# Continuum modelling of shear start-up in soft glassy materials

Roberto Benzi  Dipartimento di Fisica, Università di Roma “Tor Vergata” and
INFN, Via della Ricerca Scientifica, 1-00133 Roma, Italy  Thibaut Divoux  Univ
Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342
Lyon, France  Catherine Barentin  Université de Lyon, Université Claude
Bernard Lyon 1, CNRS, Institut Lumière Matière, F-69622 Villeurbanne, France
Sébastien Manneville  Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS,
Laboratoire de Physique, F-69342 Lyon, France  Mauro Sbragaglia  Dipartimento
di Fisica, Università di Roma “Tor Vergata” and INFN, Via della Ricerca
Scientifica, 1-00133 Roma, Italy  Federico Toschi  Department of Applied
Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven,
The Netherlands and CNR-IAC, Rome, Italy

###### Abstract

Yield stress fluids (YSFs) display a dual nature highlighted by the existence
of a critical stress $\sigma_{\text{y}}$ such that YSFs are solid for stresses
$\sigma$ imposed below $\sigma_{\text{y}}$, whereas they flow like liquids for
$\sigma>\sigma_{\text{y}}$. Under an applied shear rate $\dot{\gamma}$, the
solid-to-liquid transition is associated with a complex spatiotemporal
scenario that depends on the microscopic details of the system, on the
boundary conditions, and on the system size. Still, the general phenomenology
reported in the literature boils down to a simple sequence that can be divided
into a short-time response characterized by the so-called “stress overshoot”,
followed by stress relaxation towards a steady state. Such relaxation can be
either ($i$) long-lasting, which usually involves the growth of a shear band
that can be only transient or that may persist at steady-state, or ($ii$)
abrupt, in which case the solid-to-liquid transition resembles the failure of
a brittle material, involving avalanches. In the present paper, we use a
continuum model based on a spatially-resolved fluidity approach to rationalize
the complete scenario associated with the shear-induced yielding of YSFs. A
key feature of our model is to provide a scaling for the coordinates of the
stress overshoot, i.e., stress $\sigma_{\text{M}}$ and strain
$\gamma_{\text{M}}$ as a function of $\dot{\gamma}$, which shows excellent
agreement with experimental and numerical data extracted from the literature.
Moreover, our approach shows that the power-law scaling
$\sigma_{\text{M}}(\dot{\gamma})$ is intimately linked to the growth dynamics
of a fluidized boundary layer in the vicinity of the moving boundary. Yet,
such scaling is independent of the fate of that layer, and of the long-term
behavior of the YSF, i.e., whether the steady-state flow profile is
homogeneous or shear-banded. Finally, when including the presence of “long-
range” correlations, we show that our model displays a ductile to brittle
transition, i.e., the stress overshoot reduces into a sharp stress drop
associated with avalanches, which impacts the scaling
$\sigma_{\text{M}}(\dot{\gamma})$. This generalized model nicely captures
subtle avalanche-like features of the transient shear banding dynamics
reported in experiments. Our work offers a unified picture of shear-induced
yielding in YSFs, whose complex spatiotemporal dynamics are deeply connected
to non-local effects.

##  I Introduction

Yield stress fluids (YSFs) encompass a wide variety of amorphous soft
materials, from soft glasses like shaving creams and mayonnaise to colloidal
gels such as fresh cement pastes and silica or alumina precursors for catalyst
supports. These materials all share the existence of a critical stress
$\sigma_{\text{y}}$, coined the yield stress, below which their mechanical
response is mainly that of a solid, and above which they flow like liquids
Bonn  _et al._ (2017); Coussot (2018). Such shear-induced solid-to-liquid
transition is “reversible” in the sense that the material’s solid-like
behavior is recovered when shear is stopped –although generally with a
different microstructure from that preceding the transition Koumakis  _et al._
(2015); Helal  _et al._ (2016); Narayanan  _et al._ (2017); Jamali  _et al._
(2020). As a consequence, the mechanical properties of YSFs at a given point
in time are shaped by their shear history, which makes modelling particularly
challenging while also yielding fruitful developments in the field Balmforth
_et al._ (2014); Fielding (2014); Nicolas  _et al._ (2018); Wei  _et al._
(2018); Larson and Wei (2019).

Due to their intrinsic viscoelastic nature, YSFs are characterized by complex
yielding dynamics in both time and space. For instance, in a shear start-up
condition, i.e., when imposing a constant shear rate $\dot{\gamma}$ from rest
at time $t=0$, YSFs display a non-monotonic stress response. At short times,
the system first shows a linear elastic stress response to the applied strain
$\gamma(t)=\int_{0}^{t}\dot{\gamma}(s)\,{\rm d}s=\dot{\gamma}t$, i.e., the
shear stress $\sigma(t)$ grows proportionally to $\gamma(t)$. The stress then
reaches a maximum $\sigma_{\text{M}}$ at time $t_{\text{M}}$, corresponding to
a strain $\gamma_{\text{M}}=\dot{\gamma}t_{\text{M}}$, and then further
decreases towards its steady-state value. This time sequence is usually
referred to as the “stress overshoot” in the literature Mewis and Wagner
(2009). At $\gamma=\gamma_{\text{M}}$, the stress maximum coincides with a
pronounced anisotropy of the YSF microstructure Mohraz and Solomon (2005),
while for $t>t_{M}$, the stress relaxation coincides with the fluidization of
the material, which translates at the microscale into local plastic events,
such as super-diffusive particle motion in glasses Zausch  _et al._ (2008);
Koumakis  _et al._ (2012) and failure of strands in gels Masschaele  _et al._
(2009); Rajaram and Mohraz (2010, 2011). Moreover, at small scales, the stress
relaxation comes with a broad variety of flow profiles, from a ductile-like
response involving shear banding, either transient or permanent Mas and Magnin
(1994); Ovarlez  _et al._ (2009); Divoux  _et al._ (2010, 2012), to a more
abrupt and brittle-like rupture that can either take place in the bulk or at
the boundary of the shearing device Magnin and Piau (1990); Persello  _et al._
(1994); Pignon  _et al._ (1996); Divoux  _et al._ (2011a). The subsequent
long-term, steady-state response corresponds to the flow curve $\sigma_{\rm
ss}(\dot{\gamma})$, which is usually well-described by the empirical Herschel-
Bulkley (HB) law:

$\sigma_{\rm ss}(\dot{\gamma})=\sigma_{\text{y}}+A\dot{\gamma}^{n}\,,$ (1)

where $A$ and $n$ respectively denote the consistency index and a shear-
thinning exponent generally close to $0.5$ Barnes and Nguyen (2001); Katgert
_et al._ (2009).

From a modeling perspective, various approaches have been explored to capture
the distinctive features of the yielding transition, i.e., the stress
overshoot and the subsequent occurrence of shear banding, either only
transient or persisting at steady state. The stress overshoot has been
observed in Molecular Dynamics (MD) and Brownian Dynamics (BD) simulations,
which have confirmed the growth of microstructural anisotropy under shear up
to the stress maximum, and provided insights on the local scenario of the
yielding transition beyond the stress maximum for both gels and glasses
Whittle and Dickinson (1997); Varnik  _et al._ (2004); Park and Ahn (2013);
Santos  _et al._ (2013); Colombo and Del Gado (2014); Park  _et al._ (2017);
Johnson  _et al._ (2018). Moreover, Mode Coupling Theory has shown the
robustness of the stress overshoot, which may arise even in the presence of
homogeneous flows in well-equilibrated, i.e., non-aging systems Zausch  _et
al._ (2008); Amann  _et al._ (2013); Amann and Fuchs (2014); Amann  _et al._
(2015). Finally, the stress overshoot was also captured by 2D mesoscopic
elastoplastic models Jagla (2007), the Soft Glassy Rheology (SGR) model
Moorcroft  _et al._ (2011), the Shear Transformation Zone (STZ) model Manning
_et al._ (2007) and fluidity models Derec  _et al._ (2003); Moorcroft  _et
al._ (2011); de Souza Mendes and Thompson (2013); Fielding (2014). This latter
class of models is also known as “structural kinetic theories”, and often
referred to as “$\lambda$-models”, named after the structural parameter
$\lambda$ of the fluid used to characterize the state of its microstructure.
Since their introduction in the late 1930s, $\lambda$-models have been refined
to encompass thixotropic elasto-viscoplastic effects, among which aging, and
also include advanced concepts from the theory of plasticity such as kinematic
hardening to capture shear-induced memory effects in YSFs in their solid state
Dimitriou  _et al._ (2013); Dimitriou and McKinley (2014); Geri  _et al._
(2017); Wei  _et al._ (2019); Larson and Wei (2019).

Except for a few contributions based on MD simulations of binary Lennard-Jones
glasses that report a logarithmic scaling for the amplitude of the stress
overshoot $\sigma_{\text{M}}$ with the applied shear rate $\dot{\gamma}$
Varnik  _et al._ (2004); Rottler and Robbins (2005); Shrivastav  _et al._
(2016), most of the models listed above conclude that $\sigma_{\text{M}}$
scales as a power law of $\dot{\gamma}$, with an exponent 0.5. However,
confronting theory, simulations and experiments is made difficult due to the
lack of consensus on the relevant quantity that allows one to compare
different types of YSFs, e.g., gels and glasses with different yield stresses.
For instance, experimental and numerical literature indiscriminately report
$\sigma_{\text{M}}$ Whittle and Dickinson (1997); Derec  _et al._ (2003);
Carrier and Petekidis (2009); Divoux  _et al._ (2011a); Koumakis and Petekidis
(2011); Park and Ahn (2013), $\sigma_{\text{M}}-\sigma_{0}$ where $\sigma_{0}$
is a reference stress chosen arbitrarily beyond the overshoot Dimitriou and
McKinley (2014), $\sigma_{\text{M}}/\sigma_{\text{y}}-1$ Letwimolnun  _et al._
(2007), or $\sigma_{\text{M}}/\sigma_{\rm ss}-1$ Koumakis  _et al._ (2012);
Amann  _et al._ (2013); Amann and Fuchs (2014); Koumakis  _et al._ (2016);
Sentjabrskaja  _et al._ (2018) as a function of $\dot{\gamma}$. As for the
flow profile, transient shear bands have been reported recently in both
fluidity and SGR models Moorcroft  _et al._ (2011), STZ models Hinkle and Falk
(2016), athermal local yield stress models derived from MCT Liu  _et al._
(2018a), and MD simulations, in which long-lasting transient heterogeneous
flows were linked to the presence of over-constrained microscopic domains
Vasisht and Del Gado (2020); Vasisht  _et al._ (2020); Golkia  _et al._
(2020). Steady-state shear banding has been extensively described by the
models discussed above. For the liquid-to-solid transition, steady-state shear
banding is observed below a critical shear rate that results from the
competition between spontaneous aging of the YSF microstructure and
rejuvenation by shear Ovarlez  _et al._ (2009); Divoux  _et al._ (2016).
However, in the case of the solid-to-liquid transition, such as in shear
start-up experiments, the steady-state flow profile strongly depends on the
sample age, the nature of the boundary conditions and the geometry Gibaud  _et
al._ (2008); Seth  _et al._ (2012); Vayssade  _et al._ (2014). These effects
have been discussed in the framework of tensorial elasto-visco-plastic
approaches Cheddadi  _et al._ (2012).

Given the complexity of the aforementioned features, it is a challenging task
to build a theoretical framework capable of providing a unified view of the
yield stress transition. Only a few theoretical approaches propose a
consistent description of the yielding transition that encompasses both the
stress overshoot and the subsequent homogeneous or heterogeneous flow profiles
Fielding (2014). Therefore, a unified description of the yielding transition
still stands out as an open challenge, including questions such as the
following ones: (i) what sets the amplitude of the stress overshoot? (ii) is
there any link between the stress overshoot and the subsequent flow dynamics,
in particular the existence of a (transient) shear band? (iii) how does the
presence of spatial correlations in the flow affect the stress overshoot and
the long-term dynamics of YSFs?

In previous works, some of us have revisited a spatially-resolved fluidity
model, devised originally in Ref. Bocquet  _et al._ (2009), that takes into
account “non-local effects” to describe the steady-state flow behavior of YSFs
Benzi  _et al._ (2016); Lulli  _et al._ (2018). Our model is built on the
assumption that the steady-state flow of the YSF is described by the HB law
[see Eq. (1)] and this is approached via a dynamical equation for the local
fluidity $f$. The fluidity is introduced as the basic coarse-grained degree of
freedom for a YSF and is related to the plastic activity in the system, i.e.,
it is non-zero only when non-negligible plastic events occurs. As discussed in
Ref. Bocquet  _et al._ (2009), the spatial heterogeneity of $f$ is controlled
by a “cooperative scale” $\xi$, which relates to the extension of the region
that is impacted by a neighboring plastic rearrangement: other plastic events
may occur nearby with $\xi$ representing the corresponding correlation length,
which is typically of the order of a few times the size of the elementary
constituents of the YSF microstructure Goyon  _et al._ (2008); Geraud  _et
al._ (2013); Géraud  _et al._ (2017); Nicolas  _et al._ (2018). This feature
is referred to as “non-local effects” in the literature. Our model assumes a
dynamical equation for $f$ where two rheological branches exist: a solid
branch corresponding to $f=0$ and a fluid branch corresponding to $f>0$.

Recently, we have made two significant steps forward in modelling the complete
yielding scenario based on the above fluidity model. First, we have shown that
our theoretical approach captures long-lasting transient shear-banded flows
for YSFs that exhibit homogeneous flow in steady state Benzi  _et al._ (2019).
For these YSFs, otherwise known as “simple” YSFs Ovarlez  _et al._ (2013), our
approach yielded a quantitative prediction for the power-law scaling of the
strain-induced fluidization time, $T_{\text{f}}\sim\dot{\gamma}^{-9/4}$, and
provided a deep connection between the HB phenomenological exponent $n$ and
the exponent for stress-induced fluidization at constant stress $\sigma$,
$T_{\text{f}}\sim(\sigma/\sigma_{\text{y}}-1)^{-9/4n}$. Second, in the
companion paper to the present manuscript Benzi  _et al._ (2020), we have used
our model to obtain scaling laws for the stress overshoot with $\dot{\gamma}$.
More specifically, we have identified that the relevant quantity for
characterizing the overshoot amplitude is
$\sigma_{\text{M}}/\sigma_{\text{y}}-1$, which shows two asymptotic scalings,
$(\sigma_{\text{M}}/\sigma_{\text{y}}-1)\sim\dot{\gamma}^{2n/3}$ at “small”
$\dot{\gamma}$ and
$(\sigma_{\text{M}}/\sigma_{\text{y}}-1)\sim\dot{\gamma}^{4n/(9-n)}$ at
“large” $\dot{\gamma}$, here again providing a connection between the steady-
state rheology and the transient response. These predictions are intimately
linked to the growth dynamics of a fluidized boundary layer in the vicinity of
the moving boundary. In that framework, the spatial coexistence of a fluidized
band and a solid region is the consequence of the first-order transition
occurring at the shear band interface, where $\nabla f$ shows a rather sharp
change.

In the present manuscript, we first generalize our model to investigate the
impact of permanent shear bands, both on the early-time response and on the
overshoot scenario described in the companion paper Benzi  _et al._ (2020).
Theoretical arguments supported by numerical computations show that the
scaling of the stress maximum $\sigma_{\text{M}}(\dot{\gamma})$ is robust and
independent of the fate of the fluidized layer. In a second step, we include
long-range correlations, i.e., correlations over length scales larger than the
cooperative scale $\xi$, and examine their influence on the yielding scenario.
We find that, the presence of long-range correlations is responsible for an
increased amount of local plasticity, yielding an avalanche-like, i.e., abrupt
shear-induced fluidization of the material. The presence of avalanches also
leads to an earlier fluidization of the sample. Yet, the fluidization occurs
over a timescale that obeys the same power-law scaling as the one measured in
the absence of long-range correlations. In other words, the presence of long-
range correlations only affects the prefactor of the fluidization law, which
appears as characteristic of the material. Furthermore, we demonstrate that,
within our generalized approach, the nature of the boundary condition for the
fluidity at the moving wall strongly affects the scaling of the stress maximum
associated with the stress overshoot. In particular, imposing a zero fluidity
gradient at the moving wall, which prevents the growth of any fluidized band
at the wall triggers, a fluidization in the bulk that involves avalanches. In
practice, the stress overshoot shows an abrupt stress drop reminiscent of a
brittle failure, while its maximum displays a logarithmic increase with the
applied shear rate that strongly contrasts with the power-law scaling observed
in the absence of long-range correlations.

The paper is organized as follows. In Section II, we recall our theoretical
framework, namely a non-local fluidity approach that predicts the scalings of
both the overshoot coordinates and the fluidization time with respect to the
applied shear rate, for “simple” YSFs that display a homogeneous flow profile
in steady state. We then present and discuss the corresponding numerical
results in details. In Section III, we discuss the scalings obtained with a
non-local fluidity model with noise, which captures the case of YSFs that
display steady-state shear banding, as well as the corresponding numerical
results. In Section IV, we illustrate the robustness of the theoretical
predictions obtained in the first two sections, as well as their limits, by
revisiting experimental and numerical data from the literature. Section V
presents a generalization of our model that includes the effects of long-range
correlations and “avalanches”, leading to a thorough understanding of the
various experimental features of shear start-up in YSFs. Finally, Section VI
gathers the discussion and conclusion.

##  II Theoretical framework for systems with homogeneous steady state

In the following, we first recall the physical ingredients for our theoretical
description of “simple” YSFs that show a transient shear-banding regime prior
to homogeneous shear flow at steady state. We then introduce the full set of
equations and derive scaling arguments for the overshoot at short times.
Finally, we describe the long-time evolution towards steady state. In all
cases, the dynamical equations are solved numerically to support the scaling
arguments and to provide a detailed view of the evolution of the local flow
field.

###  II.1 Non-local fluidity model for transient shear banding

####  II.1.1 Physical ingredients and assumptions underlying the model

Physically, we aim at modelling the situation where a constant shear rate
$\dot{\gamma}$ is applied to a YSF confined between two parallel walls
separated by a distance $L$. Such a one-dimensional shear flow is
characterized by a velocity field $\mathbf{v}(x,y,t)=v(y,t)\mathbf{e}_{x}$,
where $x$ is the velocity direction and $y$ is the velocity gradient
direction. It is obtained by imposing a constant velocity $v_{0}$ to the
moving boundary located at $y=0$, i.e., $v(0,t)=v_{0}$, while the boundary at
$y=L$ is held fixed, i.e., $v(L,t)=0$ at all times $t$. As recalled in the
introduction, the fluidity $f(y,t)$ relates to the local plastic activity in
the system and its dynamics involve the cooperativity length $\xi$ as a key
physical parameter. In steady state, the fluidity coincides with the inverse
of the local viscosity, i.e., $f=\dot{\gamma}/\sigma=(\partial v/\partial
y)/\sigma$. In our model, we focus on a dynamical equation for $f(y,t)$ so
that, rather than imposing the velocity at the walls, we will devise boundary
conditions for $f$ at $y=0$ and $y=L$ together with an initial condition
$f(y,0)$ that accounts for the initial preparation of the system, another key
ingredient in modelling shear start-up of YSFs. Moreover, the dynamical
equation for $f$ accounts for a steady-state rheology described by the HB law
[see Eq. (1)] through two rheological branches, namely a solid branch
corresponding to $f=0$ and a fluid branch corresponding to $f>0$. Finally, in
order to model the viscoelastic response of the material under start-up of
shear, an elastic modulus $G_{0}$ should be considered in the whole dynamical
scenario. The stress $\sigma$ is assumed to be spatially homogeneous and its
dynamics to depend only on the imposed shear rate $\dot{\gamma}$, the elastic
modulus $G_{0}$ and the space-averaged fluidity $\langle
f\rangle=(1/L)\int_{0}^{L}f(y,t)\,\text{d}y$, which is a function of time $t$
only.

####  II.1.2 Set of coupled equations for fluidity and stress, & boundary and
initial conditions

We consider a one-dimensional system of size $L$, with $y\in[0,L]$ the spatial
coordinate along the velocity gradient direction. We assume that the
constitutive rheological law is given by the HB relation [see Eq. (1)].
Furthermore, we work with the dimensionless stress and shear rate,
respectively defined as $\Sigma=\sigma/\sigma_{\text{y}}$ and
$\dot{\Gamma}=\dot{\gamma}/(\sigma_{\text{y}}/A)^{1/n}$, resulting in the
dimensionless HB law:

$\Sigma(\dot{\Gamma})=1+\dot{\Gamma}^{n}\,.$ (2)

As introduced in Benzi  _et al._ (2016), it is assumed that the flow
properties of the system are governed by a functional
$F[f]=\int_{0}^{L}\Phi(f,m,\xi)\,\mathrm{d}y$ with

$\Phi(f,m,\xi)=\frac{1}{2}\xi^{2}(\nabla
f)^{2}-\frac{1}{2}mf^{2}+\frac{2}{5}f^{5/2}\,,$ (3)

where $f=f(y,t)$ is the dimensionless fluidity, $\xi$ the cooperativity
length, and

$m=m(\Sigma)=\frac{(\Sigma-1)^{\frac{1}{2n}}}{\sqrt{\Sigma}}\,\Theta(\Sigma-1)\,,$
(4)

with $\Theta$ the Heaviside function. The HB law (2) is recovered as the
minimum of the bulk contribution $-\frac{1}{2}mf^{2}+\frac{2}{5}f^{5/2}$ in
Eq. (3), which corresponds to $f=m^{2}$.

In the case of shear-induced fluidization, i.e., when the system is driven by
a constant shear rate $\dot{\Gamma}$, we have argued in Benzi  _et al._ (2019)
that one should consider the rescaled variable $\tilde{f}=f/\dot{\Gamma}$
rather than $f$. This rescaling, together with
$\tilde{m}=m/\dot{\Gamma}^{1/2}$ and $\tilde{y}=\dot{\Gamma}^{1/4}y/\xi$
allows for a homogeneous rescaling of the functional
$\Phi(f,m,\xi)=\dot{\Gamma}^{5/2}\Phi(\tilde{f},\tilde{m})$, where
$\Phi(\tilde{f},\tilde{m})=\frac{1}{2}(\tilde{\nabla}\tilde{f})^{2}-\frac{1}{2}\tilde{m}\tilde{f}^{2}+\frac{2}{5}\tilde{f}^{5/2}$.
Assuming that the system reaches the stable configuration corresponding to the
minimum of $F[\tilde{f}]$, we introduce a mobility function $k(\tilde{f})$
that drives the dynamics of $\tilde{f}$, leading to

$\frac{\partial\tilde{f}}{\partial
t}=-{\dot{\Gamma}}^{5/2}k(\tilde{f})\frac{\delta
F[\tilde{f}]}{\delta\tilde{f}}\,.$ (5)

We now take the simplest non-trivial form for the mobility,
$k(\tilde{f})=\tilde{f}$ and also use the rescaled time
$\tilde{t}\equiv\dot{\gamma}t$. Under a constant shear rate, $\tilde{t}$
simply corresponds to the strain $\gamma(t)$ applied to our system. Coming
back to $f=\tilde{f}\dot{\Gamma}$, we obtain:

$\frac{\partial f}{\partial\tilde{t}}=f\left[\xi^{2}\Delta
f+mf-f^{3/2}\right]\,,$ (6)

where $m(\Sigma)$ is given by Eq. (4). To close our set of equations for
shear-induced fluidization, we need an evolution equation for the stress
$\Sigma$, which is assumed to be homogeneous in space as posited before. Again
for the sake of simplicity, we assume that the stress dynamics follow a
Maxwell equation:

$\frac{\mathrm{d}\Sigma}{\mathrm{d}\tilde{t}}=\frac{1}{\dot{\Gamma}\tau}\left(\dot{\Gamma}-{\langle
f\rangle\Sigma}\right)\,,$ (7)

where $\tau$ is the dimensionless relaxation time. Physically, $\tau$ is
inversely proportional to the elastic modulus $G_{0}$ and is linked to the
material stiffness such that, for a given shear rate, decreasing $\tau$
corresponds to increasing the stiffness.

Finally, having defined the equations of motion, we need to set the boundary
conditions as well as the initial conditions. In all our numerical
simulations, we assume a spatially homogeneous initial condition
$f(y,0)=f(0)$. Moreover, we deduce from Eq. (7) that the minimum shear rate
for the overshoot to occur reads $\dot{\Gamma}_{0}=f(0)$. We choose
$f(0)=10^{-4}$ in all numerical simulations and thus perform shear start-up
flows for $\dot{\Gamma}\geq 2\times 10^{-4}$ to trigger overshoots (except in
Sect. V.3.2 where other values of $f(0)$ will be specified). As for the
boundary conditions, we assume $\partial_{y}f|_{y=L}=0$ at the fixed wall for
any $\Sigma$. At the moving wall, i.e., $y=0$, we assume that, for $\Sigma>1$,
the fluidity $f(0,\tilde{t})$ is equal to some “wall fluidity” $f_{w}$. In the
following, we make the simplest assumption $f_{w}=m^{2}$, which guarantees the
existence of a stationary solution $f=m^{2}$ when the system reaches complete
fluidization. We shall emphasize that the results discussed in the following
sections are still valid as long as the wall fluidity is proportional to
$m^{2}$. Finally, in Section V, we discuss a different boundary condition at
the moving wall, namely $\partial_{y}f|_{y=0}=0$. The physical meaning of the
two boundary conditions is as follows. On the one hand, $\partial_{y}f=0$
implies that the boundary does not change the rate of plastic events with
respect to the bulk of the system. On the other hand, $f=f_{w}$ states that
the rate of plastic events is fixed by the boundary independently of the bulk
dynamics. In both cases, the shear rate at the boundary is fixed by the
external forcing [see Eq. (9) below].

####  II.1.3 Scale invariance

An interesting property of Eqs. (6) and (7) is the invariance under the scale
transformation:

$\displaystyle m\rightarrow\lambda m$ (8) $\displaystyle
f\rightarrow\lambda^{2}f$ $\displaystyle\Sigma\rightarrow\lambda^{2}\Sigma$
$\displaystyle\dot{\Gamma}\rightarrow\lambda^{4}\dot{\Gamma}$
$\displaystyle\tilde{t}\rightarrow\lambda^{-3}\tilde{t}$
$\displaystyle\tau\rightarrow\lambda^{-5}\tau$
$\displaystyle\frac{\xi}{y}\rightarrow\lambda^{1/2}\frac{\xi}{y}$

In the following, we will use the above scale invariance property of Eqs. (6)
and (7) to discuss the scaling behaviors of the various observables.

####  II.1.4 Numerical implementation and first results

In the next sections, we shall validate our theoretical analysis against
numerical computations. To this aim, we discretize Eq. (6) on $N_{y}=512$ grid
points choosing $L=1$, $\xi=0.04$ and different values of $\tau$ ranging from
0.01 to 10. The computations are performed using finite-difference techniques
with fourth-order accuracy in space and time, solving simultaneously Eqs. (6)
and (7). Beside the initial conditions discussed above, we assume an HB index
$n=1/2$, except for some specific cases in Sec. II.3.4 where we will also
consider values of $n$ between 0.3 and 0.7 as observed in experiments.

Figure 1: (a) Stress response $\Sigma$ as a function of $\tilde{t}/\tau$ for
$\dot{\Gamma}=0.004$ ( ), $0.02$ ( ) and $5.68$ ( ). (b) $m^{2}$ as a function
of $\tilde{t}/\tau$ for $\dot{\Gamma}=0.004$. The red line shows an affine
behavior consistent with $m^{2}\sim(\tilde{t}-\tilde{t}_{\text{1}})/\tau$
(solid line) [see Eq. (12)]. (c) Size of the fluidized band $\ell_{b}$ as a
function of $\tilde{t}/\tau$ for $\dot{\Gamma}=0.02$. The solid line
corresponds to the affine scaling predicted in the low-shear (diffusive)
regime [see Eq. (16)]. (d) $\ell_{b}$ as a function of $\tilde{t}/\tau$ for
$\dot{\Gamma}=5.68$. The solid line shows the power-law scaling expected in
the high-shear regime with exponent $9/4$ [see Eq. (18)]. Numerical results
obtained with $n=1/2$, $\tau=10$, $\xi=0.04$ and $L=1$.

Moreover, in order to compute velocity profiles, we need to connect the
fluidity inferred from the numerical resolution of Eqs. (6) and (7) to the
local shear rate $\dot{\Gamma}_{\rm loc}$. The latter quantity is defined as:

$\dot{\Gamma}_{\rm loc}(y,\tilde{t})=\dot{\Gamma}+\left[f(y,\tilde{t})-\langle
f\rangle\right]\Sigma\,.$ (9)

Such a definition is justified by the fact that it leads to the following
local version of Eq. (7):

$\frac{\mathrm{d}\Sigma}{\mathrm{d}\tilde{t}}=\frac{1}{\dot{\Gamma}\tau}[\dot{\Gamma}_{\rm
loc}-f(y,\tilde{t})\Sigma]\,.$ (10)

Velocity profiles $v(y,t)$ are then computed by integration of
$\dot{\Gamma}_{\rm loc}(y,\tilde{t})$. Note that Eq. (9) implies that the
material velocity at $y=0$ is exactly equal to that of the moving wall,
$v_{0}=\dot{\Gamma}L$, at all dimensionless times $\tilde{t}$.

As a first example of our numerical computations, Fig. 1(a) shows the stress
responses $\Sigma$ versus $\tilde{t}/\tau$ for $\tau=10$ and for three
different values of the shear rate $\dot{\Gamma}$. All responses show a clear
stress overshoot that reaches larger values $\Sigma_{\text{M}}$ at later times
for increasing values of $\dot{\Gamma}$. We introduce $\tilde{t}_{\text{1}}$
as the dimensionless time at which $\Sigma(\tilde{t}_{\text{1}})=1$, and
$\tilde{t}_{\text{M}}$ the dimensionless time at which the stress reaches its
maximum value, i.e., $\Sigma(\tilde{t}_{\text{M}})=\Sigma_{\text{M}}$ [see
Fig. 1(a)]. For $\tilde{t}\in[0,\tilde{t}_{\text{1}}]$ the stress remains
smaller than the yield stress ($\Sigma<1$) and $m(\Sigma)=0$. During this time
interval, the fluidity at the wall $f_{w}=m^{2}$ is equal to zero [see Fig.
1(b)]. Beyond $\tilde{t}=\tilde{t}_{\text{1}}$, $m^{2}$ increases up to a
maximum reached at $\tilde{t}=\tilde{t}_{\text{M}}$, which coincides with the
maximum of the stress overshoot. Finally, for
$\tilde{t}\gtrsim\tilde{t}_{\text{M}}$, both the stress $\Sigma$ and the
fluidity at the wall $f_{w}=m^{2}$ decrease towards their steady-state values.
Computing the velocity profiles during shear start-up reveals that an unstable
shear band develops near the moving wall, as discussed in the companion paper
Benzi  _et al._ (2020). The size $\ell_{b}$ of the sheared region increases
with time well beyond $\tilde{t}=\tilde{t}_{\text{M}}$ [see Fig. 1(c,d)]. The
shear band occupies a significant fraction of the gap width $L$, before the
sample eventually experiences complete fluidization, i.e., $\ell_{b}=L$ Benzi
_et al._ (2019).

###  II.2 Short-time local dynamics: growth of a fluidized band

Let us now focus on the dynamics for $\tilde{t}\geq\tilde{t}_{\text{1}}$. At
$\tilde{t}=\tilde{t}_{\text{1}}$ the value of $m$ becomes positive and a thin
layer of non-zero fluidity grows near the moving wall ($y=0$). If we assume
that the term $\langle f\rangle\Sigma$ in the r.h.s. of Eq. (7) is small, then
the stress grows linearly with time above $\tilde{t}_{\text{1}}$, i.e.,

$\Sigma(\tilde{t})-1=\left(\frac{\tilde{t}-\tilde{t}_{\text{1}}}{\tau}\right).$
(11)

Obviously, this approximation does not hold for all $\tilde{t}$ above
$\tilde{t}_{\text{1}}$, for which $\Sigma(\tilde{t})$ should eventually
increase in a non-linear way to satisfy
$\mathrm{d}\Sigma/\mathrm{d}\tilde{t}=0$ at the stress maximum. Furthermore,
for $n=1/2$ and assuming that $\Sigma$ is reasonably larger than the yield
stress, i.e., $\Sigma\gg 1$, during the time interval
$[\tilde{t}_{\text{1}},\tilde{t}_{\text{M}}]$, Eq. (4) reads
$m(\Sigma)=(\Sigma-1)/\Sigma^{1/2}\sim\Sigma^{1/2}\sim(\Sigma-1)^{1/2}$. Based
on Eq. (11), we obtain the following scaling

$m(\tilde{t})\sim\left(\frac{\tilde{t}-\tilde{t}_{\text{1}}}{\tau}\right)^{1/2}.$
(12)

Equation (12) is numerically checked for $\dot{\Gamma}=0.004$ in Fig. 1(b),
where the fluidity at the wall $f_{w}=m^{2}$ indeed shows a linear growth as a
function of $\tilde{t}-\tilde{t}_{\text{1}}$. At the stress maximum
($\mathrm{d}\Sigma/\mathrm{d}\tilde{t}=0$), the r.h.s. of Eq. (7) is zero,
which yields

$\dot{\Gamma}=\left[\langle
f\rangle\Sigma\right]_{\tilde{t}=\tilde{t}_{\text{M}}}\,.$ (13)

Moreover, the spatial average $\langle f\rangle$ is dominated by a small
fluidized region of width $\ell_{b}$ near the boundary, which is forced by the
boundary condition $f_{w}=m^{2}$. Therefore, we may write $\langle
f\rangle\simeq m^{2}\,\ell_{b}\,/L$, which allows us to infer
$\ell_{b}(\tilde{t})$ from our numerical computations. If we further estimate
$m$ and $\Sigma$ using Eqs. (11) and (12) and consistently assume
$\Sigma_{\text{M}}\gg 1$, such that
$\Sigma_{\text{M}}\sim\Sigma_{\text{M}}-1$, the stress overshoot condition in
Eq. (13) becomes

$\dot{\Gamma}\sim\left(\Sigma_{\text{M}}-1\right)^{2}\,\ell_{b}(\tilde{t}_{\text{M}})\,.$
(14)

To further extract the functional relation $\Sigma_{\text{M}}(\dot{\Gamma})$,
we need to specify the time evolution of the band size $\ell_{b}$ and to
evaluate it at $\tilde{t}=\tilde{t}_{\text{M}}$. We shall now discuss the
growth dynamics of the shear band depending on the applied shear rate, based
on the two asymptotic regimes $\dot{\Gamma}\ll 1$ and $\dot{\Gamma}\gg 1$.

####  II.2.1 Band growth controlled by diffusion

For $\dot{\Gamma}\ll 1$, one may argue that in the early stages of the
dynamics, the system has to “adapt” to the boundary condition and has to
“shape up” the fluidized front that will later propagate through the gap. As
discussed in the Appendix of Sect. VII in full details, assuming that the
diffusive term in Eq. (6) plays a dominant role leads to the following scaling
prediction for the size of the band:

$\ell_{b}(\tilde{t})\sim\sqrt{m^{2}(\tilde{t})\,\xi^{2}\,\tilde{t}}\,.$ (15)

Using Eqs. (11), (12) and (15), consistently with the condition
$\tilde{t}\gg\tilde{t}_{\text{1}}$, we obtain

$\ell_{b}(\tilde{t})\sim\xi\tau^{1/2}\left(\frac{\tilde{t}-\tilde{t}_{\text{1}}}{\tau}\right)\sim\xi\tau^{1/2}\left(\Sigma(\tilde{t})-1\right)\,.$
(16)

Fig. 1(c) shows that for $\dot{\Gamma}=0.02$, $\ell_{b}$ grows linearly with
$\tilde{t}/\tau$ around the overshoot as predicted by Eq. (16), which supports
the diffusive approximation for $\dot{\Gamma}\ll 1$.

####  II.2.2 Band growth controlled by front propagation

For $\dot{\Gamma}\gg 1$, a propagation front emerges from Eq. (6). Useful
information on the scaling of the front speed as a function of $m$ can be
inferred from the scaling properties listed in Eq. (8). Indeed, Eq. (8)
implies that $\mathrm{d}\ell_{b}/\mathrm{d}\tilde{t}$ must be proportional to
the ratio of the characteristic scale $\xi/m^{1/2}$ to the time scale $m^{-3}$
so that

$\frac{\mathrm{d}\ell_{b}}{\mathrm{d}\tilde{t}}\sim\xi m^{5/2}\,.$ (17)

The validity of Eq. (17) is discussed in more details in the Appendix of Sect.
VII based on a linear stability analysis. Using Eqs. (12) and (17), we obtain

$\ell_{b}(\tilde{t})\sim\xi\tau\left(\frac{\tilde{t}-\tilde{t}_{\text{1}}}{\tau}\right)^{9/4}\sim\xi\tau\left(\Sigma(\tilde{t})-1\right)^{9/4}\,.$
(18)

In Fig. 1(d), we show that the size of the shear band indeed follows a power-
law scaling with exponent 9/4 for $\dot{\Gamma}=5.68$ around the stress
overshoot.

###  II.3 Scaling of the stress overshoot coordinates with applied shear rate

We now combine the condition for the stress overshoot given in Eq. (14) with
the scalings for $\ell_{b}$ found for $\dot{\Gamma}\ll 1$ and $\dot{\Gamma}\gg
1$ in Eqs. (16) and (18) respectively. We recall that these two predictions
are obtained under radically different scenarios: for $\dot{\Gamma}\ll 1$, the
imposed shear is small and the unstable shear band does not propagate as a
front but the system rather rearranges close to the boundaries, while for
$\dot{\Gamma}\gg 1$, we used the physical ingredient of a propagating front.

####  II.3.1 Diffusive regime at low shear rate

For $\dot{\Gamma}\ll 1$, we combine Eq. (14) with the scaling relation in Eq.
(16) evaluated at $\tilde{t}_{\text{M}}$, which leads to

$\dot{\Gamma}\sim(\Sigma_{\text{M}}-1)^{2}\ell_{b}\sim\xi\tau^{1/2}(\Sigma_{\text{M}}-1)^{3}\,,$
(19)

so that

$\Sigma_{\text{M}}-1\sim\left(\frac{\dot{\Gamma}}{\xi\tau^{1/2}}\right)^{1/3}\,.$
(20)

####  II.3.2 Asympototic regime at high shear rate

For $\dot{\Gamma}\gg 1$, combining Eq. (14) with the scaling relation in Eq.
(18) evaluated at $\tilde{t}_{\text{M}}$ leads to

$\dot{\Gamma}\sim(\Sigma_{\text{M}}-1)^{2}\ell_{b}\sim\xi\tau(\Sigma_{\text{M}}-1)^{17/4}\,,$
(21)

so that

$\Sigma_{\text{M}}-1\sim\left(\frac{\dot{\Gamma}}{\xi\tau}\right)^{4/17}\,.$
(22)

####  II.3.3 Complete scaling for $n=1/2$

By combining the two different scaling regimes predicted above for
$\dot{\Gamma}\ll 1$ and $\dot{\Gamma}\gg 1$, we may propose the following
general expression for the stress maximum $\Sigma_{\text{M}}$ reached during
the overshoot as a function of the applied shear rate $\dot{\Gamma}$:

$\Sigma_{\text{M}}-1\sim
B\left(\frac{\dot{\Gamma}}{\xi\tau^{1/2}}\right)^{1/3}+C\left(\frac{\dot{\Gamma}}{\xi\tau}\right)^{4/17}\,,$
(23)

where $B$ and $C$ are two constants independent of the model parameters. For a
given $\xi$, the intersection between the two regimes occurs at a
characteristic shear rate $\dot{\Gamma}^{*}$, whose scaling relation with
$\tau$ can be found by balancing the two terms
$(\dot{\Gamma}/\xi\tau^{1/2})^{1/3}$ and $(\dot{\Gamma}/\xi\tau)^{4/17}$ in
Eq. (23). This yields

$\dot{\Gamma}^{*}\sim\xi\tau^{-0.7}\,.$ (24)

When $\dot{\Gamma}=\dot{\Gamma}^{*}$, both terms in Eq. (23) lead to the same
scaling $\Sigma_{\text{M}}^{*}-1\sim\tau^{-0.4}$. Thus, we expect the
overshoot data to collapse onto a master curve when plotting
$(\Sigma_{\text{M}}-1)\tau^{0.4}$ as a function of
$\dot{\Gamma}/\dot{\Gamma}^{*}$. In other words, this means that

$\Sigma_{\text{M}}-1=\tau^{-0.4}\mathcal{G}[\dot{\Gamma}/\dot{\Gamma}^{*}]\,,$
(25)

where $\mathcal{G}(x)$ is a universal function such that $\mathcal{G}(x)\sim
x^{1/3}$ for $x\ll 1$ and $\mathcal{G}(x)\sim x^{4/17}$ for $x\gg 1$.

####  II.3.4 Generalization for any HB exponent $n$

Figure 2: Stress maximum $\Sigma_{\text{M}}-1$ as a function of $\dot{\Gamma}$
for three different values of the exponent $n$ in the HB constitutive relation
$\Sigma=1+\dot{\Gamma}^{n}$, namely $n=1/3$ ( ) $n=1/2$ ( ) and $n=2/3$ ( ).
The straight lines show the different scaling predictions, respectively at low
shear rates with exponent $\beta(n)$ (diffusive regime) and at high shear
rates with exponent $\alpha(n)$ (asymptotic regime) [see Eq. (27)]. The
numerical computations are performed with $\tau=10$, $\xi=0.04$ and $L=1$ for
all cases.

So far, we have assumed that the HB exponent $n$ takes the specific value
$n=1/2$ to derive Eq. (25) from Eqs. (6) and (7). Here, we generalize the
above results for any value of $n$. Starting from Eq. (4), which reads
$m^{2}(\Sigma)=(\Sigma-1)^{1/n}/\Sigma$ for $\Sigma>1$, and following the same
steps as in the previous paragraphs, it is easily shown that Eq. (23) is
changed to:

$\Sigma_{\text{M}}-1\sim
B\left(\frac{\dot{\Gamma}}{\xi\tau^{1/2}}\right)^{\beta(n)}+C\left(\frac{\dot{\Gamma}}{\xi\tau}\right)^{\alpha(n)}\,,$
(26)

where

$\beta(n)=\frac{2n}{3}\,\,\,\,\text{and}\,\,\,\,\alpha(n)=\frac{4n}{9-n}\,.$
(27)

Thus, the generalized version of Eq. (25) takes the form:

$\Sigma_{\text{M}}-1=\tau^{-\alpha(n)[1+\lambda(n)]}\,\mathcal{G}_{n}\left[\frac{\dot{\Gamma}\tau^{\lambda(n)}}{\xi}\right]\,,$
(28)

where

$\lambda(n)=\frac{\alpha(n)-\frac{\beta(n)}{2}}{\beta(n)-\alpha(n)}=\frac{3+n}{6-2n}\,,$
(29)

and $\mathcal{G}_{n}$ is the equivalent of $\mathcal{G}$ in Eq. (25) such that
$\mathcal{G}_{n}(x)\sim x^{\beta(n)}$ for $x\ll 1$ (diffusive regime) and
$\mathcal{G}(x)\sim x^{\alpha(n)}$ for $x\gg 1$ (asymptotic regime). For
$n=1/2$, we check that $\beta(1/2)=1/3$ and $\alpha(1/2)=4/17$, so that we
recover $\lambda(1/2)=0.7$ as in Eq. (24). Finally, to make the link with the
additional exponent $\mu$ used in the companion paper Benzi  _et al._ (2020),
one has: $\mu(n)=\alpha(n)[1+\lambda(n)]=2n/(3-n).$

In the companion paper, the validity of Eq. (25) has been ascertained for
$n=1/2$ and various values of $\tau$ (see Fig. 2 in Ref. Benzi  _et al._
(2020)). Here, in Fig. 2, we compare $\Sigma_{\text{M}}-1$ versus
$\dot{\Gamma}$ computed for $n=1/3$, $n=1/2$ and $n=2/3$, and for a given
$\tau=10$. As shown by the straight lines, the scalings of Eq. (29) provide
excellent predictions for the numerical results both in the diffusive regime,
$\dot{\Gamma}\ll\dot{\Gamma}^{*}$, and in the asymptotic regime,
$\dot{\Gamma}\gg\dot{\Gamma}^{*}$, where
$\dot{\Gamma}^{*}\sim\xi\tau^{-\lambda(n)}\simeq 0.01$. Note that the
robustness of our general approach is further tested against experimental
results on Carbopol microgels with $n=0.56$–0.59 in the companion paper (see
Fig. 3 in Ref. Benzi  _et al._ (2020)).

###  II.4 Transient dynamics up to full fluidization

####  II.4.1 Stress evolution towards equilibrium

Figure 3: Shear stress $\Sigma-1$ as a function of the effective shear rate
within the shear band ${\dot{\Gamma}}L/\ell_{b}$, where $\ell_{b}$ is the
thickness of the (unstable) shear band, for various global shear rates
$\dot{\Gamma}=8\,10^{-4}$ ( ), $\dot{\Gamma}=9\,10^{-3}$ ( ) and
$\dot{\Gamma}=10^{-2}$ ( ). The solid line is the HB prediction
$\Sigma=1+(\dot{\Gamma}L/\ell_{b})^{1/2}$ written in terms of the effective
shear rate. Numerical results obtained with $n=1/2$, $\tau=10$, $\xi=0.04$ and
$L=1$.

Based on our theoretical discussion, we expect that the effective shear rate
increases quite rapidly at short time scales. Once the stress maximum is
reached, a slower decaying dynamics occurs with a balance between $\Sigma$,
$\dot{\Gamma}$ and $\ell_{b}$ such that the r.h.s. of Eq. (7) is close to
zero. These qualitative expectations are actually borne out by the analysis of
the “flow phase diagram” presented in Fig. 3, where we report
$\Sigma(\tilde{t})-1$ as a function of $\dot{\Gamma}L/\ell_{b}(\tilde{t})$
during the whole fluidization process. $\dot{\Gamma}L/\ell_{b}(\tilde{t})$
represents the effective shear rate within the shear band and the red
continuous line shows the corresponding HB prediction

$\Sigma=1+\left(\frac{\dot{\Gamma}L}{\ell_{b}}\right)^{1/2}.$ (30)

Such a representation of the flow dynamics highlights a clear-cut separation
between two different dynamical regimes: one regime where the data fall way
below the HB curve and another regime where they all collapse onto the HB
curve. The former regime corresponds to the initial dynamics where $\ell_{b}$
is small, i.e., $\dot{\Gamma}L/\ell_{b}$ is large, and the stress grows
rapidly. This “unsteady” dynamical regime brings the system from the initial
time up to the stress overshoot. Once the stress maximum is reached, the
system enters a second dynamical regime where
$\mathrm{d}\Sigma/\mathrm{d}\tilde{t}\sim 0$. During this “steady” dynamical
regime, Eq. (7) implies that $\dot{\Gamma}\sim\langle f\rangle\Sigma$. Using
the estimation $\langle f\rangle\sim m^{2}\ell_{b}/L\sim\Sigma\,\ell_{b}/L$ as
in Sect. II.2, we get that $(\Sigma-1)^{2}\sim\dot{\Gamma}L/\ell_{b}$, which
boils down the HB equation.

####  II.4.2 Early-time velocity profiles and elastic recoil around the
overshoot

The temporal evolution of the normalized velocity $v(y_{0},t)/v_{0}$ computed
near the moving wall at $y_{0}=L/16$ is displayed in Fig. 4 for four different
values of $\dot{\Gamma}$ and up to $\tilde{t}=2.5\tau$, i.e., just after the
stress overshoot (see inset of Fig. 4). During, the very early stage of the
stress response, the material undergoes elastic loading and its deformation
remains affine. Thus, velocity profiles are linear as long as
$\tilde{t}\lesssim\tau$, as shown in the companion paper Benzi  _et al._
(2020). This translates in Fig. 4 into a constant velocity given by the affine
displacement $v(y_{0}=L/16,t)=15v_{0}/16$ for $\tilde{t}\lesssim\tau$,
independent of the shear rate $\dot{\Gamma}$.

Just before the overshoot, the shear band starts to nucleate and once the
overshoot is reached, as $\mathrm{d}\Sigma/\mathrm{d}\tilde{t}<0$, the local
velocity close to the moving boundary strongly decreases until a small but
detectable negative velocity is measured at $y_{0}$ near the shear band
interface. This elastic recoil is observed typically for $\tilde{t}/\tau\simeq
1.5$–3 depending on the shear rate. For larger $\tilde{t}/\tau$, the velocity
at $y_{0}$ tends to zero, indicative of the presence of an arrested region.
This behavior is in very close correspondence with the one observed
experimentally in Ref. Divoux  _et al._ (2010), as discussed in the companion
paper (see Fig. 4 in Ref. Benzi  _et al._ (2020)).

Figure 4: Normalized velocity $v(y_{0},t)/v_{0}$ computed near the moving wall
at $y_{0}=L/16$ as a function of $\tilde{t}/\tau$ for
$\dot{\Gamma}=7.61\,10^{-4}$ ( ), $1.14\,10^{-3}$ ( ), $1.71\,10^{-3}$ ( ),
and $2.56\,10^{-3}$ ( ). $v_{0}$ is the velocity of the moving wall. Inset:
corresponding stress response $\Sigma$ versus $\tilde{t}/\tau$. Numerical
results obtained with $n=1/2$, $\tau=10$, $\xi=0.04$ and $L=1$.

####  II.4.3 Shear band evolution at long times

Figure 5: Normalized velocity profile $v(y)/v_{0}$ for
$\dot{\Gamma}=1.7\,10^{-3}$ at times $\tilde{t}/\tau=2.5$ ( ), $10$ ( ), $50$
( ) and $180$ ( ). The dashed line shows the velocity profiles recorded at
steady state. Inset: normalized velocity $v(y_{0},t)/v_{0}$ computed near the
moving wall at $y_{0}=L/16$ (see arrow in the main panel) as a function of
$\tilde{t}/\tau$. Numerical results obtained with $n=1/2$, $\tau=10$,
$\xi=0.04$ and $L=1$.

We now turn to the long-term evolution of the velocity field. In Fig. 5, we
show a selection of normalized velocity profiles $v(y)/v_{0}$ for
$\dot{\Gamma}=1.7\,10^{-3}$ at times ranging from just after the overshoot
($\tilde{t}=2.5\tau$) up to much longer times ($\tilde{t}=180\,\tau$). After
the shear band nucleation and the elastic recoil discussed above, the shear
band is observed to slowly grow while the arrested region shrinks over time.
This growth of the shear band leads to a slow yet steady increase of the local
velocity $v(y_{0}=L/16)$ past the overshoot (see inset of Fig. 5). As shown in
our previous work focused on the long-time shear-induced fluidization Benzi
_et al._ (2019), this long-lived transient shear-banded state eventually gives
way to a fully fluidized state (see dashed lines in Fig. 5), i.e., the shear
band is unstable. Such a fluidization scenario associated with a well-defined
“fluidization time” $T_{\text{f}}$ is in excellent agreement with the one
observed experimentally (see also Fig. 4 in the companion paper Benzi  _et
al._ (2020)).

Finally, let us discuss in more detail how the scaling law for the
fluidization time $T_{\text{f}}$ can be obtained in the theoretical framework
of Eqs. (6) and (7). During the “steady” dynamical regime discussed in Sec.
II.4.1, the relation given in Eq. (30) holds. Consequently, the front velocity
can be computed from Eq. (17) with $m=(\Sigma-1)/\Sigma^{1/2}$ and $\Sigma$
given by Eq. (30). A Taylor expansion series in $\dot{\Gamma}$ has a leading
order in $\dot{\Gamma}$ that results in the following scaling law:

$\frac{\mathrm{d}\ell_{b}}{\mathrm{d}\tilde{t}}\sim\xi\left(\frac{\dot{\Gamma}}{\ell_{b}}\right)^{5/4}\,,$
(31)

leading to

$\ell_{b}(\tilde{t})\sim\dot{\Gamma}^{5/9}(\xi\tilde{t})^{4/9}\,.$ (32)

Upon complete fluidization, i.e., for $\tilde{t}>\tilde{T_{\text{f}}}$,
$\ell_{b}(\tilde{t})=L$ does not change anymore in time. We recall that
$\tilde{t}$ corresponds to the strain and is therefore itself proportional to
the shear rate $\dot{\Gamma}$. Thus, we obtain the scaling behavior of the
fluidization time $T_{\text{f}}$ with the shear rate $\dot{\Gamma}$ as

$T_{\text{f}}\sim\frac{1}{\xi\dot{\Gamma}^{9/4}}\,,$ (33)

which coincides with the scaling of the fluidization time discussed in Ref.
Benzi  _et al._ (2019). In other words, the scaling for the fluidization time
hinges on the dynamics after the overshoot maximum, where the system is
approaching the asymptotic stress value and the late dynamics of the process
are almost stationary.

##  III Theoretical framework for systems with permanent shear-banding

In the following, we first briefly recall the physical ingredients for our
theoretical description of permanent shear bands in YSFs at low enough shear
rates, as well as the corresponding set of equations first introduced in Ref.
Benzi  _et al._ (2016). We then derive scaling arguments for the stress
maximum reached during the overshoot at short times. Finally, we describe the
long-time evolution of the velocity profiles towards steady state.

###  III.1 Non-local fluidity model with noise

Our theoretical framework to account for permanent shear bands consists in
adding a stochastic noise of small amplitude to the previous non-local
fluidity model. Stochastic noise introduces some decorrelation between regions
of different fluidity, such that an unsheared region characterized by a vacuum
solution ($f_{0}=0$) can coexist spatially with a sheared region characterized
by non-zero fluidity in steady state Benzi  _et al._ (2016). In the absence of
noise, the vacuum state ($f_{0}=0$), i.e., the solution of Eq. (6), is
unstable and spatial coexistence of the two solutions takes place over a
finite duration. As a result, the shear band is only transient and disappears
asymptotically as seen above in Sect. II.4.3. Here, stochastic noise is added
to the dynamics by modifying Eq. (6) to :

$\frac{\partial f}{\partial\tilde{t}}=f\left[\xi^{2}\Delta
f+mf-f^{3/2}+\epsilon_{0}/4\right]+\sqrt{\epsilon_{0}f}\,w(y,\tilde{t})\,,$
(34)

where $w$ is a Gaussian random process $\delta$-correlated in space and time,
and $\epsilon_{0}$ is a material parameter that allows us to tune the level of
noise. Note that the limit $\epsilon_{0}\rightarrow 0$ is not singular, in the
sense that we recover the results discussed in the previous section when the
noise goes down to zero.

To study the properties of Eq. (34), it is useful to change variable and
introduce the field $\phi$ defined by $f=\phi^{2}$. Equation (34) then
becomes:

$\frac{\partial\phi}{\partial\tilde{t}}=-\frac{\delta\mathcal{F}[\phi]}{\delta\phi}+\sqrt{\epsilon_{0}}w(y,\tilde{t})\,,$
(35)

where the functional $\mathcal{F}[\phi]$ is given by

$\mathcal{F}[\phi]=2\int_{0}^{L}\mathrm{d}y\left[\xi^{2}\phi^{2}(\nabla\phi)^{2}-\frac{1}{4}m\phi^{4}+\frac{1}{5}\phi^{4}|\phi|\right]\,.$
(36)

For the vacuum state $\phi\simeq 0$, it was shown in Ref. Benzi  _et al._
(2016) that Eq. (35) reduces to:

$\frac{\partial\phi}{\partial\tilde{t}}=D\Delta\phi-R\phi+m\phi^{3}-\phi^{3}|\phi|+\sqrt{\epsilon_{0}}w(y,\tilde{t})\,,$
(37)

where $D$ and $R$ are renormalized coefficients induced by fluctuations that
both depend on the level of noise $\epsilon_{0}$ and on the cooperativity
length $\xi$. Due to the term $R\phi$, the noise promotes the stability of the
previously unstable vacuum solution. Moreover, Eq. (37) displays the
characteristic features of a first-order phase transition, which occurs for
$m=m(\Sigma_{c})\sim R^{1/3}$, where $\Sigma_{c}=1+\dot{\Gamma}_{c}^{1/2}$.
Consequently, for stresses such that $m^{3}<R$, i.e., shear rates lower than
the critical shear rate $\dot{\Gamma}_{c}$, a flowing region ($f\sim m^{2}$)
and a non-flowing region ($f\simeq 0$) coexist at steady state, which
corresponds to permanent shear bands Benzi  _et al._ (2016).

The condition $m^{3}<R$ for the formation of a stable shear band can also be
understood in terms of competing time scales. Indeed, the noise introduces a
new time scale $1/R$, which sets the characteristic time for a perturbation to
decay to zero. The other time scale in the model is related to $m$ and is
embedded in the term $m\phi^{3}$ in Eq. (37). Such a term is by construction
related to the bulk rheology above the yield stress, and following the scaling
arguments in Eq. (8), the associated time scale is $m^{3}$. Hence, $1/m^{3}$
represents the characteristic time scale of the shear band instability, which
can also be considered as the correlation time of the system during the
fluidization phase. In this framework, the criterion $1/R<1/m^{3}$ for
permanent shear banding implies that the ratio between the perturbation
relaxation time and the fluid correlation time is smaller than 1, i.e., that
any attempt of the system to flow will be damped after a relatively short
time. Note that the condition of competing time scales is also a key feature
of other theoretical approaches discussing the emergence of permanent shear
banding Picard  _et al._ (2005); Coussot and Ovarlez (2010); Martens  _et al._
(2012).

To summarize, noise leads to the existence of a critical shear rate
$\dot{\Gamma}_{c}$, below which shear bands are observed in steady state and
above which shear bands are only transient. The value of $\dot{\Gamma}_{c}$
depends both on the noise amplitude $\epsilon_{0}$ and on the correlation
length $\xi$. We now turn to the short-time response of the system in the
presence of such a random noise.

###  III.2 Overshoot scaling

In this section, we examine how the scaling of the stress maximum
$\Sigma_{\text{M}}(\dot{\Gamma})$ during the overshoot may be affected by the
presence of stochastic noise and of stable shear bands. First, because of the
noise, the initial condition of the space-averaged fluidity cannot be
arbitrarily small. Rather, the initial fluidity is controlled by the noise,
i.e., at $\tilde{t}=0$, we have $\langle f\rangle=f_{0}(\epsilon_{0})$. As
already mentioned in Sect. II.1.2, this defines a characteristic shear rate
$\dot{\Gamma}_{0}\equiv f_{0}(\epsilon_{0})$ below which there cannot be any
overshoot. Second, above the critical shear rate $\dot{\Gamma}_{c}$, i.e.,
when shear bands are only transient like in the absence of noise, we expect
that the theoretical arguments developed in Sect. II remain valid. Therefore,
the only case relevant to the present discussion corresponds to
$\dot{\Gamma}_{0}\ll\dot{\Gamma}\leq\dot{\Gamma}_{c}$.

To proceed, let us assume that
$\dot{\Gamma}_{0}\ll\dot{\Gamma}\leq\dot{\Gamma}_{c}$ and that the system
shows a stress overshoot of amplitude $\Sigma_{\text{M}}$ at
$\tilde{t}=\tilde{t}_{M}$. Since for $\dot{\Gamma}\leq\dot{\Gamma}_{c}$, the
equilibrium value of $\Sigma$ is $\Sigma_{c}$ Benzi  _et al._ (2016), it
follows that $\Sigma_{\text{M}}>\Sigma_{c}$. The latter inequality also
implies that the size of the permanent shear band $\ell_{b}^{s}(\dot{\Gamma})$
must be larger than that of the shear band $\ell_{b}$ at
$\tilde{t}=\tilde{t}_{M}$. Over the interval
$[\tilde{t}_{1},{\tilde{t}}_{M}]$, the shear band that nucleates at the wall
is driven by the forcing acting at the boundary, and one expects that the
growing rate of $\ell_{b}$ is close to the “deterministic” behaviour
previously discussed in Eqs. (15) and (18). Once the stress reaches its
maximum value $\Sigma_{\text{M}}$, the system tends to balance $\dot{\Gamma}$
with $\langle f\rangle\Sigma$ as in Eq. (13). After the overshoot, the
velocity of the propagating front remains controlled by Eq. (17) as long as
$\Sigma(\tilde{t})\gg\Sigma_{c}$. However, it must slow down when
$\Sigma(\tilde{t})$ becomes comparable to $\Sigma_{c}$ and the shear band
approaches its equilibrium value $\ell_{b}^{s}(\dot{\Gamma})$. As discussed
below in Sect. III.3, during this late stage, the detailed dynamics depend on
the model parameters $\xi,\epsilon_{0}$, and $\dot{\Gamma}$.

Figure 6:  Stress maximum $\Sigma_{\text{M}}-1$ as a function of the applied
shear rate $\dot{\Gamma}$ in the presence of noise. Permanent shear bands are
observed for $\dot{\Gamma}<\dot{\Gamma}_{c}=4\,10^{-2}$, while shear bands are
only transient for $\dot{\Gamma}>\dot{\Gamma}_{c}$. The red dashed line shows
the diffusive scaling, $\Sigma_{\text{M}}-1\sim\dot{\Gamma}^{1/3}$, while the
red solid line shows the asymptotic scaling,
$\Sigma_{\text{M}}-1\sim\dot{\Gamma}^{4/17}$. The dash-dotted line corresponds
to Eq. (40) where $\dot{\Gamma}_{0}=3.4\,10^{-4}$ is inferred numerically from
the initial fluidity through $\dot{\Gamma}_{0}=f_{0}(\epsilon_{0})=\langle
f\rangle(\tilde{t}=0)$. Numerical results obtained by solving Eq. (34) for
$\epsilon_{0}=10^{-7}$, $\tau=10$, and $\xi=0.04$. Inset: stress $\Sigma$ as a
function of $t/\tau$ for five different shear rates: $\dot{\Gamma}=$0.00385 (
), 0.00865 ( ), 0.0195 ( ), 0.0438 ( ), and 0.0985 ( ) highlighted by the same
symbols in the main panel.

To sum up, the scaling laws describing the locus of the stress maximum during
the overshoot are not affected by the physical ingredients controlling the
existence of stable or transient shear bands. In order to confirm this
picture, we integrate the equation for the fluidity including noise, Eq. (34),
for $\dot{\Gamma}$ in the range $[10^{-3},10]$. As in Sect. II, we couple Eq.
(34) with the Maxwell-like evolution of the stress $\Sigma(\tilde{t})$ [see
Eq. (7)], and solve both equations. The characteristic time $\tau$ is set to
10, and the noise amplitude $\epsilon_{0}$ is fixed to $10^{-7}$. According to
Ref. Benzi  _et al._ (2016), for $\epsilon_{0}=10^{-7}$ a stable shear band is
observed for $3.4\,10^{-4}\ll\dot{\Gamma}\leq\dot{\Gamma}_{c}=4\,10^{-2}$. The
corresponding evolution of the stress is reported as an inset in Fig. 6 for
five shear rates on both sides of the critical shear rate $\dot{\Gamma}_{c}$.
A clear overshoot is observed in all cases, whether shear bands are permanent
or only transient. Moreover, we report in Fig. 6 the stress maximum
$\Sigma_{\text{M}}-1$ as a function of the shear rate $\dot{\Gamma}$. At high
shear rates, the stress maximum is well described by the scaling of the
asymptotic regime, $\Sigma_{\text{M}}-1\sim\dot{\Gamma}^{4/17}$, whereas at
low shear rates, the stress maximum follows the scaling of the diffusive
regime, $\Sigma_{\text{M}}-1\sim\dot{\Gamma}^{1/3}$, as derived in the absence
of noise [see Eq. (23) and Fig. 2]. Therefore, numerical computations in the
presence of noise confirm that the scaling of the stress maximum is not
sensitive to the type of shear banding, be it permanent or transient. Finally,
in the limit of extremely low values of $\dot{\Gamma}$, we note that the
stress maximum lies slightly below the one predicted by Eq. (25) in the
diffusive regime (see two leftmost points in Fig. 6). Such a deviation is due
to the noise, which generates a non-zero fluidity $f_{0}(\epsilon_{0})$ even
for $\dot{\Gamma}=0$. In particular, for very low
$\dot{\Gamma}\gtrsim\dot{\Gamma}_{0}$, the stress overshoot condition of Eq.
(13) should be corrected by a term $f_{0}(\epsilon_{0})\Sigma_{\text{M}}$.
Equation (14) then becomes:

$\dot{\Gamma}\sim
f_{0}(\epsilon_{0})\Sigma_{\text{M}}+\ell_{b}(\Sigma_{\text{M}}-1)^{2}\,.$
(38)

Using the diffusive scaling in Eq. (16), we rewrite

$\dot{\Gamma}-f_{0}(\epsilon_{0})\sim(\Sigma_{\text{M}}-1)f_{0}(\epsilon_{0})+\left(\Sigma_{\text{M}}-1\right)^{3}\xi\tau^{1/2}\,.$
(39)

For small $f_{0}(\epsilon_{0})$ and small $\dot{\Gamma}$, solving Eq. (39) up
to first order in $\dot{\Gamma}_{0}=f_{0}(\epsilon_{0})$ leads to

$\Sigma_{\text{M}}-1=\left(\frac{\dot{\Gamma}-\dot{\Gamma}_{0}}{\xi\tau^{1/2}}\right)^{1/3}\,.$
(40)

Thus, Eq. (40) predicts a deviation from the diffusive regime, which is all
the more pronounced that the shear rate is close to $\dot{\Gamma}_{0}$. As
shown by the dash-dotted line in Fig. 6, Eq. (40) accounts very well for
$\Sigma_{\text{M}}-1$ at the lowest shear rates with no fitting parameter
since $\dot{\Gamma}_{0}$ is directly extracted numerically from the initial
fluidity.

###  III.3 Asymptotic evolution towards steady state

Figure 7: (a) Steady-state normalized fluidity $f(y)/m^{2}$ as a function of
$y/L$ achieved for the five shear rates reported in the inset of Fig. 6:
$\dot{\Gamma}=$0.00385 ( ), 0.00865 ( ), 0.0195 ( ), 0.0438 ( ), and 0.0985 (
), knowing that $\dot{\Gamma}_{c}=0.04$. (b,c) Temporal evolution of the size
$\ell_{b}$ of the fluidized band in the case of (b) transient shear bands
($\dot{\Gamma}<\dot{\Gamma}_{c}$) and (c) permanent shear bands
($\dot{\Gamma}>\dot{\Gamma}_{c}$) for the same shear rates as in (a). The red
solid lines represent a logarithmic trend in (b) and the power-law scalings
with exponents $1$ and $4/9$ in (c).

We now discuss the long-time evolution of the stress and velocity profile for
the five selected values of $\dot{\Gamma}$ on both sides of the critical shear
rate $\dot{\Gamma}_{c}$. As shown in the inset of Fig. 6, after reaching a
maximum at times $\tilde{t}\simeq 2$–4$\tau$, the stress slowly decreases
toward a stationary regime at long times ($\tilde{t}>10\tau$). In Fig. 7(a),
we further report the normalized fluidity $f(y)/m^{2}$ achieved at long time
once a stationary regime is reached, as a function of the normalized position
$y/L$. For the three lowest shear rates, which verify
$\dot{\Gamma}<\dot{\Gamma}_{c}$, the shear rate is heterogeneous showing a
permanent shear band, i.e., the coexistence between a vacuum and a compact
solution Benzi  _et al._ (2016). For shear rates much larger than
$\dot{\Gamma}_{c}$, the fluidity is spatially homogeneous [ in Fig. 7(a)]
indicating a homogeneous velocity profile. For shear rates closer to the
critical shear rate $\dot{\Gamma}_{c}$ [ in Fig. 7(a)], there is no shear
band, but the calculated fluidity is not equal to $m^{2}$ throughout the whole
system, which is a signature of the competition between a stable state (here,
a homogeneous flow) and an unstable state (shear bands).

Finally, in Figs. 7(b) and 7(c), we report the size of the fluidized band
$\ell_{b}$ as a function of time for shear rates respectively below and above
$\dot{\Gamma}_{c}$. For $\dot{\Gamma}\geq\dot{\Gamma}_{c}$, Fig. 7(c) shows
that $\ell_{b}(\tilde{t})$ follows the same scalings as those predicted in
Sect. II in the absence of noise, namely
$\ell_{b}(\tilde{t})\sim(\tilde{t}-\tilde{t}_{\text{1}})$ at short times in
the diffusive regime [see Eq. (16) and Fig. 1(c)] and
$\ell_{b}(\tilde{t})\sim\tilde{t}^{4/9}$ at long times [see Eq. (32)]. On the
other hand, for $\dot{\Gamma}\leq\dot{\Gamma}_{c}$, the shear band tends to
grow very slowly after its nucleation, almost logarithmically in time, up to
the point where it reaches its asymptotic value [see Fig. 7(b)]. In summary,
the temporal evolution of the system towards its steady state is not affected
by the existence of a stochastic noise in the dynamics as long as
$\dot{\Gamma}\geq\dot{\Gamma}_{c}$.

##  IV Comparison with literature data

In this section, we revisit some previously published data on stress
overshoots in a wide variety of glassy systems in light of the present general
theory. In the companion paper (see Fig. 3 in Ref. Benzi  _et al._ (2020)), we
successfully compared our theory to experiments on Carbopol microgels that
show transient shear banding Divoux  _et al._ (2010, 2011a, 2011b, 2012).
Equations (26) and (27) were shown to provide an excellent framework to
rescale experimental data onto the two predicted behaviors, namely (i) the
diffusive low-shear scaling for $\dot{\Gamma}\ll 1$:
$\Sigma_{\text{M}}-1\sim\dot{\Gamma}^{\beta(n)}$ with $\beta(n)=2n/3$, and
(ii) the asymptotic high-shear scaling for $\dot{\Gamma}\gg 1$:
$\Sigma_{\text{M}}-1\sim\dot{\Gamma}^{\alpha(n)}$ with $\alpha(n)=4n/(9-n)$.

###  IV.1 Overshoot scaling

Figure 8: Analysis of stress overshoots reported in the literature for (a) experiments on a dense assembly of thermosensitive soft core-shell colloids (data from Ref. Carrier and Petekidis (2009)), (b) experiments on a commercial hair gel (data from Ref. Fernandes  _et al._ (2017)), (c) numerical simulations of a Lennard-Jones glass (data for temperature $T=0.2$ from Ref. Varnik  _et al._ (2004)), and (d) numerical simulations of Brownian hard disks (data for volume fraction $\phi=0.81$ from Ref. Amann  _et al._ (2013)). Left column: stress maximum $\sigma_{\text{M}}$ vs shear rate $\dot{\gamma}$. The dotted lines show the yield stresses. The solid lines in (c) correspond to logarithmic fits $\sigma_{\text{M}}=a+b\log\dot{\gamma}$ at low and high $\dot{\Gamma}$. Right column: rescaled stress maximum as a function of $\dot{\gamma}$. The dotted lines correspond to the scaling predicted in the low-shear (diffusive) regime with exponent $\beta(n)=2n/3$ while the solid lines show the scaling expected in the high-shear regime with exponent $\alpha(n)=4n/(9-n)$, where $n$ is the HB exponent. See Table 1 for a list of all various parameters. Figure | Ref. | System |  $\sigma_{\text{y}}$ (Pa or a.u.) | $n$ |  $A$ (Pa.sn or a.u.)
---|---|---|---|---|---
8(a) |  Carrier and Petekidis (2009) | core-shell PNIPAM-PS colloids | 4.7 | 0.43 | 2.8
8(b) |  Fernandes  _et al._ (2017) | hair gel (mostly Carbopol) | 112.6 | 0.39 | 90.1
8(c) |  Varnik  _et al._ (2004) | 3D 80:20 binary Lennard-Jones mixture | 0.47 | 0.445 | 2.6
8(d) |  Amann  _et al._ (2013) | 2D Brownian hard disks | 0.8 | 0.57 | 4.8
Table 1: Parameters used in Fig. 8: the yield stress $\sigma_{\text{y}}$, the
shear-thinning exponent $n$ and the consistency index $A$ are extracted from
HB fits of the data available in Refs. Carrier and Petekidis (2009); Fernandes
_et al._ (2017); Varnik  _et al._ (2004); Amann  _et al._ (2013).

We further test the robustness of our modelling of shear start-up based on
four additional data sets extracted from the literature, respectively
experiments on a dense assembly of soft care-shell colloids Carrier and
Petekidis (2009) and on a commercial hair gel Fernandes  _et al._ (2017) in
Fig. 8(a,b) and numerical simulations of a Lennard-Jones glass Varnik  _et
al._ (2004) and of Brownian hard disks Amann  _et al._ (2013) in Fig. 8(c,d).
In all cases, the stress maximum increases mildly with the applied shear rate
that spans at least three orders of magnitude (left column of Fig. 8). Fitting
the steady-state flow curves reported for these various systems by the HB
model allows us to extract the corresponding values of the yield stresses
$\sigma_{\text{y}}$ and of the HB exponents $n$, as reported in Table 1. When
one considers the distance to the yield stress
$\sigma_{\text{M}}/\sigma_{\text{y}}-1$, rather than the absolute stress
maximum $\sigma_{\text{M}}$ clear power-law behaviors are recovered as a
function of the shear rate $\dot{\gamma}$ (right column of Fig. 8).
Strikingly, as shown by the lines in Fig. 8, these power laws are in very good
agreement with the scalings predicted by Eqs. (26) and (27), either with
exponent $\alpha(n)$ (solid lines) or with exponent $\beta(n)$ (dashed lines),
depending on the system and on the shear rate range.

Comparing the power-law fits for the various data sets, our rescaling shows
that the experimental data mostly fall into the diffusive regime [Fig.
8(a,b)], except perhaps for the hair gel at the highest shear rates. On the
other hand, the numerical data for the Brownian hard disks are all consistent
with the high-shear asymptotic regime [Fig. 8(d)], while those for the
Lennard-Jones show a transition from the diffusive to the high-shear scaling
for $\dot{\gamma}\simeq 10^{-4}$ [Fig. 8(c,right)]. The latter result offers a
radically different interpretation of the data presented in Ref. Varnik  _et
al._ (2004). There, in a case that exhibits steady-state shear banding, the
increase of the stress maximum was originally described by two consecutive
logarithmic growths $\sigma_{\text{M}}\sim\log\dot{\gamma}$ separated by a
critical shear rate that corresponds to the inverse of the structural
relaxation time [see solid lines in Fig. 8(c,left)]. Such logarithmic
behaviors of $\sigma_{\text{M}}$ with $\dot{\gamma}$ were tentatively related
to activated dynamics and shear-induced hopping. In our framework, the same
data are rather interpreted as two power-law scalings of $\Sigma_{\text{M}}-1$
vs $\dot{\Gamma}$ in connection with the growth dynamics of the shear band.
Such a strong apparent contradiction is resolved in Fig. 9 where the data of
Fig. 6 are shown as $\Sigma_{\text{M}}$ vs $\dot{\Gamma}$ in semilogarithmic
scales: two different regimes with apparent logarithmic trends clearly emerge
at low and high $\dot{\Gamma}$, in agreement with the results of Ref. Varnik
_et al._ (2004). Therefore, we may conclude that without any theoretical hint,
one can hardly make any strong statement about the overshoot scaling when
focusing on $\Sigma_{\text{M}}$ vs $\log\dot{\Gamma}$ or rather on
$\Sigma_{\text{M}}-1$ vs $\dot{\Gamma}$. Yet, by accounting for a large amount
of experimental and numerical data through power-law scalings, our theory
provides an alternative to the interpretation proposed in Ref. Varnik  _et
al._ (2004).

Figure 9: Same data $\Sigma_{\text{M}}$ versus $\dot{\Gamma}$ as in Fig. 6
plotted in semilogarithmic scales. The solid lines are logarithmic fits
$\Sigma_{\text{M}}=a+b\log\dot{\Gamma}$ at high and low $\dot{\Gamma}$,
respectively. The values of $a$, $b$ are: $a=3.04$, $b=0.26$ (low
$\dot{\Gamma}$) and $a=3.88$, $b=0.90$ (high $\dot{\Gamma}$).

###  IV.2 Shear band dynamics and fluidization

Figure 10: Transient shear banding in a 1% wt. carbopol microgel. Velocity
profiles $v(y)/v_{0}$ measured by ultrasound velocimetry at different times
after a constant shear rate $\dot{\gamma}=0.7$ s-1 is applied at $t=0$, which
corresponds to a constant velocity $v_{0}=0.77$ mm s-1 of the moving wall.
Inset: corresponding stress response $\sigma(t)$. The colored symbols show the
times at which the velocity profiles in the main graph are recorded,
respectively $t/\tau_{\text{f}}=0.01$, 0.44, 0.75, 0.98, and 2.9, with
$\tau_{\text{f}}=2300$ s. Experiments performed in a concentric-cylinder
geometry of gap $L=1.1$ mm covered with sandpaper of roughness 60 $\mu$m (data
adapted from Ref. Divoux  _et al._ (2010)).  Figure 11: Normalized width
$\ell_{b}/L$ of the fluidized band measured as a function of
$t/\tau_{\text{f}}$ in a 1% wt. Carbopol microgel during shear start-up at
$\dot{\gamma}=0.7$ ( ), 1 ( ) and 3 s-1 ( ), respectively with
$\tau_{\text{f}}=2300$, 700 and 23 s. Shear is imposed at $t=0$ in a rough
concentric-cylinder geometry of gap $L=1.1$ mm covered with sandpaper of
roughness 60 $\mu$m (data adapted from Ref. Divoux  _et al._ (2010)).

The theoretical scenario described so far not only predicts scaling laws for
the stress maximum $\Sigma_{\text{M}}$ at the overshoot but also for the
temporal growth of the band $\ell_{b}(\tilde{t})$. The whole approach hinges
on the existence of the cooperative scale $\xi$ that drives the diffusive term
$\xi^{2}\Delta f$ in Eq. (6). In Figs. 10 and 11, however, we report some
experimental evidence pointing to a more complex scenario than the one solely
described in terms of the characteristic scale $\xi$. This will prompt us to
generalize our model in the next section in order to account for these
experimental observations.

First, Fig. 10 shows the experimental velocity profiles recorded in a Carbopol
microgel after the stress overshoot, while approaching the steady fluidized
state. The fluidized region slowly grows in time, while coexisting with a
solid like region where the local shear rate is essentially zero. At some
point, just before a small yet detectable drop in the stress response (see
inset of Fig. 10 for $t\lesssim 2000$ s), we observe a spatial coexistence of
the fluidized band with a region with a non-zero shear rate (see brown circles
in Fig. 10). Such spatial coexistence is somehow possible only if the
fluidized band can propagate its effect on a scale of the order of the system
size.

Second, Fig. 11 reports the temporal evolution of the size $\ell_{b}$ of the
fluidized band during shear start-up in a Carbopol microgel for various shear
rates, including the one used in Fig. 10. Just prior to full fluidization at
time $\tau_{\text{f}}$ and in strong correlation with the stress drop revealed
in the inset of Fig. 10, $\ell_{b}$ is observed to abruptly increase. While
the shear band growth displays indications of power-law scaling early after
the stress overshoot, such a sudden acceleration is not compatible with the
scaling laws of Eqs. (16) and (18). Note that the experimental fluidization
time $\tau_{f}$ is dimensional, whereas the corresponding theoretical
fluidization time $\tilde{T_{\text{f}}}$ is dimensionless.

The experimental results in Figs. 10 and 11 hint at the presence of “long-
range” correlations in the system, whose manifestation is an abrupt
“avalanche-like” change in $\ell_{b}$ at the end of the fluidization process.
At this stage, there is no way for our model for transient shear banding
described in Sect. II to reproduce these features. To better emphasize this
point, let us consider that shear is started at $t=0$ with the initial
condition $f(y,0)=f(0)\ll 1$ that the fluidity remains homogeneous, so that
the diffusive term in Eq. (6) is always zero. Further neglecting the $f^{5/2}$
term and integrating Eq. (6) leads to

$f(\tilde{t})=\frac{f(0)}{1-f(0)\int_{0}^{\tilde{t}}m(s)\,\mathrm{d}s}\,.$
(41)

Therefore, $f(\tilde{t})/m^{2}$ becomes of order $1$ on a time scale
$(mf(0))^{-1}$, which is extremely large time for a small value of $f(0)$. In
other words, the current version of our model misses the possibility for the
system to undergo avalanche-like events as shown from the experimental data in
Figs. 10 and 11. This sets a compelling case for generalizing our model. In
the following, we shall add the possibility of avalanches to the fluidization
scenario described in the previous sections and explore their consequences on
the stress overshoot and on the shear band dynamics.

##  V Model Generalization

###  V.1 Non-local fluidity model with noise and avalanches

In the model of Sec. III, the introduction of noise was rooted in the presence
of plastic rearrangements [see Eqs. (35)–(37)]. However, it appears too naive
to consider the level of noise as independent of the fluidity. Instead, one
expects that once a fluidity fluctuation occurs, this fluctuation could
enhance the stochastic perturbations with a positive feedback, which would
eventually produce an avalanche-like behavior. We hasten to remark that there
are many different ways to modify the noise term in order to account for such
a positive feedback. Here, rather than discussing mathematical aspects in
detail, we wish to provide simple arguments on how to include avalanche-like
behaviour in the dynamical equations.

For simplicity, we shall assume a spatially homogeneous configuration
$f(y,\tilde{t})=\phi^{2}(\tilde{t})$. In the framework of Eq. (37), we
consider the following multiplicative noise term:

$\sqrt{\epsilon_{0}+\tilde{\epsilon}_{1}\phi^{2}(\tilde{t})}\,w(\tilde{t})\,,$
(42)

where $w(\tilde{t})$ is a white noise $\delta$-correlated in time and
$\tilde{\epsilon}_{1}$ is a positive parameter.

Using the tools of stochastic differential equations, one can show that the
effect of such multiplicative noise is to introduce, on average, an
instability in the fluidity $f=\phi^{2}$. More precisely, the noise term given
by Eq. (42) leads to an additional contribution to the r.h.s. of Eq. (34) that
is proportional to $\tilde{\epsilon}_{1}f$, and therefore to an instability,
whose growth rate is independent of the initial condition $f(0)$. This result
may also be obtained through simple physical considerations by introducing
pair-wise fluidity interactions in Eq. (34):

$mf^{2}(y)\rightarrow mf^{2}(y)+mf(y)\int
G(y,y^{\prime})f(y^{\prime})\mathrm{d}y^{\prime}\,,$ (43)

where $G(y,y^{\prime})$ can be thought of as a random function taking care of
the non-trivial long-range interactions in the system. Next, in the mean-field
approximation, we assume that $mf(y)\int
G(y,y^{\prime})f(y^{\prime})\mathrm{d}y^{\prime}\sim mf(y)\langle\int
G(y,y^{\prime})f(y^{\prime})\mathrm{d}y^{\prime}\rangle$, where
$\langle\dots\rangle$ stands for the average over all possible random
realisations. We further estimate the average by requiring that it is
homogeneous in $y$ and satisfies the scaling transformation given by Eq. (8).
This leads to

$mf(y)\langle\int
G(y,y^{\prime})f(y^{\prime})\mathrm{d}y^{\prime}\rangle\sim\epsilon_{1}m^{3}f\,,$
(44)

where $\epsilon_{1}$ is a positive parameter that accounts for the averaging
procedure. Note that starting from Eq. (42) and using Itô calculus Schuss
(1980), we would obtain, on average, the exact same result upon identifying
$\tilde{\epsilon}_{1}=\epsilon_{1}m^{3}$.

To summarize, whether one thinks in terms of self-sustained noise or in terms
of long-range pair-wise fluidity interactions, a simple way to account for
avalanche-like behavior is to modify Eq. (34) into

$\displaystyle\frac{\partial f}{\partial\tilde{t}}$ $\displaystyle=$
$\displaystyle f\left[\xi^{2}\Delta f+mf-f^{3/2}+\epsilon_{0}/4\right]$
$\displaystyle+$ $\displaystyle\epsilon_{1}m^{3}f\
+\sqrt{\epsilon_{0}f}w(y,\tilde{t})\,.$

The parameter $\epsilon_{1}$ is related to a new time scale, which corresponds
to the time needed for long-range interactions to modify the dynamics. In
particular, Eq. (V.1) shows that $f/m^{2}$ becomes of order $1$ on a time
scale $(\epsilon_{1}m^{3})^{-1}$, which is now independent of the initial
condition $f(0)$, contrary to our previous approach [see Eq. (41)]. In the
following, we mostly focus on the effects of long-range interactions in the
case of transient shear banding, i.e. by solving Eq. (V.1) numerically in the
case $\epsilon_{0}=0$. Equation (V.1) then reduces to

$\frac{\partial f}{\partial\tilde{t}}=f\left[\xi^{2}\Delta
f+mf-f^{3/2}\right]+\epsilon_{1}m^{3}f\,,$ (46)

where the case $\epsilon_{1}=0$ simply corresponds to the model for transient
shear banding studied in Sect. II [Eq. 6]. The full equation (V.1) with
$\epsilon_{0}\neq 0$ and $\epsilon_{1}\neq 0$ will be briefly investigated in
Sect. V.3.3. As in the previous sections, the main dynamical equation for the
fluidity is supplemented with Eqs. (4) and (7).

###  V.2 Avalanche-like dynamics of the fluidized band

####  V.2.1 Working definition of $\ell_{b}(t)$

In Sects. II and III, the size $\ell_{b}(\tilde{t})$ of the fluidized band
could be easily determined at all times as $\ell_{b}(\tilde{t})=\langle
f\rangle(\tilde{t})L/m^{2}(\tilde{t})$ since the fluidity remains vanishingly
small in the solid region. Indeed, in the absence of long-range interactions,
i.e., for $\epsilon_{1}=0$, shear bands, either permanent or transient, imply
a “phase separation” in terms of fluidity between a “fluid phase” where $f\sim
m^{2}$, and a “solid phase” where $f\ll m^{2}$, that are always separated by a
sharp interface. However, in the presence of long-range interactions, i.e.,
when $\epsilon_{1}>0$, the fluidity outside the fluidized band increases
exponentially at some point, which reduces the difference between the local
shear rates in the two “phases”, up to the point when a clear separation into
two different “phases” cannot be recognized any more. To determine such a
point and devise an operative definition of $\ell_{b}(\tilde{t})$, we consider
the ratio

$R_{\gamma}(\tilde{t})=\frac{\dot{\Gamma}_{\rm
fluid}(\tilde{t})}{\dot{\Gamma}_{\rm
solid}(\tilde{t})}=\frac{m^{2}(\tilde{t})}{f(L,\tilde{t})}\,,$ (47)

where $\dot{\Gamma}_{\rm fluid}(\tilde{t})\equiv
f(0,\tilde{t})\Sigma(\tilde{t})=m^{2}(\tilde{t})\Sigma(\tilde{t})$ denotes the
local shear rate in the shear band and $\dot{\Gamma}_{\rm solid}\equiv
f(y=L,\tilde{t})\Sigma(\tilde{t})$ the local shear rate in the solid phase.
Such expressions for the local shear rates in terms of the fluidity are
essentially valid after the overshoot (see Fig. 3). $R_{\gamma}(\tilde{t})$
characterizes the spatial phase separation at a given normalized time
$\tilde{t}$. In particular, we may consider the system as composed of two
different phases as long as $R_{\gamma}$ is larger than some value
$R^{\star}$. For $R_{\gamma}\leq R^{\star}$, the system should be considered
as an heterogeneous flow relaxing to its mechanical equilibrium within the
fluidized phases.

The above criterion on $R_{\gamma}$ is equivalent to stating that a clear
phase separation occurs as long as $f(L,\tilde{t})\leq
f^{\star}m^{2}(\tilde{t})$ where $f^{\star}=1/R^{\star}$. A simple way to
estimate $f^{\star}$ is to consider the r.h.s. of Eq. (46): outside the shear
band, the fluidity is homogeneous and the local growth rate of $f$ is given by
$2mf+\epsilon_{1}m^{3}-2.5\,f^{3/2}$. The maximum value of the growth rate is
attained for $f=64/225\,m^{2}+O(\epsilon_{1}m^{2})\sim 0.3m^{2}$. Therefore,
we may choose $f^{\star}=0.3$ and compute $\ell_{b}$ from the fluidity profile
$f(y,\tilde{t})$ by solving numerically
$f(\ell_{b}(\tilde{t}),\tilde{t})=f^{\star}m^{2}(\tilde{t})$ at each time step
of our computations.

Figure 12 shows the temporal evolution of $R_{\gamma}$ obtained by solving Eq.
(46) with $\epsilon_{1}=0.1$ for $\dot{\Gamma}=0.03,0.1,0.3,0.7$ (continuous
lines) and compared to the case of transient shear banding without avalanches,
i.e. $\epsilon_{1}=0$, for $\dot{\Gamma}=0.3$ (dash-dotted line). In the
latter case, once the shear band has nucleated, $R_{\gamma}$ remains always
very large, above $10^{3}$, up to an abrupt jump to $R_{\gamma}=1$ at
$\tilde{t}=\tilde{T_{\text{f}}}$ upon fluidization. In the presence of
avalanches, however, $R_{\gamma}$ decreases continuously with a rate that gets
faster and faster before reaching $1/f^{\star}$ at the fluidization time.
Indeed, for $\epsilon_{1}>0$, the model includes two competing mechanisms: the
growth of the shear band and the exponential growth of the fluidity outside
the shear band. As long as the fluidity outside the shear band remains small,
the shear band instability is the driving relaxation mechanism. However, once
the fluidity outside the shear band becomes non-negligible, i.e. of the order
of $f^{\star}m^{2}$ with $f^{\star}=0.3$, the avalanche-like event becomes
relevant and the system reaches its final equilibrium in a very short time.
This implies that most of the time needed for fluidization is controlled by
the time scale of the shear band growth, whereas the final stage occurs, in
comparison, on a rather short time.

Figure 12: Temporal evolution of the ratio $R_{\gamma}$ of the local shear
rates as defined by Eq. (47) and computed in the presence of avalanche-like
dynamics for $\dot{\Gamma}=0.03$, 0.1, 0.3, and 0.7 from left to right
(lighter to darker colors). Numerical results obtained by solving Eq. (46)
with $\epsilon_{1}=0.1$, $\tau=1$, and $\xi=0.04$. The dashed-dotted line
corresponds to $R_{\gamma}(\tilde{t})$ computed for $\epsilon_{1}=0$ and
$\dot{\Gamma}=0.3$. Above the horizontal line at $R_{\gamma}=R^{\star}=1/0.3$,
the system separates into a solid phase and a fluid phase. $R_{\gamma}=1$
corresponds to a linear velocity profile. All curves are plotted as a function
of the normalized time $\tilde{t}/\tilde{T_{\text{f}}}$ where
$\tilde{T_{\text{f}}}$ is the fluidization strain. Figure 13: Velocity
profiles computed at $\tilde{t}/\tilde{T_{\text{f}}}=0.26$ ( ), $0.52$ ( ) and
$0.79$ ( ) in the case of a transient shear band. $\tilde{T_{\text{f}}}=380$
is the fluidization strain. Profiles are obtained by solving Eq. (46) for
$\epsilon_{1}=0.1$, $\tau=1$, $\xi=0.04$ and $\dot{\Gamma}=0.1$. Inset:
$\ell_{b}$ vs $\tilde{t}/\tilde{T_{\text{f}}}$.

Figure 13 displays the velocity profiles at three different times in the
presence of avalanche-like events for $\epsilon_{1}=0.1$ and
$\dot{\Gamma}=0.03$. Prior to complete fluidization, the velocity profile
develops a non-negligible shear rate outside the shear band, although we can
still distinguish between two regions of different fluidity (see velocity
profile for $\tilde{t}/\tilde{T_{\text{f}}}=0.79$ in brown symbols). This
behavior is qualitatively very similar to the one reported in experiments
(compare with experimental velocity profile for $t/\tau_{\text{f}}=0.75$ in
Fig. 10). Moreover, as seen in the inset of Fig. 13, the size of the shear
band $\ell_{b}(\tilde{t})$ abruptly jumps to $\ell_{b}=L$ at
$\tilde{t}=\tilde{T_{\text{f}}}$. Because of our definition of $\ell_{b}$,
such a jump is expected and corresponds to the point where the fluidity
$f(L,\tilde{t})$ reaches $f^{\star}m^{2}$.

In order to fully justify our generalized model, we still need to check (i)
whether the fluidization time $T_{\text{f}}$ shows the same scaling behaviors
for $\epsilon_{1}>0$ and for $\epsilon_{1}=0$ and (ii) whether there is any
evidence for two different fluidization mechanisms, beside our theoretical
arguments. We proceed to address these questions in the next two subsections.

Figure 14: Size of the shear band $\ell_{b}$ as a function of
$\tilde{t}/\tilde{T_{\text{f}}}$, where $\tilde{T_{\text{f}}}$ is the
fluidization strain for $\dot{\Gamma}=0.1$ ( ), $0.3$ ( ), and $0.7$ ( )
versus $\tilde{t}/\tilde{T_{\text{f}}}$. Numerical results obtained by solving
Eq. (46) for $\epsilon_{1}=0.1$, $\tau=1$, and $\xi=0.04$. Inset: fluidization
time $T_{\text{f}}$ computed for $\epsilon_{1}=0$ ( ) and $\epsilon_{1}=0.1$ (
) versus $\dot{\Gamma}$. The red line shows the power-law scaling
$T_{\text{f}}\sim\dot{\Gamma}^{-9/4}$.

####  V.2.2 Fluidization time with avalanches

Figure 14 shows the temporal evolution of $\ell_{b}$ computed for
$\dot{\Gamma}=0.1$, 0.3, and $0.7$, which all display avalanche-like behavior
prior to full fluidization, in qualitative agreement with the experimental
data in Fig. 11. Note that our numerical computations cannot account for
random-like oscillations as observed experimentally since long-range
correlations are parametrized by $\epsilon_{1}$ using a mean-field approach.

Furthermore, the fluidization times $T_{\text{f}}$ extracted from our
computations are plotted against the imposed shear rate $\dot{\Gamma}$ in the
inset of Fig. 14 for $\epsilon_{1}=0$ and $\epsilon_{1}=0.1$. Obviously, the
value of $T_{\text{f}}$ is smaller for $\epsilon_{1}>0$ due to avalanches.
Yet, in both cases, $T_{\text{f}}$ is in excellent agreement with the
theoretical prediction $T_{\text{f}}\sim\dot{\Gamma}^{-9/4}$ reported in Benzi
_et al._ (2019) and discussed in Sect. II.4.3. We conclude that the scaling of
$T_{\text{f}}$ with respect to $\dot{\Gamma}$ is not sensitive to the presence
of multiplicative noise and avalanches. This confirms our previous argument
based on two widely different competing time scales, which characterize the
fast final stage of the fluidization process.

####  V.2.3 Signature of avalanches in the stress response

In order to identify a change in the fluidization mechanism associated with
avalanches, we focus on the stress response $\Sigma(\tilde{t})$. Figure 15
shows $\Sigma-1$ together with $\ell_{b}/L$ as a function of $\tilde{t}/\tau$
for two different applied shear rates. Interestingly, $\Sigma-1$ shows a sharp
change of slope around a small kink highlighted with a circle. While $\Sigma$
smoothly decays after the overshoot, it drops much faster and rapidly
converges to its steady-state value just after the kink. Such a sudden change
and the subsequent quasi-discontinuous transition towards the fully fluidized
steady-state stress, for which $\ell_{b}/L=1$, are also observed in the
experimental stress response in the inset of Fig. 10. This hints at a common
physical interpretation of the kink in the stress response as the signature of
the onset of a new, faster relaxation mechanism in the system, namely the
avalanche-like behavior, which leads to a rapid complete fluidization.
Therefore, although based on a mean-field approximation, the generalized
version of our model introduced in Eq. (46) captures subtle, yet noticeable
and so far unexplained, features of the experiments of Ref. Divoux  _et al._
(2010).

Figure 15: Stress response $\Sigma-1$ (solid line) and size of the shear band
$\ell_{b}/L$ (dashed line) versus $\tilde{t}/\tau$ for (a) $\dot{\Gamma}=0.3$
and (b) $\dot{\Gamma}=0.7$. The circle highlights a kink after which a fast
stress drop occurs. Numerical results obtained by solving Eq. (46) for
$\epsilon_{1}=0.1$, $\tau=1$, and $\xi=0.04$.

###  V.3 Stress overshoot with avalanches: towards brittle-like response

####  V.3.1 Influence of boundary conditions in the presence of avalanches

In this Section, we explore the role of boundary conditions on our generalized
model as defined in Eq. (46). We recall that so far we have used the boundary
conditions $f(0,\tilde{t})=m^{2}$ at the shearing wall and
$\partial_{y}f|_{y=L}=0$ at the fixed wall. The boundary condition $f=m^{2}$
at the moving wall hinges on the physical assumption that, upon increasing the
stress, plastic events are triggered by the external driving leading to an
increase of the fluidity near the wall. One may, however, consider another
situation at $y=0$ where $\partial_{y}f|_{y=0}=0$. Physically, this boundary
condition is equivalent to assume that, in the initial stage, the system is so
rigid that the driving does not trigger any additional plastic events, even if
the stress $\Sigma$ becomes relatively large. At long enough times, if long-
range correlations can build up across the system, the growth of the stress
may eventually stop.

Figure 16: Stress response $\Sigma(\tilde{t})$ computed for $\dot{\Gamma}=0.1$
in the presence of avalanches by solving Eq. (46) with $\epsilon_{1}=0.1$,
$\tau=1$, $\xi=0.04$, and the boundary condition $\partial_{y}f|_{y=0}=0$. The
horizontal dashed line corresponds to the steady-state HB value $\Sigma_{\rm
ss}=1+\dot{\Gamma}^{1/2}$. Inset: same numerical integration with the boundary
condition $f(0,\tilde{t})=m^{2}$ for $\epsilon_{1}=0.1$ ( ) and for
$\epsilon_{1}=0$ (solid line).

Figure 16 compares the predictions of Eq. (46) for the two above boundary
conditions at the moving wall. As emphasized in the previous section, in the
case where the fluidity at the wall is fixed, avalanche-like events become
relevant only during the late stages of the shear band evolution, so that we
do not expect $\epsilon_{1}$ to have any significant effect on the scaling of
$\Sigma_{\text{M}}$ with $\dot{\Gamma}$. The inset in Fig. 16 shows that, when
$f(0,\tilde{t})=m^{2}$, the overshoot is indeed reached for the same value
$\Sigma_{M}$ and at the same time $\tilde{t}_{M}$ for both $\epsilon_{1}=0$
and $\epsilon_{1}=0.1$. We have checked numerically that this result holds for
all $\dot{\Gamma}$ and that the scaling of $\Sigma_{\text{M}}$ is still given
by Eq.(25) whatever $\epsilon_{1}$ for this choice of boundary condition.

Remarkably, however, when solving Eq. (46) under the boundary condition
$\partial_{y}f|_{y=0}=0$ (main panel in Fig. 16), the stress maximum
$\Sigma_{\text{M}}$ is much larger than when $f(0,\tilde{t})=m^{2}$. Moreover,
in the former case, the maximum is immediately followed by a very sharp stress
drop. This occurs because, for $\partial_{y}f|_{y=0}=0$, there cannot be any
nucleation of a shear band at $y=0$ so that the stress $\Sigma$ grows to
rather large values until the whole system is fluidized due to the exponential
growth of the initial state, i.e., to the term proportional to $\epsilon_{1}$
in Eq. (46). Just after the overshoot, the abrupt decrease in the stress
corresponds to a sharp transition to a fluidized state. Such a behavior is
characteristic of a “brittle” yielding transition and qualitatively close to
the one discussed recently in Ozawa  _et al._ (2018); Singh  _et al._ (2020);
Barlow  _et al._ (2020).

Note that in principle one could also explore the effects of the boundary
condition $\partial_{y}f|_{y=0}=0$ on the model discussed in Sect. II, i.e.,
with $\epsilon_{1}=0$. However, as discussed in Sect. IV.2, with a homogeneous
initial condition $f(y,0)=f(0)\ll 0$, the time to reach a fluidity that is
large enough for the stress overshoot to occur is proportional to $1/f(0)$
[see Eq. (41)]. Imposing $\partial_{y}f|_{y=0}=0$ in this case would thus lead
to a rather non-physical situation characterized by extremely large values of
the stress overshoot occurring at extremely large values of the strain (i.e.,
of the time $\tilde{t}$). Therefore, in the following, we shall not
investigate this possibility.

####  V.3.2 Scaling of the stress overshoot

Figure 17: Stress response $\Sigma(\tilde{t})$ computed for
$\dot{\Gamma}=10^{-3}$ and $f(0)=10^{-4}$, $10^{-6}$, $10^{-8}$, $10^{-10}$,
$10^{-12}$, and $10^{-14}$ from left to right (lighter to darker colors). The
horizontal dashed line is the steady-state HB value, i.e.,
$\Sigma_{\text{ss}}=1+\dot{\Gamma}^{1/2}$. Numerical results obtained by
solving Eq. (46) for $\epsilon_{1}=0.1$, $\tau=1$, $\xi=0.04$, and the
boundary condition $\partial_{y}f|_{y=0}=0$. The computation is stopped when
the HB value is reached. Inset: $\Sigma_{\text{M}}^{5/2}$ as a function of
$\log[\dot{\Gamma}/f(0)]$. The red solid line shows the linear behavior
predicted by Eq. (52).

Let us now investigate the scaling properties of the stress overshoot when
$\epsilon_{1}>0$ and $\partial_{y}f|_{y=0}=\partial_{y}f|_{y=L}=0$. Starting
with a homogeneous state $f(y,0)=f(0)\ll 1$, the system cannot develop any
spatial heterogeneity in $f(y,\tilde{t})$. Therefore, we can rewrite Eqs. (46)
and (7) respectively as:

$\displaystyle\frac{\mathrm{d}f}{\mathrm{d}\tilde{t}}$ $\displaystyle=$
$\displaystyle mf^{2}+\epsilon_{1}m^{3}f-f^{5/2}\,,$ (48)
$\displaystyle\frac{\mathrm{d}\Sigma}{\mathrm{d}\tilde{t}}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau}\left(1-\frac{f\Sigma}{\dot{\Gamma}}\right)\,.$
(49)

Assuming $f\ll 1$, we may neglect the terms $mf^{2}$ and $f^{5/2}$ in Eq.
(48). The expression of $m(\tilde{t})$ given by Eq. (12) then leads to

$f(\tilde{t})\sim
f(0)\exp\left[\frac{2\epsilon_{1}\tau}{5}\left(\frac{\tilde{t}-\tilde{t}_{\text{1}}}{\tau}\right)^{5/2}\right]\,.$
(50)

The balance equation $f(\tilde{t}_{\text{M}})\Sigma_{\text{M}}=\dot{\Gamma}$
in Eq. (13) further yields

$\displaystyle\left(\frac{\tilde{t}_{\text{M}}-\tilde{t}_{\text{1}}}{\tau}\right)$
$\displaystyle\sim$
$\displaystyle\left[\frac{5}{2\tau\epsilon_{1}}\log\left(\frac{\dot{\Gamma}}{f(0)\Sigma_{\text{M}}}\right)\right]^{2/5}$
(51) $\displaystyle\sim$
$\displaystyle\left[\frac{5}{2\tau\epsilon_{1}}\log\left(\frac{\dot{\Gamma}}{f(0)}\right)\right]^{2/5}\,.$

where the last expression neglects a logarithmic correction in
$\Sigma_{\text{M}}$. Finally, Eq.(51) implies

$\Sigma_{\text{M}}\sim\left[\frac{5}{2\tau\epsilon_{1}}\log\left(\frac{\dot{\Gamma}}{f(0)}\right)\right]^{2/5}\,.$
(52)

The above expression allows us to compute the rate of the stress drop
$-\mathrm{d}\Sigma^{\star}/\mathrm{d}\tilde{t}=\sup_{\tilde{t}>\tilde{t}_{\text{M}}}[-\mathrm{d}\Sigma/\mathrm{d}\tilde{t}\,]$
after the overshoot. Since the stress drop is of the order of
$\Sigma_{\text{M}}$ and occurs on a time scale $m^{-3}$, using Eq. (12), (51),
and (52) leads to

$-\frac{\mathrm{d}\Sigma^{\star}}{\mathrm{d}\tilde{t}}\sim\Sigma_{\text{M}}m^{3}\sim\log\left(\frac{\dot{\Gamma}}{f(0)}\right)\,.$
(53)

Figure 17 presents the stress responses $\Sigma$ as a function of the
normalized strain $\tilde{t}/\tau$ for different values of
$f(0)\in[10^{-14},10^{-4}]$ and for a given shear rate $\dot{\Gamma}=10^{-3}$.
A smaller initial fluidity leads to a larger stress overshoot that occurs at
larger strains and with larger decay rate. Such an enhancement of the
“brittleness” can be rationalized qualitatively if one interprets $f(0)$ as
linked to the system rigidity at $\tilde{t}=0$. Indeed, the smaller $f(0)$,
the smaller the rate of plastic events at $\tilde{t}=0$. For a system where
plastic events are triggered by a (real or effective) temperature $T$, one
expects $f(0)\sim\exp(-E/T)$, where $E$ is the typical energy barrier for
plastic events to occur. In this case, $\log(1/f(0))\sim 1/T$ so that
decreasing $f(0)$ corresponds to lowering the temperature. For a system
showing physical aging, one rather expects $f(0)\sim t_{w}^{-a}$, where
$t_{w}$ is the waiting time and $0<a<1$ Joshi and Petekidis (2018). In this
case, $\log(1/f(0))\sim a\log(t_{w})$ and the limit $f(0)\rightarrow 0$
corresponds to infinite waiting times. In both cases, a decrease in $f(0)$
amounts to an increase in the initial “stiffness” or “rigidity” of the system.

Thus, Fig. 17 is consistent with the fact that both the stress overshoot
$\Sigma_{\text{M}}$ and its sharpness
$-\mathrm{d}\Sigma^{\star}/\mathrm{d}\tilde{t}$ increase with increasing
initial rigidity for a given $\dot{\Gamma}$. While the stress drop is very
sharp and typical of a “brittle” transition for small $f(0)$, the stress shows
a rather smooth decay after the overshoot for the largest initial fluidity
$f(0)=10^{-4}$, which indicative of a “ductile” transition. More
quantitatively, the inset in Fig. 17 shows that Eq. (52) provides an excellent
prediction for the power-law behavior of $\Sigma_{\text{M}}$ as a function of
$\log[\dot{\Gamma}/f(0)]$.

Figure 18: Stress response $\Sigma(\tilde{t})$ computed for $f(0)=10^{-4}$ and
$\dot{\Gamma}=0.001$, $0.005$, $0.025$, $0.13$, and $0.66$ from left to right
(lighter to darker colors). The horizontal dashed line is the steady-state HB
value, i.e., $\Sigma_{\text{ss}}=1+\dot{\Gamma}^{1/2}$. Numerical results
obtained by solving Eq. (46) for $\epsilon_{1}=0.1$, $\tau=1$, $\xi=0.04$, and
the boundary condition $\partial_{y}f|_{y=0}=0$. Inset:
$\Sigma_{\text{M}}^{5/2}$ as a function of $\log(\dot{\Gamma}/f(0))$. The red
solid line shows the linear behavior predicted by Eq. (52).

To complete our analysis, we report in Fig. 18 the evolution of $\Sigma$ as a
function of strain for different values of $\dot{\Gamma}$ and for a given
initial fluidity $f(0)=10^{-4}$. An excellent agreement with Eq. (52) is
observed over the whole range of shear rates under study for the scaling of
$\Sigma_{\text{M}}^{5/2}$ as a function of $\log[\dot{\dot{\Gamma}}/f(0)]$
(see inset in Fig. 18). We emphasize that all the results discussed in the
previous sections do not depend on the choice of $f(0)$ as long as $f(0)>0$.
Here, the situation is quite different: when shear band nucleation due to
external forcing cannot occur, the dependence of the stress maximum
$\Sigma_{\text{M}}$ on both the initial condition $f(0)$ and the shear rate
$\dot{\Gamma}$ strongly differs from the one predicted by Eq. (29).

Figure 19: (a) Stress response $\Sigma(\tilde{t})$ computed for
$\dot{\Gamma}=0.02$ by solving Eq. (V.1) with $\epsilon_{0}=10^{-6}$,
$\epsilon_{1}=0.1$, $\tau=1$, $\xi=0.04$, and the boundary condition
$\partial_{y}f|_{y=0}=0$. Permanent shear bands exist for
$\dot{\Gamma}\leq\dot{\Gamma}_{c}\simeq 0.1$ (see section III). The horizontal
dashed line is the steady-state HB value, i.e.,
$\Sigma_{\text{ss}}=1+\dot{\Gamma}^{1/2}$. Inset: velocity profiles at
$\tilde{t}/\tau=1$ ( ) and $\tilde{t}/\tau=50$ ( ). (b) Spatiotemporal diagram
of $f(y,\tilde{t})/m^{2}$ rescaled between 0 and 1 (see color bar).

####  V.3.3 Combined effect of avalanches and boundary conditions in the case
of steady-state shear banding

Let us close this section by investigating the effect of both $\epsilon_{1}$
and the boundary conditions on the formation of a permanent shear band, i.e.,
on the model introduced in Sect. III where $\epsilon_{0}>0$. To this aim, Eq.
(V.1) is supplemented with the boundary conditions
$\partial_{y}f|_{y=0}=\partial_{y}f|_{y=L}=0$. Because of the space-dependent
noise, the effect of cooperativity, i.e., the Laplacian term in the fluidity
equation, can no longer be disregarded. Figure 19 displays the results of the
full model for $\dot{\Gamma}=0.02$, $\epsilon_{0}=10^{-6}$, and
$\epsilon_{1}=0.1$. The stress response $\Sigma(\tilde{t})$ shows a large
overshoot followed by a quite sharp stress drop [Fig. 19(a)]. The velocity
profiles remain linear up to the stress overshoot, after which the system
cannot reach a fully fluidized state. Because $\dot{\Gamma}<\dot{\Gamma}_{c}$,
the system rather develops a steady shear band at $y/L\simeq 0.4$ at long
times [inset of Fig. 19(a)]. As shown in the spatiotemporal diagram of Fig.
19(b), the fluidity field $f(y,t)$ normalized by $m^{2}$ is constituted of a
large number of small shear bands (with large fluidity) that spontaneously
emerge just after the overshoot and progressively give way to only one large
shear band that survives in the middle of the system at long enough times. In
this case, our model leads to a clear phase separation long after the stress
overshoot, associated with the formation of a permanent shear band. It is
interesting to notice that similar observations were reported in the numerical
simulations of Ref. Ozawa  _et al._ (2018).

To summarize, although extremely simplified, the model generalization
introduced in Eq. (46) seems to capture most of the phenomenology reported in
the experiments of Figs. 10 and 11. Moreover, within a single theoretical
framework, our model accounts qualitatively for “brittle” and “ductile”
yielding transitions depending on different choices of the boundary
conditions. Remarkably, rather different behaviors, including stress
overshoots, transient or steady-state shear banding, avalanche-like
fluidization, and brittle-like yielding may thus be explained based on the
same theoretical approach.

##  VI Discussion and conclusion

In this manuscript, we have used a continuum model to describe the shear-
induced solid-to-liquid transition of YSFs. The present work builds upon a
kinetic elastoplastic model, originally derived by Bocquet et al. in Bocquet
_et al._ (2009) to describe the stationary flow of soft glassy materials. Our
results constitute a significant improvement over the original model, as our
approach accounts for the transient response towards steady state, including
the presence of shear bands, i.e., the coexistence in space and time of two
different phases, namely a solid phase and a fluidized phase. Our fluidity
approach, which is based on a dynamical equation coupled to a Maxwell model,
includes four control parameters, namely the cooperativity length $\xi$, the
Maxwell relaxation time $\tau$, and the amplitudes of two stochastic noises
$\epsilon_{0}$ and $\epsilon_{1}$, which correspond to short-range and long-
range fluidity fluctuations, respectively.

In practice, starting from the solid state, the solid-to-liquid transition
under applied shear rate is associated with a stress overshoot. At the
mesoscale, the yielding of the material corresponds to a phase transition that
comes in two flavors depending on the boundary conditions at the moving wall
and on the noise amplitude.

On the one hand, fixing the fluidity at the moving wall, the short-term
rheological response is dictated by the nucleation and growth of a fluidized
band in the vicinity of that boundary. This result is robust, independent of
the presence of both short-range and long-range noises. In other words, the
stress overshoot characteristics are essentially independent of the fate of
the shear band, whether stable or unstable. Moreover, the locus of the stress
maximum scales as a power-law with the applied shear rate, whose exponent
depends on both the HB exponent $n$ and on the short-time fluidization
dynamics, which involves either a diffusive growth at small shear rates or a
front propagation at large shear rates. In case the shear band is transient,
its lifespan decreases for increasing shear rates and scales as
$T_{\text{f}}\sim\dot{\gamma}^{-9/4}$, which exponent is universal and
insensitive to both parameters $\epsilon_{0}$ and $\epsilon_{1}$.

On the other hand, setting the fluidity gradient to zero at the moving wall
prevents the growth of a shear band at this boundary. In the presence of
short-range and long-range noises, this condition results in diffuse plastic
events in the bulk, leading to multiple shear bands that compete with one
another. The macroscopic signature of this scenario is seen in both the stress
maximum that shows a smoother logarithmic dependence with the applied shear
rate and in the subsequent stress relaxation, which is characterized by a
sharp and almost discontinuous drop that resembles that of a brittle failure.
The presence of long-range correlations in the fluidity via the parameter
$\epsilon_{1}$ also results in a sudden, avalanche-like fluidization at long
times.

These scenarios show that ductile and brittle transitions are two natural
outcomes of our fluidity model, whose main feature is to allow for the
coexistence of two phases. Our model’s predictions are in quantitative
agreement with a broad collection of numerical and experimental results,
encompassing colloidal gels and soft glasses. In that sense, our approach
unifies many different features of transient flows in soft glassy materials.
Nonetheless, our approach neglects some features that may play an important
role during shear start-up flows, including lubrication forces. Such
elastohydrodynamic (EHD) interactions Seth  _et al._ (2008, 2011); Liu  _et
al._ (2018b) were recently shown to affect the scaling of the stress maximum
reached during the stress overshoot Khabaz  _et al._ (2021). Our model is
versatile enough that an EHD contribution could be easily included in our
approach, which is left for future work.

Finally, with respect to previous experimental and numerical work on the
ductile-to-brittle transition in soft materials Picallo  _et al._ (2010);
Tixier  _et al._ (2010); Ramos  _et al._ (2011); Erk  _et al._ (2012); Arif
_et al._ (2012); Kovács  _et al._ (2013); Barlow  _et al._ (2020), a novel
feature highlighted by our study is that boundary conditions may control such
a transition in soft glassy materials. We have used the simplest boundary
conditions that allow or suppress phase nucleation at the moving wall. Yet, a
clear physical connection between boundary conditions and the microscopic
processes at the walls is still missing. Indeed, in experiments, such
processes may depend on numerous factors such as the wall surface roughness or
the interaction between the sample and the wall. Therefore, future work should
focus on providing a microscopic justification for the various types of
boundary conditions used in the present work.

##  VII Appendix

In this Appendix, we provide supplemental information on the velocity of the
fluidized band and on its time dependence. The strategy is the following: (i)
we first recall some basic features on a relevant reaction-diffusion equation
and the velocity of its travelling fronts; (ii) we then use these results to
capture some interesting information about the scaling of the front velocity
$\mathrm{d}\ell_{b}/\mathrm{d}\tilde{t}$ as a function of $m$ and $\tilde{t}$.

###  VII.1 Travelling Fronts in case of a linear reaction-diffusion equation

We consider the scalar field $f$ obeying the following linear reaction-
diffusion equation

$\frac{\partial f}{\partial\tilde{t}}=D\Delta f+mf\,,$ (54)

where $D$ is a diffusion constant and $m$ is a reaction constant. It is
convenient to rewrite the equation as

$\frac{\partial}{\partial\tilde{t}}\left[\exp\left(-m\tilde{t}\right)f\right]=D\Delta\left[\exp\left(-m\tilde{t}\right)f\right]\,,$
(55)

where $\exp\left(-m\tilde{t}\right)f$ is solution of the diffusion equation,
i.e.,
$\exp\left(-m\tilde{t}\right)f\sim\exp\left(-\frac{y^{2}}{4D\tilde{t}}\right)$,
hence

$\begin{split}f&\sim\exp\left[m\tilde{t}-\frac{y^{2}}{4D\tilde{t}}\right]\\\
&=\exp\left[\frac{1}{\tilde{t}}\left(\sqrt{m}\tilde{t}+\frac{y}{\sqrt{4D}}\right)\left(\sqrt{m}\tilde{t}-\frac{y}{\sqrt{4D}}\right)\right].\end{split}$
(56)

These calculations show that the solution of Eq. (54) exhibits a travelling-
wave behaviour with a characteristic velocity

$|v_{f}|\sim\sqrt{Dm}\,.$ (57)

If the reaction constant $m$ now depends on time and if the characteristic
time scale of the evolution of $m$ is large compared to the one of $f$, we can
use a quasi-stationary approximation and extend the previous result such as
the time evolution of the front velocity is given by

$|v_{f}|\sim\sqrt{Dm(\tilde{t})}\,.$ (58)

###  VII.2 Travelling Fronts in case of the fluidity model

Let us now consider the following equation for the fluidity

$\frac{\partial f}{\partial\tilde{t}}=f\left[\xi^{2}\Delta
f+mf-f^{3/2}\right]\,,$ (59)

where $\xi$ is constant and $m$ time-dependent. Let us now assume that
$m(\tilde{t})$ is a “mild” function of time $\tilde{t}$ in comparison to the
velocity of the fluidized band $\mathrm{d}\ell_{b}/\mathrm{d}\tilde{t}$. This
assumption allows us to solve Eq. (59) with $m$ constant and to determine the
scaling of the front velocity $d\ell_{b}/d\tilde{t}$ as a function of $m$. We
assume that at some reference time (say $\tilde{t}=0$), the system is
characterized by the following initial profile

$f(y,0)=\begin{cases}f_{\mbox{\tiny 0}}(y)&\hskip 14.45377pt0\leq
y\leq\ell_{b}(0)\\\ 0&\hskip 14.45377pt\ell_{b}(0)\leq y\leq L\,,\end{cases}$
(60)

where $\ell_{b}(0)=\ell_{b}(\tilde{t}=0)$ is the initial width of the
fluidized band and where $f_{\mbox{\tiny 0}}(y)$ is some space dependent
solution which satisfies the stationary equation:

$f_{\mbox{\tiny 0}}\left[\xi^{2}\Delta f_{\mbox{\tiny 0}}+mf_{\mbox{\tiny
0}}-f_{\mbox{\tiny 0}}^{3/2}\right]=0\,.$ (61)

We then look for the dynamical equation of a perturbation field that is space-
and time-dependent, $\delta f(y,\tilde{t})$, and which is non zero outside the
shear band. To this aim, we use the following profile
$f(y,\tilde{t})=f(y,0)+\delta f(y,\tilde{t})$ in Eq. (59).

Using Eq. (61), we find at first order in $\delta f$ that the perturbation
field satisfies

$\frac{\partial(\delta f)}{\partial\tilde{t}}=\xi^{2}f_{\mbox{\tiny
0}}\Delta\delta f+\left(mf_{\mbox{\tiny 0}}-\frac{3}{2}f^{3/2}_{\mbox{\tiny
0}}\right)\delta f\,,$ (62)

where $f_{\mbox{\tiny 0}}$ is a space-dependent function. Hence, both terms
$\xi^{2}f_{\mbox{\tiny 0}}$ and $\left(mf_{\mbox{\tiny
0}}-\frac{3}{2}f^{3/2}_{\mbox{\tiny 0}}\right)$ in Eq. (62) are non constant
in space. This implies that the front velocity is also space-dependent [see
Eq. (58)]. In order to go one step further, we assume that the velocity of the
fluidized band is controlled by the maximum front velocity. This is a
reasonable approximation but that we expect not to impact the scaling
properties of the front velocity as a function of $m$. Let us denote the
maximum fluidity $f^{(M)}_{\mbox{\tiny 0}}$ such as the new reaction-diffusion
equation to solve is

$\frac{\partial(\delta f)}{\partial\tilde{t}}=\xi^{2}f^{(M)}_{\mbox{\tiny
0}}\Delta\delta f+\left(mf^{(M)}_{\mbox{\tiny
0}}-\frac{3}{2}(f^{(M)}_{\mbox{\tiny 0}})^{3/2}\right)\delta f\,.$ (63)

Based on the results of Eq. (58) and assuming that $m$ does not change much at
the onset of the instability, we find that the front velocity is given by

$\frac{\mathrm{d}\ell_{b}}{\mathrm{d}\tilde{t}}\sim\xi\sqrt{f^{(M)}_{\mbox{\tiny
0}}\left(mf^{(M)}_{\mbox{\tiny 0}}-\frac{3}{2}(f^{(M)}_{\mbox{\tiny
0}})^{3/2}\right)}\,.$ (64)

Hence, assuming that $f^{(M)}_{\mbox{\tiny 0}}\sim m^{2}$, we get the
following scaling

$\frac{\mathrm{d}\ell_{b}}{\mathrm{d}\tilde{t}}\sim\xi m^{5/2}\,.$ (65)

Note that the initial condition [Eq.(60)] can also be supplemented with a
constant fluidity in space $f_{0}\ll 1$, which may be related to the way the
sample is prepared. Numerical simulations showed that the result reported in
Eq. (64) is robust with respect to changes in the value of $f_{0}$. In
particular, we found that the prediction for the front velocity [Eq. (64)]
works fine up to logarithmically small corrections.

In the above derivations, we assume the existence of a propagating front, and
consequently we relate the front velocity to $m$ by a suitable scaling.
However, the solution may not necessarily be that of a propagating front. As
discussed in the text, the early time dynamics may result from a diffusive
process allowing the system to adapt to the boundary conditions. If diffusion
takes place, the scaling for $\ell_{b}(\tilde{t})$ is different. Let us look
at Eq. (59) and concentrate only on the balance of

$\frac{\partial f}{\partial\tilde{t}}\sim f\xi^{2}\Delta f\,,$ (66)

where the term $f\xi^{2}$ acts as a diffusion coefficient. If we assume $f\sim
m^{2}$ we get the following diffusion prediction for the growth of the
fluidized band

$\ell_{b}(\tilde{t})\sim\xi\,m\,\tilde{t}^{1/2}\,.$ (67)

Thus, depending on the physical mechanism responsible for the propagation of
the fluidized band, we find different behaviours.

###### Acknowledgements.

This research was supported in part by the National Science Foundation under
Grant No. NSF PHY-1748958 through the KITP program on the Physics of Dense
Suspensions.

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