11institutetext: Central South University, Changsha 410083, China
22institutetext: Theoretische Physik, Universität Tübingen,72076 Tübingen,
Germany

#  Why Granular Media Are Thermal,
and Quite Normal, After All

Yimin Jiang 11 Mario Liu 22<EMAIL_ADDRESS>
###### Abstract

Two approaches exist to account for granular dynamics: The athermal one takes
grains as elementary, the thermal one considers the total entropy that
includes microscopic degrees of freedom such as phonons and electrons.
Discrete element method (DEM), granular kinetic theory and athermal
statistical mechanics (ASM) belong to the first, granular solid hydrodynamics
(GSH) to the second one. A discussion of the conceptual differences between
both is given here, leading, among others, to the following insights: • While
DEM and granular kinetic theory are well justified to take grains as athermal,
any entropic consideration is far less likely to succeed. • In addition to
modeling grains as a gas of dissipative, rigid mass points, it is very helpful
take grains as a thermal solid that has been sliced and diced. • General
principles that appear invalid in granular media are repaired and restored
once the true entropy is included. These abnormalities [such as invalidity of
the fluctuation-dissipation theorem, granular temperatures failing to
equilibrate, and grains at rest unable to explore the phase space] are
consequences of the athermal approximation, not properties of granular media.

published in: Eur. Phys. J. E (2017) 40: 10; DOI 10.1140/epje/i2017-11497-4

###### Contents

  1. 1 Introduction
  2. 2 Two Opposite Models for Grains
    1. 2.1 Is a Pendulum athermal?
    2. 2.2 Athermal Gas versus Thermal Solid
    3. 2.3 Thermal and Athermal Explanation of Tapping
    4. 2.4 The Concerns of the Athermal Community
  3. 3 Granular Thermodynamics
    1. 3.1 The Granular Temperature ${T_{g}}$
    2. 3.2 More on the Relation between $S_{g}$ and $S_{Ed}$
    3. 3.3 The Elastic Strain Field ${u_{ij}}$
    4. 3.4  Granular Equilibrium Conditions
  4. 4 Conclusions
  5. A The Expressions of GSH
    1. A.1 Relaxation equation for ${u_{ij}}$
    2. A.2 The Elastic Energy ${w_{\Delta}(u_{ij},\rho)}$
    3. A.3 Granular Heat ${w_{T}(s_{g},\rho)}$
    4. A.4 Relaxation Equation for ${T_{g}}$
    5. A.5 Mass and Momentum Conservation
    6. A.6 Summary
  6. B Tapping and Compaction
    1. B.1 Reversible and Irreversible Compaction
    2. B.2 Memory Effects versus Hidden Variables
    3. B.3 Critical State Under Perturbations
    4. B.4 Tapping
  7. C Constant Rate Experiments
    1. C.1 The Hypoplastic Model
    2. C.2 Load and Unload
    3. C.3 Approach to the Critical State
    4. C.4 Shear Jamming
  8. D  Constant Stress Experiments
    1. D.1 Diverging Shear Strain
    2. D.2 Angle of Repose $\varphi_{re}$ versus Angle of Stabilty $\varphi_{st}$
    3. D.3 Shear Bands
  9. E Fast Dense Flow
    1. E.1 The ${\mu(I)}$-Rheology
    2. E.2 The Solid-Liquid Boundary
    3. E.3 Kamrin’s nonlocal constitutive relation

##  1 Introduction

Taking grains as elementary particles interacting via the Newtonian law, the
discrete element method (DEM) is the tool of choice for many coming to terms
with granular behavior CundallStrack ; dem1 ; dem2 . Similarly, granular
kinetic theory also assumes that grains are structureless particles undergoing
dissipative collisions kin0 ; kin1 ; kin2 ; kin3 ; kin4 . Both results have
consolidated the wide-spread believe of the physics community that grains may
generally be approximated as elementary. Starting from this belief, “athermal
statistical mechanics” (ASM) defines a reduced entropy $S_{g}$ that does not
contain any microscopic degrees of freedom, only those of granular
configurations, and assumes it is maximal in equilibrium. One example is the
Edward entropy, $S_{Ed}$, given by the number of possibilities grains may be
stably packed Edw ; raphi ; 1nico ; 1 ; 2 .

This is a leap of faith. The irrelevance of microscopics for DEM or the
kinetic theory does not imply that the true entropy $S$, of the microscopic
degrees of freedom such as phonons and free electrons, is always irrelevant.
Brownian motions are negligible because grains are macroscopically large, each
containing many microscopic degrees of freedom, such that $S\gg S_{g}$. Taking
$S+S_{g}$ to be maximal implies equilibrium holds as long as $S$ is maximal,
quite irrespective of the value of $S_{g}$.

The lack of Brownian motion shows that phonons and electrons do not move
grains. But this is insufficient for the conclusion of athermality. Relevant
is the question: Which of the two entropies, $S_{g}$ or $S$, is being
increased by dissipation off equilibrium, during collisions and relaxations,
and becomes maximal in equilibrium? Grains are athermal only if $S_{g}$ is the
answer. However, whenever a system loses energy and heats up measurably –
finger or thermometer, it is $S$ that is being increased. This is what grains
do in experiments, and if one would trace the lost energy, also in DEM. Note
DEM does not need any entropic considerations, because it already possesses
the dissipative terms that push the system toward equilibrium.

ASM takes the reduced entropy $S_{g}$ to be maximal, and as yet shuns
discussion of dissipation. GSH takes $S$ to be maximal in equilibrium, with
dissipative terms constructed to increase $S$ off equilibrium. Only GSH
successfully describes a wide range of granular phenomena, see granR1 ; granR2
; granR3 ; granR4 ; granRgudehus ; granL1 ; granL2 ; granL3 ; exp1. ; exp2. ;
exp3. ; exp4. ; exp5. ; exp6. ; exp7. ; exp8. ; exp9. ; PG2013 ; granRexp .
They include fast dense flow, elaso-plastic motion, static stress
distribution, propagation of elastic waves, and compaction, And GSH reduces,
in appropriate limits, to the hypoplasticity kolymbas1 ; kolymbas2 , Kamrin’s
nonlocal constitutive relation kamrin ; kamrin2 , the $\mu(I)$-rheology
pouliquen1 ; pouliquen2 ; pouliquen3 , and the kinetic-theory-based
hydrodynamic equations of granular gases. These results should suffice to
convince that the true entropy is a useful quantity, and that GSH presents the
appropriate macroscopic framework for understanding the multitude of granular
behavior.

GSH is still a qualitative theory, providing a bird’s eye view of granular
behavior, and approaching a quantitative status only in some select
experiments. We are working hard to calibrate the theory’s coefficients,
making it more realistic.

Its starting postulates – what we take to be the basic physics underlying
granular behavior – are “two-stage irreversibility” and “variable transient
elasticity”. The first addresses the three length scales of granular media –
macroscopic, granular and microscopic, and the fact that energy in the
macroscopic degrees of freedom first decays into the granular, then the
microscopic ones. The second addresses the fact that stresses relax when
grains jiggle – faster the stronger the jiggling is.

These two concepts introduce the granular temperature $T_{g}$ and the elastic
strain $u_{ij}$ as state variables. Hereby, $T_{g}$ quantifies the quickly
fluctuating elastic and kinetic energy of the grains, while $u_{ij}$ is static
or slowly varying, as it accounts for the grains’ coarse-grained elastic
deformation. Both $T_{g},u_{ij}$ obey equations accounting for the relaxation
toward the equilibrium characterized by maximal true entropy $S$. These
relaxation equations are therefore what directly link the existence of phonons
and free electrons to the flow and stacking of the grains: A flow comes to a
halt because there are phonons and electrons to excite, and heat to generate;
the stacking is stable because it is the minimum energy state, with all
available energy already transfered into heat. In addition, GSH contains the
conservation equations for mass, momentum and energy, and the balance equation
for the true entropy.

In Sec.2, we start by considering a pendulum – which is, same as a grain, a
macroscopic object that may be taken as a rigid mass particle. We ask the
question whether it is thermal, using the answer to draw some conclusions.
Then we discuss two models for grains: an athermal gas versus a thermal solid
that has been sliced and diced, identifying the first with ASM and the second
with GSH, and compare both in the context of compaction and tapping, as this
is the one subject believed to be well accounted for by ASM. We point to an
experiment that agrees with the results of GSH, but would be hard to account
for employing ASM.

In Sec.3, we first point out that the set of variables of granular
thermodynamics includes the elastic strain $u_{ij}$, the true temperature $T$,
and the granular temperature $T_{g}$. We carefully show how they are defined,
and what conceptual pitfalls they entail. Maximizing $S$, we find that
granular equilibrium is given by vanishing $T_{g}$, uniform $T$, and the
validity of force equilibrium.

Force equilibrium is especially noteworthy, because it shows that all stable
configurations of grains at rest – those counted by the Edward entropy – are
in equilibrium. They are stable because the true entropy $S$ has a local
maximum, while unstable ones in the neighborhood have smaller entropies.

We also discuss the failure of $T_{g}$ to equilibrate, showing that given two
systems, each with a granular and a true temperature, the behavior is
completely analogous to four systems with four temperatures connected by the
same heat currents. There is therefore little unique or incomprehensible about
the behavior of $T_{g}$.

A conclusion ends this paper, though the appendices also warrant closer
attention. A simplified, minimalist version of GSH is provided there, which
streamlines the arguments and expressions, and stresses easy comprehension. It
shows why GSH’s basic structure is natural, even necessary, and works out its
most important ramifications. This renders the paper self-contained, backing
up claims staked in the main text. Therefore, the present paper not only
deflects concerns people accustomed to the athermal model have, it also eases
their introduction to GSH.

The simplification of GSH consists of one main point – taking all transport
coefficients to be constant. Generally, they are functions of the density, and
constant only if the density is unchanged. If instead the pressure is
constant, circumstances are more complicated, because the pressure $P$ is a
function of $\rho,u_{ij},T_{g}$. If $u_{ij}$ and $T_{g}$ change with time, the
density $\rho$ needs to compensate, rendering the transport coefficients also
changing with time. This simplification makes simple, analytical solutions
possible, at the price of marring the realism of $P=$ const. experiments.

After a presentation of GSH, we go on to consider some of its ramifications.
We start with compaction, including the reversible and irreversible branch,
also the memory effect. Then we examine elasto-plastic motion at given shear
rates, especially the approach to the critical state – a classic experiment in
soil mechanics. Next, we discuss shear jamming – the fact that a granular
system becomes jammed upon shearing SJ1 ; SJ2 ; SJ3  – showing that it is
described by the same general solution as the approach to the critical state,
albeit with altered initial conditions.

Elasto-plastic motion at given shear stresses is equally interesting. The
phenomena accounted for include the difference between the angle of repose and
stability, shear bands, and the observation of a divergent shear strain. To
account for the latter, Nguyen et al. aging  borrowed concepts such as
fluidity, aging and rejuvenation parameter from the glassy dynamics, although
these are undefined, at most vague notions in granular media. As will be
shown, they are, respectively, $T_{g}$, its relaxation rate, and its
production rate, and the glassy dynamics turns out to be a reduced, scalar
version of GSH in the limit of constant stresses.

Finally, considering fast dense flow, we show that both the $\mu(I)$-rheology
and Kamrin’s nonlocal constitutive relation are natural consequences of GSH.
All these demonstrate the ease and usefulness of a comprehensive, thermal
approach.

A word on the hydrodynamic formalism, with the help of which GSH was derived.
Developed by Landau in the context of superfluidity Khal ; LL6  and introduced
to complex fluids by de Gennes deGennes , it holds a special place in
modeling, for which the dichotomy between realism and comprehension exists.
Constitutive relations typically focus on the first, while hydrodynamic
theories usually enable the latter. This is because hydrodynamic theories are
set up starting from comparatively few initial inputs, all derived from the
understanding of the system’s basic physics. (Examples are the quantities
being conserved and continuous symmetries being spontaneously broken. In the
case of GSH, as explained, it is two-stage irreversibility and variable
transient elasticity.) One then employs general principles – conservation
laws, Galilean invariance, and the second law of thermodynamics – to confine
the structure of the theory. As a result, a hydrodynamic theory leaves little
leeway (typically a handful of scalar functions) to fit the wide range of
experiments of a given system. When constructing a hydrodynamic theory, it is
therefore quickly obvious if the initial inputs are wrong – as is frequently
the case. If not, the theory develops predictive power and the capability to
become realistic. In comparison, constitutive models are constructed relying
on the accumulated data from specific experiments. It is highly realistic,
though less predictive in untested circumstances.

Notations:
We take $\partial_{t}a\equiv\frac{\partial}{\partial t}a$ for any $a$, the
velocity as $v_{i}$,
and denote the strain rate as
$v_{ij}\equiv\frac{1}{2}(\nabla_{i}v_{j}+\nabla_{j}v_{i})$,
its traceless part as $v^{*}_{ij}$, with
$v_{s}\equiv\sqrt{v^{*}_{ij}v^{*}_{ij}}\,\,\,(\equiv||v^{*}_{ij}||$).
In addition, we denote the elastic strain $\varepsilon^{ela}_{ij}$ as
$u_{ij}$,
the elastic stress as $\pi_{ij}$, the Cauchy or total stress as $\sigma_{ij}$.
Finally, with $u^{*}_{ij},\pi^{*}_{ij},\sigma^{*}_{ij}$ traceless, we take
$u_{s}\equiv\sqrt{u^{*}_{ij}u^{*}_{ij}}$,
$\quad\pi_{s}\equiv\sqrt{\pi^{*}_{ij}\pi^{*}_{ij}}$,
$\quad\sigma_{s}\equiv\sqrt{\sigma^{*}_{ij}\sigma^{*}_{ij}}$,
and $\Delta\equiv-u_{\ell\ell}$, $P_{\Delta}\equiv\pi_{\ell\ell}/3$,
$P\equiv\sigma_{\ell\ell}/3$.

##  2 Two Opposite Models for Grains

There are two pictures that we associate with grains, a gas of elementary but
macroscopic particles, and a block of rock that has been sliced and diced. The
first is athermal, the second thermal. Both work well within their respective
range of validity. The simple example of a pendulum is helpful to find out
what they are.

###  2.1 Is a Pendulum athermal?

A pendulum is, like a grain, a macroscopic object. Its linearized equation of
motion, appropriate for small amplitudes, reads (with $\theta$ the pendulum
angle, $g$ the gravitational constant, $l$ the length of the string, and
$\alpha$ the friction coefficient)

$\ddot{\theta}+\alpha\dot{\theta}+\theta g/l=0.$ (1)

Given this equation, one can calculate the pendulum’s motion, its return to
equilibrium hanging down, with no need to ever consider its entropy $S$.

Nevertheless, a pendulum is not athermal, because the frictional force
$\alpha\dot{\theta}$ increases $S$, of the pendulum itself and the surrounding
air. In fact, one can derive this force by starting from the second law of
thermodynamics, that $S$ can only increase, until it is maximal in
equilibrium. The total and conserved energy $E$ is given by the potential,
kinetic and heat contributions,
$E=\frac{1}{2}Mgl\theta^{2}+\frac{1}{2}Ml^{2}\dot{\theta}^{2}+\int T{\rm d}S$,
or $\dot{E}=0=Mgl\theta\dot{\theta}+Ml^{2}\dot{\theta}\ddot{\theta}+T\dot{S}$.
Inserting the pendulum equation, $\ddot{\theta}+Y+\theta g/l=0$, with an
unspecified force $Y$, we find

$\dot{S}=(Ml^{2}/T)\dot{\theta}\,Y.$ (2)

Since $Ml^{2}/T>0$, but not necessarily $\dot{\theta}$, the requirement
$\dot{S}>0$ confines the form for $Y$, of which the simplest is:
$Y=\alpha\dot{\theta}$, $\alpha>0$. This force acts until
$\theta,\dot{\theta}=0$, implying $E=\int T{\rm d}S$, or $S=$ max.

Grains are not different in this respect. Because DEM possesses the proper
dissipative forces that drive the system toward equilibrium, $S=$ max, it may
treat grains as elementary. But if one needs to derive dissipative terms
setting up a continuum-mechanical theory, it is not clear how one can possibly
avoid $S$.

###  2.2 Athermal Gas versus Thermal Solid

Faced with the task to account for granular behavior, it may seem natural to
always model the medium as a gas of elementary grains. Yet, within the
entropic context, the analogy between flying grains and gaseous atoms is not
close: Energy conservation holds only for atoms, not for elementary grains.
Accordingly, thermodynamics and statistical mechanics work only for the
former.

ASM tries to ameliorate this by replacing the energy with volume or stress,
taking the latter to be conserved. Yet energy conservation, related to time
translational symmetry, is a fundamental property of matter; constancy of
volume or stress are merely experimental prescriptions.

There are more problems: Failure of the temperatures to equilibrate,
invalidity of the fluctuation-dissipation theorem, jammed grains lacking the
possibility to explore the phase space, … All these in addition to the basic
problem, $S\gg S_{g}$. Now, if the gas model poses such difficulties, the
conjecture that a grain is further away from an atom then a block of rock that
has been sliced and diced, naturally arises.

Generally speaking, of the three phases of matter, the gaseous one is the
simplest to describe. The solid phase became more easily accountable only
after it was realized that, at low enough temperatures, it may be modeled as a
gas of free quasi-particles: mainly phonons, and in conductors, also free
electrons. Considering a block of solid including them, the dissipated energy
is not lost. With the total energy conserved, classical thermodynamics holds,
and statistical mechanics (of phonons and electrons) is valid. Crucially, same
holds for a stack of two blocks – or a pile of grains, as long as they are
macroscopic.

The notion that grains at rest are jammed, in need of shaking for phase space
exploration, is now inappropriate: Phonons and electrons roam nearly as freely
in a pile of grains as in a block of rock. The fluctuation-dissipation
theorem, dealing with thermal fluctuations of phonons and electrons kubo ,
holds also in granular media.

With granular microscopics reassuringly healthy, one confidently proceeds to
consider the macroscopic description of granular dynamics. A block of solid is
typically an elastic medium. Cutting the block in half, with one part on top
of the other, we expect them to again be elastic under shear if they do not
slip. If they do, we subtract the slipping portion from the total displacement
to obtain the deforming one. Further dicing the block to eventually arrive at
many (macroscopic) pieces, the system is still elastic – though we need to
keep track of the deforming displacement. We call the associated strain field
elastic, denote it as $u_{ij}$, and use it to account for the coarse-grained
elastic deformation of the system.

Since the elastic stress $\pi_{ij}$ stems from the deformation of the grains,
we have $\pi_{ij}=\pi_{ij}(u_{kl})$. Or more completely,
$\pi_{ij}=\pi_{ij}(u_{kl},\rho)$.

Force equilibrium among grains is the condition for maximal entropy $S$ with
respect to variation of $u_{ij}$, see Sec.3.4. Therefore, any stable
configuration of grains at rest is in equilibrium. Including phonons and
electrons that explore the phase space, grains at rest are indeed in the most
conventional of equilibria.

Off-equilibrium, granular dynamics is operative. To set up a continuum-
mechanical theory, and derive its many dissipative terms, an entropic
consideration that includes both $S$ and $S_{g}$ is necessary – as has been
done to derive GSH. Equilibrium is characterized by $S_{tot}=S+S_{g}$ being
maximal, or approximately $S=$ max. Taking $S_{g}=$ max is correct only if
$S=$ const. Yet since dissipation is ubiquitous among grains, and any
dissipation heats up the grains, this is not at all a likely scenario.

###  2.3 Thermal and Athermal Explanation of Tapping

A heap of grains has many stable configurations, many local maxima of $S$. The
logarithm of this number, typically an increasing function of the density, is
the Edward entropy, $S_{Ed}(\rho)$. Assuming it is $S_{Ed}(\rho)$ that is
maximal in equilibrium implies that the density increases under tapping,
simply because this increases $S_{Ed}$.

Since $S_{g}$ counts all granular states, in and off equilibrium, including
the much more numerous ones with grains flying and jiggling, $S_{Ed}$ contains
only a very small subclass of the states in $S_{g}$, or

$S_{Ed}\ll S_{g}\ll S.$ (3)

Drawing any conclusions from $S_{Ed}$ is justified only if $S$ and $S_{g}$ do
not depend on the density. Yet we know they do.

The alternative explanation emplyong GSH involves $S$, rather than $S_{Ed}$.
In a free column of grains, every layer (carrying the same load from above)
has a given pressure. In GSH, as mentioned, the pressure is a function of the
density and granular deformation,
$P(\rho,u_{ij})\equiv\frac{1}{3}\pi_{\ell\ell}$. Under tapping, because grains
periodically lose or loosen contact with one another, their deformation is
slowly lost, $u_{ij}\to 0$. As $u_{ij}$ relaxes, the density $\rho$ increases
to maintain $P$. Underlying this train of arguments is an increase of $S$: As
$u_{ij}$ relaxes, the associated elastic energy dissipates into heat. This
increase is incomparably larger than any change in $S_{Ed}$.

Looking for an experiment to discriminate between both explanations, we note
that the first is independent of the pressure, while compaction results from
constant pressure in GSH. If grains are submerged in a liquid of the same
density, gravitation does not produce any pressure. Starting from an initial
stress, tapping will diminish $u_{ij}$, and with it also the stress, but will
not increase the density.

According to GSH, small-amplitude cyclic shear jiggles grains and has
essentially the same effect as tapping. Doing this at constant pressure,
compaction takes place. But at constant volume, we only achieve $u_{ij}\to 0$,
causing the pressure and shear stress to also vanish. A GSH-calculation of the
second case is rendered in Fig.2, the associated experiment is rendered in
Fig.2. Details of the calculation may be found in granRgudehus , though its
relevance to tapping was not realized then. See also appendix B for more
results on compaction, both the reversible and irreversible branch, and the
explanation of the memory effect.

Figure 1: GSH calculation depicting the relaxation of the shear stress
$q\equiv\sigma_{zz}-\sigma_{xx}$ and the pressure $P$, in a triaxial geometry,
for given shear oscillation, $\varepsilon_{zz}$, but no compression,
$\varepsilon_{v}=$ const. (Inset amplifies the last calculated cycle.)

Figure 2: Imposing isochoric deformation cycles to saturated sand in undrained
triaxial tests, Wichtmann et al. Wichtmann09  obtained results qualitatively
similar to that of Fig.2.

###  2.4 The Concerns of the Athermal Community

A large fraction of the granular physics community believes granular media
cannot be treated by conventional tools of theoretical physics, because
general principles, including energy conservation, thermodynamics and the
concept of equilibrium, are invalid. As should be obvious by now, these are
consequences of the athermal model, not abnormalities of granular media as
such.

Then there are those who do accept that equilibrium is given by $S=$ max, but
take this as a result of equilibrium statistical mechanics alone, lacking any
relevance off-equilibria. This is a partial and erroneous view. First of all,
the entropy is also defined in local equilibrium and in generalized
equilibrium, in which a few slowly relaxing variables are off their
equilibrium values. Starting off-equilibrium, the entropy grows continuously,
by changing the value of slow variables, and by distributing conserved
quantities, until it is maximal, and the system in equilibrium. For instance,
a pendulum comes to a standstill hanging down, and the energy redistributes to
achieve a uniform temperature. This evolution is accounted for by dissipative
terms, which are derived by requiring that $S$ always increases – as we did
for the pendulum around Eq.(2), and as was done setting up GSH.

##  3 Granular Thermodynamics

###  3.1 The Granular Temperature ${T_{g}}$

In any uniform medium such as water or air, there are two length scales,
macro- and microscopic. All degrees of freedom may be divided into either of
these two groups. A hydrodynamic theory takes the degrees of freedom from the
first as explicit variables, each with an equation of motion. These including
mass, momentum and energy density. Those from the second group are taken
summarily, with their contribution to the energy lumped together as heat, and
characterized by the temperature $T$. Irreversibility is caused by the
macroscopic energy decaying into heat.

In granular media, there is an intermediate, mesoscopic group of degrees –
momentum and deformation of individual grains. In DEM, these are explicit
variables. But for a hydrodynamic theory, a summary inclusion again suffices,
with their energy lumped into granular heat, quantified by $T_{g}$.
Irreversibility is now caused by the macroscopic energy decaying into granular
heat, and then on to true heat. This is what we term two-stage irreversibilty.

We separate mesoscopic from microscopic degrees, with $T$ and $T_{g}$, instead
of lumping them into one group and one temperature, not only because of the
different length scales. Equally important is the fact that $T_{g}$ is an
independent state variable, on which the dynamics critically depends: The
elastic stress relaxes when the grains jiggle, when $T_{g}\not=0$, while it is
(within limits) independent of $T$. Lumping both into one temperature obscures
this difference.

Extending this argument, we see that a third temperature is superfluous. For
instance, it is not useful to introduce a temperature for those degrees
characterizing the surface roughness of grains. Though the length scale is
distinctly smaller, no aspect of macroscopic granular dynamics depends
critically on an associated temperature. These degrees are simply part of
$T_{g}$. Similarly, it is not useful to introduce a configuration temperature
$T_{Ed}$ associated with the Edward entropy, as it does not possess any
significance independent from $T_{g}$, see also the discussion in Sec.3.2.

Both $T_{g}$ and $T$ are genuine temperatures, as each characterizes the
energy of a group of degrees of freedom. Same holds for the granular and true
entropy, $S_{g}$ and $S$: Each is the logarithm of the number of states in the
associated group. Any difficulties treating $T_{g}$ as a temperature arises
only because $T$ is ignored. For instance, the granular temperatures $T_{g}$
of two systems in contact are typically different – seemingly a failure of
$T_{g}$ to equilibrate. Given two granular systems, 1 and 2, with only 1 being
excited, there are, in the steady state, four generally unequal temperatures:
$T^{1},T_{g}^{1},T^{2},T_{g}^{2}$. And there are three ongoing energy fluxes:
$(T^{1}_{g}\to T^{1})$, $(T^{1}_{g}\to T_{g}^{2})$, $(T_{g}^{2}\to T^{2})$.
This is in complete analogy to four conventional thermal systems, (1, 1a, 2,
2a), with only (1a) being heated by an external energy flux, and (1a,1), (1a,
2a), (2a, 2) in pairwise thermal contact. We expect all four temperatures:
$T_{1},T_{1a},T_{2},T_{2a}$ to be different.

Having stated this, we note a practical difference. Taking the energy density
as a function of the two entropy densities, $w=w(s,s_{g})$, the conjugate
variables are:

$T\equiv{\partial w}/{\partial s},\qquad T_{g}\equiv{\partial w}/{\partial
s_{g}}.$ (4)

Denoting $s_{tot}=s+s_{g}$, we may write

${\rm d}w=T{\rm d}s+T_{g}{\rm d}s_{g}=T{\rm d}s_{tot}+(T_{g}-T){\rm d}s_{g},$
(5)

and identify $T{\rm d}s_{tot}$ as the equilibrium energy change for changes of
the total entropy, and $(T_{g}-T){\rm d}s_{g}$ as the extra energy
contribution for $T_{g}\not=T$. With $(T_{g}-T)$ characterizing the non-
optimal energy distribution between the granular and microscopic degrees of
freedom, the energy $w$ has a minimum at $T_{g}=T$, and is, expanded, given
as: $w\propto(T_{g}-T)^{2}$. This also implies $(T_{g}-T)$ relaxes until it
vanishes.

Now, since $s\gg s_{g}$, and any granular motion at all occurs at $T_{g}\gg
T$, we have $T_{g}-T\approx T_{g}$, $s_{tot}\approx s$, and the rewriting of
Eq.(5) did not change anything. Therefore, we may simply take $w\propto
T_{g}^{2}$, with $T_{g}$ relaxing until equilibrium, $T_{g}=0$. And since $T$
does not relax, we have $T=$ uniform in equilibrium, see more in Sec.3.4.

A division into three scales works when they are well separated – though this
is a problem of accuracy, not viability. Scale separation is well satisfied in
large-scaled, engineering-type experiments, less so in small-scaled ones.
Using glass or steel beads (typically larger) aggravates the problem.
Nevertheless, when the system is too small for spatial averaging, one may
still average over time and runs to get rid of the fluctuations not contained
in a hydrodynamic theory.

Finally, it is useful to realize that since granular heat $\int T_{g}{\rm
d}s_{g}\propto T_{g}^{2}$ denotes the elastic and kinetic energy of the
grains, it reduces, in the limit of vanishing $\rho$, in which only the
kinetic part remains, to the energy or temperature of the kinetic theory
$T_{k}$, implying $T_{k}\propto T_{g}^{2}$. For the same reason, Eq.(35) for
$T_{g}$ is, in this limit, the same as the energy balance of the kinetic
theory.

###  3.2 More on the Relation between $S_{g}$ and $S_{Ed}$

The Edwards entropy $S_{Ed}$, a function of the volume $V$, is employed with

${\rm d}S_{Ed}={\rm d}V/X$ (6)

as the basic thermodynamic relation for a “mechanically stable agglomerate of
infinitely rigid grains at rest” Edw . This ansatz is better appreciated by
taking the granular entropy as $S_{g}(E,V)$ (while still neglecting phonons
and electrons), writing

${\rm d}S_{g}=(1/T_{g}){\rm d}E+(P/T_{g}){\rm d}V.$ (7)

For infinitely rigid grains at rest, as there is no kinetic or deformation
energy, we have $E\equiv 0$, which remains zero however these grains are
arranged, ${\rm d}E\equiv 0$, hence

${\rm d}S_{g}=(P/T_{g}){\rm d}V\equiv{\rm d}V/X,$ (8)

reducing $S_{g}$ to $S_{Ed}$. More generally, because grains are elastic and
frequently in motion, ${\rm d}E\not=0$, and Eq.(7) holds.

There is no independent Edward temperature $T_{Ed}\sim X$. It is simply what
$T_{g}$ reduces to if one chooses to take grains as infinitely rigid and
perennially at rest.

We also note that infinite rigidity is not a realistic limit in sand: Because
of the Hertz-like contact between grains, very little material is deformed at
first contact, and the compressibility diverges at the jamming point, for
$u_{ij}\to 0$, see Eq.(28). This is a geometric fact independent of the
material’s rigidity.

###  3.3 The Elastic Strain Field ${u_{ij}}$

A block of rock, if elastic, is well accounted for by the elastic energy, the
elastic stress, and the equation of motion of the strain $\varepsilon_{ij}$,

$\displaystyle w=w(\varepsilon_{ij}),\quad\pi_{ij}=-\partial
w/\partial\varepsilon_{ij},$ (9)
$\displaystyle\partial_{t}\varepsilon_{ij}=v_{ij}\equiv(\nabla_{i}v_{j}+\nabla_{j}v_{i})/2.$

Slicing the block in half, with one part on top of the other, we expect them
to again be elastic under shear if they do not slip. If they do, we subtract
the slipping portion from the total displacement to obtain the deforming one.
If one uses it to calculate the strain field, the first two above relations
remain valid – as only the deforming field leads to an elastic energy, which
in turn leads to a restoring force that is the elastic stress. We call the new
strain field elastic, denote it as $u_{ij}$, and write

$w=w(u_{ij}),\quad\pi_{ij}(u_{ij})=-\partial w/\partial u_{ij}.$ (10)

Further cutting the block to eventually arrive at many macroscopic pieces,
this consideration still holds. We again take the portion of the strain that
deforms the grains as $u_{ij}$, and again, only this field senses the
restoring stress.

An analogy should make the last half sentence clearer. The wheels of a car
driving up a slippery slope have a gripping portion $\theta_{g}$, and a
slipping portion $\theta_{s}$. The force on the car is $\partial
w_{g}/\partial\ell$, with $w_{g}$ the gravitational energy and $\ell$ the
distance traversed, implying that the torque on the wheels is $\partial
w_{g}/\partial\theta_{g}$.

Eqs.(10) are useful in two aspects. First, given an evolution equation for
$u_{ij}$ (that is yet to be found, see appendix A.1), we have a granular
theory as mathematically complete as ideal elasticity. And there is no need to
consider, in addition, the plastic strain rate
$\partial_{t}\varepsilon_{ij}^{p}\equiv\partial_{t}\varepsilon_{ij}-\partial_{t}u_{ij}$.
Second, given the elastic stress-strain relation $\pi_{ij}=\pi_{ij}(u_{kl})$,
any information on $\pi_{ij}$ is equivalent to that on $u_{ij}$. Only the
first is directly measurable. Note for given $\rho$, the stress
$\pi_{ij}\equiv-\partial w/\partial u_{ij}$ is a monotonic function of
$u_{ij}$, as any stable energy is convex, $\partial^{2}w/\partial
u_{ij}\partial u_{k\ell}>0$.

In soil mechanics, the plastic strain rate is related to flow rules and deemed
important. Moreover, the energy is often taken as a function of both $u_{ij}$
and $\varepsilon_{ij}^{p}$ Houlsby2 . Yet since granular energy is either
elastic or kinetic, stored either in the deformation or motion of the grains,
it is not clear what the energy contribution of $\varepsilon_{ij}^{p}$ is.

Finally, we note that the elastic strain is not obtained by coarse-graining
its mesoscopic counterpart, $u_{ij}\not=\langle u_{ij}^{mes}\rangle$. The
reason is, both the energy and stress are given by averaging: $w=\langle
w^{mes}\rangle$, $\pi_{ij}=\langle\pi_{ij}^{mes}\rangle$. Since ${\rm
d}w=\langle{\rm d}w^{mes}\rangle=-\langle\pi_{ij}^{mes}{\rm
d}u_{ij}^{mes}\rangle=-\langle\pi_{ij}^{mes}\rangle{\rm d}u_{ij}$ and
$\langle\pi_{ij}^{mes}{\rm
d}u_{ij}^{mes}\rangle\not=\langle\pi_{ij}^{mes}\rangle{\rm d}\langle
u_{ij}^{mes}\rangle$, we have ${\rm d}u_{ij}\not={\rm d}\langle
u_{ij}^{mes}\rangle$.

###  3.4  Granular Equilibrium Conditions

The state variables of any granular media are the density $\rho$, the momentum
density $\rho v_{i}$, the two entropy densities $s,s_{g}$, and the elastic
strain $u_{ij}$. Denoting the energy density in the rest frame ($v_{i}=0$) as
$w=w(s,s_{g},\rho,u_{ij})$, with $T\equiv{\partial w}/{\partial s}$,
$T_{g}\equiv{\partial w}/{\partial s_{g}}$, $\mu\equiv{\partial
w}/{\partial\rho}$ and $\pi_{ij}\equiv-{\partial w}/{\partial u_{ij}}$, we
have

${\rm d}w=T{\rm d}s+T_{g}{\rm d}s_{g}+\mu{\rm d}\rho-\pi_{ij}{\rm d}u_{ij}.$
(11)

Another useful conjugate variable, the fluid pressure $P_{T}$, belongs to the
energy density per unit mass, $w/\rho$, and is given (with $V$ denoting the
volume) as

$P_{T}\equiv-w+sT+s_{g}T_{g}+\mu\rho=-\frac{\partial(wV)}{\partial
V}=-\frac{\partial(w/\rho)}{\partial(1/\rho)}.$ (12)

In dry granular media, $P_{T}$ is the pressure exerted by jiggling grains, see
Eq.(33).

Equilibrium conditions, valid irrespective of the energy expression
$w(s,s_{g},\rho,u_{ij})$, are obtained by formerly requiring $S=\int s\,{\rm
d}^{3}r=$ max, for given energy $\int w\,{\rm d}^{3}r$ and mass
$\int\rho\,{\rm d}^{3}r$, with $T_{g}$ allowed to relax. Employing Eq.(11), we
first obtain

$\nabla_{i}T=0,\quad T_{g}=0.$ (13)

Usually, $T_{g}$ vanishes quickly. After this has happened, $d\rho$ and
$du_{\ell\ell}=-d\rho/\rho$ no longer vary independently. They therefore share
a solid-like equilibrium condition,

$\nabla_{i}(\pi_{ij}+P_{T}\delta_{ij})=\rho\,{g}_{i}.$ (14)

This is the expression of force equilibrium for the jammed state.

If $T_{g}$ is kept finite, say by tapping, the system may further increase its
entropy by independently varying $\rho$ and $u_{ij}$. It then arrives at two
fluid-like equilibrium conditions, the first of which requires the shear
stress to vanish, and a free surface to be horizontal,

$\pi_{ij}=0,\quad\nabla_{i}P_{T}=\rho\,{g}_{i}.$ (15)

##  4 Conclusions

As grains are too large to display thermal fluctuations, they are widely taken
as particles without any internal structure, and considered “athermal.” Though
an excellent approximation for DEM and the kinetic theory, it fails in any
entropic considerations, because the inner-granular, microscopic degrees of
freedom are then the dominating ones. We have clarified the reasons why and
when an athermal approach works, and when it fails.

Embracing the notion that the macroscopic theory for granular media is that of
a thermal solid that has been sliced and diced, and introducing two
temperatures, true and granular, one arrives at GSH, a comprehensive yet
simple hydrodynamic theory capable of accounting for a wide range of granular
phenomena.

As this paper serves a dual purpose, to deflect concerns people accustomed to
the athermal model have, and to ease their introduction to GSH, we present a
minimalist version of GSH in appendix A, complete with a number of analytic
solutions. (The original version, more complex and realistic, was derived a
decade ago granR1 , and has since been employed to account for many
experiments granR4 ; granRexp .)

A summary of this version of GSH is given here, to display its mathematical
structure. Explanations are found in appendix A. The state variables of any
granular system are the density $\rho$, the momentum density $\rho v_{i}$, the
granular entropy density $s_{g}$, and the elastic strain $u_{ij}$, see
Sec.3.4. (The entropy $s$ is excluded here, as we do not aim to consider
effects such as temperature diffusion or thermal expansion – though $s$ was of
course included in deriving GSH.) Their evolution equations are: the
continuity equation, momentum balance, balance of the granular entropy, and
the evolution equations for the elastic strain. Denoting $\Delta\equiv-
u_{kk}$, $u^{*}_{ij}\equiv u_{ij}+\frac{1}{3}\Delta$,
$v_{ij}\equiv\frac{1}{2}(\nabla_{i}v_{j}+\nabla_{j}v_{i})$ as the strain rate,
and $v^{*}_{ij}$ again traceless, we have

$\displaystyle\partial_{t}$ $\displaystyle\rho+\nabla_{i}(\rho v_{i})=0,$ (16)
$\displaystyle\partial_{t}$ $\displaystyle(\rho
v_{i})+\nabla_{i}(\sigma_{ij}+\rho v_{i}v_{j})=\rho g_{i},$ (17)
$\displaystyle\,\sigma_{ij}=\pi_{ij}+P_{T}\delta_{ij}-\eta_{1}T_{g}v^{*}_{ij},$
(18) $\displaystyle\,\pi_{ij}(u_{ij})\equiv-{\partial w\,\,}/{\partial
u_{ij}},\,\,P_{T}\equiv-{\partial(w/\rho)}/{\partial(1/\rho)},$ (19)
$\displaystyle\partial_{t}$ $\displaystyle
T_{g}=-R_{T}[T_{g}(1-\xi_{T}^{2}\nabla^{2}_{i})T_{g}-f^{2}v^{*}_{ij}v^{*}_{ij}],$
(20) $\displaystyle\partial_{t}$ $\displaystyle u^{*}_{ij}=v^{*}_{ij}-\lambda
T_{g}u^{*}_{ij},$ (21) $\displaystyle\partial_{t}$
$\displaystyle\Delta+v_{\ell\ell}=\alpha_{1}u^{*}_{ij}v^{*}_{ij}-\lambda_{1}T_{g}\Delta.$
(22)

(Note that $\sigma_{ij}$ is fixed if $w$ is given. In constitutive models, it
is typically taken as an expression with little constraints, to be extracted
from data alone.) This set of partial differential equations is closed if the
energy density $w$ and the seven transport coefficients:
$\lambda,\lambda_{1},\alpha_{1},R_{T},\xi_{T},f,\eta_{1}$ are known [$f$ is
not independent, cf Eq.(37)]. These scalars provide the leeway GSH has for
fitting experiments, from elasto-plastic motion to fast dense flow. A
simplified expression for $w$ is given in Eqs.(26,27,32). The transport
coefficients, all functions of the density, are taken as constant here.

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##  Appendix A The Expressions of GSH

We provide a brief derivation of Eqs.(16-22) in this section.

###  A.1 Relaxation equation for ${u_{ij}}$

First, we consider what $\partial_{t}u_{ij}$ is: The free surface of a
granular system at rest is frequently tilted. When perturbed, when the grains
jiggle and $T_{g}\not=0$, the tilted surface will decay and become horizontal.
The stronger the grains jiggle, the faster the decay is. We take this as
indicative of a system that is elastic for $T_{g}=0$, transiently elastic for
$T_{g}\not=0$, with a stress relaxation rate that increases with $T_{g}$. We
take the rate $\propto T_{g}$, as this yields rate-independence for the
elasto-plastic regime.

A relaxing stress is typical of any visco-elastic system such as polymers
polymer-1 ; polymer-2 ; polymer-3 ; polymer-4 . In granular media, the
relaxation rate is not a material constant, but a function of the state
variable $T_{g}$ – a behavior that we call variable transient elasticity.
Remarkably, by taking the evolution equation for $u_{ij}$ to reflect variable
transient elasticity, one finds that the result suffices to capture elasto-
plasticity.

For the traceless part $u^{*}_{ij}$, we start with the equation of elasticity,
then add a relaxation term $\propto T_{g}$. The next line is a rewrite that
clarifies the stationary state,

$\displaystyle\partial_{t}u^{*}_{ij}$ $\displaystyle=v^{*}_{ij}-\lambda
T_{g}u^{*}_{ij}\qquad(\text{with}\,\,u_{c}\equiv 1/\lambda f)$ (23)
$\displaystyle=-\lambda
T_{g}[u_{ij}^{*}-u_{c}\frac{v_{ij}^{*}}{v_{s}}\frac{fv_{s}}{T_{g}}],$

where $v_{s}\equiv\sqrt{v^{*}_{ij}v^{*}_{ij}}>0$. The negative trace of the
elastic strain, $\Delta\equiv-u_{kk}$, obeys a slightly more complicated
equation, with a term preceded by $\alpha_{1}$ accounting for dilatancy. With
$u_{s}\equiv\sqrt{u^{*}_{ij}u^{*}_{ij}}>0$, it is

$\displaystyle\partial_{t}\Delta+v_{\ell\ell}$
$\displaystyle=\alpha_{1}u^{*}_{ij}v^{*}_{ij}-\lambda_{1}T_{g}\Delta\qquad(\text{with}\,\,\Delta_{c}/{u_{c}}\equiv{\alpha_{1}}/{\lambda_{1}f})$
(24)
$\displaystyle=-\lambda_{1}T_{g}\left[\Delta-\Delta_{c}\frac{u_{s}}{u_{c}}\frac{fv_{s}}{T_{g}}\frac{v^{*}_{ij}}{v_{s}}\frac{u^{*}_{ij}}{u_{s}}\right].$

As we shall see, in the rate-driven stationary state, we have $fv_{s}=T_{g}$,
$u_{s}=u_{c}$, and ${v^{*}_{ij}}{u^{*}_{ij}}=v_{s}u_{s}$, reducing the right
hand side to $-\lambda_{1}T_{g}\left[\Delta-\Delta_{c}\right]$.

If the anisotropy of the stress changes appreciably, convective terms (or the
objective time derivative) become important, then one needs the substitution
polymer-1

$\partial_{t}u_{ij}\to(\partial_{t}+v_{k}\nabla_{k})u_{ij}+\Omega_{ik}u_{kj}-u_{ik}\Omega_{kj},$
(25)

where
$\Omega_{ik}\equiv\textstyle\frac{1}{2}(\nabla_{i}v_{k}-\nabla_{k}v_{i})$.

###  A.2 The Elastic Energy ${w_{\Delta}(u_{ij},\rho)}$

As a first step to specify $w=w(s,s_{g},\rho,u_{ij})$, we note that due to the
lack of interaction among grains, $w$ vanishes when the grains are neither
deformed nor jiggling. Taking

$w(\rho,s_{g},u_{ij})=w_{\Delta}(\rho,u_{ij})+w_{T}(\rho,s_{g}),$ (26)

we therefore require $w_{T}\to 0$ for $s_{g}\to 0$, and $w_{\Delta}\to 0$ for
$u_{ij}\to 0$. The dependence on the true entropy density $s$ is not
specified, because we are not at present dealing with any temperature related
phenomena. But this is easily ameliorated if needed.

The elastic energy $w_{\Delta}$ needs to be chosen such that it yields the
right static stress distribution and elastic wave propagation. Equally
important, it needs to account for the fact that elastic solutions are
untenable if (1) the density is too small and the grains loose contact with
one another, or if (2) the shear stress is too large, say when a slope is too
steep, and the grains start slipping. Because both instabilities may happen in
equilibrium, when grains are at rest, they have to be encoded in the energy,
not in evolution equations or transport coefficients. This is done most easily
by a transition of the energy from convex to concave at the instabilities,
because elastic solutions are stable only if the energy is convex. Defining
the random loose packing density $\rho_{lp}$ as the minimal one sustaining an
elastic solution, we require the energy to possess a convexity transition at
$\rho_{lp}$, such that no elastic solution is stable for $\rho<\rho_{lp}$.
Similarly, we define a yield stress $\pi_{yie}$ such that the energy is, at
given pressure $P$, convex for
${\pi_{s}}\equiv\sqrt{\pi^{*}_{ij}\pi^{*}_{ij}}<\pi_{yie}$ and concave for
${\pi_{s}}>\pi_{yie}$.

This implies that the angle of stability $\varphi_{st}$ is – for an idealized,
uniform and infinite slope – given by
$\tan\varphi_{st}\equiv{\pi_{yie}}/\sqrt{2}P$: On an infinite plane inclined
by the angle $\varphi$, with $y$ the depth of the granular layer on the plane,
and $x$ along the slope, we take the stress to be
$\pi_{xx},\pi_{yy},\pi_{zz}=P_{\Delta}$, $\pi_{xy}=\pi_{s}/\sqrt{2}$,
$\pi_{yz},\pi_{xz}=0$. Integrating $\nabla_{j}\pi_{ij}=g_{i}\rho$ assuming a
variation only along $y$, we find $\pi_{xy}=g\sin\varphi\int\rho(y)dy$ and
$\pi_{yy}=\pi_{xy}/\tan\varphi$. The angle of stability $\varphi_{st}$ is
reached when the energetic instability of Eqs.(30) is breached. (Effects from
proximity to the wall or floor, such as clogging, or the fact that the angle
increases when the layer is very thin, are not considered here.) The smaller
angle of repose $\varphi_{re}$ – at which any granular flow first comes to an
halt – is a property of granular dynamics, and as discussed in appendix D.2,
given by $\tan\varphi_{re}={\pi_{c}}/\sqrt{2}P$, with $\pi_{c}$ the critical
shear stress and $\pi_{c}/P$ the critical friction.

A simple expression satisfying all the above points is, with ${\cal A,B}>0$,

$\displaystyle w_{\Delta}=\sqrt{\Delta}$
$\displaystyle[2{\mathcal{B}}(\rho)\Delta^{2}/5+{\mathcal{A}}(\rho)u_{s}^{2}],$
(27) $\displaystyle P_{\Delta}=\sqrt{\Delta}$ $\displaystyle({\cal
B}\Delta+{\cal A}{u_{s}^{2}}/{2\Delta}),\,\,\pi_{ij}^{*}=-2{\cal
A}\sqrt{\Delta}\,u_{ij}^{*},$ (28) $\displaystyle{4P_{\Delta}}/{\pi_{s}}$
$\displaystyle=2({\cal B}/{\cal
A})(\Delta/u_{s})+{u_{s}}/{\Delta},\qquad\qquad$ (29)

where $P_{\Delta}\equiv\pi_{\ell\ell}/3$. These expressions have been
validated for: (1) static stress distributions in silo, sand pile, point load
on a granular sheet, calculated employing $\nabla_{i}\pi_{ij}=\rho g_{i}$, see
granR1 ; exp1. ; exp2. ; (2) incremental stress-strain relation from varying
static stresses exp3. ; (3) propagation of elastic waves at varying static
stresses exp5. .

Moreover, the energy is convex only for

$u_{s}/\Delta\leq\sqrt{2{\cal B}/{\cal
A}},\quad\text{or}\,\,\,\pi_{s}\leq\pi_{yie}\equiv P_{\Delta}\sqrt{2{\cal
A}/{\cal B}},$ (30)

turning concave if this condition is violated. Because $\pi_{yie}/P_{\Delta}$
is observed to be not or only weakly density dependent, we take ${\cal
B}/{\cal A}=$ const, and assume

${\cal B}={\cal
B}_{0}\left[\frac{\rho-\bar{\rho}}{\rho_{cp}-\rho}\right]^{0.15},$ (31)

where ${\cal B}_{0}>0$ is a constant, and
$\bar{\rho}\equiv\frac{1}{9}(20\rho_{\ell p}-11\rho_{cp})$. ($\rho_{cp}$ is
the random-close packing density, the highest one at which grains may remain
uncompressed. For lack of space, grains cannot rearrange at $\rho_{cp}$,
diminishing plasticity. Note $\bar{\rho}<\rho_{\ell p}<\rho_{cp}$, with
$\rho_{cp}-\rho_{\ell p}\approx\rho_{\ell p}-\bar{\rho}$.) This expression is
constructed to account for four granular characteristics: (1) It is concave
for any density smaller than $\rho_{\ell p}$, such that no elastic state is
stable. (2) It is convex between $\rho_{\ell p}$ and $\rho_{cp}$, ensuring the
stability of elastic solutions in this region. (3) The density dependence of
sound velocities (as measured by Hardin and Richart hardin ) is well rendered
by $\sqrt{{\cal B}/\rho}$. (4) The slow divergence at $\rho_{cp}$ approximates
the fact that the system is much stiffer for $\rho=\rho_{cp}$.

The above expressions may be amended for more realism. First, including the
third invariant, $u_{3}\equiv\sqrt[3]{u^{*}_{ij}u^{*}_{jk}u^{*}_{ki}}$, see
exp7. , $u_{ij}$ and $\pi_{ij}$ are no longer colinear,
$u_{ij}^{*}/u_{s}\not=\pi_{ij}^{*}/\pi_{s}$, though they still share the same
principal axes. As a result, Coulomb and Lade-Duncan yield laws may be
accounted for – in addition to the Drucker-Prager one. Second, including
higher order strain terms allows one to account for compressional
instabilities, the fact that a granular state can only sustain a maximal
static pressure granR2 ; granR4 .

###  A.3 Granular Heat ${w_{T}(s_{g},\rho)}$

In this section, we specify granular heat ${w_{T}(s_{g},\rho)}$, and derive
the fluid pressure $P_{T}$ from it. In the gaseous phase, grains have only
kinetic energy. With $n=\rho/M$ the number of grains per unit volume (and $M$
the average mass of the grain), the energy density is
$w_{T}=\frac{3}{2}T_{k}n$. ($T_{k}$ is the granular temperature for granular
gas.) At lower granular temperatures, when enduring contacts dominate, this
formula is invalid. Going to the small-temperature limit, we expand $w_{T}$ in
$s_{g}$, requiring it to be minimal for $T_{g}=0$, see the consideration in
Sec.3.1, below Eq.(5),

$w_{T}=\frac{s_{g}^{2}}{2\rho b(\rho)}=\frac{\rho b}{2}T_{g}^{2},\quad
T_{g}\equiv\frac{\partial w_{T}}{\partial s_{g}}=\frac{s_{g}}{\rho b}.$ (32)

Assuming only analyticity of $w_{T}$ (ie. that it may be expanded in a power
series), this expression is force-independent and fairly general.

Instead of looking for a formula interpolating between $T_{g}$ and $T_{k}$, we
employ Eq.(32) for all values of $T_{g}$. For large $T_{g}$, this means we
identify $w=\frac{1}{2}\rho b{T_{g}^{2}}$ with $w=\frac{3}{2}T_{k}\rho/M$, or
$T_{k}=bMT_{g}^{2}/3$. Of course, only one temperature can be the actual
temperature, which equilibrates with the true temperature $T$ of the grains.
Strictly, it is $T_{k}$ in the gaseous, and $T_{g}$ in the solid phase. Yet
pragmatically, equilibration occurs only in the solid phase, for vanishing
$T_{g}$. In the gaseous phase, equilibration would falsely imply the existence
of super hot grains of more than million degrees.

Eq.(32) works well: For instance, in the kinetic theory, the pressure is found
$\propto T_{k}$, see luding2009 ; Bocquet , while it is $\propto T_{g}^{2}$ in
GSH,

$-P_{T}\equiv\left.\frac{\partial(w_{T}/\rho)}{\partial
1/\rho}\right|_{s_{g}}=\left.\frac{\partial[(w_{T}-T_{g}s_{g})/\rho]}{\partial
1/\rho}\right|_{T_{g}}=\frac{T_{g}^{2}\rho^{2}}{2}\frac{\partial
b}{\partial\rho}.$ (33)

We choose $b=b(\rho)$ such that it is $P_{T}\propto w_{T}$ for $\rho\to 0$,
and $P_{T}\propto w_{T}/(\rho_{cp}-\rho)$ for $\rho\to\rho_{cp}$, see
luding2009 ; Bocquet ,

$b=\frac{b_{1}}{\rho}+b_{0}\left[1-\frac{\rho}{\rho_{cp}}\right]^{a},\quad
P_{T}=\frac{T_{g}^{2}\rho^{2}}{2}\left[\frac{b_{1}}{\rho^{2}}+\frac{b_{0}}{(\rho_{cp}-\rho)^{1-a}}\right]\equiv
g_{p}T_{g}^{2},$ (34)

with $a\approx 0.1$. [The quantity $g_{p}$ in Eqs.(22) of granRexp  contains a
misprint.] Given ${\cal B}(\rho)$ of Eq.(31), there is also a contribution
$\propto\Delta^{2.5}$ to $P_{T}$ from $w_{\Delta}$. It is neglected because it
is much smaller than the elastic one, $P_{\Delta}\propto\Delta^{1.5}$ for
$\Delta$ small.

###  A.4 Relaxation Equation for ${T_{g}}$

Starting from the balance equation for $s_{g}\propto T_{g}$, we can easily
write down the equation for $T_{g}$, which is increased by viscous heating
$(fv_{s})^{2}$ and decreased by relaxation $\propto T_{g}^{2}$,

$\partial_{t}T_{g}/R_{T}=-T_{g}(1-\xi_{T}^{2}\nabla^{2}_{i})T_{g}+(fv_{s})^{2}+T_{e}^{2}.$
(35)

The relaxation rate $R_{T}T_{g}$ is of order $10^{3}$/s in dense media; the
diffusion length $\xi_{T}$ is a few granular diameters. For $v_{s}=0$ and
$T_{g}$ uniform, we have, in the steady state, $T_{g}=T_{e}$, where $T_{e}$ is
the granular temperature produced by external perturbations, say a sound field
or tapping. We take $T_{e}=0$ here, but shall return in appendix B to discuss
its significance. For granular gas, $T_{g}\propto\sqrt{T_{k}}$, Eq.(35)
reduces to the energy balance of the kinetic theory. For steady and uniform
$v_{s}$, we quickly arrive at

$T_{g}=fv_{s}.$ (36)

With $\eta_{1}T_{g}$ the shear viscosity, see Eq.(39), we have (cf. granR4 )

$f\equiv\sqrt{\eta_{1}/R_{T}b\rho},$ (37)

which is therefore not independent.

For fast dense flow, vigorous agitation of the grains reflects a large
$T_{g}$. For slow, elasto-plastic motion, $T_{g}$ is too small to be directly
observable. Aside from an occasional slip, grains essentially participate in
the macroscopic motion of the given shear rate, with no perceptible deviations
that would contribute to $T_{g}$. Nevertheless, such a slip leads to
vibrations of the gains, implying a $T_{g}$ still many orders of magnitude
larger than the true temperature $T$. As this changes the elastic property by
enabling stress relaxation, $T_{g}$ remains relevant in the elasto-plastic
regime.

###  A.5 Mass and Momentum Conservation

Mass and momentum conservation,

$\partial_{t}\rho+\nabla_{i}(\rho v_{i})=0,\qquad\partial_{t}(\rho
v_{i})+\nabla_{i}(\sigma_{ij}+\rho v_{i}v_{j})=\rho g_{i},$ (38)

are part of GSH; they are to be solved in conjunction with Eqs.(23,24,35), the
evolution equations for $\Delta,u^{*}_{ij}$and $T_{g}$. The total (or Cauchy)
stress is

$\sigma_{ij}=\pi_{ij}+P_{T}\delta_{ij}-\eta_{1}T_{g}v^{*}_{ij},$ (39)

with the elastic stress $\pi_{ij}$ from Eq.(28) and the fluid pressure
$P_{T}\propto T_{g}^{2}$ from Eqs.(34). The third term is the viscous stress,
with the shear viscosity $\eta_{1}T_{g}$.

###  A.6 Summary

The variables of GSH are: $u_{ij},T_{g},\rho,v_{i}$, their evolution equations
form a closed set of nonlinear, partial differential equations, which need to
be solved for a range of boundary conditions. Below, we shall be solving it in
various simple limits.

The statics of GSH is given by the elastic energy $w_{\Delta}$, with the two
coefficients, $\cal A,B$, and the thermal energy $w_{T}$, with the coefficient
$b$. Their density dependence is specified in Eqs(31,34). The dynamics
consists of the Cauchy stress, $\sigma_{ij}$ of Eq.(39), and the relaxation
equations for $T_{g}$, $u^{*}_{ij}$ and $\Delta$, which share six transport
coefficients:

$\Delta_{c},\,\lambda,\,\lambda_{1},\,R_{T},\,\xi_{T},\,\eta_{1},$ (40)

all functions of $\rho$, left unspecified here. (With
$f\equiv\sqrt{\eta_{1}/R_{T}b\rho}$ and $u_{c}\equiv 1/\lambda f$, neither is
independent.) In evaluating GSH, we take $\rho=$ const, such that all
coefficients also are. This renders a number of analytic solutions possible.
Frequently, experiments are performed at constant pressure P, or a stress
component. Then the density may change, and the coefficients with it. This
algebraically more involved case usually need to be solved numerically,
something we shall eschew in this paper.

The stationary solutions of $u^{*}_{ij}$ and $\Delta$, see Eqs.(23,24), are
respectively

$\frac{u_{s}}{u_{c}}=\frac{fv_{s}}{T_{g}},\quad\frac{\Delta}{\Delta_{c}}=\frac{u_{s}}{u_{c}}\frac{fv_{s}}{T_{g}}.$
(41)

Employing in addition
$\frac{\pi_{s}}{\pi_{c}}=\frac{\sqrt{\Delta}\,u_{s}}{\sqrt{\Delta_{c}}\,u_{c}}$
from Eq.(28), we have

$\frac{\pi_{s}}{\pi_{c}}=\frac{\Delta}{\Delta_{c}}=\frac{u_{s}^{2}}{u_{c}^{2}}=\frac{(fv_{s})^{2}}{T_{g}^{2}},\quad\frac{\pi_{ij}^{*}}{\pi_{s}}=\frac{u_{ij}^{*}}{u_{s}}=\frac{v_{ij}^{*}}{v_{s}}.$
(42)

In addition to $T_{g}=fv_{s}$, see Eq.(36), we have $u_{s}=u_{c}$,
$\Delta=\Delta_{c}$, and the critical stress,

$\pi_{c}=\pi_{s}(\Delta_{c},u_{c}),\quad P_{c}=P_{\Delta}(\Delta_{c},u_{c}).$
(43)

This is the state in which all three variables are stationary, with
$\Delta_{c},u_{c},P_{c},\pi_{c}$ rate-independent – ie. possessing values that
are independent of the shear rate $v_{s}$.

Driving a granular system at a slow shear rate $v_{s}$, the system executes
the rate-independent elasto-plastic motion, and the stress is given by the
first term of Eq.(39), $\sigma_{ij}=\pi_{ij}$. This is where the hypoplastic
model holds. In steady state, we have the critical stress, $P=P_{c}$ and
$\sigma_{s}=\pi_{c}$. At higher rates, the next two terms can no longer be
neglected, and we enter the regime of fast dense flow and the
$\mu(I)$-rheology. For steady flow, with $P_{T}\propto T_{g}^{2}=(fv_{s})^{2}$
and $\eta_{1}T_{g}v^{*}_{ij}=\eta_{1}fv_{s}v^{*}_{ij}$, we have,

$P=P_{c}+e_{p}(\rho)v_{s}^{2},\quad\sigma_{s}=\pi_{c}+e_{s}(\rho)v_{s}^{2}.$
(44)

Being quadratic, the corrections $\sim v_{s}^{2}$ come on slowly, leaving the
rate-independent regime, $\sigma_{ij}=\pi_{ij}$, to persist for many orders of
magnitude in the rate.

In studying the relaxation dynamics, we may differentiate between two cases.
First, driving a granular system at a given shear rate, with $T_{g}$ quickly
settling into the stationary solution, $T_{g}=fv_{s}$, the two remaining
relaxation Eqs.(23,24) are

$\displaystyle\partial_{t}u^{*}_{ij}$ $\displaystyle=v^{*}_{ij}-\lambda
fu^{*}_{ij}v_{s},$ (45) $\displaystyle\partial_{t}\Delta+v_{\ell\ell}$
$\displaystyle=\alpha_{1}u^{*}_{ij}v^{*}_{ij}-\lambda_{1}\Delta fv_{s}.$

Second, to study the dynamics at given stress, or equivalently, at given
elastic strain, $\partial_{t}u_{s},\partial_{t}\Delta=0$, we insert Eq.(42)
into Eq.(35) to find

$\displaystyle\partial_{t}T_{g}$
$\displaystyle/(R_{T}T_{g})+\xi_{T}^{2}\nabla^{2}_{i}T_{g}=-T_{g}(1-\pi_{s}/\pi_{c}),$
(46) $\displaystyle fv_{s}$ $\displaystyle=T_{g}\sqrt{\pi_{s}/\pi_{c}}.$

Both equations are very useful. The rate-independent Eqs.(45) account for the
approach to the critical state at given rate, and for load/unload. It
possesses the same mathematical structure as the hypoplastic model kolymbas1 ;
kolymbas2 , and yields very comparable results for elasto-plastic motion.
Equations.(46), on the other hand, describe creep, angle of repose, shear
band, and it reduces to Kamrin’s nonlocal constitutive relation kamrin ;
kamrin2  for stationary flows, $\partial_{t}T_{g}\propto\partial_{t}v_{s}=0$.

##  Appendix B Tapping and Compaction

Numerous experiments employing varying external perturbations show a
compaction of the packing of grains, see the review article 1nico . This
phenomenon is widely accounted for by employing the Edwards entropy $S_{Ed}$,
or some generalization of it. We have reasons to doubt its basic assumption:
Considering $S_{Ed}$ implies that all other, much larger entropy contributions
remain constant, see the discussion in the introduction, also at the end of
Sec.3.1. Moreover, this understanding makes compaction a singular effect, in
need of a special treatment not apparently useful for any other granular
phenomena. In contrast, we believe that compaction is well-embedded into other
granular phenomena, and closely related to an observation familiar to
engineers – density increase at given pressure under cyclic shear.  The point
is, shear rates produce $T_{g}$, and $\Delta$ relaxes if $T_{g}\not=0$, see
Eq.(24). Since the pressure $P_{\Delta}={\cal B}(\rho)\Delta^{1.5}$, see
Eq.(28), is kept constant, the density $\rho$ increases to compensate. (We
neglect the deviatory stress for simplification.)

The effect of compaction remains the same however $T_{g}$ is generated, via a
shear rate $fv_{s}$, or by an external perturbation $T_{e}$ (generated by
sound field, periodic water injection, or tapping), because including the
external perturbation $T_{e}$, the uniform, stationary solution of Eq.(35) is

$T_{g}^{2}={(fv_{s})^{2}+T_{e}^{2}}.$ (47)

###  B.1 Reversible and Irreversible Compaction

We consider the total pressure $P=P_{\Delta}+P_{T}$, see Eqs.(28,31,34,39),

$P_{\Delta}={\cal B}(\rho)\Delta^{1.5},\quad P_{T}=g_{p}(\rho)T_{g}^{2},$ (48)

with $\cal B$ and $g_{p}$ monotonically increasing functions of $\rho$. For
$T_{g}$ small, the seismic pressure $P_{T}$ may be neglected, and $\rho$
increases as $\Delta$ relaxes, because $P=P_{\Delta}=$ const. This portion of
the compaction is irreversible. Most soil-mechanical experiments are in this
limit.

For larger $T_{g}$, the seismic pressure $P_{T}$ cannot be neglected. So
$\Delta$ relaxes with the density increasing irreversibly and
$P_{\Delta}+P_{T}=$ const. Since $g_{p}$ is a more sensitive function of
$\rho$ than $\cal B$, mainly $P_{T}$ is increased. After the relaxation has
run its course, we have $\Delta,P_{\Delta}=0$, $P_{T}=$ const. If one modifies
$T_{g}$ now, the density will change in response, see Eq.(34), as

$\frac{\rho^{2}}{\rho_{cp}-\rho}=\frac{2P}{ab_{0}T_{g}^{2}},$ (49)

if $b_{1}$ is neglected. This happens reversibly, in both directions.

To calculate the dynamics of irreversible compaction, we start with the
relation

$\left.\frac{\partial\rho}{\partial\Delta}\right|_{T_{g},P}=\frac{\partial_{t}\rho}{\partial_{t}\Delta}=\frac{\partial
P/\partial\Delta}{\partial P/\partial\rho}\equiv A.$ (50)

Because
$\partial_{t}\Delta+\lambda_{1}T_{g}\Delta=-v_{kk}=\partial_{t}\rho/\rho$, see
Eq.(24), the relaxation of $\Delta$ is given by

$\partial_{t}\Delta=-\Delta/\tau_{\Delta},\quad\tau_{\Delta}\equiv(1+A/\rho)/\lambda_{1}T_{g}.$
(51)

Note $\rho=\rho(\Delta)$ and $\tau_{\Delta}=\tau_{\Delta}(\Delta)$ for
$P,T_{g}=$ const, hence the dynamics is not exponential. (We assume vanishing
shear strain $v_{s},u_{s}\equiv 0$.)

###  B.2 Memory Effects versus Hidden Variables

Changing $T_{g}$ midway, at constant $P=P_{\Delta}+P_{T}$, with $\Delta$ still
finite, will mainly lead to a change in $\Delta$ (as $\rho$ responds more
slowly). This disrupts the relaxation of $\Delta$, in essence resetting its
initial condition. This was observed in mem  and interpreted as a memory
effect. Such “memory” is usually a result of hidden variables: When the system
behaves differently in two cases, although all state variables appear to be
the same, we speak of memory or history-dependence. But an overlooked variable
that has different values for the two cases will naturally explain the
difference. In the present case, the manifest and hidden variables are $\rho$
and $\Delta$, respectively.

###  B.3 Critical State Under Perturbations

If we want to render the above considerations quantitative, we need to relate
the amplitude of external perturbations to the temperature $T_{e}$. One clean
way to do this is to observe the critical state exposed to external
perturbations. Inserting Eq.(47) into (42), we find

$\pi_{s}(\Delta_{c},u_{c})=\frac{\pi_{c}}{1+(T_{e}/fv_{s})^{2}},$ (52)

instead of Eqs.(43). Clearly, the critical shear stress vanishes at slow
rates, $T_{e}\gg fv_{s}$, and is unchanged at higher ones, $T_{e}\ll fv_{s}$.
This strongly rate-dependent behavior has variably been observed, see eg.
vHecke2011 . Especially, we have $\pi_{s}=\frac{1}{2}\pi_{c}$ for
$T_{e}=fv_{s}$. Therefore, varying the amplitude of an external perturbation
at a given shear rate, we may identify the amplitude at which
$\pi_{s}=\frac{1}{2}\pi_{c}$ holds as equivalent to $fv_{s}$.

###  B.4 Tapping

Next, we consider tapping in greater detail, aiming to relate it to the above
presented, generally valid mechanism of compaction. Gentle tapping leads to
both $\Delta$ and $T_{g}$ fluctuating. As long as grains loosen but do not
loose contact with one another, $\Delta$ remains finite at all time, $P_{T}$
may be neglected, and $\bar{\Delta}$ (averaged in time) will relax
monotonically, as accounted for by Eq.(51). Because a given layer in a
granular column with a free upper surface is subject to a constant pressure,
the density increases to compensate for the diminishing $\Delta$.

Stronger tapping leads to a higher $T_{g}$, with (1) grains losing contact and
$\Delta$ vanishing periodically on one end, and (2) $T_{g}$ vanishing with
$\Delta$ maximal on the other, when grains come down to a crushing stop. This
is a hard case to account for, because the system undergoes the transition
from solid to liquid, further on to gas and collisionless free flight, then
all the way back again, after every tap, see lupou . And it raises the
question whether the system, when being tapped again, will pick up the
relaxation of $\Delta$ where it was left at, when the system last crushed to a
stop. Now, given the fact that tapping is but one way to achieve compaction,
leading to results very similar to that of many other methods 1nico , it does
appear that grains in free flight remember the average packing efficiency when
they were at rest. Possibly, this memory is the basic effect, while a finite
$T_{g}$ and a further reduction of $\Delta$ a small perturbation, which
becomes evident only after the accumulation of many taps.

This is only a surmise, but if proven true, we may take tapping as driven by
the same compaction mechanism as described above. Then GSH provides an
alternative understanding for compaction that is transparent, conventional and
demystified.

##  Appendix C Constant Rate Experiments

Driving a granular system at a given uniform shear rate, with $T_{g}$ quickly
settling into the stationary solution, $T_{g}=fv_{S}$, the two relevant
relaxation equations are given by Eqs.(45).

###  C.1 The Hypoplastic Model

The hypoplastic model is a state-of-the-art constitutive relation that
achieves considerable realism in the rate-independent regime of elasto-plastic
motion, especially in triaxial experiments. For $v_{kk}=0$, and since
$\sigma_{ij}=\pi_{ij}$, it has the form

$\partial_{t}\pi_{k\ell}=H_{k\ell ij}v^{*}_{ij}+\Lambda_{k\ell}v_{s},$ (53)

as postulated by Kolymbas kolymbas1 , where $H_{k\ell ij},\Lambda_{k\ell}$ are
functions of the stress and density. Clearly, it has very similar dependence
on the shear rate as Eqs.(45). In fact, taking

$\displaystyle\partial_{t}\pi_{k\ell}(\Delta,u_{s})=\frac{\partial\pi_{k\ell}}{\partial
u_{ij}}\partial_{t}u_{ij}=-\frac{\partial^{2}w_{\Delta}}{\partial
u_{ij}\partial
u_{k\ell}}(\partial_{t}u^{*}_{ij}-\frac{\delta_{ij}}{3}\partial_{t}\Delta),$

with Eqs.(27,45), we can easily deduce the expressions for $H_{ijk\ell}$ and
$\Lambda_{ij}$. (The equivalence between GSH and hypoplasticity also holds for
$v_{kk},\partial_{t}\rho\not=0$.)

The name of “hypoplasticity” arose originally because it was believed that
$H_{ijk\ell}$ and $\Lambda_{ij}$ cannot be obtained from any potential
(contrary to what is done here). As a result, hypoplasticity was a very
flexible theory, containing $42$ functions as adjustable parameters. Great
efforts have been, and are being, invested in finding accurate expressions for
them. The results of GSH have started to change this among modern
practitioners of hypoplasticity. Recently, Niemunis, Grandas Tavera and
Wichtmann use a Gibbs potential to obtain $H_{ijkl}$ – an improvement that
they call neo-hypoplasticity. Calibrating $H_{ijkl}$ using incremental stress-
strain relations, they found excellent agreement with observation of elasto-
plastic motion 99 . Subsequently, we showed that this Gibbs potential is,
legendre transformed, rather similar to $w_{\Delta}$ of Eq.(27), see
ActaGeoMechanica . This is reassuring, as it implies that all results of
hypoplasticity – including various butterfly-curves, the approach to the
critical state and different slopes at load and unload – can also be produced
by GSH.

###  C.2 Load and Unload

One big advantage of GSH is its simple structure that enables analytic
solutions for many experimental situations. Therefore, in spite of the general
agreement with hypoplasticity, we shall examine Eqs.(45) more closely. For
load and unload, the equations are, respectively,

$\partial_{t}u_{s}=v_{s}-\lambda
fu_{s}v_{s},\quad\partial_{t}u_{s}=-v_{s}-\lambda fu_{s}v_{s}.$ (54)

Note, first of all, that the described behavior is elastic for vanishing shear
stress, $u_{s}\to 0$. The shear strain is maximal in the stationary, critical
state, in which both terms cancel, and dissipation, produced by the relaxing
second term, is considerable. (The shear strain may well be non-monotonic,
passing a maximum before assuming the critical value, see next section.) Next,
note that the slope $u_{s}$ versus $v_{s}$ is, respectively for load and
unload, $1-\lambda fu_{s}$ and $-(1+\lambda fu_{s})$, a simple fact not
related to any “history-dependence.”

###  C.3 Approach to the Critical State

Solving Eqs (45) for constant $v_{s}$, with the initial conditions:
$\Delta=\Delta_{0},u_{s}=0$, the relaxation into the critical state is an
exponential decay for $u_{s}$, and a sum of two for $\Delta$,

$\displaystyle u_{s}(t)=u_{c}(1-e^{-\lambda
f\varepsilon_{s}}),\quad\varepsilon_{s}\equiv v_{s}t,$ (55)
$\displaystyle\Delta(t)=\Delta_{c}(1+f_{1}\,e^{-\lambda
f\varepsilon_{s}}+f_{2}e^{-\lambda_{1}f\varepsilon_{s}}),$ $\displaystyle
f_{1}\equiv\frac{\lambda_{1}}{\lambda-\lambda_{1}},\quad
f_{2}\equiv\frac{\Delta_{0}}{\Delta_{c}}-\frac{\lambda}{\lambda-\lambda_{1}}.$

It is useful that a simple, analytical solution for this signature experiment
of soil mechanics exists. Typically, $\lambda$ is larger than $\lambda_{1}$,
and the decay of $u_{s}$ and $f_{1}$ faster than that of $f_{2}$. Note $f_{2}$
may be negative, and $\Delta(t)$ is then non-monotonic. The associated
pressure $P_{\Delta}$ and shear stress $\pi_{s}$ are those of Eqs (28),
neither is monotonic for a negative $f_{2}$.

###  C.4 Shear Jamming

Generally speaking, that grains jam when the volume is reduced seems obvious,
less so when they are sheared. Yet this has been observed SJ1 . Further, the
density at which this happens seems to memorize the deformation history. After
a cyclic compression, the density at which the grains unjam is significantly
larger SJ2 ; SJ3 .

In GSH, a state is jammed if it stably sustains an elastic stress,
$\Delta,u_{s}\not=0$. Between the random lose and random close values,
$\rho_{lp}\leq\rho\leq\rho_{cp}$, the system is unjammed if $\Delta,u_{s}$
vanish – independent of the density. Note that for $\rho<\rho_{lp}$, we always
have $\Delta,u_{s}\equiv 0$, and the system is never jammed; while for
$\rho>\rho_{cp}$, $\Delta$ is always finite, and the system cannot be
unjammed.

Shear jamming is a special case of the approach to the critical state, as
accounted for by Eqs.(55). Starting from $\Delta_{0}=0$ (rather than
$\Delta_{0}\not=0$ as usual), a steady shear rate $v_{s}$ will (for
$\rho_{lp}\leq\rho\leq\rho_{cp}$) increase $\Delta,u_{s}$, jam the system, and
push it eventually into the critical state. This is the simple physics of
shear-jamming.

The memorized jamming density is easily explained by granular compaction in
the presence of $T_{g}$, as considered in appendix B. A cyclic compression
increases the packing efficiency, because it produces a $T_{g}$ that lets
$\Delta,u_{s}$ relax. When the system becomes unjammed again, with
$\Delta,u_{s}=0$, the actual density is larger than the initial one. Calling a
density at which $\Delta,u_{s}$ become zero the jamming density, we find it to
possess memory. As in appendix B.2, this is due to the hiden variables,
$\Delta,u_{s}$.

Finally, a word on the intermediate so-called fragile state that is sandwiched
between the unjammed and stably jammed one. Starting from $\Delta_{0}=0$ and
$u_{s}=0$, the growth of $u_{s}$ is initially faster than that of $\Delta$,
see Eqs.(45), such that the stability condition $u_{s}/\Delta\leq\sqrt{2{\cal
B}/{\cal A}}$ of Eqs.(30) is breached. Only after $u_{s}$ has reached a
certain size, does $\Delta$ catch up, to satisfy Eq.(30) and jam the system
stably. [In usual geotechnical experiments, the initial condition always has a
sufficiently large $\Delta_{0}$ such that Eqs.(30) is never breached.] GSH at
present does not provides an answer for how a granular system behaves when
stability conditions are breached, but we are working to include this aspect.

##  Appendix D  Constant Stress Experiments

We now consider the case of constant stress and density, accounted for by
Eqs.(46), a relaxation-diffusion equation for
$fv_{s}=T_{g}\sqrt{\pi_{s}/\pi_{c}}$.

In this context, it is useful to realize that stress-controlled experiments
cannot be performed in simple triaxial apparatuses with stiff steel walls,
because the correcting rates employed by the feedback loop to keep the stress
constant (or slowly changing) are usually too strong. As a result, $T_{g}$ is
excited that distorts, even overwhelms, its relaxation. The situation is then
more one of consecutive constant rates, less one of constant stress. Employing
soft springs to couple the granular system to its driving device makes small-
amplitude stress corrections possible without exciting much $T_{g}$.

###  D.1 Diverging Shear Strain

First, consider uniform solutions of Eqs.(46). With the initial conditions
$T_{g}=T_{0}$, $v_{s}=v_{0}$ and $T_{0}\sqrt{\pi_{s}/\pi_{c}}=fv_{0}$ at
$t=0$, we find

$T_{g}={T_{0}}/({1+{r_{T}}{T_{0}}t}),\quad r_{T}\equiv
R_{T}[1-\pi_{s}/\pi_{c}].$ (56)

The solution for the shear rate, $v_{s}={v_{0}}/(1+r_{v}v_{0}t)$, with
$r_{v}\equiv{r_{T}}f\sqrt{\pi_{c}/\pi_{s}}$, implies a logarithmically
divergent total shear strain,

$\varepsilon_{s}-\varepsilon_{0}\equiv\int v_{s}{\rm
d}t=\ln(1+r_{v}v_{0}t)/r_{v}.$ (57)

Note that $\varepsilon_{s}$ does not actually diverge, because as $T_{g}\to
0$, the system enters the quasi-elastic regime, where its relaxation becomes
exponential, see granR4 . So even if the creep $\varepsilon_{s}$ is large
close to $\pi_{c}$, it comes to a halt eventually, and the system is
mechanically stable.

The $T_{g}$-relaxation is slower the closer $\pi_{s}$ is to $\pi_{c}$,
infinitely so for $\pi_{s}=\pi_{c}$. The state is then indistinguishable from
the rate-controlled, critical state, which, clearly, may be maintained also at
given stress. For $\pi_{s}>\pi_{c}$, the relaxation rate is negative, and
$T_{g}\propto v_{s}$ will grow without bound. The system then has two
possibilities to become stable again, either a non-uniform shear band, or
accelerating to the rate-dependent regime of the $\mu(I)$-rheology, see the
sections below.

In an experiment involving a fan submerged in sand and coupled to the motor
with a very soft spring, Nguyen et al. aging  first pushed the system to a
certain shear stress at a given rate, producing a $T_{g}$. Then, switching to
maintaining the shear stress, they observed a shear strain
$\varepsilon_{s}(t)$ that appears to diverge logarithmically.

Ignoring the strongly non-uniform stress distribution, they carried out an
analysis using two scalar equations borrowed from glassy media that may be
roughly mapped to the ones employed above, where the granular temperature
$T_{g}$, its relaxation rate $R_{T}$, and its production rate $R_{T}f^{2}$
were replaced by fluidity, aging and rejuvenation parameter. This is certainly
a sensible approach when one lacks a granular theory to interpret the data.
But less so given the fact that GSH provides a unified, tensorial and
realistic treatment tailored to granular media, thus affording a transparent,
well-founded and innate understanding. One circumstantial support for GSH are
hot spots – localized regions of strong, temporary deformations that the
authors observed. They identified the rate of hot spots as fluidity, but could
more directly have taken them as $T_{g}$.

Finally, we note that an infinitesimal $T_{g}$ will not destabilize a static
shear stress exceeding $\sigma_{c}$, only a $T_{g}$ sufficiently large will
grow without bound. This is because the critical stress diverges for $T_{g}\to
0$ and the window between $\sigma_{c}$ and $\sigma_{s}^{yield}$ vanishes
granR4 ; granRexp . Therefore, a static shear stress remains metastable for
$T_{g}=0$, turning instable only at the yield stress,
$\pi_{s}^{y}/\sqrt{2}P=\sqrt{{\cal A}/{\cal B}}$, as given in Eq.(30).

###  D.2 Angle of Repose $\varphi_{re}$ versus Angle of Stabilty
$\varphi_{st}$

Aranson and Tsimring were the first to construct a theory for these two angles
aranson ; Aranson1 . Taking the stress as the sum of a solid- and a fluid-like
part, they define an order parameter $\hat{\varrho}$ that is 1 for solid, and
0 for dense flow. They then postulate a free energy $f(\hat{\varrho})$ such
that it is stable with $\hat{\varrho}=1$ only for $\varphi<\varphi_{st}$, with
$\hat{\varrho}=0$ only for $\varphi>\varphi_{re}$, and
$\varphi_{st}>\varphi>\varphi_{re}$ as the bi-stable region. In GSH, these two
angles need not be postulated ad hoc, as they are already given by the yield
and critical stress, from Eqs.(30,43):

$\displaystyle\tan\varphi_{st}$
$\displaystyle=\pi_{yie}/\sqrt{2}P_{\Delta}=\sqrt{{\cal A}/{\cal B}},$ (58)
$\displaystyle\tan\varphi_{re}$
$\displaystyle={\sigma_{c}}/\sqrt{2}\,P_{c},\quad\varphi_{re}<\varphi_{st}.$
(59)

The collapse that occurs when one slowly tilts a plate supporting a layer of
grains at $\varphi_{st}$, is a process that happens at $T_{g}=0$ and hence
encoded in $w_{\Delta}$.

The angle of repose $\varphi_{re}$ is related to the results of the last
section: As long as the shear stress is held below the critical one,
$\pi_{s}<\pi_{c}$, the $T_{g}$-relaxation will run its course, and the system
is in a static, mechanically stable state afterwards. At $\pi_{s}=\pi_{c}$,
the system becomes critical, and no longer comes to a standstill. Therefore,
$\varphi_{re}$ is given by the critical stress.

The inequality $\varphi_{re}<\varphi_{st}$ holds because the critical state is
an elastic solution, cf. Eq.(43), and $\varphi_{st}$ the angle at which all
elastic solutions become unstable.

###  D.3 Shear Bands

For $\pi_{s}>\pi_{c}$, no uniform stationary solution of Eq.(46) exists, but a
nonuniform one does, with $T_{g}\equiv 0$ for $x\leq 0$ and $x\geq\xi_{sb}$,
and a shear band in between:

$\displaystyle\nabla^{2}T_{g}$
$\displaystyle=-T_{g}/\xi^{2}_{sb},\quad{\xi}^{2}_{sb}\equiv\xi_{T}^{2}/[\pi_{s}/\pi_{c}-1],$
$\displaystyle v_{s}/v_{s}^{0}$ $\displaystyle=T_{g}/T_{g}^{0}=\sin(\pi
x/\xi_{sb}).$ (60)

The velocity difference across the band is $\Delta v=\int v_{s}{\rm d}x=\int
v_{s}^{0}\sin(\pi x/\xi_{sb}){\rm d}x$, or

$v_{s}^{0}=\Delta v/(2\xi_{sb})=\sqrt{\pi_{s}/\pi_{c}}\,T_{g}^{0}/f.$ (61)

That the correlation length $\xi_{sb}$ diverges for $\pi_{s}=\pi_{c}$ gives a
justification for the term critical. Note the density $\rho$ must be lower in
the shear band than in the quiescent region, $\rho_{sb}<\rho_{q}$, because
only then can the shear stress $\pi_{s}$ (in 1D a constant quantity) be
smaller in the quiescent region, but larger in the shear band, than the
critical stress, $\pi_{c}(\rho_{q})>\pi_{s}>\pi_{c}(\rho_{sb})$.

Due to the lacks of a length scale, constitutive models such as
hypoplasticicty or barodesy cannot account for shear bands. There are various
approaches to ameliorate it, by introducing gradient terms wu2  or adding
state variables to account for the couple stress and the Crosserat rotation
wu1 . The latter method works, but at the price of a far more complex theory,
constructed for the sole purpose of describing shear bands. And it poses
questions about the physics: If the couple stress is important in the shear
band, because it is fluid, why is it not in the uniformly fluid and gaseous
state of granular media?

##  Appendix E Fast Dense Flow

Eqs.(46) state that for subcritical stresses: $\sigma_{s}=\pi_{s}<\pi_{c}$,
the granular temperature and the shear rate, $T_{g}=fv_{s}$, will relax to
zero; for super-critical stresses, $\sigma_{s}=\pi_{s}>\pi_{c}$, they will
accelerate, until Eqs.(44), again a stable solution, are satisfied.

###  E.1 The ${\mu(I)}$-Rheology

Although $\sigma_{s}$ and $P$ are functions of $\rho,v_{s}^{2}$, we can
rewrite Eq.(44), with

$\mu\equiv\frac{\sigma_{s}}{P},\quad\mu_{1}\equiv\frac{\pi_{c}(\rho)}{P_{c}(\rho)},\quad\mu_{2}\equiv\frac{e_{s}(\rho)}{e_{p}(\rho)}$
(62)

as

$\mu=\mu_{1}+(\mu_{2}-\mu_{1})\hat{I},\quad\hat{I}\equiv\frac{e_{p}v_{s}^{2}}{P}=\frac{P_{T}}{P_{T}+P_{c}},$
(63)

where both $\mu_{1}$ and $\mu_{2}$ have been observed as independent of the
density, see Bagnold ; schofield ; wood1990 . [Note
$e_{p}=\frac{1}{2}f^{2}\rho^{2}\partial b/\partial\rho$, see Eqs.(33,44).]

Stating that $\mu$ varies between two density-independent plateaus with a
single variable $\hat{I}$ that vanishes with $v_{s}$, and goes to one as
$v_{s}\to\infty$, this GSH-formula is close to the well-known
$\mu(I)$-rheology pouliquen1 ; pouliquen2 ; pouliquen3 ; DEM-mu1 ; DEM-mu2 :
$\mu=\mu_{1}+({\mu_{2}-\mu_{1}})I/({I+I_{0}})$, with $\mu_{1},\mu_{2}$
density-independent, and $I/(I+I_{0})$ going from zero to one.

But there is the difference that $\hat{I}\propto v_{s}^{2}/P\propto I^{2}$.
Supporting GSH is the rate-independence of the critical state,

$\partial\sigma_{s}/\partial v_{s}\stackrel{{\scriptstyle v_{s}\to
0}}{{\equiv}}0,$

valid for many orders of magnitude of $v_{s}$. If the stress is an analytic
function of $v_{s}$, with no kink, linear dependence
$\partial\sigma_{s}/\partial v_{s}=$ const. is ruled out, but not a quadratic
one. [Dimensional arguments lead to both the $\mu(I)$-rheology and to
Eq.(63).]

For $\rho<\rho_{\ell p}$, there is no elastic solution, $P_{c},\pi_{c}=0$.
Then $\mu\equiv\mu_{2}$ irrespective how large the shear rate $v_{s}$ is. When
studying granular rheology, one varies $v_{s}$ keeping either the density or
the pressure constant. Since $P_{c}\gg e_{p}(\rho)v_{s}^{2}$ and $\pi_{c}\gg
e_{s}(\rho)v_{s}^{2}$ for the usual rates, it is not easy to go beyond the
limit of $\mu=\mu_{1}$ for given $\rho$. Not so for given pressure, because
the density decreases for increasing $v_{s}$, and a discontinuous transition
from Eqs.(63) to $\mu\equiv\mu_{2}$ takes place at $\rho=\rho_{\ell p}$, where
$P,\sigma_{s}$ decrease by three orders of magnitude, see campbell . The
$\mu(I)$-relation, lacking this bit of information, is in fact valid only for
$\rho>\rho_{\ell p}$.

On the other hand, the second part of $\mu(I)$-rheology, concerning the
packing fraction $\phi\equiv\rho/\rho_{g}$ and taken as $\phi=\phi(I)$, holds
only for $\rho<\rho_{\ell p}$, because $\phi=f(I)$, or $f^{-1}(\phi)=I\propto
v_{s}^{2}/P$, implies $P_{c}=0$.

For given density $\rho$, the shear stress
$\sigma_{s}=\mu_{1}P^{c}(\rho)+\mu_{2}e_{p}(\rho)v_{s}^{2}$ is a monotonic
function of the rate $v_{s}$. For constant pressure, with $\rho=\rho(P,v_{s})$
given by $P=P^{c}(\rho)+e_{p}(\rho)v_{s}^{2}$, the shear stress $\sigma_{s}$
may well be non-monotonic exp6. .

###  E.2 The Solid-Liquid Boundary

One frequently observes granular liquid executing fast dense flow, bordering
on a quiescent, or solid, region. More careful observation reveals that the
velocity is continuous, with a quickly decaying creep taking place in the
solid, see Komatsu et al komatsu , Crassous et al crassous . GSH accounts for
this as a result of $T_{g}$ diffusing from the fluid region into the solid
one.

We take the solid-liquid boundary at $x=0$, and only variations perpendicular
to it. For such a one-dimensional geometry, the pressure $P$ and shear stress
$\sigma_{s}$ are necessarily uniform. The density is discontinuous at $x=0$,
being larger on the solid side, $\rho_{s}>\rho_{\ell}$. Hence
$\sigma_{s}>\pi_{c}$ in liquid, executing fast dense flow according to
Eq.(63), and $\sigma_{s}=\pi_{s}<\pi_{c}$ in solid.

The shear rate $T_{g}=fv_{s}$ is uniform on the liquid side, for $x<0$. On the
solid side, $x>0$, with the liquid maintaining the shear stress $\sigma_{s}$,
Eqs.(46) hold. Therefore

$\displaystyle\nabla^{2}T_{g}=T_{g}/\xi^{2}_{cr},\quad\xi_{cr}^{2}\equiv\xi_{T}^{2}/[1-\pi_{s}/\pi_{c}]$
(64) $\displaystyle\text{implying}\quad
v_{s}/v_{s}^{0}=T_{g}/T_{g}^{0}=\exp(-x/\xi_{cr}),$ (65)

where $T_{g}^{0}$ is the fluid value, and
$v_{s}^{0}=[T_{g}^{0}/f(\rho_{s})]\sqrt{\pi_{s}/\pi_{c}(\rho_{s})}$. (The
velocity and $T_{g}$ are continuous, the rate $v_{s}$ is not.) We require
$T_{g}\to 0$ for $x\to\infty$.

It is not surprising that the decay length $\xi_{cr}$ diverges for
$\pi_{s}=\pi_{c}$, because the solid region turns critical then, and ceases to
exist.

###  E.3 Kamrin’s nonlocal constitutive relation

In two recent papers kamrin ; kamrin2 , Kamrin et al propose a nonlocal
constitutive relation (KCR) well capable of accounting for the steady flows in
the split-bottom cell fenistein . A key ingredient is the fluidity $g\equiv
v_{s}/\mu$. Denoting $\mu\equiv\sigma_{s}/P$, $\mu_{s}\equiv\sigma_{c}/P_{c}$,
it is taken to obey

$\xi^{2}_{cr}\nabla^{2}g=g-g_{loc},\quad\xi_{cr}\propto
1/\sqrt{|\mu-\mu_{s}|}.$ (66)

Because $g_{loc}=0$ for $\mu<\mu_{s}$, this relations is rather similar to
Eq.(64), with $g$ assuming the role of $T_{g}$, and the two decay lengths
diverging at the same stress value. For $\mu\geq\mu_{s}$, the system is fluid
and $g=g_{loc}$ essentially constant. With
$g_{loc}\propto\sqrt{P}(1-\mu_{s}/\mu)$, KCR is consistent with a first-order
expansion in the inertial number of the $\mu(I)$-rheology. Therefore, GSH is,
in this limit, also similar to KCR.

This is fortunate, because again, there is a symbiotic relation between GSH
and KCR, similar to that between GSH and hypoplasticity, which gives the
constitutive relations a solid foundation in physics, and GSH a robust
connection to reality.

More recently, Kamrin and Bouchbinder constructed a “two-temperature
continuous mechanics,” with $\theta_{c}$ and $\theta_{v}$, the first a
configuration temperature, the second for the vibrational degrees of freedom
ttcm . (Microscopic degrees of freedom are not considered. There would be
three temperatures otherwise.) They hypothesize that the fluidity $g$ may be
related to $\theta_{c}$. We believe that $g$ is, as discussed above,
essentially $T_{g}$ of GSH, which (quantifying the fluctuating granular
degrees of freedom) is closer to $\theta_{v}$.