# Mechanical Behaviour of Glasses and Amorphous Materials

Anshul D. S. Parmar  School of Advanced Materials, Jawaharlal Nehru Center for
Advanced Scientific Research, Jakkur Campus, Bengaluru 560064, India.
Srikanth Sastry<EMAIL_ADDRESS>
Advances in the Chemistry and Physics of Materials: Overview of Selected
Topics, World Scientific (2020)

###### Abstract

A wide range of materials can exist in microscopically disordered solid forms,
referred to as amorphous solids or glasses. Such materials – oxide glasses and
metallic glasses, to polymer glasses, and soft solids such as colloidal
glasses, emulsions and granular packings – are useful as structural materials
in a variety of contexts. Their deformation and flow behaviour is relevant for
many others. Apart from fundamental questions associated with the formation of
these solids, comprehending their mechanical behaviour is thus of interest,
and of significance for their use as materials. In particular, the nature of
plasticity and yielding behaviour in amorphous solids has been actively
investigated. Different amorphous solids exhibit behaviour that is apparently
diverse and qualitatively different from those of crystalline materials. A
goal of recent investigations has been to comprehend the unifying
characteristics of amorphous plasticity and to understand the apparent
differences among them. We summarise some of the recent progress in this
direction. We focus on insights obtained from computer simulation studies, and
in particular those employing oscillatory shear deformation of model glasses.

##  I Introduction

Amorphous materials are ubiquitous in nature, and arise as among the common
solid forms of a wide range of substances. The most familiar are various forms
of silica glass, but chalcogenide glasses and amorphous metallic alloys or
metallic glassesSchuh  _et al._ (2007a) are other common examples of what are
some times termed hard glasses. At the other end of the spectrum are a variety
of soft materials, which superficially form a very distinct class, but share
some fundamental characteristics with hard glassesBonn  _et al._ (2017);
Nicolas  _et al._ (2018). Some examples are dense colloidal assemblies, gels,
emulsions, pastes and foams. These materials are typically studied for their
flow or rheological properties, but many of them are observed to possess a
yield stress below which they exhibit solid-like response and above which they
exhibit viscous flow. Other common materials that may deserve mention are
polymer glasses and granular materials. Many biological and geophysical
formations on relevant scales are well described as amorphous solids, and
their analysis is therefore informed by the understanding of amorphous solids
(statistics of earthquakes is a familiar and oft-quoted example).

Figure 1: Examples of amorphous solids and deformations. (a) Image of a crack
with atomic resolution in a planar $MoS_{2}$ sample Ly  _et al._ (2017). (b)
Bulk metallic glass pillars under uniaxial stress. The yield behaviour of a
composite glass reinforced with dendrites becomes more ductile Hofmann  _et
al._ (2008). (c) The plastic events in a slowly sheared colloidal system is
represented in terms of a change in the neighbours of colloids. The color map
and the size scale represent the number of nearest neighbors lost during a
plastic event. Schall  _et al._ (2007). (d) The strain localisation in a
granular pack of micrometer size glass beads under biaxial compression Le
Bouil  _et al._ (2014). (e) Localized rearrangements in a bubble raft. The
interface between neighboring bubbles exhibits a $T1$ event, triggered by
applied strain Biance  _et al._ (2009). (f) The stretching of a polymer melt.
At sufficiently high strain rates the polymer melt behaves in a way similar to
a solid Huang  _et al._ (2016). (g) Landslide hit highway in Malaysia (source:
China Press, The Straits Times). (h) The mechanical response to applied
deformation is illustrated to involve local rearrangements. Bocquet  _et al._
(2009).

A common feature of these substances is the lack of microscopic order, unlike
crystalline materials. In terms of their mechanical and flow behaviour, all
these materials exhibit solid-like behaviour, in that upon the application of
small external stresses, they respond elastically, with deformations being
proportional to the applied stresses, and reversible, so that they return to
the original state when the external stresses are removed. At larger applied
stresses, they display plastic deformation, which remain when the applied
stresses are removed. The point of onset of plasticity, the yield point, is
thus an important material characteristic, which precedes various forms of
failure of these materials. This broad and simplified description of
mechanical behaviour is superficially the same as that for crystalline
materials, and thus one must clarify what distinguishes amorphous materials
from crystalline solids. A key difference is the translational order present
in crystals, and the consequent possibility to have well defined structural
defects. In particular, the plastic response of crystals is described in terms
of the presence and movement of dislocation defects. In amorphous solids,
there is no translational order, and thus no well defined notion of defects
that are instrumental in plastic deformation. A related key distinction is
that since amorphous solids are typically obtained as preparation protocol
dependent, out of equilibrium, materials, the amorphous state is not unique
for a given substance. The preparation method plays a key role in their
observed behaviour, and a modification of the structure is often a key element
in the response to applied stresses (see Figure 1) Argon (1979); Schuh  _et
al._ (2007b); Schall and van Hecke (2010); Falk and Langer (2011); Barrat and
Lemaître (2011); Bonn  _et al._ (2017); Nicolas  _et al._ (2018).

Studies of mechanical behaviour of amorphous solids focus on these key
distinctions from the crystalline state, but it may be noted that the
description of crystalline solids on large scales shares some of these
features Sethna  _et al._ (2017). A second important general consideration
pertains to the notion of solidity of either crystalline or amorphous
materials. On general grounds, it has been argued that a solid at finite
temperature, under finite external stresses, will always deform and flow at
finite shear rates and the distinction between solids with a finite shear
modulus and viscoelastic fluids must be made with some care Sausset  _et al._
(2010); Yoshino and Mézard (2010). Thus, many attempts to understand yielding
as a well defined transition focus on the limit of zero temperature, as
discussed below. On the other hand, an understanding of behaviour in limiting
cases forms the basis for describing finite temperature and shear rate
behaviour Nath  _et al._ (2018).

Figure 2 shows an illustration of stress-strain curves of amorphous solids,
displaying some typical behaviours observed. In some materials, the dependence
of stress on the strain is gradual, and smoothly approaches an asymptotic
steady flow value. In other cases, one observes a stress overshoot, followed
by a drop, before approaching the steady flow value. It is typically observed
that in the cases where one observes a stress overshoot, yielding is
accompanied by the presence of strain localisation, i. e., the spatial
localisation of plastic strain within narrow regions of the sample
(illustrated in Figure 3 (a)). Such localisation of plastic strain is, for
example, observed in metallic glasses, and leads to brittle failure (Figure 3
(b)). The stress overshoot and the flow stress also depend on the rate at
which strain is applied. The yielding and flow behaviour can also be
represented in terms of steady state flow stresses as a function of strain
rates, as shown in Figure 2 (c). The steady flow stresses typically increase
with strain rates but in some cases, the flow curves are non-monotonic
(illustrated in Figure 2 (c)). The non-monotonic flow curves are observed to
correspond to the presence of shear bands within which the bulk of the flow
becomes localised (illustrated in Figure 3)Coussot and Ovarlez (2010). The
presence of strain localisation and shear banding represent a key feature of
yielding behaviour that requires explanation.

Figure 2: An illustrative image showing stress strain curves with and without
overshoots, and flow curves. (a) A sketch of the stress-strain curves for a
glass sheared at different strain rates. Larger rates
($\dot{\gamma}_{1}<\dot{\gamma}_{2}<\dot{\gamma}_{3}<\dot{\gamma}_{4}<...$)
lead to larger yield stresses and larger steady state stresses Falk and Langer
(2011). (b) A sketch of stress-strain curves for glasses prepared at different
cooling rates. The lower cooling rates
($\Gamma_{1}>\Gamma_{2}>\Gamma_{3}>\Gamma_{4}...$) result in better annealed
glasses. More annealed glasses show larger maximum stresses. (c) Steady-state
flow curves, i.e., dependence of the steady-state shear stress $\sigma_{ss}$
on the shear rate $\dot{\gamma}$ for a yield stress fluid. Homogeneous flow
corresponds to a monotonic flow curve, of the Herschel-Bulkley form. Herschel
and Bulkley (1926). A regime with a negative slope in the flow curve is
generally associated with instability and the presence of shear bands Coussot
and Ovarlez (2010). Figure 3: Illustrative images illustrating (left) strain
localization and (right) shear banding under flow. (left) The blue region
corresponds to lower strain and the red regime corresponds to localization of
strain. (right) The velocity profile in a shear banding system, compared with
uniform flow.

Thus, some of the key questions to address in discussing the mechanical
behaviour of amorphous solids pertains to the nature of elementary processes
of plasticity that occur in amorphous solids, the nature of interactions among
them, the nature of the yielding transition, the dependence of the mechanical
response on the preparation history and structural properties of the amorphous
solids, the nature of structural change during deformation, and the presence
or otherwise of strain localisation and shear banding. We address these
questions in this review, providing a summary of progress in answering them.
Only some selected themes are summarised, and many omitted, to permit an
adequate discussion of the topics chosen. Hence, the overview presented should
be treated more as a perspective than a comprehensive review. In Section 2, we
provide a general background and overview of theoretical and computational
approaches to develop a description of mechanical behaviour and yielding in
amorphous solids. We do not discuss relevant experimental results
specifically, but these are well summarised in various reviews we mention
Schuh  _et al._ (2007a); Bonn  _et al._ (2017); Nicolas  _et al._ (2018).
Section 3 discusses selected recent results, that address the nature of the
yielding transition, the influence of sample preparation and annealing, and
shear banding. Section 4 contains a summary of current status and outlook for
further investigations.

##  II Background and Previous Work

Figure 4: An illustration of plastic rearrangements at the particle scale:
Particle positions before and after a shear transformation event. The darker
area represents the region that deviated from the affine deformation field.
During this event, the indicated cluster of one large and three small
particles changes orientation. (source Falk and Langer (1998))

The nature of rearrangements leading to plastic deformation have been
investigated over several years, as a step in developing a description of
plasticity in amorphous solids. Early work by Argon and co-workers Argon
(1979) attempted to understand such rearrangements by considering disordered
bubble rafts as tractable model systems for glasses. These investigations
indicated that under shear deformation, plastic rearrangements involved the
participation of a small number of bubbles. These localised rearrangements
have been termed shear transformations, and have been subsequently observed in
numerous computer simulations and experiments Falk and Langer (1998); Maloney
and Lemaître (2006); Schuh  _et al._ (2007b); Chikkadi and Schall (2012);
Jensen  _et al._ (2014). A schematic view, obtained from computer simulations
of a two dimensional glass model Falk and Langer (1998) is shown in Figure 4.
Although one may envisage circumstances where plastic deformations may not be
so localised, and will not occur as isolated events (see Ref. Nicolas  _et
al._ (2018) for a recent review and detailed discussion), the idea of
localised shear transformations or shear transformation zones (STZ) underlies
many lines of modeling plasticity. Unlike dislocations in crystals, such STZs
do not have a well defined structural characterization, nor does one a priori
have the ability to identify locations where plastic events will occur.
Considerable effort has been directed therefore at arriving at ways of
identifying the locations of plastic events, based on an analysis of local
structure, energies, local yield stress, soft modes, etc. Shi  _et al._
(2007); Ding  _et al._ (2014); Manning and Liu (2011); Patinet  _et al._
(2016); Tyukodi  _et al._ (2016), including the deployment of machine learning
approaches Cubuk  _et al._ (2015, 2016). However, the possibility of
predicting with high reliability the locations of plastic events remains a
subject of ongoing exploration.

Figure 5: Comparison of simulation results and STZ theory. (a) Simulated
position (Y) dependent strain rates, as functions of the scaled position for
various strains, when the system is subjected to a global strain at a fixed
rate. The dark dashed line is the imposed strain rate. (b) Corresponding
estimates of the strain rates from the STZ theory. (c) Simulated potential
energy per atom as a function of position for various strains. (d) The STZ
prediction of effective temperature as a function of position. The dark dashed
lines in panels (c) and (d) show the initial values for the potential energy
and effective temperature. (source Manning  _et al._ (2007))

Based on the idea that STZs exist, may be created or destroyed, and can
undergo transitions between different states in response to applied shear,
Falk and Langer Falk and Langer (1998, 2011) proposed a description of
plasticity termed STZ theory. In this description, the population of STZs is
described to be governed by an effective or mechanical temperature $\chi$.
Based on a nonequilibrium thermodynamic formulation of the STZs Ashwin  _et
al._ (2013); Falk and Langer (2011), expressions for the evolution of the STZ
population, the rate of plastic deformation, and the effective temperature
itself $\chi$ (which evolves during the deformation towards a steady state
value governed by the shear rate) have been proposed. Yielding as a function
of applied stress arises as a dynamical instability beyond a critical applied
stress. In the original formulation, the STZ theory has no spatial dependence,
but it does so in an extended form where the mechanical temperature is treated
as varying as a function of space and time Manning  _et al._ (2007). These
analyses manage to capture many aspects of mechanical response in amorphous
solids. An example is shown in Figure 5, which compares computer simulation
results of the formation and spreading of shear bands with the predictions of
STZ theoryShi  _et al._ (2007); Manning  _et al._ (2007). In the mentioned
work, the local average potential energies from simulations are compared with
the effective temperatures. Although the notion of a mechanical temperature
and the comparison with energies are both reasonable and well motivated, there
is so far no concrete prescription for measuring this temperature. The other
key notion, of an STZ population, with distinct states of occupation, also
requires a concrete prescription to directly access it which is currently
missing, and has indeed motivated some of the research to identify locations
in a glass that are involved in plasticity Shi  _et al._ (2007); Ding  _et
al._ (2014); Manning and Liu (2011); Patinet  _et al._ (2016).

Figure 6: (a) Schematic representation of an Eshelby inclusion. (b) The
tangential non-affine displacement field during a plastic event from computer
simulations. (source Maloney and Lemaître (2006))

A more elaborate understanding of plastic events is provided by considering
the nature of the perturbation they cause to the surrounding solid.
Considering an idealisation of a point shear transformation, under other
simplifying assumptions, Picard et al Picard  _et al._ (2004) derived an
expression for the displacement field (correspondingly the strain field, and
assuming linear elasticity, the stress field) in two dimensions, which
exhibits a quadrupolar form:

$\sigma_{xy}({\bf r})\propto\int d{\bf r{{}^{\prime}}}G({\bf r}-{\bf
r{{}^{\prime}}})\epsilon^{pl}_{xy}({\bf r{{}^{\prime}}})$

where $\epsilon^{pl}_{xy}$ is the plastic shear strain at some point, which
causes an incremental stress $\sigma_{xy}$ elsewhere in the system, with the
propagator $G$ given by

$G(r,\theta)\propto{cos(4\theta)\over r^{2}}.$

This result carries over to three dimensions, with the propagator varying as
$1/r^{3}$. The analysis by Picard et al is related to earlier work by Eshelby
Eshelby (1957), on the elastic field caused by an inclusion in an elastic
medium that undergoes a change of form, illustrated schematically in Figure 6
(a). Signatures of non-affine displacements arising from a localised plastic
event have been investigated in a number of numerical and experimental studies
Falk and Langer (1998); Maloney and Lemaître (2006); Schuh  _et al._ (2007b);
Chikkadi and Schall (2012); Jensen  _et al._ (2014). An early example is shown
in Figure 6 (b). These analyses reveal that the effect of a localised plastic
event is long ranged, bearing a characteristic anisotropic form, which has
been found to be a valid description for amorphous solids.

Such propagation of stresses constitutes a mechanism for long ranged and
anisotropic interactions between past and prospective regions of plasticity,
and has consequences for theoretical descriptions and modeling. On the one
hand, some investigators have argued that the long range nature of the
interactions between plastic events implies that mean field descriptions
should have a high degree of validity for real systems Dahmen  _et al._
(2009). On the other hand, a class of models, termed elastoplastic models,
have been developed taking cognizance of the explicit nature of stress
propagation Nicolas  _et al._ (2018). We describe the general characteristics
of elastoplastic models and indicate some developments based on their study
first.

Elastoplastic models represent an amorphous solid as being composed of cells,
each of which may be thought of as representing a subvolume of the size of a
plastic event. Each cell has a stress value, and a local yield stress value
that is drawn from a distribution that forms part of the description of the
model. A binary variable represents whether a plastic event is triggered at
the cell or not. The stress at all sites increases monotonically with time
with the increase of strain (typically at a fixed strain rate). A plastic
event is triggered when the stress at a given site exceeds the local yield
stress at that site. The stress then is propagated to other sites according to
the propagator discussed earlier (or simpler ones) at a specified rate, and it
relaxes at the site of the plastic events. Many variations of this basic
strategy can be considered, including varying rates at which a plastic event
is triggered, inclusion of thermal activation, etc. The expected variation of
stress at a given site is shown in Figure 7. In addition to the fully explicit
models of the kind described, various works have also considered mean field
versions, such as the Hebraud-Lequeux model. This model describes the time
evolution of the distribution of local stresses, dependent on a parameter
$\alpha$ that controls stress diffusion, in proportion to the number of sites
that display plastic activity Hébraud and Lequeux (1998). One obtains a
transition from Newtonian fluid behaviour at low shear rates to yield stress
fluid behaviour obeying the Herschel-Bulkley constitutive law, as $\alpha$ is
varied. Elastoplastic models have also been employed to study avalanches,
correlated plastic events that occur in tandem, typically in the steady flow
regime Talamali  _et al._ (2012); Liu  _et al._ (2016). Avalanches in the
steady state are relevant if one is interested in the dynamics of plasticity
in that regimeSalerno and Robbins (2013) (for example, strain localisation,
which we discuss in the next section), and has been of interest in the context
of intermittent dynamics in earthquakes and other driven systems.

Figure 7: Schematic representation of the variation of local stress in an
elastoplastic model. For stresses larger than the threshold $\sigma_{Y}$, a
site yields with a probability $1/\tau_{pl}$ (plastic rate). The site is
active for a a duration given by the restructuring time $\tau_{el}$. (source
Martens  _et al._ (2012))

The nature of avalanches approaching yielding have also been addressed by many
studies, motivated by analogies with related systems. We mention two such
cases, which will be referred to in discussion of recent results in the next
section. Dahmen and co-workers Dahmen  _et al._ (2009) have proposed a model
for avalanches and yielding, motivated by ideas emerging from the study of the
movement of interfaces in the presence of pinning sites, that predicts
avalanches with a power law distribution with a cut off, with a universal
power law exponent of $-3/2$, and a mean avalanche size that diverges as the
yielding point is approached. While a variety of results have been compiled to
show the validity of this description Dahmen  _et al._ (2011); Uhl  _et al._
(2015), recent results described in the next section raise questions on some
aspects of such a description of yielding. Another model that has been used as
a reference is the random field Ising model (RFIM), in which, in addition to
the external field $H$, a random field is applied at each site. The strength
of the random field $R$ is a key control parameter, and in the $H$-$R$ plane,
the zero temperature phase diagram of this system exhibits a critical point at
a critical disorder $R_{c}$, and at lower $R$, a spinodal limit defines the
equivalent of the yielding point. Analysis in Ref. Nandi  _et al._ (2016)
indicates that at the transition point, the susceptibility (associated with
avalanche size) does not diverge, unlike the mean field case. The implications
of this picture to yielding in amorphous solids are discussed in the next
section.

Another perspective from which plasticity and yielding in glasses and
amorphous solids has been analysed is to focus on the energy landscape
properties of the deformed glasses. Such analysis has typically been conducted
in the athermal quasi-static limit (AQS), in which the solids are studied in
the zero temperature limit, subject to quasistatic strain, such that the
glasses always reside in energy minima Malandro and Lacks (1999); Lacks and
Osborne (2004); Karmakar  _et al._ (2010); Hentschel  _et al._ (2011);
Dasgupta  _et al._ (2012, 2013); Jaiswal  _et al._ (2016); Parisi  _et al._
(2017); Lerner  _et al._ (2018). Plastic rearrangements arise as a result of
the loss of stability of the energy minimum that the system resides in, which
constitutes a saddle-node bifurcation Karmakar  _et al._ (2010); Hentschel
_et al._ (2011). Such analyses have been extensively conducted through
computer simulations of atomic glasses and some results will be discussed in
the next section.

##  III Recent Investigations

A key question that has been addressed by many workers is the nature of the
avalanches and how they may behave as the yielding transition is approached.
We begin by a discussion of work in this regard, based on an energy landscape
perspective. Procaccia and co-workers Karmakar  _et al._ (2010); Lerner and
Procaccia (2009) have analysed the scaling of the stress drop $\Delta\sigma$
and energy drops $\Delta U$ during steady state avalanches, observing that
they both scale with N, with $\Delta\sigma\sim N^{\beta}$, $\Delta U\sim
N^{\alpha}$, with $\alpha-\beta=1$, and it has been found that for both two
and three dimensions, $\alpha=1/3$, and $\beta=-2/3$. On the other hand, it
was found that $\beta\approx-0.62$ for the first plastic event in a freshly
prepared glass (which is not universal but appears to depend on the glass
preparation), and is related to the behaviour at small values of the
distribution of distances to the next plastic event, $\Delta\gamma$, which is
generally expected to vary as $P(\Delta\gamma)\sim\Delta\gamma^{\eta}$.
Analysing the first plastic event of a freshly quenched glass, Karmakar et al
Karmakar  _et al._ (2010) argued (and showed) that the full distribution
agrees well with the Weibull distribution, leading to a relationship
$\beta=-1/(1+\eta)$. Hentschel et al Hentschel  _et al._ (2015) considered the
statistics of intervals to yield in finite strain windows below the yielding
point, and made the interesting observation that $\beta$ values, from
$\beta=-0.62$ at $\gamma=0$ to $\beta=-2/3$ for strain $\gamma$ above the
yield strain, vary non-monotonically, approaching lower values in between
(Figure 8 (a)). Based on the analysis of a Fokker-Planck equation for the
evolution of the Hessian eigenvalues, they argued that for finite strain,
$\eta=0$, and if the relation $\beta=-1/(1+\eta)$ remains valid in this
regime, one would expect $\beta=-1$, which would correspond to having
avalanches of unit size all the way to the yielding transition.

A similar analysis, on the basis of elasto-plastic models and scaling analysis
was performed by Jin et al Lin  _et al._ (2015); Lin and Wyart (2016), who
concluded that for a given value of $\eta$ ($\theta$ in their notation), the
mean avalanche size would scale as $<S>\sim N^{\theta/(1+\theta)}$. In this
case, exponent values are greater than zero and vary continuously with stress
(Figure 8 (b)), as in Ref. Hentschel  _et al._ (2015). The non-zero values of
the observed exponent are argued to imply system-spanning avalanches for all
stresses (strains) below yielding. Similar observations are made by Ozawa et
al Ozawa  _et al._ (2018) (discussed further below) albeit with the variation
that with annealing of the glasses, the exponent $\theta$ approaches zero in
the pre-yield regime. In attempting to find a reconciliation, Lerner et al
Lerner  _et al._ (2018) have suggested that system sizes may matter strongly
in accessing the relevant regime of the distributions of distances to local
yield, infeasible at the present.

Figure 8: (a) The exponent $\eta=-(1+\frac{1}{\beta})$ for different strain
windows from an AQS simulation of a two dimensional atomic glass for fast and
slowly quenched samples. The inset shows the theoretical expectation based on
the analysis in Hentschel  _et al._ (2015) (source Hentschel  _et al._
(2015)). (b) The exponent $\theta$ ($\eta$) for an elastoplastic model
exhibits similar behaviour and has been interpreted to suggest criticality of
avalanches at all strains below yielding. (source Lin and Wyart (2016)).

We next discuss briefly some developments in discussing the nature of the
yielding transition, before returning to the question of avalanches. A number
of recent studies have addressed the nature of the yielding transition, with
an emerging consensus that one must understand it as a spinodal limit. Based
on a mean field infinite dimensional description of the glass transition for
hard spheres, yielding has been described as a spinodal point Rainone  _et
al._ (2015); Urbani and Zamponi (2017), and subsequent simulation studies have
worked out some of the details of these calculations for finite dimensional
systems Jin  _et al._ (2018). Jaiswal et al Jaiswal  _et al._ (2016) explored
the idea that yielding was associated with a delocalisation of glass
configurations in configuration space. They considered an ensemble of
initially nearby configurations, and computed the distribution of overlaps
between them as a function of the imposed strain. This distribution is
initially peaked at large overlaps, but becomes bimodal as the strain is
increased. Yielding is identified as a first order transition, associated with
the low overlap peak of the distribution becoming larger in weight (Figure 9).
This formulation is familiar in replica theories of the glass transition, and
in Ref. Parisi  _et al._ (2017); Procaccia  _et al._ (2017), a replica
analysis of this problem is carried out, with a description of yielding as a
spinodal instability, and is associated with the formation of shear bands.
Application of random first order transition theory (RFOT) to the strength of
glasses and shear banding has also been discussed by Wolynes and co-workers
Wisitsorasak and Wolynes (2012, 2017). To understand the origins of shear
bands, Dasgupta et alDasgupta  _et al._ (2012, 2013) considered the energetics
of arrangements of Eshelby-like inclusions in two dimensions, and showed that
an organization of a large density of aligned inclusions along a line/plane
become lower energy solutions than an inclusion-free solid, discontinuously at
a yielding density. Such a calculation is informative, but does not provide a
full analysis of under what conditions a shear banding instability may be
observed in an amorphous solid. Shear bands form in some sheared glasses but
not all. The evidence for the presence of avalanches that grow in size, or are
system spanning, as yielding is approached is correspondingly varied. Some
investigations that address and clarify these points are discussed below.

Figure 9: (a) Averaged stress vs. strain shown along with the overlap function
$<Q_{12}>$ for the same of strain $\gamma$ (source Jaiswal  _et al._ (2016)).
(b) The averaged probability distribution function $P_{\gamma}(Q_{12})$ at
$\gamma_{y}=0.088$ . At the “yield” strain the distribution has two peaks with
equal weights.

Many investigations of mechanical response have been conducted employing
cyclic or oscillatory deformation of glasses, either under the AQS protocol,
or at finite rates and/or temperatures Lacks and Osborne (2004); Fiocco  _et
al._ (2013); Regev  _et al._ (2013); Priezjev (2013); Regev  _et al._ (2015);
Leishangthem  _et al._ (2017); Das  _et al._ (2018); Priezjev (2018); Parmar
_et al._ (2019). Under cyclic deformation, with the strain amplitude as the
control parameter, it is found that the glasses evolve in properties, but in
distinct ways above and below a threshold amplitude that is identified as the
yielding amplitude Fiocco  _et al._ (2013); Leishangthem  _et al._ (2017);
Parmar  _et al._ (2019). Glasses anneal (i.e., reach lower energy states) with
repeated cycles below the yielding strain, with the degree of annealing being
the highest at the yielding transition (Figure 10). Glasses with different
degrees of annealing behave very differently with uniform shear (poorly
annealed glasses show a monotonic increase in stress vs. strain, whereas
better annealed glasses show a stress overshoot), but considering the maximum
stress as a function of the strain amplitude $\gamma_{max}$, a very different
picture emerges. The stress strain curves show a discontinuous jump at the
yielding amplitude. The nature of avalanches below and above yielding are
qualitatively different. Avalanches below yielding are finite, and do not grow
upon approach to the yielding point, or with system size, whereas avalanches
above yielding are system spanning, and show the $N^{1/3}$ scaling previously
described. Thus, the conclusion from this work is that yielding occurs as a
discontinuous transition, and avalanche sizes below yielding show no
indication of growth upon approaching the yielding point, nor with system
size. The nature of the yielding transition in cyclic shear is strongly
influenced by the annealing of the glasses that is observed. Cyclic
deformation has been employed, in addition to studies of yielding, in
understanding transitions from reversible to irreversible behaviourPine  _et
al._ (2005); Corte  _et al._ (2008), and memory formation Keim  _et al._
(2019), which we do not discuss here, but details of which may be found in
Ref. Keim  _et al._ (2019).

Figure 10: (a) Averaged stress-strain curves for uniform shear and cyclic
shear for a 3D model glass (Kob-Andersen BMLJ), instantly quenched from the
liquid at two different temperatures, $T=1.00$ and $T=0.466$. Maximum stress
$\sigma_{max}$ versus $\gamma_{max}$ is shown for cyclic shear. The vertical
line at $g_{max}=0.08$ indicates the sharp yielding transition observed. (b)
Asymptotic energy per particle at $\gamma=0$ plotted against the corresponding
strain amplitude. Energies decrease with $\gamma_{max}$ until the yield strain
amplitude is reached ($\gamma_{max}\leq\gamma_{y}$), after which they increase
with $\gamma_{max}$. (c) Mean avalanche size versus system size $N$ shows no
significant system size dependence for $\gamma_{max}\leq\gamma_{y}$ but a
clear $N^{1/3}$ dependence above. The discontinuous change in behaviour marks
the yielding transition, seen at $\gamma_{max}=0.08$. (source Leishangthem
_et al._ (2017))

For amplitudes larger than the yielding amplitude, the cyclically sheared
glasses exhibit strain localisation, that is stable in characteristics for a
given amplitude (the width, for example) but some interesting features are
observed (Figure 11) – The energies of the particles within the shear band
become the highest possible (top of the landscape, as discussed in the context
of inherent structures (energy minima) in glass forming liquids Sastry  _et
al._ (1998); Angell (1998); Utz  _et al._ (2000)) while the energies outside
the shear band continue to decrease with increase of strain amplitude, and
display logarithmic relaxation in cycle number as also seen in liquid
simulations Parmar  _et al._ (2019); Das  _et al._ (2018) and for homogeneous
glasses below the yielding amplitude. These observations are relevant for our
discussion of shear banding and strain localisation below.

Figure 11: (a) A sample configuration of particles from the steady state, for
a cyclially sheared glass, at strain amplitude $\gamma_{max}=0.09$ above the
yield strain. The color map is based on the displacement of particles between
two consecutive cycles at zero strain. Highly mobile particles (Particles with
squared displacement $>0.2\sigma^{2}_{AA}$) are colored in red, whereas
particles in blue move considerably less. (b) Mean squared displacement
between two cycles is shown as a function of the coordinate $z$ in the shear
direction, along with a Gaussian fit. (c) The mean energy changes
discontinuously across the yielding transition. The subsystem outside the
shear band is annealed for amplitudes even beyond the yield amplitude, whereas
particles within the shear band have higher energies (source Parmar  _et al._
(2019)).

The implications of annealing on the nature of yielding has been considered by
Ozawa et al Ozawa  _et al._ (2018), who considered glasses that were annealed
to different degrees, obtained by equilibrating a model liquid over a wide
range of temperatures, and quenching them rapidly to zero temperature. They
find that for low enough initial temperatures (or degree of annealing), the
glasses exhibit yielding associated with a sudden drop in stress. Such sudden
stress jumps are associated with strong strain localisation. But as the degree
of annealing reduces, a more gradual transition with and without stress
overshoots is found (Figure 12). These authors interpret their results in
terms of the behaviour seen in the random field Ising model (providing
evidence for such a comparison), and argue that the spectrum of yielding
behaviours observed can be rationalised in terms of the zero temperature phase
diagram of the RFIM, with a random critical point separating brittle and
ductile yielding behaviours.

Figure 12: (a) The stress-strain curves for samples prepared at several
temperatures ($T_{ini}$), which determine their degree of annealing. The role
of annealing in the response is evident, with larger discontinuous jumps with
higher annealing, below a critical $T_{ini}$, above which the stress-strain
curves are continuous. (b & c) Snapshots of the system at the yield point for
cases with continuous (b) and discontinuous (c) stress-strain curves. The
color code is based on non-affine displacements between the initial
configuration and the configuration at the yield strain. (source Ozawa  _et
al._ (2018)).

Ozawa et al Ozawa  _et al._ (2018) also analyse a mean field elastoplastic
model, paying particular attention to the initial distribution of stresses
that captures the degree of annealing. In such a calculation also, the trend
observed with annealing above is recaptured. A very similar investigation is
also found in Ref. Popović  _et al._ (2018) who draw similar conclusions. It
is interesting that the mean field model in Dahmen  _et al._ (2009) with
weakening produces similar results. A somewhat intriguing feature of all these
analyses is that they capture, at the mean field level, behaviour that is
supposed to be associated with strong spatial correlations (strain
localisation), and is also expected to arise from a specific form of the
stress redistribution.

The presence of shear bands in flow or strain localisation have been studied
extensively in the past, both experimentally and in computer simulations, in
soft glassy materials and hard glasses, as described in Ref. Divoux  _et al._
(2010); Moorcroft  _et al._ (2011); Moorcroft and Fielding (2013); Fielding
(2014); Divoux  _et al._ (2016); Bonn  _et al._ (2017); Chaudhuri and Horbach
(2013); Shrivastav  _et al._ (2016); Vasisht  _et al._ (2017); Nicolas  _et
al._ (2018). In the context of hard glasses, strain localisation has been
observed and discussed in the context of yielding, as already mentioned
several times. Apart from the work mentioned above, there have been some
observations, starting from Shi  _et al._ (2007), that a correlation exists
with the annealing of glasses and strain localisation during yield. However, a
physical understanding of the connection is only now beginning to emerge. On
the other hand, one may also inquire as to whether the strain localisation
observed at yielding will persist or the shear band will grow and engulf the
system as the strain increases. In the rheological context of mainly soft
glasses, the persistence or otherwise of shear bands, and dependence on shear
rates and other parameters, have been analysed by a variety of investigations
Jagla (2007); Coussot and Ovarlez (2010); Jagla (2010); Martens  _et al._
(2012); Radhakrishnan and Fielding (2016); Shrivastav  _et al._ (2016); Alix-
Williams and Falk (2018); Vasisht  _et al._ (2017, 2018). The physical picture
of the theoretical analyses invoke some form of competition between rates of
aging, which is treated as a spontaneous process, and rejuvenation, which is
often associated with shearing. Following Coussot and Ovarlez Coussot and
Ovarlez (2010), Martens et al Martens  _et al._ (2012) analysed a related
mechanism, where the relaxation time taken by an activated region to relax
stresses plays a key role (refer to Figure 7). When this time scale
$\tau_{el}$ gets long enough, permanent shear bands arise. In a mean field
calculation, this can be seen in non-monotonic flow curves (Figure 13) and in
a simulation, in the form of localisation of shear. In the latter case, shear
banding is observed only when an Eshelby-type stress propagator is used but
not when a short range propagator is used. This latter observation seems to
support the idea that the nature of stress propagation is of key importance,
as in the analysis in Ref. Dasgupta  _et al._ (2012, 2013). But once again,
mean field or other abstract models Martens  _et al._ (2012); Radhakrishnan
and Fielding (2016) appear to capture shear banding even without explicit
consideration of the nature of stress propagation in space. Further, aging and
rejuvenation rates are not available when one considers AQS dynamics, and
where the effects of annealing appear in the rheological models is also not
clear. In the context of cyclic shear discussed above, even though performed
under AQS conditions, aging and rejuvenation effects are present and play a
role, but a key difference from some of the model studies mentioned is that
both aging and rejuvenation are induced by the shear deformation itself,
heterogeneously in space. It would be interesting to have theoretical analyses
that will encompass these scenarios.

Figure 13: (a) Flow curves from a mean field calculation, and (b) shear band
formation or otherwise in an elasto-plastic model simulation employing an
Eshelby-type vs short range propagator. (source Martens  _et al._ (2012))

##  IV Outlook

The discussion in the previous two sections clearly indicates that much
progress has been made in recent times in answering some of the key questions
related to yielding in amorphous solids, concerning the nature of elementary
processes, their interactions, the nature of the yielding transition, the
dependence of the mechanical response on the preparation history and
structural properties of the amorphous solids, and the presence or otherwise
of strain localisation and shear banding during yielding and flow. There is a
convergence of different approaches that at the outset are not very related
(e. g., elasto-plastic models vs. glass transition theory), and increasingly,
a convergence of what the nature of the yielding transition is. The degree of
annealing of glasses has been seen as a key factor, and for well annealed
glasses, a sudden, discontinuous yielding transition without precursors
appears to be the commonly observed behaviour. Annealing is expected to change
the structure, but whether one can structurally identify locations where
plastic deformation events can take place is an interesting subject of ongoing
investigation. A discontinuous yielding transition resulting from a spinodal
instability have been arrived at in a variety of ways. But the nature of the
spinodal (whether critical, not, and what kind of divergences may be expected,
if any) will continue to be investigated in the near future, but the questions
are sharper than they were before. There isn’t yet fully clarity or consensus
on the question of the nature of avalanches before yielding, in particular
whether they would be system spanning or not. The origins, nature and role of
aging and rejuvenation effects is another area where open questions remain to
be answered. As described, cyclic shear offers an interesting avenue to
investigate key aspects of yielding but current modeling approaches are
limited in this regard. For example, elastoplastic models are rule based
automata, which do not easily permit the notion of reversing the strain.
Energy or free energy based formulations, such as those of Jagla Jagla (2007,
2010, 2017) are very interesting in this context to explore as starting
points. At finite shear rates and finite temperatures, the location and nature
of yielding can change substantially (e. g. Parmar  _et al._ (2019)).
Systematically revisiting those features will be important after the athermal
limit is well understood.

After this survey of theoretical approaches, it may well be appropriate to end
by returning to practical matters, and ask if these ideas will in anyway “help
make better glass”, paraphrasing P. W. Anderson Anderson (1995). Indeed, a key
feature of the behaviour of well annealed glasses is brittle failure, which is
not a desirable feature, while other properties of such annealed glasses are.
Are there ideas for how such failure can be avoided? Based on the
understanding of the nature of the yielding instability, various investigators
have proposed the inclusion of pinning centers and micro-alloying as a way of
curtailing run away instabilities Hentschel  _et al._ (2016); Garrett  _et
al._ (2012); Tsai  _et al._ (2015); González (2016); Jiang  _et al._ (2014);
Bhowmik  _et al._ (2018, 2019) and thereby make annealed glasses more
resistant to brittle failure, and more ductile. More such insights may arise
from the development of a unified view of yielding and flow of amorphous
solids in the coming years.

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