# Brittle to ductile transitions in glasses: Roles of soft defects and loading
geometry

David Richard1<EMAIL_ADDRESS>Edan Lerner1 Corresponding author:
<EMAIL_ADDRESS>Eran Bouchbinder2 Corresponding author:
<EMAIL_ADDRESS>1Institute of Theoretical Physics, University
of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands
2Chemical and Biological Physics Department, Weizmann Institute of Science,
Rehovot 7610001, Israel

###### Abstract

Understanding the fracture toughness of glasses is of prime importance for
science and technology. We study it here using extensive atomistic simulations
in which the interaction potential, glass transition cooling rate and loading
geometry are systematically varied, mimicking a broad range of experimentally
accessible properties. Glasses’ nonequilibrium mechanical disorder is
quantified through $A_{\rm g}$, the dimensionless prefactor of the universal
spectrum of nonphononic excitations, which measures the abundance of soft
glassy defects that affect plastic deformability. We show that while a
brittle-to-ductile transition might be induced by reducing the cooling rate,
leading to a reduction in $A_{\rm g}$, iso-$\\!A_{\rm g}$ glasses are either
brittle or ductile depending on the degree of Poisson contraction under
unconstrained uniaxial tension. Eliminating Poisson contraction using
constrained tension reveals that iso-$\\!A_{\rm g}$ glasses feature similar
toughness, and that varying $A_{\rm g}$ under these conditions results in
significant toughness variation. Our results highlight the roles played by
both soft defects and loading geometry (which affects the activation of
defects) in the toughness of glasses.

##  I Introduction

Glasses are intrinsically nonequilibrium materials whose properties may vary
vastly with their thermomechanical history, e.g. the cooling rate at which
they are formed from a melt [1]. The nonequilibrium nature of glasses is
mesoscopically manifested through a broad range of disordered structures,
which qualitatively differ from the corresponding ordered structures featured
by their crystalline counterparts. Macroscopically, the nonequilibrium nature
of glasses may be manifested in a substantial variability in various material
properties, even for the very same composition.

One such important material property is the ability of a glass to resist
catastrophic failure. The latter is commonly quantified by the fracture
toughness, which measures the material resistance to failure in the presence
of an initial crack [2]. The value of the fracture toughness reflects
multitude of spatiotemporal processes that take place in the material prior to
catastrophic failure. In one limit, failure is accompanied by rather localized
plastic deformation (cf. Fig. 1a, right) and is generally quite abrupt. The
fracture toughness is typically small in this case, and failure is commonly
termed _brittle-like_. In the opposite limit, failure is accompanied by
significant spatially-extended plastic deformation (cf. Fig. 1a, left) and is
generally more gradual. The fracture toughness is typically larger in this
case, and failure is commonly termed _ductile-like_.

Figure 1: Computer glasses span a broad range of macroscopic and mesoscopic
material properties. (a) Snapshots of the non-affine plastic deformation
during fracture tests with an initial diamond crack for ductile-like (left)
and brittle-like (right) failure. The non-affine deformation is quantified by
the widely-used $D^{2}_{\rm min}$ field [3], here and throughout this paper,
where a darker color is indicative of more intense plastic deformation. (b)
The modified Lennard-Jones type potential $\varphi(r)$ used in this work (see
_Methods_). $\varphi$ is plotted vs. the normalized interparticle distance
$r/r_{\rm min}$, where $r_{\rm min}$ is the minimum of the potential. The
parameter $r_{\rm c}$ controls the strength and range of the attractive term.
(c) The shear modulus $\mu$ (reported in simulational units, see _Methods_)
vs. the quench rate $\dot{T}$ for various $r_{\rm c}$ values. The color code
for the latter is detailed in the legend, and will be adopted throughout the
paper. (d) Poisson’s ratio $\nu$ vs. $\dot{T}$. The meaning of the open orange
squares is explained in panel (f) below. (e) The vibrational density of states
(vDOS) of QLMs for $\dot{T}\\!=\\!0.05$ and two values of $r_{\rm c}$ (cf.
color code in panel (c)). The peak in the vDOS corresponds to the first
(phononic) shear wave. (f) The prefactor $A_{\rm g}$ of the universal
nonphononic vDOS $D(\omega)\\!=\\!A_{\rm g}\,\omega^{4}$, made dimensionless
through scaling by $\omega_{0}^{5}$, as a function of $\dot{T}$. A narrow band
of nearly equal $A_{\rm g}$ values is highlighted by a blurry orange region,
and the glasses within this narrow band are marked by open orange squares (the
same glasses are marked in panel (d)). (g) The sample-to-sample shear modulus
$\mu_{\rm s}$ distributions $P(\mu_{\rm s}/\mu)$ for the same glasses as in
(e). $\mu$ is the mean (plotted in panel (c)) and the width of one of the
distributions, indicative of the magnitude of shear modulus fluctuations, is
marked by $\Delta\mu$. (h) The dimensionless quantifier of shear modulus
fluctuations $\chi$ (see _Methods_ for exact definition) is plotted against
$\dot{T}$ for various $r_{\rm c}$’s. The blurry orange region marks iso-$\chi$
states (similarly to the iso-$\\!A_{\rm g}$ states in panel (f)). As stated
above, the same color code describing different values of $r_{\rm c}$ is
maintained throughout this paper.

Understanding the transition between these two modes of failure is an
important challenge in materials science [4, 5, 6, 7, 8, 9]. Establishing,
understanding and predicting structure-dynamics-properties relations in
glasses, e.g. in the context of material failure, involve various challenges.
First, one should develop tools to quantify the disordered structures featured
by glasses. Second, one should identify a certain sub-class of structural
degrees of freedom that is relevant for a given macroscopic material property.
Third, one should understand the dynamic evolution of these relevant degrees
of freedom under some prescribed conditions. Finally, one should be able to
coarse-grain over mesoscopic structural degrees of freedom in order to
understand how collective dynamics control the macroscopic material property.

Quantifying mesoscopic structural and mechanical disorder in glasses is
challenging. One aspect of the challenge is that we generally lack tools and
concepts to distinguish one disordered state from another, in sharp contrast
to well established “order parameters” in other condensed-matter systems [10].
Another aspect of the challenge is the associated length scales; glassy
disorder typically manifests itself on small length scales, usually
corresponding to a few atomic distances, which are not directly accessible to
currently available experimental techniques in molecular glasses.
Consequently, computer glasses — where such length scales are readily
accessible — have played important roles in our understanding of glassy
disorder [11, 12, 13, 14].

Recently, computer studies revealed and substantiated the existence of low-
frequency (soft) nonphononic vibrational modes in glasses [16, 17, 18, 19, 20,
15, 14, 21]. These soft vibrational modes (or excitations) have been shown to
be quasilocalized in space [16, 17, 22], as opposed to the spatially-extended
nature of low-frequency phonons (i.e. plane waves), hence they are termed
hereafter quasilocalized modes (QLMs). Moreover, QLMs have been shown to
follow a universal vibrational density of states (vDOS) ${\cal
D}(\omega)\\!\sim\\!\omega^{4}$, where $\omega$ is the angular vibrational
frequency [23, 24, 19, 20, 25, 15, 14, 26, 27, 28] (cf. examples in Fig. 1e).
As the $\omega^{4}$ law is universal — i.e. independent of the glass
composition, interatomic interactions and dimensionality — all of the glass
history dependence is encapsulated in the non-universal prefactor $A_{\rm g}$,
defined through ${\cal D}(\omega)\\!\equiv\\!A_{\rm g}\,\omega^{4}$. $A_{\rm
g}$, once properly non-dimensionalized (see below), has been shown to provide
a measure of the number density of QLMs in a glass [15, 14, 30, 26, 29].

QLMs, in turn, have been shown to correlate with the spatial loci of
irreversible rearrangements in glasses, i.e. they statistically correlate with
Shear-Transformation-Zones (STZs), the carriers of plasticity in glassy
materials [31, 32, 33, 18, 34, 35, 36, 21]. As such, the properly non-
dimensionalized $A_{\rm g}$ is directly relevant for plastic deformability,
and consequently to the fracture toughness, as it provides a measure of the
number of soft defects embedded inside glassy structures [15, 14, 30, 29].
Another measure of mesocopic mechanical disorder in glasses can be defined
using the fluctuations of the shear modulus $\mu$ [37, 38, 39, 29, 7]. This
measure, denoted hereafter by $\chi$ (see _Methods_ and Fig. 1g), has been
recently shown to control the rate of wave attenuation in computer glasses
[41, 42, 39], i.e. the amplitude of Rayleigh scattering rates in the low-
frequency (long-wavelength) limit. In this paper, we will be using $A_{\rm g}$
and $\chi$ as quantifiers of mesoscopic glassy disorder, which are directly
relevant to the fracture toughness.

$A_{\rm g}$ and $\chi$ allow one to compare on equal footing different
glasses, either glasses of the same interatomic interactions/composition
formed by different thermal protocols or glasses of different interatomic
interactions/composition. While existing evidence clearly indicates that
$A_{\rm g}$ and $\chi$ must play important roles in determining the fracture
toughness of these materials, one obviously cannot exclude the possibility
that other physical quantities and factors play a role as well. Indeed, some
experimental and computational studies [43, 4, 44, 45, 5, 6, 46] suggested
that Poisson’s ratio $\nu$ plays an important role in determining the fracture
toughness of glasses (though some other studies challenged this view [47, 29,
48]). A possible interpretation is that $\nu$ is sensitive to the structure of
a glass, and in particular that it may be indirectly related to plastic
deformability and its sensitivity to shear versus tensile deformation [49, 6],
and hence affects the toughness. As such, one can speculate that the observed
effect of $\nu$ is not qualitatively different from that of $A_{\rm g}$ and
$\chi$ — indeed we show below that well-defined relations between $\nu$ and
$A_{\rm g}$ (and $\chi$) exist. It remains unclear, however, whether $\nu$ —
or more precisely Poisson contraction/effect and other loading geometry
effects — also play distinct roles in determining the toughness.

Our goal in this paper is to understand the roles played by soft defects, as
quantified by $A_{\rm g}$ and indirectly by $\chi$, and by the loading
geometry employed in mechanical tests on the fracture toughness of glasses, in
particular on brittle-to-ductile transitions. This goal is achieved by
employing extensive computer glass simulations in 3D, which offer a powerful
and flexible platform to address the posed questions. To mimic glasses’
compositional variability we employ computer models based on Lennard-Jones
interactomic pairwise potentials with a _tunable_ parameter $r_{\rm c}$ (to be
accurately defined below), shown recently to give rise to glasses of widely
variable properties [50, 29, 51, 52]. We also apply a broad range of cooling
rates $\dot{T}$ across the glass transition (to be accurately defined below).
In addition, we employ cutting-edge algorithms that allow to generate deeply
supercooled glasses [7], to a degree that is comparable or even surpasses that
of laboratory glasses. Overall, the range of variability of the macroscopic
properties (e.g. $\mu$ and $\nu$, cf. Fig. 1c-d) of our computer glasses is
comparable to that observed in laboratory glasses. Finally, we carefully and
systematically vary the imposed loading geometry in the fracture toughness
tests in order to quantify its effect on the toughness.

We find that the fracture toughness, and in particular brittle-to-ductile
transitions, are controlled by _both_ the abundance of soft defects — as
quantified by $A_{\rm g}$ (and indirectly by $\chi$) — and by the loading
geometry of the fracture test. The latter affects the relative magnitude of
shear and tensile deformation experienced by the material, and consequently
the emerging plastic dissipation, for a fixed $A_{\rm g}$. That is, the
loading geometry controls the activation of soft defects, whose abundance is
controlled by $A_{\rm g}$. Only under a certain choice of loading geometry,
where Poisson contraction can take place, Poisson’s ratio $\nu$ can be
sensibly used to quantify the fracture toughness _together_ with $A_{\rm g}$
(or $\chi$). These results provide basic insights into the physical origin of
the failure resistance of glasses.

##  II Results

In order to study the origin of brittle-to-ductile transitions in glasses,
such as the one illustrated in Fig. 1a, we first aim at generating computer
glasses featuring a broad range of properties. This is achieved by employing
the computer glass model put forward in [50], where particles interact via a
tunable Lennard-Jones-like pairwise potential (a similar approach was taken in
[53]). In particular, the strength and range of the attractive term of the
pairwise potential are controlled by a parameter $r_{\rm c}$, as shown in Fig.
1b and accurately defined in _Methods_. Varying $r_{\rm c}$, which may
qualitatively correspond to varying the glass composition, results in dramatic
changes in emergent material properties, as will be discussed and demonstrated
soon (see also [29, 51]). In addition, we generate different glasses by
quenching equilibrium liquids at a broad range of rates into their arrested
glass phase. The cooling rate is quantified by $\dot{T}$, the dimensionless
absolute value of the rate of change of temperature during the quench (see
_Methods_ for precise definition). Yet another glass formation protocol, which
is not quantified by $\dot{T}$, is employed and discussed below.

In Fig. 1c, we present the shear modulus $\mu$ (in simulation units, see
_Methods_) as a function of $\dot{T}$ for various values of $r_{\rm c}$. The
corresponding results for Poisson’s ratio $\nu$ are shown in Fig. 1d. The
range of variation in $\nu$ accessed by tuning $r_{\rm c}$ compares well with
the one for which the so-called ductile-to-brittle transition is observed in
bulk metallic glasses [54], and see _Supplementary material_ for further
comparisons between our model’s elastic properties and typical laboratory
glasses’ elastic properties. While our model’s macroscopic linear response
coefficients make direct contact with laboratory glasses, computer glasses
offer unique access to various physical quantities that are not directly
accessible to experiments, as highlighted above. In Fig. 1e, we present the
vDOS of QLMs for two values of $r_{\rm c}$ and a fixed $\dot{T}$, both
revealing the universal ${\cal D}(\omega)\\!=\\!A_{\rm g}\,\omega^{4}$ law.
Here and elsewhere, we report frequencies in terms of
$\omega_{0}\\!\equiv\\!c_{s}/a_{0}$, where $c_{s}$ is the (zero frequency)
shear wave speed and $a_{0}\\!\equiv\\!(V/N)^{1/3}$ ($V$ is the system’s
volume containing $N$ particles). The observed prefactor $A_{\rm g}$ vastly
varies between the two cases. $A_{\rm g}$, made dimensionless through
multiplication by $\omega_{0}^{5}$, is presented in Fig. 1f for all $\dot{T}$
and $r_{\rm c}$ values used in Fig. 1c-d. The dimensionless $A_{\rm g}$
(hereafter we exclusively refer to the dimensionless one) indeed reveals large
variability. Interestingly, different combinations of $\dot{T}$ and $r_{\rm
c}$ result in nearly identical $A_{\rm g}$ values (marked by open orange
squares), a property that will be used below.

Figure 2: A ductile-to-brittle transition induced by varying the cooling rate
through the glass transition. (a) Tensile stress-strain curves (unconstrained
uniaxial tension) of glasses with $r_{\rm c}\\!=\\!1.3$, prepared at various
quench rates $\dot{T}$ (see legend). (b) The corresponding non-affine plastic
deformation computed at $\epsilon\\!=\\!0.15$.

$A_{\rm g}$, providing a measure of the abundance of soft defects inside a
glass [15], serves as a disorder quantifier that is relevant for plastic
deformability and fracture toughness. Another quantifier of mechanical
disorder in glasses can be obtained by considering the distribution of the
local shear modulus, probed by studying a large number of systems of size $N$
[39, 52]. In Fig. 1h we present the shear modulus distributions corresponding
to the two case shown in Fig. 1e. It is observed that smaller $A_{\rm g}$
glasses feature reduced shear modulus fluctuations $\Delta\mu/\mu$. The latter
can be used to construct an $N$-independent quantifier of mechanical disorder
in the form $\chi\\!\equiv\\!(\Delta\mu/\mu)\sqrt{N}$ [41, 42, 52, 39]. The
dependence of $\chi$ on both $r_{\rm c}$ and $\dot{T}$ is presented Fig. 1g,
and the relation between $\chi$ and $A_{\rm g}$ is further discussed below
(cf. Fig. 3d and _Supplementary material_). Next, we start considering the
fracture toughness of the glasses at hand.

###  II.1 Reduction in cooling rate can induce a ductile-to-brittle transition

In order to probe the fracture toughness we consider 3D glass samples
containing an initial central crack of fixed geometry that cuts through the
sample. In particular, we choose a diamond-shaped crack of fixed length and
main vertex angle, which give rise to traction-free crack surfaces. The main
vertex angle is chosen to be sufficiently small in order to reproduce the
classical square root singularity of linear elastic fracture mechanics [2],
see _Supplementary material_ for additional details. The initial crack length
is chosen to be sufficiently larger than the particle size and sufficiently
smaller than the sample’s dimensions in order to ensure reasonable scale
separation, though we cannot entirely exclude any finite size effects (see
_Supplementary material_). All fracture toughness calculations are performed
under quasi-static athermal conditions, i.e. vanishing applied strain-rate and
zero temperature, as our focus is on the effect of glass structure and loading
geometry on the toughness.

Figure 3: The fracture toughness depends on both the number of soft defects,
quantified by $A_{\rm g}$, and on Poisson contraction, quantified by $\nu$,
under unconstrained uniaxial tension. (a) Snapshots of the non-affine plastic
deformation at $15\%$ strain in the $r_{\rm c}\\!-\\!\dot{T}$ parametric
plane. (b) Strain-strain curve for a poorly-annealed (high $\dot{T}$, thin
curves) and well-annealed (low $\dot{T}$, thick curves) glasses with various
$r_{c}$. (c) $A_{\rm g}$ vs. $\nu$, where ductile-like glasses correspond to
full symbols and brittle-like to empty ones. The orange squares and the blurry
region indicate glasses with a nearly identical density of OLMs
(iso-$\\!A_{\rm g}$). (d) $A_{\rm g}$ vs. $\chi$, roughly following a power-
law relation $A_{\rm g}\\!\sim\\!\chi^{10/3}$ for all glasses (see discussion
in the _Supplementary material_).

We first consider the effect of varying the cooling rate $\dot{T}$ on the
glass resistance to failure under unconstrained uniaxial tension, where
‘unconstrained’ means that the lateral edges (perpendicular to the tensile
axis) are traction-free and hence free to contract. We focus here on the
$r_{\rm c}\\!=\\!1.3$ glasses ensemble, and plot in Fig. 2a the tensile
stress-strain curves for four different cooling rates $\dot{T}$ (see legend).
It is observed that for the two largest $\dot{T}$ values, the corresponding
glasses feature gradual stress relaxation after reaching a peak stress, with
no clear sign of catastrophic failure. For the two smallest $\dot{T}$ values,
however, the glass appears to lose its load-bearing capacity at a well-defined
strain level, indicating catastrophic failure. Consequently, there appears to
exist a cooling rate that induces a ductile-to-brittle transition, as was also
demonstrated in previous work [55, 56, 9, 57]. This is corroborated by the
spatiotemporal dynamics of the deformed samples, as demonstrated in Fig. 2b,
where snapshots at $15\%$ strain are shown. Consistent with the stress-strain
curves, we observe that while glasses formed under the two highest cooling
rates feature a blunted crack accompanied by large scale plastic deformation
(darker regions, see figure caption), glasses cooled at the two lowest rates
feature a crack that propagates through the samples, accompanied by rather
localized plastic deformation.

It is known that varying the cooling rate $\dot{T}$ leads to varying glass
structures and consequently to variation in many material properties of a
glass [58, 59, 14]. Our next goal is to understand which physical properties
control the ductile-to-brittle transition observed in Fig. 2, when $\dot{T}$
is varied.

###  II.2 Do soft defects exclusively control the ductile-to-brittle
transition? The role of Poisson contraction under unconstrained uniaxial
tension

As explained in the Introduction, $A_{\rm g}$ — whose $\dot{T}$ dependence is
presented in Fig. 1f — is a natural candidate for controlling the observed
ductile-to-brittle transition. To test this natural expectation, we cannot
restrict ourselves to a single $r_{\rm c}$ value; rather, we employ glasses
cooled at different quench rates $\dot{T}$ _and_ of different interaction
parameters $r_{\rm c}$ that nevertheless share nearly the _same_
(dimensionless) $A_{\rm g}$. These glasses, which are marked by open orange
squares in Fig. 1f, are expected to feature similar resistance to failure — if
indeed $A_{\rm g}$ fully controls the fracture toughness of glasses. In Fig.
3a we plot $15\%$-strain snapshots in the $r_{\rm c}\\!-\\!\dot{T}$ parametric
plane (covering the whole range of employed parameters) — similarly to Fig. 2b
(in fact, the latter is reproduced here in the third row from the top) —,
where the nearly iso-$\\!A_{\rm g}$ are highlighted by open orange rectangles.
It is observed that iso-$\\!A_{\rm g}$ glasses are either brittle-like or
ductile-like, indicating that $\\!A_{\rm g}$ does not exclusively control the
fracture toughness under the present conditions (i.e. under unconstrained
uniaxial tension). This conclusion is corroborated by the stress-strain curves
presented in Fig. 3b.

Figure 4: Brittle-ductile phase diagram in the $\chi$-$\nu$ plane under
unconstrained uniaxial tension. (a) $\chi$ vs. the stiffness parameter
$k_{\lambda}$ of FSP glasses. (b) The corresponding $\nu(k_{\lambda})$. (c)
Unconstrained uniaxial tension stress-strain curves for glasses with $r_{\rm
c}=1.5$, prepared with different stiffness $k_{\lambda}$ as indicated in the
legend. Snapshots show the non-affine deformation at $15\%$ strain. See text
for additional discussion. (d) Ductile-brittle phase diagram in the
$\chi\\!-\\!\nu$ plane for FSP glasses, where ductile-like glasses correspond
to full symbols and brittle-like to empty ones. (inset) the same, but for
glasses prepared via a conventional MD quench. The results presented in the
inset correspond to those already presented in Figs. 3c-d.

What additional physical quantities/factors/processes, which also vary with
$\dot{T}$, play a role in ductile-to-brittle transitions in glasses? These,
once identified, should allow to differentiate between the qualitatively
different behaviors observed among the iso-$\\!A_{\rm g}$ glasses, highlighted
by open orange rectangles in Fig. 3a. As explained in the Introduction, there
exists some evidence [43, 4, 44, 45, 5, 6, 46] that Poisson’s ratio $\nu$
might affect the fracture toughness of glasses. The $\dot{T}$ dependence of
$\nu$ for our glasses is presented in Fig. 1d, clearly indicating that
iso-$\\!A_{\rm g}$ glasses feature different $\nu$’s (marked by open orange
squares). Consequently, we present in Fig. 3c a brittle-ductile phase diagram
in the $A_{\rm g}\\!-\\!\nu$ plane, where ductile-like glasses correspond to
full symbols and brittle-like to empty ones (we include data for all of our
glasses, across the full range of $\dot{T}$ and $r_{\rm c}$ values employed,
as in Fig. 1). It is observed that iso-$\\!A_{\rm g}$ glasses of relatively
high $\nu$ are ductile-like, while those with smaller $\nu$ are brittle-like.
This indicates that $\nu$ might indeed plays a role in the fracture toughness
of glasses.

What is the physical (and causal) effect of $\nu$ on the fracture toughness?
Before setting out to address this important question, we aim at further
substantiating the claim that both glassy disorder — as quantified by either
$A_{\rm g}$ or $\chi$ — and some effect that is related to $\nu$ play a role
in determining the fracture toughness of glasses. The results presented in
Fig. 3c clearly support this claim, yet for relatively large values of $r_{\rm
c}$ it is impossible to reach sufficiently small $A_{\rm g}$ levels to probe
the brittle regime using conventional Molecular Dynamics (in which glasses are
formed by quenching their corresponding liquids). Consequently, we employ
below recently introduced computer algorithms that allow the generation of
glassy solids featuring $A_{\rm g}\\!\to\\!0$, qualitatively representing
extreme supercooling conditions.

To this aim, we first note that as shown in Fig. 3d, glasses form by quenching
a liquid feature a clear relation between $A_{\rm g}$ and $\chi$ (see also
_Supplementary material_). The existence of such a relation, which appears to
be weakly dependent on $r_{\rm c}$, suggests that for these glasses one can
approximately use $A_{\rm g}$ and $\chi$ interchangeably. This is not the case
for systems featuring $A_{\rm g}\\!\to\\!0$. The latter can be generated by a
protocol that is referred to in what follows as the _FSP algorithm_ ,
described in detail in _Methods_ and in [7]; forming a computer glass using
the FSP algorithm amounts to minimizing an augmented potential parameterized
by a stiffness $k_{\lambda}$ that controls the mechanical noise of the
resulting glasses (while $r_{\rm c}$ can still be varied, see _Methods_), with
lower-$k_{\lambda}$ FSP glasses featuring less mechanical fluctuations [7]. As
is now established, FSP glasses can feature a gap in their nonphononic vDOS
[7] for sufficiently small $k_{\lambda}$, and in particular in this limit they
do not feature the $\omega^{4}$ law. That is, lower-$k_{\lambda}$ FSP glasses
essentially feature $A_{\rm g}\\!\to\\!0$. Yet, these glassy solids still
exhibit a finite and well-defined $\chi$, as demonstrated in Fig. 4a.
Moreover, $\nu$ also varies systematically with $k_{\lambda}$ and $r_{\rm c}$,
as shown in Fig. 4b.

The merit of FSP glasses in the present context is evident from panels (a) and
(b) of Fig. 4. That is, we have at hand glasses with $A_{\rm g}\\!\to\\!0$ (in
the low $k_{\lambda}$ regime), whose $\chi$ and $\nu$ can be systematically
varied. Consequently, FSP glasses are most suitable for exploring the ductile-
to-brittle transition in terms of both $\chi$ and $\nu$, providing access to
broader region of the brittle-like phase. In Fig. 4c, we present the stress-
strain curves of FSP glasses with $r_{\rm c}\\!=\\!1.5$ for three value of
$k_{\lambda}$, together with spatial snapshots of each glass at a late stage
in the loading process. For the largest $k_{\lambda}$ value
($k_{\lambda}\\!=\\!1000$), the behavior is ductile-like, exhibiting extensive
(system covering) plastic deformation and steady-state flow, essentially being
insensitive to the initial crack. For the smallest $k_{\lambda}$ value
($k_{\lambda}\\!=\\!50$), we enter into the brittle-like regime, where
catastrophic failure takes place, accompanied by localized plastic deformation
near the crack (which is perpendicular to the tensile axis) and an abrupt drop
in the stress when the sample loses its load-bearing capacity. Interestingly,
for the intermediate $k_{\lambda}$ value ($k_{\lambda}\\!=\\!100$), the
initial crack started growing along an inclined shear-band (resulting in a
finite abrupt stress drop), but eventually plastic deformation in the shear-
band took over (resulting in a gradual stress relaxation towards a plateau),
with no catastrophic failure.

Figure 5: Different loading geometries and their effect on the imposed
deformation. (a) An illustration of a 2D projection of a 3D system (top),
where the grey square of dimensions $L_{0}\\!\times\\!L_{0}$ represents the
undeformed state. Under unconstrained uniaxial tension loading (applied strain
$\epsilon_{\|}$ in the vertical direction), the system is free to contract in
the two transverse/lateral directions (the out-of-plane direction is not
shown), resulting in strain of magnitude $\epsilon_{\perp}\\!<\\!0$, see the
empty rectangle (illustrating the deformed state). In this loading geometry,
one has $R\\!=\\!-\epsilon_{\perp}/\epsilon_{\|}\\!=\\!\nu\\!\leq\\!1/2$, i.e.
the relative importance of tensile and shear deformation is determined in this
loading geometry by Poisson’s ratio $\nu$. (b) The same as panel (a), but for
constrained tension loading. In this loading geometry, one has
$\epsilon_{\perp}\\!=\\!0$, resulting in
$R\\!=\\!-\epsilon_{\perp}/\epsilon_{\|}\\!=\\!0$ (see empty rectangle, and
note that the grey $L_{0}\\!\times\\!L_{0}$ square is the same as in panel
(a)). (c) The same as panels (a) and (b), but for pure shear loading. In this
loading geometry, one has $\epsilon_{\perp}\\!=\\!-1/2\epsilon_{\|}$,
resulting in $R\\!=\\!-\epsilon_{\perp}/\epsilon_{\|}\\!=\\!1/2$ (see empty
rectangle, and note that again the grey $L_{0}\\!\times\\!L_{0}$ square is the
same as in panels (a) and (b)). At the bottom of each panel we present a pair
of $\epsilon_{\|}\\!=\\!0.15$ snapshots, with $\nu\\!=\\!0.367$ (red, left)
and $\nu\\!=\\!0.275$ (blue, right), deformed under the stated $R$ value. It
is observed that the smaller $R$, the more localized plastic deformation is,
and correspondingly the more brittle-like failure is.

We performed an extensive analysis of all FSP glasses at hand (the four
$r_{\rm c}$ values studied earlier and a broad range of $k_{\lambda}$ values)
and present the results in Fig. 4d in a phase-diagram in the $\chi$-$\nu$
plane, where a ductile-like behavior corresponds to full squares (color code
as in Fig. 1) and a brittle-like behavior corresponds to empty squares. First,
it is observed that indeed FSP glasses allow to explore a broader region of
the brittle-like phase. Second, and most importantly, it is observed that both
$\chi$ and $\nu$ (more precisely some effect that is related to $\nu$) play a
role in the ductile-to-brittle transition, as is evident from the curved phase
boundary (a blurry region is added as a guide to the eye).

Finally, we add in the inset the results for the glasses formed by quenching a
liquid — presented earlier in Fig. 3 (but not in the $\chi$-$\nu$ plane).
Interestingly, despite the widely different glass formation protocols
involved, the two classes of glasses appear to reveal a qualitatively similar
ductile-brittle phase diagram in the $\chi$-$\nu$ plane, where both decreasing
$\chi$ and $\nu$ promote a brittle-like behavior. It is important to note
again that the results presented in Fig. 4d were obtained for our choice of
initial crack of fixed geometry and dimensions (see also _Supplementary
material_); varying the latter may change quantitative aspects of the results,
but the combined effect of $\chi$ and $\nu$ on the fracture toughness under
unconstrained uniaxial tension conditions is general. Once this is
established, we focus our attention on the question posed above regarding the
physical (and causal) effect of $\nu$ on the fracture toughness, which is
addressed next.

###  II.3 The interplay of loading geometry and plastic deformability:
Constrained tension and soft defects again

Poisson’s ratio $\nu$ is a linear elastic (reversible) response coefficient
that quantifies the relative transverse contraction of a material under
unconstrained (transversely-free) uniaxial tension. In assessing the relevance
of $\nu$ to the fracture toughness of glasses, one can a priori distinguish
two qualitatively different classes of physical effects. First, $\nu$ can be
viewed as a directly and easily measurable quantity that allows to indirectly
probe nonlinear irreversible response coefficients that are relevant for
plastic deformability and toughness. Indeed, $\nu$ is expected to vary when
the material structure and interactions vary, and in particular it exhibits
systematic variations with $A_{\rm g}$ and $\chi$, as demonstrated in Figs.
3-4. Second, $\nu$ — as a linear elastic coefficient — may affect the geometry
of deformation glasses experience under external driving forces. In
particular, in the unconstrained uniaxial tension tests exclusively considered
up to now, $\nu$ controls the degree of Poisson contraction, which in turn
determines the relative magnitude of tensile and shear deformation. That is,
$\nu$ in this loading geometry will affect the activation of soft defects
(whose number is controlled by $A_{\rm g}$), which are generally more
sensitive to shear deformation.

As physically relevant disorder measures such as $A_{\rm g}$ and $\chi$ have
been identified and quantified, we hypothesize that a linear elastic response
coefficient such as $\nu$ does not contain _additional_ physical information
about plastic deformability, on top of $A_{\rm g}$ and/or $\chi$.
Consequently, while in the absence of well-founded quantifiers of glassy
disorder one may use $\nu$ as some rough proxy of disorder, when these are at
hand and taken into account — as is the case here — the effect of $\nu$ in the
phase diagram of Fig. 4d strongly points towards the second possibility
discussed above. That is, one needs to understand the role of the
deformation/loading geometry on the fracture toughness, both in relation to
$\nu$ and in more general terms.

To address the latter, we introduce here the biaxiality ratio $R$ (cf. Fig.
5), defined as $R\\!\equiv\\!-\epsilon_{\perp}/\epsilon_{\|}$, where
$\epsilon_{\|}$ is the tensile/extensional strain and $\epsilon_{\perp}$
corresponds to the transverse strains, i.e. the strain components in the
directions perpendicular to the applied tension axis. The main utility of $R$
is that it quantifies the relative importance of tensile and shear deformation
imposed by the loading geometry, and since soft defects (STZs) are most
sensitive to shear deformation [3], $R$ offers a simple measure of the
fraction of soft defects being activated under a certain loading geometry
(while their number is controlled by $A_{\rm g}$). Since, in principle,
$\epsilon_{\|}$ and $\epsilon_{\perp}$ can be independently controlled, $R$
can take any value (and can even evolve during a mechanical test). Yet, we
focus here on a few cases that are most relevant to our discussion. For pure
shear deformation, defined by the absence of linear dilatational strain
($\epsilon_{\|}+2\epsilon_{\perp}\\!=\\!0$), we have $R\\!=\\!1/2$. As plastic
deformation in glasses is most sensitive to shearing [3], this value of $R$ is
expected to facilitate a more plastic response, see Fig. 5c. For the
unconstrained uniaxial tension loading geometry considered up to now, we have
$R\\!=\\!\nu\\!\leq\\!1/2$; that is, $R$ in this loading geometry is
determined by a material property, quantified by $\nu$. Therefore, we see that
reducing $\nu$ in the unconstrained loading configuration tends to increase
the relative importance of tensile deformation (cf. Fig. 5a), which may
facilitate more brittle-like behavior, in qualitative agreement with the
results of Fig. 4d.

As $R$ is in general not a material property, but can rather be controlled
through the imposed loading geometry, one can use it to disentangle the roles
of $A_{\rm g}$ (or $\chi$) — i.e. the role of soft defects — and of the
loading geometry on the fracture toughness. To that aim, we performed
constrained tension tests, where the transverse/lateral boundaries
(perpendicular to the main tensile axis) are held fixed, i.e.
$\epsilon_{\perp}\\!=\\!0$. This case corresponds to $R\\!=\\!0$, see Fig. 5b
and _Methods_ for additional discussion. The loading geometry corresponding to
$R\\!=\\!0$ allows to eliminate the role of $\nu$ as a linear response
coefficient that controls Poisson contraction. Consequently, we expect the
fracture toughness in this case to be determined by $A_{\rm g}$. Since for
$R\\!=\\!0$, which obviously favors tensile deformation, catastrophic failure
is unavoidable, $A_{\rm g}$ is expected to affect the value of the fracture
toughness, not strictly a brittle-to-ductile transition. In particular, we
expect iso-$\\!A_{\rm g}$ glasses to feature very similar toughness under
constrained tension, despite having different $\nu$’s, and the toughness to
systematically increase with $A_{\rm g}$ for the same glass cooled at
different rates $\dot{T}$.

Figure 6: Disentangling the effect of soft defects and loading geometry on the
toughness. (a) Stress-strain curves under constrained tension, for glasses
prepared at a high quench rate $\dot{T}\\!=\\!0.02$ and different $r_{\rm
c}$’s. See text and _Methods_ for the exact definitions of $\sigma_{\|}$,
$\epsilon_{\|}$ and $E_{\|}$. (b) The same as in panel (a), but for
iso-$\\!A_{\rm g}$ glasses, featuring approximate collapse. (c) The Toughness
(computed as the integral over the stress-strain curve and normalized by
$E_{\|}$, see text for details) vs. $A_{\rm g}$. (d) The Toughness vs.
$\dot{T}$.

These predictions are tested in Fig. 6, where we first show in panel (a) that
different glasses (i.e. different $r_{\rm c}$ values) feature rather large
variability in their failure behavior under constrained tension for a fixed
cooling rate $\dot{T}$ (here we present stress-strain curves; a quantitative
measure of the fracture resistance is presented in panels (c) and (d)). The
discussion above predicts that if one repeats the calculations of Fig. 6a for
different glasses featuring nearly the same $A_{\rm g}$, under the same
constrained tension loading geometry ($R\\!=\\!0$), most of the variability in
the failure behavior would be gone. The results of these calculations are
presented in Fig. 6b, where it is indeed observed that the stress-strain
curves of iso-$\\!A_{\rm g}$ glasses featuring different $\nu$ values,
approximately collapse, as predicted. These results should be compared to —
and contrasted with — the results shown in Fig. 3c obtained under
unconstrained uniaxial tension. We stress that these results strongly suggest
that once the geometrical Poisson contraction is eliminated (or more generally
when $R$ is prescribed), Poisson’s ratio $\nu$ does not play a major role in
determining the toughness of glasses.

Another prediction discussed above is that the fracture toughness is expected
to systematically increase with $A_{\rm g}$ under constrained tension. In
order to test this prediction, we quantify the fracture resistance through the
“Toughness” [60], which is defined as the integral under the stress-strain
curve up to failure (it has the dimensions of energy density, and should be
distinguished from the “Fracture toughness” $K_{\rm c}$ [2], which has the
dimensions of stress times the square root of length, see _Supplementary
material_). Note that the Toughness is usually employed for samples that do
not contain an initial crack (as it obviously depends on the crack geometry
and dimensions); here we do apply it to such samples, but as done throughout
this work, the initial crack geometry and dimensions are strictly kept fixed.
In Fig. 6c, we plot the Toughness vs. $A_{\rm g}$ for our entire glasses
ensemble, calculated under constrained tension. As predicted, we observe
systematic increase of the Toughness with $A_{\rm g}$. Finally, we expect that
the $\dot{T}$ dependence of $A_{\rm g}$, previously presented in Fig. 1f, is
fully and transparently mirrored in the dependence of the Toughness on
$\dot{T}$. This is strongly supported by the results shown in Fig. 6d.

##  III Discussion and conclusion

In this work, employing extensive atomistic simulations and recently developed
concepts, we studied the physical origin of the failure resistance of glasses,
and in particular the emerging brittle-to-ductile transitions. We showed that
these are controlled by both the abundance of soft defects — as quantified by
$A_{\rm g}$, the prefactor of the universal $\omega^{4}$ vDOS of nonphononic
excitations in glasses (and indirectly by $\chi$) — and by the loading
geometry of the fracture test employed to extract the toughness. The loading
geometry/configuration is shown to affect the relative magnitude of shear and
tensile deformation experienced by the material near the initial crack, and
consequently the emerging plastic dissipation, for a fixed $A_{\rm g}$.
Roughly speaking, $A_{\rm g}$ controls the number of soft defects/STZs in a
glass, and the loading geometry controls the fraction of soft defects/STZs
being actually activated, and hence both affect plastic deformation and
consequently the fracture toughness.

We find that only under a certain choice of loading geometry, where Poisson
contraction can take place, Poisson’s ratio $\nu$ can be sensibly used to
quantify the fracture toughness _together_ with $A_{\rm g}$ (or $\chi$). These
results suggest that brittle-to-ductile transitions in glasses are not
controlled by a critical Poisson’s ratio, as previously proposed, and
elucidate the physical role $\nu$ might play in affecting the fracture
toughness of glasses.

Our findings thus provide basic insights into the physical origin of the
failure resistance of glasses, and should serve as important input for
theories that aim at predicting it. Such theories should include the number of
soft defects, as quantified here by $A_{\rm g}$, and its deformation-induced
spatiotemporal evolution as important ingredients. Our findings also call for
additional future investigations, along the lines we delineate next.

In this work we employed an athermal ($T\\!=\\!0$) and quasistatic (i.e. we
approach the limit of vanishing deformation rates) glass-deformation protocol
[61, 31]. While this choice is suitable for identifying structural and loading
geometry effects on the fracture toughness, it completely suppresses thermal
and rate effects, and might therefore introduce uncontrolled artefacts. It is
obviously desirable to extend the present study to more physically realistic
loading protocols that involve both finite temperatures and strain-rates.

Equally important for future work are mechanical tests on computer glasses
that trace out the role of sample size and thickness in determining the
failure resistance to failure; in this work we employed glassy slabs of small
thickness (see _Methods_), hence systematically studying possible thickness
effects is in place. In the _Supplementary material_ we show that increasing
the glass in-plane dimensions — other than the slab thickness — does not
appear to affect the fracture behavior. In addition, as our results were
obtained for an initial crack of fixed geometry and dimensions (but see also
_Supplementary material_), it would be important in future work to
sufficiently increase both the system size and the initial crack size to
obtain quantitative results that are entirely independent of both.

We selected to study a rather flexible computer glass model system —
introduced first in [50] — that features a very large variability of glasses’
mechanical properties [29, 51]. While we are unable at this point to single
out a particular deficiency of this model, the employed pairwise interactions
might be too simple to capture the emergent effects of more complex and
realistic particle interactions of laboratory glasses in their entirety. For
example, the Poisson ratio of our computer glasses does not fall below
$\nu\\!=\\!1/4$, while other (network) computer glasses (e.g. those employed
in [6]) can feature $\nu\\!<\\!1/4$.

At the same time, the power of computer simulations allows us to probe and
measure generally applicable quantifiers of mechanical disorder such as
$A_{\rm g}$ and $\chi$; while the former does not appear to be directly
accessible experimentally, the latter could be — in principle — indirectly
accessed in future experiments via transverse sound attenuation measurements
at wavenumbers $\lesssim\\!1$nm-1, or via other experimental techniques [62,
63]. As new and more informative quantifiers of STZ-densities are being
developed (cf., e.g., [36, 64, 21]), we cannot rule out that more informative
dimensionless quantifiers of mechanical disorder can shed further light on the
relation between mechanical disorder and plastic deformability, and hence on
the fracture toughness.

In our analysis of the abundance of STZs as captured by $A_{\rm g}$, we have
not considered the effect of geometric/mechanical coupling of STZs to
different deformation modes. In [29] it was shown that varying $r_{\rm c}$ can
dramatically affect the ratio of dilatational to shear eigenstrains associated
with STZs viewed as Eshelby inclusions. These changes in the intrinsic
geometric properties of soft defects can potentially play a role in the
failure of glasses. For example, the coupling of soft defects to different
deformation modes can affect the susceptibility to cavitation, which is in
fact observed in our simulations (but will be discussed elsewhere) as
predicted in [7].

Finally, our work constitutes a step towards a quantitative understanding of
deformation-induced failure in disordered solids. Future investigations should
aim at better understanding the role of deformation-induced rejuvenation of
glassy structures — going beyond $A_{\rm g}$ that characterizes the _initial_
glass structure — in controlling the ductile or brittle nature of failure.
That is, plastic deformation itself could be limited — in some cases — in its
ability to generate new soft defects that facilitate further plastic flow in
the material, as typically happens within shear bands [65].

##  IV Materials and Methods

Here we provide details about the computer glass models employed, about the
methods used to deform the glass samples, and about the observables
considered.

###  IV.1 Computer glass models

We employ a glass model put forward in [50]; in this model, particles of equal
mass $m$ interact via the following pairwise potential

$\varphi(r_{ij},\lambda_{ij})\\!=\\!\left\\{\begin{array}[]{cc}\\!\\!4\varepsilon\bigg{[}\big{(}\frac{\lambda_{ij}}{r_{ij}}\big{)}^{12}-\big{(}\frac{\lambda_{ij}}{r_{ij}}\big{)}^{6}\bigg{]},&\frac{r_{ij}}{\lambda_{ij}}<x_{\mbox{\tiny
min}}\\\
\varepsilon\bigg{[}a\big{(}\frac{\lambda_{ij}}{r_{ij}}\big{)}^{12}-b\big{(}\frac{\lambda_{ij}}{r_{ij}}\big{)}^{6}+\sum\limits_{\ell=0}^{3}c_{\mbox{\tiny$2\ell$}}\big{(}\frac{r_{ij}}{\lambda_{ij}}\big{)}^{2\ell}\bigg{]},&x_{\mbox{\tiny
min}}\\!\leq\\!\frac{r_{ij}}{\lambda_{ij}}<x_{c}\\\
0\,,&x_{c}\leq\frac{r_{ij}}{\lambda_{ij}}\end{array}\right.,$ (1)

where $\varepsilon$ is a microscopic energy scale, $x_{\mbox{\tiny
min}},x_{c}$ are the (dimensionless) locations of the minimum of the Lennard-
Jones potential and modified cutoff, respectively, and the $\lambda_{ij}$’s
are the length parameters, described further below. We express the
dimensionless cutoff $x_{c}$ in terms of $x_{\mbox{\tiny min}}\\!=\\!2^{1/6}$,
for simplicity, by defining $r_{c}\\!\equiv\\!x_{c}/x_{\mbox{\tiny min}}$.
$r_{c}$ serves as one of the key control parameters in our study; we refer
readers to [29, 51] for a comprehensive study of the _elastic_ properties of
glasses formed with this model, under variations of $r_{c}$ and of glass
preparation protocol. The coefficients $a,b,\\{c_{\mbox{\tiny$2\ell$}}\\}$ are
chosen such that the attractive and repulsive parts of $\varphi$, and its
first two derivatives, are continuous at $x_{\mbox{\tiny min}}$ and at
$x_{c}$, see Table 1 for the coefficients’ numerical values. Unless specified
otherwise, we employ simulational units, where energies are expressed in terms
of $\varepsilon$, temperature in terms of $\varepsilon/k_{B}$, lengths in
terms of $\lambdabar$ (see further discussion below), elastic moduli in terms
of $\epsilon/\lambdabar^{3}$, and times in terms of
$(m\lambdabar^{2}/\varepsilon)^{1/2}$.

Table 1:  Pairwise potential coefficients coefficient | $r_{c}=1.2$ | $r_{c}=1.3$ | $r_{c}=1.4$ | $r_{c}=1.5$
---|---|---|---|---
a | -106.991613526652 | -17.7556513878655 | -2.942014535960528 | 1.1582440286928275
b | -304.918469059567 | -50.37332289908061 | -12.11892507229410 | -2.2619482444770567
$c_{\mbox{\tiny$0$}}$ | -939.388037994211 | -138.58271673010657 | -35.72455291073821740 | -12.414700446492716
$c_{\mbox{\tiny$2$}}$ | 1190.70962256002 | 161.71576064627635 | 38.70071979329345996 | 12.584354590303674
$c_{\mbox{\tiny$4$}}$ | -541.3001315875512 | -66.7252832098764 | -14.5415594738601088 | -4.320508006050397
$c_{\mbox{\tiny$6$}}$ | 85.86849369147127 | 9.50283097488097 | 1.86201465049568722 | 0.49862551162881885

We employ the pairwise potential in Eq. (1) in two distinct protocols for
creating glasses. For the first procedure, we follow the conventional computer
glass formation route: high-temperature equilibrium liquid configurations at
pressure $P\\!=\\!1$ are generated at an initial temperature $T_{\mbox{\tiny
initial}}\\!=\\!0.7$. Those equilibrium configurations are then quenched at
fixed zero pressure $P\\!=\\!0$, and at a constant cooling rate $\dot{T}$ (as
specified in the figure legends) until a temperature $T_{\mbox{\tiny
final}}\\!=\\!0.05$ is reached. Any residual heat is subsequently removed
using a standard energy minimization algorithm. For this protocol, we employ a
50:50 binary mixture of ‘large’ (l) and ‘small’ (s) particles, and fix the
effective size parameters $\lambda_{\rm ss}\\!=\\!\lambdabar$, $\lambda_{\rm
sl}\\!=\\!\lambda_{\rm ls}\\!=\\!1.18\lambdabar$, and $\lambda_{\rm
sl}\\!=\\!1.4\lambdabar$, for small-small, small-large, and large-large
interactions, respectively.

Glasses formed with the second procedure — introduced next — are referred to
as _Fluctuating-Size-Particles_ (FSP) glasses. They were created by employing
the algorithm put forward in [7]; within this algorithm, an augmented
potential energy is employed for creating glasses with very high variability
of their mechanical stability. This variability is achieved during glass
formation by allowing particle sizes $\lambda_{i}$ to fluctuate about a
preferred effective size $\lambda_{i}^{(0)}$ at an energetic penalty
determined by a potential of the form

$\tilde{\varphi}(\lambda_{i})=\frac{k_{\lambda}}{2}\big{(}\lambda_{i}-\lambda_{i}^{(0)}\big{)}^{2}\bigg{[}\bigg{(}\frac{\lambda_{i}^{(0)}}{\lambda_{i}}\bigg{)}^{2}+\frac{1}{4}\bigg{]}\
,$ (2)

where the stiffness $k_{\lambda}$ constitutes the main control parameter of
FSP glasses. Notice that this form is slightly different from the effective-
size potential used in [7]; the latter turns out to be unstable for small
$k_{\lambda}$ and small confining pressures. We set the preferred sizes of
half the particles at $\lambda_{i}^{(0)}\\!=\\!\lambdabar/2$ (as before,
$\lambdabar$ forms the microscopic units of length), the other half at
$\lambda_{i}^{(0)}\\!=\\!7\lambdabar/10$, and used the convention
$\lambda_{ij}\\!=\\!\lambda_{i}\\!+\lambda_{j}$. Crucially, once a glass is
created with the FSP algorithm, all subsequent analyses and mechanical tests
are performed under _fixed_ particle sizes $\lambda_{i}$, namely using the
exact same pairwise potential as given by Eq. (1). Importantly, local minima
(glassy states in mechanical equilibrium) of the augmented potential also
constitute local minima of the potential given by Eq. (1). Finally, we note
that the particles of FSP glasses are polydispersed in size, since during
glass formation the effective particle sizes fluctuate, see [7] for further
discussion and details. We employ the FSP algorithm to create zero pressure
glasses, starting from high temperature liquid states and minimizing the
augmented potential using a variety of stiffnesses $k_{\lambda}$ (and various
values of $r_{c}$) as reported in the figure legends. In general, decreasing
the stiffness parameter $k_{\lambda}$ results in stiffer glasses featuring
less mechanical fluctuations.

Using the two protocols described above we created two sets of ensembles: one
set was created for extracting the micromechanical properties of each ensemble
as reported in Fig. 1 for glasses formed with a finite cooling-rate, and in
Fig. 4 for glasses formed using the FSP protocol. For these calculation we
prepared between 256 to 5000 glasses of $N\\!=\\!4096$ particles (except for
lowest cooling rate $\dot{T}\\!=\\!2.10^{-5}$ for which $N\\!=\\!2197$). The
second set of ensembles — prepared for our mechanical tests — were larger,
rectangular glass slabs of dimensions
$L_{x}\\!\times\\!L_{y}\\!\times\\!L_{z}$, with
$L_{x}\\!=\\!L_{y}\\!=\\!L_{0}=60a_{0}$ and $L_{z}\\!=\\!15a_{0}$ (recall that
$a_{0}\\!\equiv\\!(V/N)^{1/3}$), containing $N\\!=\\!54000$ particles in
total.

###  IV.2 Mechanical loading simulations

Mechanical loading is carried out via athermal quasistatic simulations, i.e.
zero strain-rate and temperature. For both unconstrained uniaxial tension and
constrained tension (see also Fig. 5 and “Mechanical and structural
observables” below), the glass is affinely deformed by imposing an extensional
strain along the $y$-axis (regarded as the parallel direction in the text). We
define the extensional strain as $\epsilon_{\|}\\!=\\!(L-L_{0})/L_{0}$, where
$L$ and $L_{0}$ are the current and initial box length along $\vec{e}_{y}$,
respectively. Particle positions are changed according to
$y_{i}\\!\to\\!y_{i}+\delta\epsilon_{\|}\,y_{i}$, where the strain step
$\delta\epsilon_{\|}$ is fixed at each step such that the accumulated
extensional strain $\epsilon_{\|}$ increases by $10^{-3}$. Subsequently, we
minimize the potential energy $U$, where periodic boundary conditions are
applied in all directions. For constrained tension, we employ the conventional
conjugate gradient algorithm. For unconstrained uniaxial tension, we combine
the FIRE minimization algorithm [66] with the Berendesen barostat. The latter
keeps the stress in the transverse directions, i.e. along the $x$\- and
$z$-axis, zero, with a time constant $\tau_{\rm Ber}\\!=\\!10.0$. For both
loading geometries, the minimization is stopped once the ratio between the
typical gradient of the potential and the typical interparticle force drops
below $10^{-10}$. During each simulations, and closely following Ref. [3], we
monitor the non-affine part of the plastic deformation based on the commonly
used $D^{2}_{\rm min}$ field [3] between the initial configuration
($\epsilon_{\|}\\!=\\!0$) and the current state.

###  IV.3 Mechanical and structural observables

The potential energy of our glasses is given by
$U\\!=\\!\sum_{i<j}\varphi_{ij}(r_{ij})$, where the pairwise potential we
employed is described in Eq. (1). As we focus on athermal conditions, the
shear modulus is defined as [67]

$\mu\equiv\frac{1}{V}\left(\frac{\partial^{2}U}{\partial\gamma^{2}}-\frac{\partial^{2}U}{\partial\gamma\partial\mathitbf
x}\cdot\mbox{\boldmath${\cal
H}$}^{-1}\cdot\frac{\partial^{2}U}{\partial\mathitbf
x\partial\gamma}\right)\,,$ (3)

where

$\mbox{\boldmath${\cal H}$}\equiv\frac{\partial^{2}U}{\partial\mathitbf
x\partial\mathitbf x}$ (4)

is the Hessian matrix of the potential $U$, and $\mathitbf x$ denotes particle
coordinates. The latter are considered to transform via $\mathitbf
x\\!\to\\!\bm{H}(\gamma)\cdot\mathitbf x$ with the parameterized shear
transformation

$\bm{H}(\gamma)=\left(\begin{array}[]{ccc}1&\gamma&0\\\ 0&1&0\\\
0&0&1\end{array}\right)\,.$ (5)

The pressure is given by

$p\equiv-\frac{1}{V{\,\mathchar 22\relax\mkern-12.0mud}}\frac{\partial
U}{\partial\eta}\,,$ (6)

where ${\,\mathchar 22\relax\mkern-12.0mud}\\!=\\!3$ is the dimension of
space, and $\eta$ is the isotropic dilatational strain that parameterizes the
suitable transformation of coordinates $\mathitbf
x\\!\to\\!\bm{H}(\eta)\cdot\mathitbf x$ as

$\bm{H}(\eta)=\left(\begin{array}[]{ccc}e^{\eta}&0&0\\\ 0&e^{\eta}&0\\\
0&0&e^{\eta}\end{array}\right)\,.$ (7)

The athermal bulk modulus $K\\!\equiv-\frac{1}{{\,\mathchar
22\relax\mkern-12.0mud}}\frac{dp}{d\eta}$ is given by

$K=\frac{1}{V{\,\mathchar
22\relax\mkern-12.0mud}^{2}}\left(\frac{\partial^{2}U}{\partial\eta^{2}}-{\,\mathchar
22\relax\mkern-12.0mud}\frac{\partial
U}{\partial\eta}-\frac{\partial^{2}U}{\partial\eta\partial\mathitbf
x}\cdot\mbox{\boldmath${\cal
H}$}^{-1}\cdot\frac{\partial^{2}U}{\partial\mathitbf x\partial\eta}\right)\,.$
(8)

With the definitions of the shear and bulk moduli in hand, the Poisson’s ratio
$\nu$ of a 3D solid is given by

$\nu\equiv\frac{3K-2\mu}{6K+2\mu}\,.$ (9)

As explained above (see “Mechanical loading simulations”) and in the main
text, we employed two different types of loading geometries, namely
unconstrained uniaxial tension and constrained tension (cf. Fig. 5). In the
former, uniaxial tension of magnitude $\sigma_{\|}$ is applied, while keeping
the transverse stresses (in the two perpendicular directions) zero, i.e.
$\sigma_{\perp}\\!=\\!0$. The resulting extensional strain $\epsilon_{\|}$
satisfies $\sigma_{\|}\\!=\\!E\epsilon_{\|}$, where $E\\!=\\!2\mu(1+\nu)$ is
the conventional Young’s modulus. In this loading geometry, the biaxiality
ratio is set by Poisson’s ratio, i.e.
$R\\!=\\!-\epsilon_{\perp}/\epsilon_{\|}\\!=\\!\nu$.

In the latter case, i.e. in the constrained tension loading configuration (cf.
Fig. 5), the transverse boundaries (perpendicular to the applied tension
direction) are held fixed, i.e. $\epsilon_{\perp}\\!=\\!0$, and consequently
the transverse stresses $\sigma_{\perp}$ are no longer zero (in fact, they are
tensile, i.e. $\sigma_{\perp}\\!>\\!0$). Note that since
$\sigma_{\perp}\\!\neq\\!0$, i.e. the resulting state of stress is no longer
uniaxial (though uniaxial stress $\sigma_{\|}\\!>\\!0$ is applied), we term
this loading geometry ‘constrained tension’, omitting ‘uniaxial’). In this
loading geometry, one can define two different moduli (linear response
coefficients) $E_{\|}$ and $E_{\perp}$, according to
$\sigma_{\|}\\!=\\!E_{\|}\epsilon_{\|}$ and
$\sigma_{\perp}\\!=\\!E_{\perp}\epsilon_{\perp}$. Using Hooke’s law, one finds

$E_{\|}=2\mu\,\frac{1-\nu}{1-2\nu}\qquad\hbox{and}\qquad
E_{\perp}=2\mu\,\frac{\nu}{1-2\nu}\ .$ (10)

These expressions are verified against numerical simulations in the
_Supplementary material_. Note that $E_{\|}(\mu,\nu)\\!\geq\\!E(\mu,\nu)$ for
every $\mu$ and $\nu$, and that in this case we have a zero biaxiality ratio,
i.e. $R\\!=\\!-\epsilon_{\perp}/\epsilon_{\|}\\!=\\!0$.

###  IV.4 Quantifiers of mechanical disorder

In our work we consider two dimensionless quantifiers of mechanical disorder.
The first quantifier is defined as

$\chi\equiv\frac{\Delta\mu}{\langle\mu\rangle}\sqrt{N}\,,$ (11)

where $\Delta\mu$ is the ensemble-standard-deviation of the shear modulus
$\mu$ (of glasses created with the exact same protocol), $N$ is the system
size, and $\langle\mu\rangle$ denotes the ensemble-average shear modulus.
$\chi$ has been studied in [29] under variations of glasses’ interparticle
interaction potential, and under different glass annealing protocols in [39,
52].

The second quantifier of mechanical disorder considered in our work is the
prefactor $A_{\rm g}$ of the nonphononic spectrum of a glass, which is of the
form ${\cal D}(\omega)\\!=\\!A_{g}\omega^{4}$, independent of spatial
dimension [25] or microscopic details [26]. The latter is obtained by
performing a partial diagonalization of the Hessian matrix ${\cal H}$, defined
in Eq. (4), and calculated for each member of our glass ensembles, to obtain
the eigenfrequencies $\omega_{\ell}$ that solve the eigenvalue problem
$\mbox{\boldmath${\cal
H}$}\cdot\bm{\psi}_{\ell}\\!=\\!\omega^{2}_{\ell}\,\bm{\psi}_{\ell}$ (all
masses are all set to unity). We then histogram the eigenfrequencies for each
glass ensemble, to obtain their distribution ${\cal D}(\omega)$ over the
frequency axis. Importantly, we note that the scaling form of ${\cal
D}(\omega)\\!\sim\\!\omega^{4}$ is most readily observable in glasses of sizes
small enough to suppress the otherwise-dominant phononic waves that emerge in
the low-frequency spectrum of any elastic solid, as explained in detail in
[19].

Acknowledgments and declaration of conflicts and other disclosures D.R.
acknowledges support of the Simons Foundation for the “Cracking the Glass
Problem Collaboration” Award No. 348126. E.L. acknowledges support from the
NWO (Vidi grant no. 680-47-554/3259). E.B. acknowledges support from the Ben
May Center for Chemical Theory and Computation and the Harold Perlman Family.

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Supplementary material: “Brittle to ductile transitions in glasses: Roles of
soft defects and loading geometry”

The goal of this document is to provide additional supporting data to the
results presented in the manuscript, as detailed in the table of contents.

##  S-I Contents

  1. 1.

Scaling arguments for the $A_{g}$-$\chi$ relation

  2. 2.

Crack geometry and validation of the tip square root singularity

  3. 3.

Constrained uniaxial tension vs. constrained tension

  4. 4.

$\chi$-$\nu$ and $\mu_{\rm na}/\mu$-$\nu$ relations for MD-quenched- and FSP-
glasses

  5. 5.

Effect of crack length on the brittle-to-ductile transition

  6. 6.

Effect of slab thickness on toughness

  7. 7.

Effect of initial crack length on glass toughness

  8. 8.

Correlations between the abundance of soft defects and the elastic-moduli
susceptibility to thermal aging

## 1\. Scaling arguments for the $A_{g}$-$\chi$ relation

In Fig. 3d of the main text we show that for our computer glasses made by
quenching liquids at finite cooling rates $\dot{T}$, the quantifiers of
mechanical disorder $A_{\rm g}$ and $\chi$ are approximately related by

$A_{\rm g}\sim\chi^{10/3}\,.$ (S1)

Here we explain the origin of this approximate scaling law. In [1] it was
shown that $\chi\\!\sim\\!\xi^{2/3}$ where $\xi$ is a glassy lengthscale that
characterizes responses to local perturbations. In [2] it was demonstrated
that $\xi\\!\sim\\!\omega_{\rm g}^{-1}$, with $\omega_{\rm g}$ being a
characteristic frequency scale of soft quasilocalized excitations. Finally, in
[3] it was shown that $A_{\rm g}\\!\sim\\!\omega_{\rm g}^{-5}$, implying
together with the rest of the aforementioned scaling laws that $A_{\rm
g}\\!\sim\\!\chi^{10/3}$, consistent with our data of Fig. 3d of the main
text.

## 2\. Crack geometry and validation of the tip square root singularity

In Fig. S1a, we present a 2D projection of the typical starting configuration
used throughout our study. The initial crack geometry follows a diamond shape
(see sketch in the inset of Fig. S1b) with semi-axis $c$ and total crack
length $2c$. The vertex angle $\beta$ is set to $18.4^{\rm o}$. For such a
geometry, the stress singularity at the crack tip will scale with $1/(2\pi
r)^{\lambda_{I}}$, with $r$ being the distance from the crack tip and
$\lambda_{I}$ being the mode-I fracture singularity order [4]. The later
depends on the vertex angle $\beta$, and can be approximated by the following
form [4]

$\lambda_{I}(\beta)\simeq
1.247\cos(\beta)-1.312\cos^{2}(\beta)+0.8532\cos^{3}(\beta)-0.2882\cos^{4}(\beta)\,,$
(S2)

which is plotted in Fig. S1b. It is clear that, for our chosen vertex angle
$\beta\\!=\\!18.4^{\rm o}$ (indicated by the vertical dashed line),
$\lambda_{I}(18.4^{\rm o})\\!\approx\\!1/2$, and so we can expect to observe
the well-known $\sim\\!1/\sqrt{r}$ singularity at distance $r$ from the crack
tip.

To confirm this expectation, we compute the local strain along the $x$-axis
(perpendicular to the loading direction) for a fixed strain of
$\epsilon\\!=\\!2\%$, and different crack lengths $2c$. In practice, the local
strain $\epsilon(x)$ is extracted from the $D^{2}_{\rm min}$ procedure
proposed by Falk and Langer in Ref. [5], and the distance $x$ is measured from
the center of the crack, as illustrated in Fig. S1a. For a thin crack, one
expects the local strain to take the following form [6]

$\epsilon(x)=\frac{\epsilon_{\infty}}{\sqrt{1-(c/x)^{2}}},$ (S3)

with $\epsilon_{\infty}$ denoting the remote loading strain. Despite the
limited size offered by atomistic simulations, we observe in Fig. S1c a good
agreement between the local strain and Eq. (S3). Shifting the $x$-axis of Fig.
S1c by the semi-crack length $c$, we indeed observe in Fig. S1d the expected
square root singularity of the strain approaching the crack tip.

Figure S1: Crack geometry and square root stress singularity. (a) Snapshot
showing the typical starting configuration of a glass with a diamond shape
crack. (b) Dependence of the singularity exponent $\lambda_{I}$ in mode-I
fracture on the vertex angle $2\beta$ (see sketch for definition). The
vertical red dashed line indicates the angle used throughout this work. (c)
Local extensional strain $\epsilon$ along the $x$-axis, as indicated in (a),
for various crack lengths $c$ and fixed global extensional strain
$\epsilon_{\infty}\\!=\\!0.02$. (d) Same data as in (c), plotted on double
logarithmic scales against $x\\!-\\!c$, demonstrating the square root
singularity of the strain close to the notch tip.

## 3\. Constrained uniaxial tension vs. constrained tension

In Fig. S2a-b, we validate our expression for the different moduli under
constrained uniaxial tension and constrained tension (see _Methods_ in the
main text). The predicted linear response (dashed lines in Fig. S2) follows
directly from computing independently the athermal shear modulus $\mu$ and the
Poisson’s ratio $\nu$.

Figure S2:  Elastic moduli: unconstrained uniaxial tension vs. constrained
tension. (a) We plot a typical stress-strain curve of a glass under
unconstrained uniaxial tension. Black dashed line indicates the linear
response controlled by the Young modulus. (b) Typical stress-strain curve
under constrained tension. The stress parallel and perpendicular to the
loading direction are colored in black and purple, respectively. Dashed lines
indicate the predicted linear response based on Hooke’s law.

## 4\. $\chi$-$\nu$ and $\mu_{\rm na}/\mu$-$\nu$ relations for MD-quenched-
and FSP-glasses

In Fig. S3a, we compare the dependence of the disorder quantifier $\chi$ on
the Poisson’s ratio $\nu$ between molecular-dynamics (MD) quenches and FSP
glasses. We observe that our the $\chi$-$\nu$ curves for FSP glasses and MD-
quenched glasses do not fall on the same curves. Instead, we find that $\chi$
measured for FSP glasses is generally lower than that measured for MD-quenched
glasses, at the same $\nu$. This discrepancy could be explained by the gapped
nonphononic vibrational spectra featured by FSP glasses, as shown in [7].

We also report in Fig. S3b the ratio $\mu_{\rm na}/\mu$ between the ‘non-
affine’ shear modulus $\mu_{\rm na}$ and the total shear modulus $\mu$,
plotted against the Poisson’s ratio $\nu$. In contrast to $\chi$, we observe
that the $\mu_{\rm na}/\mu$-$\nu$ relation featured by MD-quenched glasses and
FSP glasses agree very well, with some small deviation observed for the
$r_{\rm c}\\!=\\!1.2$ glasses. Interestingly, for $r_{\rm c}=1.3$ we observe
that some MD-quenched and FSP glasses do not fail in the same way despite
having the same $\mu_{\rm na}/\mu$ and $\nu$. We leave for future
investigation the use of the SWAP-Monte-Carlo algorithm [8] to create deeply
aged glasses, which do not exhibit a gapped nonphononic vibrational spectra,
in contrast to FSP glasses.

Figure S3: $\chi$ vs. $\nu$ and $\mu_{\rm na}/\mu$ vs. $\nu$ relations. (a)
Disorder quantifier $\chi$ that encapsulates the shear modulus fluctuations,
plotted against the Poisson’s ratio $\nu$. (b) The ratio $\mu_{\rm na}/\mu$ is
plotted against $\nu$. Filled and empty symbols correspond to ductile-like and
brittle-like failure, in glass samples with an initial crack length
$c\\!=\\!6$. Circles and squares correspond to MD-quenched and FSP glasses,
respectively.

## 5\. Effect of crack length on the brittle-to-ductile transition

The ductile-to-brittle transition presented in Fig. 4 of the main text under
unconstrained uniaxial tension pertains to glasses with an initial crack
length of $c\\!=\\!6$. Here we present additional data obtained under
variations of the crack length. For MD-quenched glasses, and in the absence of
an initial crack, only samples of the $r_{\rm c}\\!=\\!1.2$ glasses break
before 15% uniaxial strain is reached. Different from our results in the main
text for $r_{\rm c}\\!=\\!1.4$ — for an initial crack length of $c\\!=\\!6$ —,
$r_{\rm c}\\!=\\!1.4$ glasses with an initial crack length of $c\\!=\\!9$ all
exhibit crack propagation.

We perform the same analysis for FSP glasses. We find that without the
presence of an initial crack, we observe cavitation for all $r_{\rm c}$’s, but
only for glasses featuring extremely low disorder, namely $\chi\\!<\\!0.8$. As
for MD-quenched glasses, the ductile-to-brittle transition moves toward higher
$\chi$ values when increasing the crack length. This trend highlights the
effect of stress focusing at the crack tip, facilitating void nucleation and
subsequently crack propagation.

Figure S4:  Effect of crack length $c$ on the ductile-to-brittle transition.
(a1-c1) Snapshots of the non-affine plastic deformation at $15\%$ uniaxial
strain for MD-quenched glasses in the $r_{\rm c}\\!-\\!\dot{T}$ plane, for
crack lengths $c\\!=\\!0$ (a1), $c\\!=\\!6$ (b1), and $c\\!=\\!9$ (c1).
(a2-c2) Ductile-brittle phase diagram in the $\chi\\!-\\!\nu$ plane for MD-
quenched glasses and crack lengths $c\\!=\\!0$ (a2), $c\\!=\\!6$ (b2), and
$c\\!=\\!9$ (c2). Glasses featuring ductile-like (brittle-like) failure are
represented by full (empty) symbols. Blurry regions are added only as guides
to the eye. (a3-c3) Same as (a2-c2), for FSP glasses. (a4-c4) Snapshots of the
non-affine plastic deformation at $15\%$ strain for FSP glasses in the $r_{\rm
c}\\!-\\!k_{\lambda}$ plane, for $c\\!=\\!0$ (a4), $c\\!=\\!6$ (b4), and
$c\\!=\\!9$ (c4).

## 6\. Effect of slab thickness on toughness

Throughout our study presented in the main text, the sample thickness was
fixed to $15$ particles diameters. It is well known that the fracture
toughness of a material does depend on the sample thickness. A complete study
of the effect of sample thickness on the toughness and failure mode is beyond
the scope of our present manuscript. We nevertheless report here the effect of
sample thickness on the toughness under unconstrained uniaxial test for one
glass realization: an FSP-glass prepared with $k_{\lambda}\\!=\\!1000$ and
potential cutoff $r_{\rm c}=1.2$. We find a noticeable increase of the
toughness for a small crack length ($c\\!=\\!6$, see Fig. S5a and Fig. S5c),
whereas we observe for thicker samples an enhanced necking during fracture
(see Fig. S5e). For a larger crack length, namely $c=9$, we find the toughness
to be nearly independent of the sample thickness (see Fig. S5b and Fig. S5d).

Figure S5: Effect of sample thickness on the toughness under unconstrained
uniaxial tension. (a) Stress-strain curves under unconstrained uniaxial
tension for FSP glasses ($r_{\rm c}=1.2$ and $k_{\lambda}=1000$) with various
slab thicknesses and a fixed initial crack length $c=6$. (b) Same data as in
(a) but for an initial crack length of $c=9$. (c) Toughness as a function of
the sample thickness for a fixed initial crack length $c\\!=\\!6$. (d) Same
data as in (c) but for a crack length of $c\\!=\\!9$. (e) Snapshots of the
non-affine plastic deformation at $15\%$ uniaxial strain for samples with
different thicknesses and a fixed initial crack of $c\\!=\\!6$.

## 7\. Effect of initial crack length on glass toughness

The results for glass toughness under constrained tension — presented in Fig.
6 of the main text — were only reported for an initial crack length with
$c\\!=\\!6$. Here we provide additional data pertaining to various initial
crack lengths. Figure S6 is organized as follows; we first show in columns
($a_{x}$) typical stress-strain curves for poorly and well-aged MD-quenched
glasses. Columns ($b_{x}$) show the influence of the crack length on the
stress-strain curves for a fixed quench rate (here $\dot{T}\\!=\\!0.02$). In
columns ($c_{x}$) we report the dimensionless toughness as a function of the
dimensionless crack length $\tilde{c}\\!=\\!c/a_{0}$ ($a_{0}$ is an
interparticle distance) for poorly (circles) and well-aged (squares) glasses.
Finally, we show that the dimensionless toughness measure times the square
root of the crack length $\sqrt{\tilde{c}}$ is largely independent of the
crack size, and thus controlled by the square root singularity of the stress
at the tip.

Figure S6: Effect of initial crack length on glass toughness under constrained
tension. (a) Stress-strain curves under constrained tension, for glasses
prepared at a high quench rate (thin curve) and low quench rate (thick curve).
See text and _Methods_ for the exact definitions of $\sigma_{\|}$,
$\epsilon_{\|}$ and $E_{\|}$. (b) The same as in panel (a), but for for
glasses prepared at a high quench rate $\dot{T}\\!=\\!0.02$ and different
crack length $c$. (c) The Toughness (computed as the integral over the stress-
strain curve, see text for details, and normalized by $E_{\|}$) as a function
of the normalized crack length $\tilde{c}=c/a_{0}$ (with $a_{0}=\rho^{-1/3}$)
for glasses prepare at a high (circles) and low (squares) quench rates. (d)
Same data as in panel (c) but where the toughness is scaled by the square root
of the crack length. Error-bars are estimated from sample-to-sample
fluctuations. From the top to the bottom row are data for $r_{\rm
c}\\!=\\!1.5$, $1.4$, $1.3$, and $1.2$, respectively.

## 8\. Correlations between the abundance of soft defects and the elastic-
moduli susceptibility to thermal aging

In this final Section we discuss the relation between macroscopic and
microscopic materials properties, and the susceptibility of a glass’s shear
modulus to thermal annealing. In particular, we consider the number density of
soft spots — as embodied in the prefactor $A_{\rm g}$ — and the Poisson’s
ratio $\nu$, in as-cast, highly disordered glasses. As explained in the
_Methods_ section of the main text, the athermal shear modulus can be
decomposed into an affine term $\mu_{\rm af}$ (Born term) and a non-affine
term $\mu_{\rm na}$ as $\mu\\!=\\!\mu_{\rm af}-\mu_{\rm na}$. The total
modulus reads

$\mu\equiv\frac{1}{V}\left(\frac{\partial^{2}U}{\partial\gamma^{2}}-\frac{\partial^{2}U}{\partial\gamma\partial\mathitbf
x}\cdot\mbox{\boldmath${\cal
H}$}^{-1}\cdot\frac{\partial^{2}U}{\partial\mathitbf
x\partial\gamma}\right)\,$ (S4)

where $\mbox{\boldmath${\cal
H}$}\\!\equiv\\!\frac{\partial^{2}U}{\partial\mathitbf x\partial\mathitbf x}$
is the Hessian matrix of the potential energy $U$, which captures the physics
of soft excitations (plasticity carriers) in the system. In the case of highly
disordered samples, ${\cal H}$ embodies a high density of soft excitations
that are known to couple with the shear-induced forces
$\frac{\partial^{2}U}{\partial\mathitbf x\partial\gamma}$. The effect of soft
modes is to increase $\mu_{\rm na}$ and decrease $\mu$, which ultimately
decreases the shear to bulk moduli ratio $\mu/K$ and therefore increases the
Poisson’s ratio $\nu$. Overall, we thus expect that samples with a high
$A_{\rm g}$ will feature a large Poisson’s ratio.

Upon thermal aging, $A_{\rm g}$ decreases (up to one order of magnitude for
$r_{\rm c}=1.5$, and a factor $2-3$ for $r_{\rm c}=1.2$), which in turn lowers
$\mu_{\rm na}$ and increases the total shear modulus. Therefore, glasses with
the largest variation in $A_{\rm g}$ are excepted to also show the largest
stiffening of $\mu$. Finally, we also expect that as-cast glasses with the
largest Poisson’s ratio will exhibit a large variability in their shear
modulus upon thermal aging. These trends are demonstrated in Fig. S7a-b with
experimental data for bulk metallic glasses (digitized from Refs. [9, 10]),
and in Fig. S7c for our computer glasses with various interaction parameters
$r_{\rm c}$ as indicated by the legend.

Figure S7: Effect of thermal aging on macroscopic elasticity: laboratory bulk
metallic glasses vs. computer glasses. (a) Correlation between the aging-
induced stiffening of the shear modulus $\mu/\mu_{0}$ and as-cast Poisson’s
ratio $\nu_{0}$ for various bulk metallic glasses (BMGs) annealed for 48 hours
(taken from Ref. [9]). (b) Rescaled shear modulus $\mu/\mu_{0}$ plotted
against $\nu$ for two BMGs, showing either a weak or substantial aging-induced
stiffening (taken from Ref. [10]). (c) Same as in (b) but for our MD-quenched
glasses with various $r_{\rm c}$, where the shear modulus is normalized by the
high-quench-rate modulus $\mu_{\infty}$.

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