# Coarse-graining amorphous plasticity: impact of rejuvenation and disorder

Botond Tyukodi  Department of Physics, Babeş-Bolyai University, 400084 Cluj-
Napoca, Romania  Martin Fisher School of Physics, Brandeis University,
Waltham, Massachusetts 02454, USA  Armand Barbot  PMMH, CNRS, ESPCI Paris,
Sorbonne Université, Université de Paris, Université PSL, F-75005 Paris France
Reinaldo Garciá-Garciá  Departamento de Física y Matemática Aplicada, Facultad
de Ciencias, Universidad de Navarra, Pamplona 31008, Spain  Matthias Lerbinger
PMMH, CNRS, ESPCI Paris, Sorbonne Université, Université de Paris, Université
PSL, F-75005 Paris France  Sylvain Patinet  PMMH, CNRS, ESPCI Paris, Sorbonne
Université, Université de Paris, Université PSL, F-75005 Paris France  Damien
Vandembroucq  PMMH, CNRS, ESPCI Paris, Sorbonne Université, Université de
Paris, Université PSL, F-75005 Paris France

###### Abstract

The coarse-graining of amorphous plasticity from the atomistic to the
mesoscopic scale is studied in the framework of a simple scalar elasto-plastic
model. Building on recent results obtained on the atomistic scale, we discuss
the interest in a disordered landscape-informed threshold disorder to
reproduce the physics of amorphous plasticity. We show that accounting for a
rejuvenation scenario allows us to reproduce quasi-quantitatively the
evolution of the mean local yield stress and the localization behavior. We
emphasize the crucial role of two dimensionless parameters: the relative
strength of the yield stress disorder with respect to the typical stress drops
associated with a plastic rearrangement, and the age parameter characterizing
the relative stability of the initial glass with respect to the rejuvenated
glass that emerges upon shear deformation.

##  I Introduction

The plastic deformation of glasses and amorphous solids is a complex
phenomenon. Dependence of the mechanical behavior (e.g. brittle or ductile) on
preparation, avalanches and shear-banding are among its classical hallmarks.
These features come together with a cohort of other puzzling properties: aging
or rejuvenation Debenedetti-PRL00 ; Egami-NatComm17 ; Egami-ActaMater18 ;
Patinet-PRE20-rejuvenation ; plasticity-induced anisotropy and Bauschinger
effect RVTBR-PRL09 ; Procaccia-PRL10 ; Patinet-PRL2020-Bauschinger ; memory
effects under cyclic loading Sastry-PRL14 ; Keim-RMP19 .

The understanding of the mechanical properties of glasses and amorphous
materials has thus motivated an increasing amount of studies in the last
decades Schuh-ActaMat07 ; RTV-MSMSE11 ; Bonn-RMP17 ; Tanguy-CRP21 . In the
absence of crystalline structure and thus of dislocations, it has appeared in
particular that plastic deformation in amorphous materials results from a
series of localized rearrangements of the amorphous structure, Shear
Transformations Argon-ActaMet79 ; FalkLanger-PRE98  that interact through the
surrounding elastic medium BulatovArgon94a ; BulatovArgon94b ; BulatovArgon94c
. The associated complex mechanical behavior naturally involves a wide range
of time and length scales RTV-MSMSE11 .

In parallel to the increasing number of atomistic simulations developed to
study and characterize these various features, the last decades have seen the
development of over-simplified lattice elasto-plastic models operating at a
mesoscopic scale (see the recent review of Nicolas et al.Nicolas-RMP18  and
references therein). Relying on the coupling of a local threshold dynamics and
a long-range elastic interaction induced by the local plastic rearrangements
of the amorphous structure, these models were able to reproduce most of the
complexity of amorphous plasticity despite the drastic reduction of degrees of
freedom. In the same spirit as the Ising-like models for magnetism or the
shell models for fluid turbulence, one may consider that mesoscopic Elasto-
Plastic Models (EPMs) belong to the class of models recently coined by
Bouchaud Bouchaud-EPN19  as _metaphoric_. Such models do not attempt to
describe reality precisely or necessarily rely on very plausible assumptions.
Instead, they aim to illustrate non-trivial mechanisms, the scope of which
goes much beyond the specifics of the model itself.

Conversely, mesoscopic EPMs can also be considered as an important step in the
framework of a multiscale modelling of amorphous plasticity, operating at an
intermediate scale between atomistic simulations and constitutive models at
continuous scale van-der-Giessen-MSME20 . Such a multiscale strategy requires
identifying the parameters to be transferred to the upper scale and the
calibration between the microscopic and mesoscopic scales. Driven by the rapid
development of new characterization techniques of local shear transformations
Puosi-SM15 ; Albaret-PRE16 ; PVF-PRL16 ; Patinet-PRE18 ; Richard-PRM20 , this
difficult question of a quantitative connection between atomistic and
mesoscopic scale has very recently been addressed in Refs. Patinet-CRP21 ;
Pardoen-IJP21 ; Liu-Martens-PRL21 ; Patinet-Acta22 .

At first sight, a complete quantitative agreement between atomistic and
mesoscopic scales may appear elusive since it requires an ever-increasing
number of parameters. Although not perfect, the above-cited works impressively
reproduce several key aspects of amorphous plasticity upon quasi-static or
finite shear rate loading: stress-strain curves, flow curves, creep Liu-
Martens-PRL21  but also some of the microscopic and macroscopic fluctuations,
as well as non-trivial emergent behaviors such as the Bauschinger effect
Patinet-CRP21 .

Here we intend to discuss coarse-graining in a slightly different spirit.
Sticking to the most simple scalar version of a mesoscopic elastoplastic
model, we highlight the importance of accounting for the structure and
evolution of the underlying disordered glassy landscape. In particular, we
explore the rejuvenation scenario proposed in Ref. Patinet-PRE20-rejuvenation
which states that upon loading, the glass is gradually transformed through
local rearrangements into a new glass phase with plastic properties that are
close to that of an inherent state of a supercooled liquid, close to the mode
coupling transition.

Building on recent atomistic studies Patinet-PRE18 ; Patinet-
PRE20-rejuvenation  we try to understand to what extent we can reproduce the
most salient features of AQS (Athermal Quasistatic) simulations in the
framework of lattice models with the sole knowledge of the initial statistical
properties grained from atomic configurations. In particular, from the
success, limitations and even failures of the different hypotheses that can be
developed within this approach, we try to get more insight into the structure
and evolution of the model’s Potential Energy Landscape (PEL) glasses upon
shearing.

In the following, we first summarize in section II recent results of AQS
simulations of a simple binary model glass PVF-PRL16 ; Patinet-PRE18 ;
Patinet-PRE20-rejuvenation . In particular, we show the contrasting
localization behavior depending on glass preparation and discuss the
statistics of local plastic thresholds in the initial state. In section III we
give a short presentation of the mesoscopic EPM. In section IV, we present our
up-scaling strategy. In section V and VI we present and discuss comparisons of
atomistic and mesoscopic results obtained for a poorly annealed glass and a
gradually quenched glass characterized by a shear-banding behavior. We finally
discuss in section VII different perspectives about the coarse-graining of
amorphous plasticity.

##  II Atomistic simulations of model glasses

###  II.1 Preparation

Three different two-dimensional binary Lennard-Jones (LJ) model glasses
obtained with contrasting thermal histories are considered in the present
article. For each glass, 100 samples of 10000 atoms are prepared. The
atomistic models and the protocols of preparation used here have been
presented in detail in Refs. Patinet-PRE18 ; Patinet-PRE20-rejuvenation .
Periodic boundary conditions are applied to the simulation box. In the
following, all the physical quantities are given in LJ units. The binary
system is prepared in the liquid state (NVT ensemble) at high temperature,
then cooled down and equilibrated to a supercooled liquid state at $T=T_{p}$.
Then the supercooled liquid at parent temperature $T_{p}$ is either abruptly
quenched by energy minimization or gradually quenched at a slow rate down to a
low-temperature glassy state ($T\approx 0.03$) before it is also abruptly
quenched by energy minimization.

A first glass is obtained from the instant quench of a supercooled liquid
equilibrated at $T_{p}=0.38$. This temperature is close to the estimated
temperature of the Mode Coupling Transition of our system $T_{MCT}=0.373\pm
0.01$ Patinet-PRE20-rejuvenation . This glass , which we will refer to as MCT
in the following, will be used as a proxy of the glass structure at Mode
Coupling Transition.

A second glass is obtained from the instant quench of a supercooled liquid
equilibrated at $T_{p}=0.351$. This temperature is still significantly higher
than the numerical temperature of glass transition, estimated at $T_{G}\approx
0.31$. This glass which we will refer to in the following as ESL (for
Equilibrated Supercooled Liquid) will be used as a typical poorly annealed
glass.

A third glass is obtained from the gradual quench of a supercooled liquid
equilibrated at $T_{p}=0.351$ and slowly quench over a period of $10^{6}$.
This glass which we will refer to in the following as GQ (for Gradual Quench)
will be used as a typical well-annealed glass. This statement is obviously to
be considered in the context of conventional molecular dynamics, i.e. in the
absence of accelerated equilibrium procedures such as swap Monte Carlo
algorithm Berthier-JStat19 .

###  II.2 Athermal Quasistatic Shear (AQS) loading

The glass samples are deformed in simple shear geometry in Athermal Quasi-
Static conditions, i.e. by series of elementary deformation operations
consisting of one small incremental affine global shear step followed by an
energy minimization step Maloney-PRL04a ; Maloney-PRE06 . The shearing
protocol is applied up to a global shear strain $\gamma=5$. Local strain
fields are computed by integration using Hencky’s finite strain definition.
Details about the computation of the local stress and strain fields can be
found in Ref. Patinet-PRE18 .

(a)

(b)

(c)

Figure 1: Plastic behavior obtained at atomistic scale for fast and slow
quenched glasses ESL and GQ respectively – a) Stress-strain curves (after
averaging over 20 realizations) – Plastic strain field after 20% of total
shear deformation in b) the fast quenched glass ESL and c) the gradually
quenched glass GQ. A shear-band is clearly visible in the latter case.

In Fig. 1a we show the stress curves obtained for the soft glass ESL and the
hard gradually quenched glass GQ. While ESL shows a quasi-monotonous stress
strain curve, GQ exhibits a stress peak followed by a softening branch.
Eventually, independently on the initial state, the two glasses reach the same
stress plateau. In Fig. 1b and c, we show strain maps of obtained after
deforming ESL and GQ by $20\%$. While mild spatial fluctuations are observed
for ESL, a well defined shear band is clearly visible for GQ.

###  II.3 Local Yield Stress

The propensity of the model glass for plastic deformation is estimated through
the frozen matrix method exposed and detailed in Refs. PVF-PRL16 ; Patinet-
PRE18 . A circular region of radius $\xi$ is isolated in the model glass, with
a surrounding shell of width $2R_{c}$, equal to twice the cut-off length of
the interaction potential. The size $\xi=5\sigma$ chosen for the radius of the
inner core is such that continuum elasticity is obeyed but with significant
fluctuations of the local moduli and anisotropy Tsamados-PRE09 . Series of
affine steps of shear deformation are imposed to the (frozen) shell. After
each step, only the inner core is allowed to relax. The deformation is applied
up to a plastic event that takes place in the core. The associated yield
stress $\sigma_{c}$ and stress drop $\Delta\sigma$ are computed. This
operation is repeated in 18 different shearing directions regularly spaced
every $10^{\circ}$. Due to the heterogeneity of local elastic properties, the
minimum stress to apply along a given direction to get a plastic rearrangement
may be obtained for an affine deformation of the frozen outside shell in nay
other direction. The final local yield stress $\sigma_{c}$ in the $xy$
direction is thus defined as the (positive) minimum of the projected stress
thresholds obtained in the 18 different directions PVF-PRL16 .

We summarize below recent results obtained in Refs. PVF-PRL16 ; Patinet-PRE18
that allow us to characterize the plastic behavior of the three glasses (ESL,
GQ, MCT) of interest at the local scale. These results will help us to develop
later our up-scaling strategy.

(a)

(b)

Figure 2: Local Yield stress results: a) distribution of local yield stress
$\sigma_{c}$; b) distribution of frozen stress fluctuations $\delta\sigma$

In Fig. 2a we show the distributions of local yield stresses $P(\sigma_{c})$
for the three glasses ESL, GQ and MCT. As already documented in Refs. PVF-
PRL16 ; Patinet-PRE18 ; Patinet-PRE20-rejuvenation  we see clearly that the
younger the glass, the lower the local yield stresses. In other terms, the
frozen matrix method developed in Ref. PVF-PRL16  allows us to give a
microscopic interpretation of the relationship between age and mechanical
stability of glasses. Note that a discrete negative tail is present in all the
three cases. This apparently surprising feature results from the existence of
a frozen stress field after the quench protocol. If we call
$\delta\sigma(\mathbf{r})$ the internal shear stress in a patch located at
$\mathbf{r}$ and $\sigma_{c}(\mathbf{r})$ its local yield stress, mechanical
stability simply requires the satisfaction of the inequality
$\delta\sigma(\mathbf{r})<\sigma_{c}(\mathbf{r})$. We can thus define the
local plastic strength
$x(\mathbf{r})=\sigma_{c}(\mathbf{r})-\delta\sigma(\mathbf{r})$ which,
contrary to the local yield stress has to remain positive to ensure mechanical
stability. This local plastic strength is nothing but the amount of external
shear stress to be applied so that the patch reaches its limit of stability
and experiences a plastic rearrangement.

In Fig. 2b we show the distributions of the local internal stress fields
$\delta\sigma$ for the three glasses. Interestingly the three distributions
fall onto the same Gaussian distribution of width $w_{g}=0.175$. Despite the
presence of a power-law tail for large fluctuations, the quench protocol thus
does not seem to alter dramatically the frozen stress fields. We note that
these internal stress fluctuations in the initial configuration amount to
about $10-20\%$ of the local yield stresses.

In Ref. Patinet-PRE18  it was moreover shown that the distribution
$P(\Delta\sigma)$ of local stress drops $\Delta\sigma$ follows an exponential
trend $P(\Delta\sigma)=\tau_{max}^{-1}\exp(-\Delta\sigma/\tau_{max})$ with the
older the glass, the larger the local yield stress and the larger the value of
$\tau_{max}$. In addition a parabolic dependence between the local stress drop
and the local yield stress was reported:
$\overline{\Delta\sigma}=a_{0}+a_{1}\sigma_{c}^{2}$.

The frozen matrix method developed in Refs.Puosi-SM15 ; PVF-PRL16 ; Patinet-
PRE18  thus appears to be a powerful tool to probe the disordered landscape of
glassy material and its evolution upon loading. In the following, we use the
new information collected through this method to inform the mesoscopic
elastoplastic models.

##  III A simple mesoscopic model of amorphous plasticity

Simulations at mesoscopic scale are performed using the scalar lattice model
developed in Refs. VR-PRB11 ; TPVR-Meso12 ; Tyukodi-PRL18 . We summarize below
its main features. We first recall the definition of the model, we then
highlight the crucial role of two dimensionless parameters respectively
associated with the amplitude of the elastic interaction and the evolution of
the disorder upon loading. We finally introduce the upscaling strategy
developed within the present study.

###  III.1 Basics of the scalar mesoscopic elastoplastic model

Discretization – A spatial discretization is performed at scale $a$ onto a
square lattice of size $N\times N$. Bi-periodic boundary conditions are
implemented. The material is considered as elastically homogeneous with a
shear modulus $\mu$ and a compressive modulus $K$. In the following we
restrict ourselves to a scalar model and only consider the shear component of
the stress tensor along the macroscopic shear loading direction. The
mechanical state of a given configuration is then given by the shear stress
field $\sigma_{ij}$ and the plastic strain field $\varepsilon^{p}_{ij}$ where
$(i,j)$ are the coordinates of the cell. The total strain field is thus the
sum of the elastic and plastic strain fields:
$\varepsilon_{ij}=\varepsilon^{el}_{ij}+\varepsilon^{p}_{ij}$ where by
definition $\varepsilon^{el}_{ij}=\sigma_{ij}/2\mu$.

Local threshold dynamics – A local scalar criterion of plasticity is
implemented: stability holds if $\sigma_{ij}<\sigma_{ij}^{c}$. Whenever the
local stress exceeds the local thresholds, the cell experiences a small slip
so that the local plastic strain is incremented by a small value
$\delta\varepsilon_{p}$. The latter estimates the plastic strain experienced
by a cell associated with a local rearrangement of the amorphous structure.
The stress landscape of an individual cell thus consists of series of elastic
branches of slope $2\mu$ terminating at a (fluctuating) threshold value along
the stress scale and separated by (fluctuating) plastic strain increments
along the strain scale. Note that here we focus on the mechanical behavior
upon a monotonous shear loading and for that reason, we consider only stress
thresholds in the forward direction and disregard thresholds in the backward
direction.

Elastic interaction – Local plastic events take place in cells which are
surrounded by other elastic cells. A local plastic strain $\delta e_{pl}$ in
an individual cell gives rise to a global plastic strain
$\delta\varepsilon_{pl}=\delta e_{pl}/N^{2}$. At constant total strain this
induces a decrease of the elastic strain of the same amount and thus a
macroscopic stress drop $\delta\Sigma=2\mu\delta e_{pl}/N^{2}$. Moreover it
induces a local stress drop within the inclusion and a long range internal
stress field of quadrupolar symmetry in the far field. In the case of two-
dimensional elasticity, we get for an Eshelby inclusion Eshelby1957  of area
${\cal A}=\pi a^{2}$ experiencing an eigenstrain $\varepsilon_{0}$:

$\displaystyle\Delta\sigma^{\mathrm{i}n}(r,\theta)$
$\displaystyle=-\frac{\mu\varepsilon_{0}}{1+\mu/K}\;,$ (1)
$\displaystyle\Delta\sigma^{\mathrm{o}ut}(r,\theta)$
$\displaystyle=\frac{\mu\varepsilon_{0}}{1+\mu/K}\frac{{\cal
A}\cos(4\theta)}{\pi r^{2}}\;.$ (2)

where the superscripts in and out correspond to the inner and outer parts of
the inclusion. Note that the amplitude of the local stress drop is directly
proportional to the eigenstrain $\varepsilon_{0}$ and that of the far field is
proportional to the product ${\cal A}\varepsilon_{0}$ of the inclusion area
and the eigenstrain. To account for bi-periodic boundary conditions, the
implementation of the Eshelby Green function $G$ is performed in the
reciprocal space. Details and discussions about different techniques of
implementing this long ranged elastic kernel can be found in Ref. TPVR-Meso12
; TPRV-PRE16 ; Tyukodi-PRL18 .

Disorder – In the initial state, the plastic thresholds $\sigma_{ij}^{c}$ are
drawn from an initial random distribution $p_{i}(\sigma_{c})$. Whenever a
plastic event occurs in a cell, the plastic slip is accompanied by a renewal
of the threshold. The new threshold is drawn from a renewal random
distribution $p_{r}(\sigma_{c})$. In the classical disordered scenario,
defined by $p_{r}=p_{i}$ the very same distribution is used for the plastic
thresholds of the initial state and the plastic thresholds obtained after
rearrangements: the materials is left statistically unchanged after a
rearrangement . In the rejuvenation scenario the two distributions $p_{r}$ and
$p_{i}$ are different. The rationale behind this hypothesis early discussed in
Ref. VR-PRB11  is that there is no particular reason that the glassy state
obtained in the initial state after a particular thermal history be the same
as the one obtained after a mechanically activated plastic event. In
particular this choice is expected to allow rejuvenation and impact shear-
banding Debenedetti-PRL00 ; Varnik-PRL03 ; Varnik-JCP04 . In the following two
variants of the model will be used depending on which of the rejuvenation or
the disordered scenario is chosen.

Elementary event \- Everytime the plastic criterion is satisfied in a cell
$(i,j)$, the basic brick of the model thus performs the following sequence of
three operations:

  * •

The strain field is incremented by a local plastic slip
$\delta\varepsilon_{p}$ drawn from a random distribution
$q(\delta\varepsilon_{p})$;

  * •

The stress field is incremented by the internal stress field
$\delta\sigma=G*\delta\varepsilon_{p}$ induced in the material by the local
slip $\delta\varepsilon_{p}$.

  * •

Finally the local plastic threshold $\sigma_{c}^{ij}$ is replaced by a new
random value drawn in a renewal distribution $p_{r}(\sigma_{c})$.

Synchronous quasi-static dynamics – The elastoplastic behavior of an amorphous
material consists of series of elastic loading branches interspersed of stress
drops. Starting from a stable configuration, the weakest site $(i^{*},j^{*})$
is identified as the one that minimizes the plastic strength field
$x_{ij}=\sigma^{c}_{ij}-\sigma_{ij}$. A plastic event is performed at
$(i^{*},j^{*})$. The elastic stress redistribution has then potentially
destabilized a set of other cells. After identification, this set of unstable
cells is updated synchronously Tyukodi-PRL18 . Again, the series of plastic
events may have destabilized other cells. This operation of identification of
unstable sites and synchronous updates is performed until the avalanche stops
and a stable configuration is recovered. The whole sequence is then repeated
until a target macroscopic strain value is reached.

This series of operations thus defines a quasi-static driving in the very same
spirit as AQS in atomistic simulations: a global homogeneous deformation is
applied up to the first instability takes place. The first plastic event
triggers series of successive plastic events (an avalanche). The global level
of deformation is thus kept constant up to the end of the avalanche. These two
steps (loading up to the end of the elastic branch, plastic avalanche) are
repeated as long as desired Tyukodi-PRL18 .

###  III.2 Two Dimensionless parameters

Before addressing in the next section the implementation of our coarse-
graining strategy we briefly discuss here the role of two dimensionless
parameters, associated to disorder strength and rejuvenation, respectively.
The stationary regime is expected to be controlled solely by the disorder
strength while the description of the transient regime (e.g. stress peak)
requires in addition the rejuvenation parameter.

Disorder strength – In analogy with depinning we can define a “disorder
strength” $s$ that balances the fluctuations of the local yield stress
$\Delta\sigma^{c}$ with the stress drop $\Delta\sigma^{in}$ induced by a
plastic event of amplitude $\varepsilon_{0}$:

$s=\frac{\Delta\sigma_{c}^{r}}{\Delta\sigma^{in}}\\\
=\frac{\Delta\sigma_{c}^{r}}{\mu^{*}\varepsilon_{0}}\;,$ (3)

where $\mu^{*}=\mu/(1+\mu/K)$ is an effective shear modulus associated with
the stress drop within a plastic inclusion.

In the depinning context, the strength disorder would control the transition
between weak and strong pinning (see Ref. PVR-PRL13  for a discussion in the
case of crack front propagating in a heterogeneous material). In particular it
controls the value of the effective toughness of the material (the depinning
threshold).

In the present case of amorphous plasticity, the effect of this strength
disorder is more subtle and is at least two-fold. If one considers plastic
deformation confined within a thin shear-band along the shear direction, one
recovers the standard case of a depinning line with a positive restoration
stress along the band. However if one considers a deformation in the full
system we get a stress interaction whose sign fluctuates depending on the
polar angle so that far from the inclusion it can be regarded as a mechanical
noise. In this context, as early discussed in Ref. TPVR-Meso12  the disorder
strength both controls the value of the flow stress $\sigma_{F}$ :

$\frac{\sigma_{F}-\overline{\sigma_{c}^{r}}}{\overline{\sigma_{c}^{r}}}=f(s)\;,$
(4)

and the diffusion properties upon deformation via the typical value of the
strain associated to a plastic event $\varepsilon_{0}$

$\delta\varepsilon^{2}_{pl}=g(s)\varepsilon_{tot}\;.$ (5)

where $\varepsilon_{tot}=\varepsilon_{el}+\varepsilon_{p}$ is thetotal strain.

Note that both the disorder strength parameter $s$ and the reduced flow stress
are defined with respect to the mean plastic threshold of the renewal
distribution. Again, this parameter is intended here to control the stationary
flow regime.

Rejuvenation/age parameter – As mentioned above, when dealing with the
disorder landscape of the amorphous solids,we have to consider i) the
distribution $p_{i}(\sigma_{c})$ of local yield stresses $\sigma_{c}^{i}$ in
the initial configuration, which depends on the particular preparation
protocol used to prepare the glass; ii) the renewal distribution
$p_{r}(\sigma_{c})$ of the new local yield stresses $\sigma_{c}^{r}$ of the
zones that rearranged. This allows to define an age (or rejuvenation)
parameter:

$r=\frac{\overline{\sigma_{c}^{i}}-\overline{\sigma_{c}^{r}}}{\overline{\sigma_{c}^{r}}}$
(6)

A positive value of $r$ corresponds to an aged material that rejuvenates upon
shear loading. In Refs. VR-PRB11  we could show that such positives values of
the age parameter were associated to shear banding: the larger $r$, the
thinner and the more persistent the shear band.

In the following, rather than performing a full parametric study, we restrict
ourselves to two limit cases. In the first one, we use for the renewal
distribution of local yield stress the very same one as in the initial state.
In the absence of rejuvenation we thus keep $r=0$ and the only expected
evolution of the actual distribution of local plastic thresholds is the
statistical hardening induced by the exhaustion of weak spots upon loading
BVR-PRL02 ; TPVR-Meso12 ; Patinet-PRE20-rejuvenation . In the second case, we
test the rejuvenation scenario proposed in Ref. Patinet-PRE20-rejuvenation ,
i.e. we use for the renewal distribution, the distribution of thresholds
associated to an instant quench from a supercooled liquid at a temperature
close to the mode coupling transition. We thus use the distribution of local
yield stresses obtained for the glass coined as MCT in section II. In this
scenario, performing the numerical application from the data obtained for the
distributions of local yield stress for the different glasses, we get for the
age parameter:

$\displaystyle r^{ESL}$
$\displaystyle=\frac{\overline{\sigma_{c}^{ESL}}-\overline{\sigma_{c}^{MCT}}}{\overline{\sigma_{c}^{MCT}}}\approx
0.14$ (7) $\displaystyle r^{GQ}$
$\displaystyle=\frac{\overline{\sigma_{c}^{ESL}}-\overline{\sigma_{c}^{MCT}}}{\overline{\sigma_{c}^{MCT}}}\approx
0.65$ (8)

##  IV Upscaling strategy

We discuss here how to connect the above defined mesoscopic model with the
atomistic simulations. We identify the parameters of the models from the
results obtained at atomic scale but we also try to make more explicit the
(numerous) hypotheses associated this upscaling operation that involves a
drastic reduction of degrees of freedom. We finally list the observables that
will be compared between the simulations operating at atomic and mesoscopic
scale repectively.

Before going further, let us first emphasize an obvious statement: the present
coarse-graining operation is intended to fail, the model being too simple and
caricatural to expect a complete quantitative agreement. Still, it is of
interest to study to what extent this class of scalar models is able to cope
for the phenomenology of amorphous plasticity. In the following we discuss the
various coarse-graining parameters that we can extract from atomistic
simulations.

###  IV.1 Coarse-graining parameters

  * •

Elastic properties – We use the mean shear and bulk moduli $\mu$ and $K$
measured in the initial state. In particular the spatial fluctuations of the
elastic properties as well as their evolution upon deformation are
disregarded.

  * •

Coarse-graining length scale – In the atomistic simulations the plastic
properties are estimated in disks of radius $\xi=5\sigma$. The systems under
study have a linear size $L\approx 100\sigma$. Here we use for the mesh size
of the lattice model $\ell=\xi\sqrt{\pi}\approx 8.86$, slightly larger than
$\xi$ such that the area of one cell coincides with the area of the disk
${\cal A}=\pi\xi^{2}$ used to estimate the plastic thresholds in atomistic
simulations PVF-PRL16 ; Patinet-PRE18 . The linear size of the box used to
perform the atomistic simulations is thus about $11\ell$. In the following
most of the mesoscopic simulations are performed on small systems of size
N=16.

  * •

Distributions of plastic thresholds – The distributions $p_{i}(\tau^{c})$ used
to model the two glasses at mesoscopic scale in their as-quenched initial
state are directly the ones computed from atomistic simulations using Refs.
PVF-PRL16 ; Patinet-PRE18 :

$p^{G}_{i}(\sigma_{c})={\cal P}^{G}(\sigma_{c})\;,$ (9)

where ${\cal P}$ stands for the distribution measured from atomistic
simulations and the superscript $G=ESL,GQ$ refers to the glass considered.

(a)

(b)

Figure 3: Stress-strain curves for the glass ESL and a mesoscopic model
obtained without (a) and with (b) assuming correlation between the
characteristic slip increment and the stress threshold.

(a)

(b)

Figure 4: Variance of strain and stress vs true strain (a) in the ESL glass
and in the meso model (b). Evolution of the mean threshold vs true strain in
the ESL glass and in the meso model with and without rejuvenation.

In the first disordered variant of the model, we assume that the distribution
of plastic thresholds remains invariant under deformation. We thus use
$p_{r}(\sigma_{c})=p_{i}(\sigma_{c})$ for the distributions of renewed plastic
thresholds that characterize the glassy state obtained after a plastic
rearrangement. In the rejuvenation variant of the model, we account for a
deformation-induced rejuvenation effect. More precisely, we assume that the
glassy local structure obtained after a plastic rearrangement is specific, and
a priori different from that of the as-quenched glass. We actually expect that
this new glassy structure contributes to the gradual formation of the
stationary glass eventually obtained after large deformation. In that
perspective, the distribution $p_{r}(\tau^{c})$ of the new plastic thresholds
should be independent on the initial condition and in the general case rather
depend on the conditions of the loading (strain rate, temperature). Following
the observation reported above we assume in this glassy variant that the
renewal distribution of thresholds is the same as the distribution of initial
thresholds in a glass abruptly quenched from a temperature close to the mode
coupling transition:

$p^{G}_{r}(\sigma_{c})={\cal P}^{MCT}(\sigma_{c})\;.$ (10)

  * •

Distributions of local slip increments – The amplitude of the slip increments
directly controls the level of elastic interactions in the system and as such,
the final value of the flow stress TPVR-Meso12  or the broadening rate of
shear-bands VR-PRB11 . This coupling parameter Budrikis-NatComm17  is thus
crucial for the plastic properties of amorphous solids. The determination of
the local slip events remains extremely challenging Rodney-PRL09 . The
successful mapping recently obtained by Albaret et al. Tanguy-PRE16  between
the stress field measured in atomistic simulation of amorphous silicon under
shear and a sum of elementary Eshelby stress fields opens the way toward a
quantitative analysis of these local events. In section 1 where we summarized
results presented in Ref. Patinet-PRE18 , the local slip amplitudes were
estimated after the measurement of the local stress drops obtained after the
relaxation of the local glassy structure following a plastic rearrangement.
Local slip amplitudes were shown to follow a power law distribution bounded by
an exponential cut-off and to be significantly correlated with the local yield
stress.

In most mesoscopic models of amorphous plasticity, slip increments are settled
at a constant value or drawn from a random distribution, generally
uncorrelated with the plastic thresholds. In contrast, models based on an
energy landscape naturally introduce a strong correlation between local slips
and thresholds. The direct relationship between the two quantities thus
depends on the precise shape of the potential energy landscape chosen in the
model Jagla-PRE17 ; Jagla-PRE18 .

In the present model we consider for the slip increments an exponential
distribution where we account for a direct dependence on the stress threshold:

$p(\delta\varepsilon_{p}|\sigma^{c})=\frac{1}{\varepsilon_{0}(\sigma_{c})}\exp\left(-\frac{\delta\varepsilon_{p}}{\varepsilon_{0}(\sigma_{c})}\right)\;,$
(11)

where the mean value of the slip increment obeys a parabolic dependence to the
local yield stress:

$\quad\varepsilon_{0}(\sigma_{c})=c_{0}+c_{1}\sigma_{c}^{2}\;.$ (12)

In the following, the coefficients $c_{0}$ and $c_{1}$ will be used as tuning
parameters to reproduce as much as possible the stress-strain curves (the
values of the maximum stress and of the flow stress). In order to test the
importance of the correlation between local stress drops and local yield
stresses, we will also try to fit only the flow stress value through the
tuning of the sole coefficient $c_{0}$, thus fixing the other coefficient
$c_{1}=0$.

As detailed below, a systematic comparison will then be made between atomistic
and mesoscopic results for different observables of interest: strain field,
stress field and their fluctuations. Note again that most of the parameter
being directly extracted from the initial configurations of the atomistic
simulations, we are left with a very constrained problem so that we deal only
with one or two fitting parameters to reproduce the results.

###  IV.2 Observables

The extreme simplicity of the mesocopic model obviously limits the extent of
observables to be compared between atomistic and mesoscopic scales. In the
following, we present the evolution of the stress and strain fields upon
shearing as well as the evolution of the local plastic properties.

In order to make the comparison as quantitative as possible, we compute
averaged behaviors (mean values or mean profiles) as well as their fluctuating
part (variance and higher moments). In addition to the stress and stress
fields we also give a special attention to the field of local yield stresses.

##  V Deformation of a soft/fresh glass

We consider here the deformation of the soft glass ESL. As discussed in the
previous section, we want to gradually enrich the description of the
disordered landscape and test the importance of two hypotheses: a rejuvenation
effect and the correlation between local thresholds and slip increments.

###  V.1 Stress-strain curve

In the most naïve version, the EPM considers no correlation and no
rejuvenation. We implement a distribution of plastic thresholds similar as the
one measured by the local yield stress method for the glass ESL. When the
plastic criterion is satisfied in a cell, the latter experiences a local slip.
The amplitude of the slip is taken from an exponential distribution of width
$c_{0}$. To account for a change of local structure a new plastic threshold is
drawn from the very same distribution as the initial one. Here $c_{0}$ is a
tunable parameter. Its value $c_{0}=0.0626$ is adjusted so that the stress
plateau of the stress-strain curves reproduces the results of the AQS
atomistic simulation.

In a second version, rejuvenation is considered: when a site has slipped, the
plastic threshold is renewed but its value is now taken from a rejuvenated
distribution. Here, following the scenario discussed in Ref. Patinet-
PRE20-rejuvenation , we use the distribution of thresholds obtained for a
glass instantly quenched (inherent structure) from a liquid equilibrated at
$T=0.38$, a temperature close to the Mode-Coupling transition. As in the
previous case, the width $c_{0}$ of the distribution of slip increment is used
as a tunable parameter. With $c_{0}=0.0506$ to reproduce the level of the
stress plateau of the stress strain curve.

We show in Fig. 3 the comparisons of the stress-strain curve obtained with
atomistic and mesoscopic simulations, respectively. In both cases, the
stationary regime is nicely reproduced. In other words, the stress plateau can
be reproduced with various distributions of local yield stresses. We see
however that in the absence of correlation between local thresholds and slip
increments (Fig. 3a), the mesoscopic models fail to reproduce the transient
part of the stress-strain curve. In particular, while a discrete peak is
observed in atomistic simulations, the stress-strain curve remains monotonic
in the mesoscopic case. The account of rejuvenation slightly improves the
results but the one-parameter fit performed here to reproduce the stress
plateau clearly does not allow us to get the transient part of the plastic
behavior.

In the presence of correlation between local thresholds and slip increments
(Fig. 3b), we observe on the contrary that the mesoscopic simulations
reproduce very nicely the atomistic stress strain curve. Here two tunable
parameters were used. The width $\varepsilon_{0}$ of the exponential
distribution used to generate a slip increment $\delta\varepsilon_{p}$ after a
threshold $\sigma_{c}$ is given by
$\varepsilon_{p}=c_{0}+c_{1}\sigma_{c}^{2}$. The additional parameter $c_{1}$
thus quantifies the level of correlation. The pair of parameters
$(c_{0},c_{1})$ is tuned so that the mesoscopic stress-strain curve reproduces
the level of the stress peak and that of the stress plateau. We observe that
the full curve is nicely fitted. This close agreement persists independently
on the choice to use or not the hypothesis of rejuvenation. We get
$(c_{0},c1)=(0.0304,0.0216)$ in the absence of rejuvenation and
$(c_{0},c1)=(0.0276,0.0206)$ with rejuvenation. These first results thus tend
to emphasize the importance of the underlying structure of the potential
energy landscape (PEL). In order to reproduce the atomistic results we need to
consider a hierarchical-like structure of the PEL such that the deeper the
well (the larger the threshold), the further the neighboring well (the larger
the slip).

Interestingly, the additional coefficient $c_{1}$ does not significantly
affect the flow stress. If one computes the averages value
$\overline{\varepsilon_{0}}=c_{0}+c_{1}\overline{\sigma_{c}^{2}}$, we get
$\overline{\varepsilon_{0}}=0.0634$ and $0.0521$ in the presence of
correlations, i.e. very close values of the ones obtained without correlation.
In spite of the additional complexity the dimensionless parameter $s$,
directly proportional to $\varepsilon_{0}$ thus fully controls the level of
the plateau stress.

###  V.2 Strain fluctuations

In Fig. 4a we show the evolution of the strain variance upon loading. Here we
considered correlation between stress drops and local yield stress and used
the coefficients $(c_{0},c_{1})$ fitted to reproduce the stress-strain with
and without rejuvenation. We observe that while the fluctuations obtained in
the mesoscopic simulations are overestimated by a factor about $1.5$ with
respect to the atomistic results, the global evolution of the variance upon
loading is very nicely reproduced in the early regime as well as in the late
regime. We remark that the level of fluctuations apparently does not depend on
the rejuvenation hypothesis: the curve almost superimpose. Again, as proposed
in Ref. TPVR-Meso12  this indicates that the diffusive behavior of the strain
field is fully controlled by the sole disorder strength parameter via the
typical value of the plastic slip $\varepsilon_{0}$

###  V.3 Local yield stress

In Fig. 4b we show the evolution of the local yield stresses upon loading. The
atomistic results show a clear hardening trend. The average local yield stress
shows an increase of about 10% upon loading. This increase takes place in the
transient regime of deformation so that a stationary value is quickly reached.
As for the strain fluctuation, we consider here the correlation case and study
the effect of rejuvenation. Without rejuvenation, the mesoscopic model
qualitatively reproduces the hardening trend, but tends to to overestimate the
plastic thresholds. However with rejuvenation, one observe a quasi-
quantitative agreement with the atomistic results. In contrast to the plateau
stress and the strain fluctuation, the reproduction of this observable, thus
crucially depends on the rejuvenation hypothesis. Accounting for the fact that
the level of rejuvenation is not a free parameter but has been arbitrarily
fixed by considering the renewal distribution as deriving from the glass MCT,
this results gives a strong argument to the validity of the rejuvenation
scenario.

##  VI Shear-banding of a hard/gradually quenched glass

We consider here the deformation of the hard gradually quenched glass GQ. As
discussed in the previous section, we want to gradually enrich the description
of the disordered landscape. We first focus on the respective effects of
correlation and rejuvenation on the stress-strain curves. Since the latter
exhibits a stress peak in association with a shear-banding behavior, we then
follow the evolution of a localization ratio based on the participation ratio
of the strain field. As for the fresh glasss ESL we also follow the evolution
of the mean value of the local yield stress field.

###  VI.1 Stress-strain curve

We show in Fig. 5 the comparisons of the stress-strain curve obtained for the
hard glass GQ with atomistic and mesoscopic simulations, respectively. As for
the soft glass ESL, Fig. 5a shows the results in the absence of correlation
between stress drop and local yield stress. In the absence of correlation and
rejuvenation ($c_{0}=0.0836$), the plateau stress is correctly reproduced but
the only trace of a stress peak is a tiny bump. Conversely, in the presence of
rejuvenation ($c_{0}=0.02$) , despite the fact that the fitting procedure
operates only on the stress plateau value, a well defined stress peak gets
clearly visible and almost reaches the atomistic value. The general shape of
the stress-strain curve shows however clear discrepancies compaed to atomistic
data.

When accounting for the correlation between stress drops and local yield
stresses and performing a two-coefficients fitting procedure, the results get
better. As required, both the value of the plateau stress and the stress peak
are nicely reproduced. In the absence of rejuvenation we get
$(c_{0},c1)=(0.0344,0.0164)$ and a very good agreement in the stress strain
curve. In the presence of rejuvenation we get $(c_{0},c1)=(0.0235,0.0163)$.
Interestingly, the global reproduction of the stress-strain curve is not so
good in this case with stress peak significantly sharper than in the atomistic
case. As for the fresh glass ESL we note that the mean value of the slip
increment $\overline{\varepsilon_{0}}$ keeps almost constant with and without
correlation. Without rejuvenation we get
$\overline{\varepsilon_{0}}=c_{0}+c_{1}\overline{\sigma_{c}^{2}}=0.0833$ and
with rejuvenation $\overline{\varepsilon_{0}}=0.0429$. Again these values are
extremely close to those obtained in the absence of correlation, which
confirms that the level of the stress plateau is fully controlled by the value
$\varepsilon_{0}$ and/or dimensionless parameter $s$ (strength of disorder).

(a)

(b)

Figure 5: Stress-strain curve in the GQ and in the meso model with and without
correlation between local stress drops and yield stress

###  VI.2 Localization

In Fig. 6a we show the evolution of a localization index $LOC$ of the strain
field upon shear loading. This index is based on the participation ratio $PR$
classically used to characterize localization phenomena. In practice, for a
strain field $\varepsilon_{ij}$ on a grid of linear size $N$, we define

$LOC(\varepsilon)=\frac{1}{N.PR(\varepsilon)}=\frac{\sum_{ij}\varepsilon_{ij}^{4}}{(\sum_{ij}\varepsilon_{ij}^{2})^{2}}$
(13)

In the case of a localization of a single band, we get $LOC(\varepsilon)=1$
while for a fully homogenous field we get $LOC(\varepsilon)=1/N$. The glass GQ
shows a strong localization behavior. The maximum value is obtained just after
the location stress peak and corresponds to the nucleation of a shear band.
The subsequent broadening of the band then induces a gradual decrease of the
localization index.

We compare these results with mesocopic simulations performed with correlation
of stress drops with local yield stress and we discuss the effect of the
rejuvenation hypothesis. The localization behavior appears to be very well
reproduced by the mesoscopic model with rejuvenation and correlation. The
global trend is recovered: a fast increase associated to the nucleation of the
band, followed by a slower decrease associated to the broadening of the band.
The maximum value of the localization index is reasonably reproduced.

Note that even in the absence of rejuvenation, we still get a significant
localization behavior. The global trend is recovered but the level of
localization remains under-estimated.

###  VI.3 Local yield stress

In Fig. 6b we show the evolution of the mean value of the local yield stress
upon shear deformation. As discussed in Ref. Patinet-PRE20-rejuvenation , a
rather complex behavior emerges. we observe first a slight increase of the
mean plastic threshold that can be associated to a statistical hardening
process. Then a shear-band nucleates and broadens. The associated replacement
of the old hard glass by a fresh softer one induces a gradual decrease of the
average local yield stress.

We compare these atomistic results with the mesoscopic ones obtained with and
without rejuvenation. Without rejuvenation, there is actually no mechanism at
play that could tend to decrease the mean local yield stress. We thus only
observe a growing trend which results from the statistical hardening discussed
in section V in the case of the glass ESL and the mesoscopic model
significantly overestimates the values of the atomistic mean yield stress.

When accounting for rejuvenation, the mesoscopic results get better. In
particular, we recover a similar trend as with the atomistic simulations: a
first increasing regime, followed by a slow decreasing regime. However form
the quantitative aspect, the agrrement is not spectacular and the mesoscopic
model tends here to slightly underestimate the atomistic results. Still, it
clearly appears here that the rejuvenation hypothesis is crucial to recover at
least qualitatively the complex evolution of the mean yield stress upon
loading.

(a)

(b)

Figure 6: Evolution of the localization ratio and of the mean threshold vs
true strain in the GQ glass and in the meso model with and without
rejuvenation.

(a)

(b)

Figure 7: Evolution of the standard deviation of the internal stress field in
the glasses ESL and GQ vs meso model.

##  VII Discussion

In this contribution, we discussed the coarse-graining of amorphous plasticity
from the atomistic to the mesoscopic scale. We deliberately restricted
ourselves to a very simplistic scalar version of a mesoscopic elasto-plastic
model. We tried to identify a few key parameters of the underlying glassy
landscape from the failures and successes encountered in this approach. In
particular, we tried to test the rejuvenation scenario proposed in Ref.
Patinet-PRE20-rejuvenation .

We emphasized the role of two dimensionless parameters, respectively
associated with the strength of the disorder with respect to the typical
stress drop and with the relative age of the initial glass with respect to the
fresher one that emerges upon loading.

Testing our approach on two glasses of contrasting soft and hard behaviors, we
could nicely reproduce the stress-strain curves, the evolution of the mean
plastic thresholds and the localization behavior. We also got a qualitative
agreement for the diffusive behavior of the strain field of the glass ESL.

###  VII.1 Structure of the disordered yield stress landscape

The results we obtain are consistent with the rejuvenation scenario proposed
in Ref. Patinet-PRE20-rejuvenation  according to which, upon shear
deformation, the initial glass is gradually replaced by a fresh new one whose
mechanical properties are close to a glass abruptly quenched from a
temperature close to the mode coupling transition. We also made evidence for
the necessity of correlating the local stress drops following a local plastic
event with the height of the stress barrier, i.e. the local yield stress. Such
a relation is not unexpected and directly derives from the structure of the
underlying potential energy landscape. Still, it is interesting to show that
accounting for such a correlation between local stress drop and yield stress
is necessary to reach a quantitative agreement between atomistic and
mesoscopic simulations. Additional features of the disordered yield stress
landscape certainly deserve further study. In particular, we did not consider
any memory. In the present implementation of the model, once a plastic event
takes place, the former yield stress is immediately forgotten, independently
of the amplitude of the local re-arrangement. This memory effect and its
dependence on the initial preparation has been recently discussed in Ref.
Patinet-Acta22 .

###  VII.2 Elastic heterogeneity and internal stress fluctuations

In the present version of the mesoscopic model, elastic properties are
homogeneous and do not evolve upon deformation. This assumption is at odds
with the actual behavior observed in atomistic simulations. The macroscopic
moduli significantly change upon shearing. In the stationary state, obtained
after very large deformation, one gets for the model glasses under study
$\mu_{ss}=13.2$ and $K_{ss}=59$ to contrast with the initial values
$\mu_{ESL}=14.7$, $K_{ESL}=52$ and $\mu_{GQ}=19$, $K_{GQ}=55$. These are
significant changes, up to a few tens of per cent for the glass GQ.

The spatial heterogeneity of the elastic properties was also completely
neglected in the present study. From the qualitative point of view, this
omission may not be dramatic since the structure, and the symmetry of the
associated internal stress fields are likely to be close to that of the
Eshelby stresses induced by plastic inclusions. From the quantitative point of
view, the impact may be more significant.

Along with the relative (and sometimes impressive) success of the coarse-
graining attempts reported so far in the present manuscript, it is essential
to insist on the apparent failure that can be identified. In Fig. 4, we
reported strain fluctuations that overestimated the atomistic results by a
factor of about 1.5. This overestimating trend of the microscopic fluctuations
is all the more impressive when we focus on the internal stress field. In Fig.
7, we report the evolution of the standard deviation of the internal stress
fields for the ESL and GQ glasses and the associated mesoscopic models. While
the latter does not account for the stresses induced by the elastic disorder,
they overestimate the fluctuations by 2 to 3. The reasons behind this large
discrepancy are not entirely clear at this stage. The scalar character of the
model may induce an effect of ”constructive interference”, artificially
enhancing fluctuation that would be partly smeared out by an orientational
disorder of the weak slip directions (see Refs Patinet-CRP21 ; Patinet-Acta22
for discussion of the effect of full tensorial disorder). Another source of
discrepancy may be the overestimation of the local yield stress by the frozen
matrix method. In the same spirit, the mesoscopic model relies on a hypothesis
of continuum elasticity in the rearranging regions and disregards non-affine
elastic effects. The impact of these different points on internal stress
fluctuations certainly deserves further study.

##  VIII Acknowledgements

BT acknowledges support from the Romanian Ministry of Education and Research,
CNCS-UEFISCDI, project no. PN-III-P1-1.1-PD-2019-0236, within PNCDI III and
the Brandeis MRSEC NSF DMR-2011846. DV and SP acknowledge countless
discussions with S. Roux on mesoscopic models of amorphous plasticity and
their connection with atomistic simulations.

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