In any case, at $t_{0}\rightarrow 0$ the integral in Eq. (3) is divergent and
infinite forces are obtained. Such a problem arises in describing the dynamics
of all disordered systems demonstrating logarithmic aging 33 ; 35 ; 36 . In
these systems generally, the logarithmic behavior is valid only after a
microscopic time scale of the system.

In the most general form, non-steady state friction under a constant shearing
may be a function of both strain or displacement and the shearing velocity.
However, dividing Eq. (2) by $v$ and making the replacement $x=vt$ denotes the
granular friction in terms of sliding distance only. This is, in fact, imposed
by our assumption that shear-induced rejuvenation is proportional to the shear
rate. We inspect this property of our granular system in the experiments
below. However, the rejuvenation, in principle, can be a more complex
function, $g(v)$, of the flow rate than assumed here. Rewriting Eq. (2) for a
rejuvenation term of the general form $\alpha g(v)(F-F_{d})$ shows no change
in the functional form of the transient stress overshoot. The linear
rejuvenation we use, or equivalently the strain-rate independent transient
friction, can be considered as the first-order term in an expansion; we limit
ourselves here to this term.

Figure 1: Schematic of the experimental apparatus. A rheometer (Anton Paar MCR
302) is used to rotate a small cut of an aluminum cylindrical tube (outer
diameter 29.7 mm, wall thickness of 2.3 mm) around its symmetry axis. The tube
is laying on a plane covered by a $\sim 1$ mm layer of PMMA spherical
particles of $\sim 40$ $\mu m$ diameter. Both the tube cross section and the
bottom plate surfaces are sandblasted to have a surface roughness of 2 -3 $\mu
m$ (measured with a Keyence optical profilometer). The contact between the
rheometer tool and the tube is through frictionless bearing so that the tube
is free to move in normal direction. In this way, the rheometer doesn’t apply
any normal load to the granular media. Figure 2: Stress overshoot of the onset
of granular flow. (a) Friction as a function of time for three typical static
to steady flow transitions of the same granular media and system with
nominally identical experimental conditions, but prepared at different times.
The constant imposed velocity is 0.86 $\mu m/s$. The red dotted lines are the
best fits of Eq. (3) with $a$, $\alpha v$, and $t_{0}$ respectively as 3.31,
0.036, 0.23 for blue; 2.99, 0.051, 0.39 for gray; and 2.67, 0.058, 0.55 for
black curve, with the units of $mN$, $s^{-1}$, $s$. (b) Friction transients as
a function of deformation at different sliding velocities for a granular
sample that is not changed anymore after the preparation, just rests for 1 min
between the tests. The insets show the details of the behaviors at short
times. (c) The parameters $a$ (left axis) and $\alpha v$ (right axis) against
the total normal load $F_{N}$ (the weight of the tube) on the granular system.
The error bars indicate the standard error of at least eight independent
measurements. The inset shows the steady-state friction as a function of the
normal load. The standard errors in this case are smaller than the marker
size.

Experiments. — Our experimental system is schematically shown in Fig. 1. The
cross section of a thick cylindrical tube and a plane perpendicular to its
axis forms a frictional interface that is separated by a medium of hard
spherical polymethyl methacrylate (PMMA) particles. The tube is rotated by a
custom-made rheometer tool, while it is free in normal direction. Therefore,
independent of the Reynolds dilatancy in the granular material (the volume
change with shear deformations) 37 ; 38 ; 39 ; 40 , the normal load on the
material is constant and given by the weight of the tube. This is in contrast
to granular flows that are constrained by a constant gap, channel, or an
opening 15 ; 17  The other advantage of our experimental setup is using a
rheometer that can move a macroscopic object smoothly with high precision and
control, while measuring the applied torque. Moreover, the rheometer is able
to switch directly between shear rate- and stress-controlled motions, without
releasing the existing stress on the material.

Examples of the measured granular friction as a function of time under a
constant imposed rotational rate are presented in Fig. 2(a). The different
curves are repeats of the same experiment at different times with nominally
identical preparations and experimental conditions; and the dotted lines are
the best fits of Eq. (3). As the graph shows, each of the friction overshoot
dynamics are consistent with our aging and rejuvenation description
separately. However, while the overshoot of the onset of granular flow always
emerges and is in agreement with Eq. (3), the peak height and width
considerably vary between different preparations of the same system: this is
the sensitive dependence on preparation mentioned earlier. Such sensitivity of
granular flows to the initial preparation has also been demonstrated in
formation of conical granular piles 41 . Importantly, the startup force
depends on the preparation, but the steady-state friction does not show this
sensitivity. Our formulation and fittings attribute the force variations in
the onset of granular flow primarily to changes in $t_{0}$ (see the figure
caption). This is a variable that takes a value close to the singular point of
integral of Eq. (3), i.e. $t=0$. Theoretically, $t_{0}$ is a constant of
integration that describes the initial state of the granular material before
it starts the aging-rejuvenation evolution.

In Fig. 2(b), to avoid the dependence on the preparation, we focus on the
friction transients from a single preparation of the tube on the granular
layer which is not modified between the consecutive sliding tests. The plot
shows the shear resistance of the granular media against the sliding distance
at different imposed sliding rates. Before any measurement, we let the system
work at rotational rate of $10^{-5}$ $s^{-1}$ (equivalent to sliding velocity
of 0.86 $\mu m/s$) for at least one hour to make sure that the granular pile
underneath and around the tube does not change during the measurements. Each
shear test starts 1 min after the previous shearing was stopped; although the
onset stress overshoot of our granular system does not show any significant
dependence on this rest time in the range of several minutes (Fig. S1). As
evident in the figure, all the friction transients collapse onto a single
curve, which is perfectly described by the solution of Eq. (3) in terms of the
sliding distance. This confirms one of our underlying assumption that the
granular mechanics (in the quasi-static regime) is only a function of sliding
distance and, notably, does not depend on the shear rate. The steady-state
granular friction in the slow shearing regime also does not change with the
sliding rate, similarly to the macroscopic friction between solids.

Figure 3: Solid-to-liquid failure or unjamming transition of granular matter.
(a) A stress ramp (black, right axis) is imposed on a granular sample, which
its transient stress overshoot under a constant shear rate (0.86 $\mu m/s$)
has been determined before (grey, right axis). The granular material flows
suddenly at a point slightly above the peak frictional strength (blue, left
axis). The inset magnifies the microscale deformations before the solid-to-
liquid failure. (b) Creep deformation of the granular material subjected to
constant shear stresses below the threshold friction. The characteristic
stress overshoot of the granular sample under a constant shearing (0.86 $\mu
m/s$) and the applied shear stresses are shown in the inset. When the applied
stress is above the steady-state friction (blue dashed line), the creep
deformation is larger. However, the granular structure remains stable and
resist the flow. (c) After the peak of the friction overshoot, at time $t=80$
s, the imposed constant shear rate is instantly switched to a constant shear
stress by the rheometer (black, right axis). The constant stress in the second
stage is slightly higher than the frictional strength at $t=80$ s. This
immediately leads to the failure and flow of the granular material (blue, left
axis). Again, the complete stress overshoot of the sample under a constant
shear has been determined before (grey, right axis).

To further test our first-principle equation for the onset of granular flow,
we repeat the shear experiment with five different normal loads on the
granular layer. Amontons’ law states that friction is proportional to the
normal load on the frictional interface. This law also applies to granular
friction 24 . In our slow shear experiments, the steady-state frictional
resistance ($F_{d}$) accurately shows this property [Fig. 2(c), inset].
Therefore, Eq. (2) scales correctly over the full range, if $a$ changes
proportional to the normal load, and $\alpha$ remains constant. Fig. 2(c)
presents $a$ and $\alpha$ as functions of the normal load for multiple
friction experiments. Although due to the nonlinearity of friction equations,
the overshoot height is very sensitive to the chosen constants, the average
results unmistakably confirm that the friction overshoot predicted by Eq. (2)
is consistent with the Amontons’ law, and $a$ and $\alpha$ are in fact the
granular system parameters.

We now discuss the consequences of this stress overshoot of the onset of
quasi-static granular flows for the (in) stability of stressed granular solids
and their transition to flowing liquid-like state. Fig. 3(a) shows the generic
experiment to measure the threshold (yield) stress: we increase the imposed
force (stress) in a stepwise manner (black points) and observe that at a
critical stress, all of a sudden, the sample starts to deform rapidly (blue
points). If we compare the result of this experiment with the friction force
under imposed shear rate, we see that the critical stress corresponds to the
maximum of the peak (gray curve). However, as the inset shows, if we zoom in
on the displacement curve, there is already significant deformation happening
below the threshold stress. This is a signature of creep under stress that is
investigated in Fig. 3(b). In this figure, we impose two different force
levels, one above and one below the steady-state friction. For both cases, the
applied force is smaller than the peak friction; and the peak friction has to
be overcome before the system can flow macroscopically. The deformations
observed here correspond to a small fraction of a particle diameter (smaller
than 1 $\mu m$, compared to the 40 $\mu m$ grain size) over time scales of
several hours, so this is not a flow. Nevertheless, it is interesting to note
that both creep curves are logarithmic and the larger force shows a
significantly larger creep. This is not unlike our initial assumption that the
aging of the system under a quasi-static flow is logarithmic also. The
discussion of the static creep however goes beyond the quasi-stationary flow
that we describe by Eq. (2) 42 .

The conclusion from the above experiments is that the system creeps, but
cannot flow for forces smaller than the peak friction. Moreover, experiments
imposing a constant strain rate are very different from experiments under an
imposed force. This has been observed previously in granular material rheology
38 : there is a large range of shear rates that cannot be attained under an
imposed stress, since for an imposed stress the system either flows rapidly,
or not at all. In the current work, this is also evident from the experiment
in Fig. 3(c), where the rheometer first imposes a constant shear rate on the
granular sample until the peak frictional resistance has passed and friction
is a decreasing function of time. This of course signals an instability, which
is triggered if we then (at time $t=80~{}s$) impose a force that slightly
exceeds the one that was measured before: the sample starts to deform rapidly.
Conversely, when the same experiment is repeated on the left side of the
friction peak, the system stops. This implies that for the same stress the
system can either not flow at all, or flow rapidly, showing the importance of
the (flow) history of the sample, as predicted also by our simple model.

In summary, we have presented a comprehensive and consistent picture of the
onset of dense granular flows, in which the catastrophic failure of the
material is linked to its continuous slow dynamics arising from aging and
shear effects. This underlines the fundamental role of aging of the granular
packing in the origin of the yield strength. Our simple analytical model
accounts for: (1) the rate-independent rheology in the slow-flow regime 18 ;
28 ; 29 , (2) creep deformation 42 , (3) the difference between the static and
dynamic yield stress 3 ; 43 , and (4) the sensitive dependence of the flow
properties on the preparation of the system 3 ; 41 ; all of which are generic
features of granular dynamics. An important implication of the proposed model
and the rheology described here is that the mechanical failure of granular
fault gouge, leading to an earthquake, could be predicted from analysis of the
slow and aseismic slip of the fault and the external stresses.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program (Grant
agreement No. 833240)

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