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\DOI{10.5802/crphys.126}
\datereceived{2022-06-17}
\daterevised{2022-09-21}
\dateaccepted{2022-10-05}
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\dateposted{2023-01-18}

\begin{noXML}

%\TopicEF{From everyday glass to unordered solids}{Du verre quotidien
%aux solides d\'esordonn\'es}

\title{Some strange things about the mechanical properties of glass}

\alttitle{Quelques étrangetés sur les propriétés mécaniques du verre}

\author{\firstname{Tanguy} \lastname{Rouxel}\CDRorcid{0000-0002-9961-2458}}
\address{Glasses and Mechanics Lab., Physics Institute (IPR), UMR
UR1-CNRS 6251, Universit\'{e} de Rennes, Campus de Beaulieu, 35042
Rennes cedex, France}
\address{Institut Universitaire de France, France}
\email{tanguy.rouxel@univ-rennes1.fr}

\keywords{\kwd{Glass}\kwd{Mechanical
properties}\kwd{Elasticity}\kwd{Hardness}\kwd{Fracture toughness}}

\altkeywords{\kwd{verre} \kwd{propriétés mécaniques} \kwd{élasticité} \kwd{dureté} \kwd{résistance à la rupture}}

\begin{abstract} 
Glasses and crystals from the same chemical system mostly share the
same interatomic bond strength. Nevertheless, they differ by the
arrangement of bonds in space, which gives birth to different atomic
packing efficiencies. We show in this review that as far as the elastic
moduli and hardness are concerned, the atomic packing density
predominates over the bond strength. The shear modulus of crystalline
phases is usually much larger than the one of glasses with the same
stoichiometric composition, thanks to a more efficient packing of atoms
in the former. In contrast, the increase in hardness is quite limited,
likely because of the additional contribution of dislocation activity
to the deformation processes beneath the indenter in the case of
crystals (shear plasticity). We also show that the occurrence of
chemical heterogeneities (weak channels) at the mesoscopic scale in
glasses, which is often associated with the lack of long range atomic
ordering, promotes easy fracture paths and is responsible for the low
toughness and fracture surface energy.
\end{abstract}

\begin{altabstract}
Les verres et les cristaux issus du même système chimique partagent généralement la même force de liaison interatomique. Néanmoins, ils diffèrent par l'arrangement des liaisons dans l'espace, ce qui donne lieu à des efficacités d'empaquetage atomique différentes. Nous montrons dans cette étude qu’en ce qui concerne les modules élastiques et la dureté, la densité d'empilement atomique prédomine sur la force de liaison. Le module de cisaillement des phases cristallines est généralement beaucoup plus élevé que celui des verres ayant la même composition stœchiométrique, grâce à un empilement plus efficace des atomes dans les premières. En revanche, l'augmentation de la dureté est assez limitée, probablement en raison de la contribution supplémentaire de l'activité des dislocations aux processus de déformation sous l'indenteur dans le cas des cristaux (plasticité de cisaillement). Nous montrons également que l'apparition d'hétérogénéités chimiques (canaux faibles) à l'échelle mésoscopique dans les verres, qui est souvent associée à l'absence d'ordonnancement atomique à longue portée, favorise les chemins de rupture facile et est responsable de la faible ténacité et de la faible énergie de surface de rupture.
\end{altabstract}


\end{noXML}
\datepublished{2024-04-26}
\begin{document}
\maketitle
\vspace*{40pt}
\section{Introduction}\label{sec1}

There are some general trends in materials science regarding the
mechanical properties. For example, the elastic moduli mostly correlate
with the melting point~\cite{1}, and ductility usually comes with
softness~\cite{2}. As a matter of fact, ductile metals such as copper,
lead, silver or aluminum are softer than beryllium, titanium, tungsten
or chromium, which behave in a more brittle manner. Besides, as
materials become stronger, they often become more brittle. The gain in
strength in high carbon steel comes at the expense of ductility.
Ceramics are strong, but brittle. Polymers are soft and easy to shape.

Nevertheless, a look at the fine details of the changes of the
properties with composition, shows that the intuitive tendencies are
not always followed and that the physics behind the mechanical behavior
is mostly incompletely taken into account. Glasses offer a unique
opportunity to follow the property change in a very smooth way, and
across chemical systems where the interatomic bonding character evolves
from metallic, to ionic and covalent. A review and a comparative
analysis of the mechanical properties of glasses, including elasticity,
hardness, strength, toughness and viscosity, across metallic,
chalcogenide, silicate, oxynitride and oxycarbide glass forming
chemical systems bring to light some striking features. Just to mention
a few of them: (1)~the stiffest glasses, that is those with the largest
elastic moduli, are not those with the strongest interatomic bonds; (2)
glasses are mostly much less rigid but nearly as hard as their
crystalline polymorphs (same stoichiometry), and the softest (less
hard) glasses are also the most brittle ones; (3)~despite the fact that
glasses break in an ideally brittle manner, with almost no visible
permanent deformation at fracture and adopt an elastic behavior at
temperatures below their glass transition temperature (say
$T<0.5T_{g}$), they can be indented and scratched, so that some
permanent deformation mechanism shows up below $T_{g}$, where viscous
flow can be neglected; (4)~although inorganic glasses are brittle and
their elastic moduli are exceeded by those of numerous metals, their
intrinsic (or ultimate) strength is exceptional and reach the values
proposed for the theoretical strength of materials; and (5) there are
``fragile'' liquids (per Angell concept) that turn out to result in
tough glasses!

Each chemical system brings some pieces to the puzzle of the global
picture. For instance, compositional effects within silicate glasses
shed light on the key role of the atomic packing density, and those
within chalcogenide glasses reflect the importance of the average
atomic network cross-linking, while metallic glasses ({MG}s), in
comparison with oxide and chalcogenide ones, show what is going on when
the interatomic bonds are losing their directionality and their
topological constraints.

\section{The elastic moduli and Poisson's ratio of glasses}\label{sec2}

The elastic moduli of metals, and especially the shear modulus, were
often reported to be correlated to the melting point, that is to the
dissociation energy ($\langle U_0\rangle $). However, an elastic
modulus is given in pascals, that is in J${\cdot}$m$^{-3}$, whereas the
dissociation energy is expressed in J${\cdot}$mol$^{-1}$. Elastic moduli,
and in the first place the bulk modulus ($K$), are governed by the
distribution of bonds in space, and thus involve the atomic packing
density ($C_{g}$). For an arbitrary oxide glass composition $\langle
U_{0}\rangle $ can be estimated from the dissociation enthalpy of each
oxide ($\mathrm{A}_{x}\mathrm{O}_{y}$) introduced in the
composition~\cite{3,4}, $\Delta H_{d}$, which is given by a classical
Born--Haber cycle: $\Delta H_d = - \Delta H_f 
(\mathrm{A}_{x}\mathrm{O}_{y}) + x\Delta H_{\mathrm{at}} (\mathrm{A})
+ y\Delta H_{\mathrm{at}} (\mathrm{O})$, where $\Delta H_f$ is an
enthalpy of formation and $\Delta H_{\mathrm{at}}$ is an atomization
enthalpy (see~\cite{4} for details):
{\begin{equation}\label{eq1}
\langle U_{0}\rangle = \displaystyle \sum _{i}x_{i}\Delta
H_{\mathrm{di}}
\end{equation}}\unskip
where $x_{i}$ and $\Delta H_{\mathrm{di}}$ are respectively the molar
fraction and the dissociation enthalpy of the ith oxide contribution to
the gram-atom of glass. In the case of a metallic glass, $\langle
U_0\rangle$ was estimated from existing thermochemistry data~\cite{5,6}
for the elements constituting the studied glasses. For example, in the
case of a binary system, 
{\begin{equation}\label{eq2}
\langle U_{0}\rangle = x \Delta H _{\mathrm{at}}(\mathrm{A,g}) +
 y \Delta H _{\mathrm{at}}(\mathrm{B,g}) -
 y \Delta H _{\mathrm{mixing}} (\mathrm{A}_{x}\mathrm{B}_{y})
\end{equation}}\unskip
were $\langle U_0\rangle$ represents the dissociation energy (standard
pressure), i.e. the energy necessary to obtained separate gaseous atoms
from the solid material (a gram-atom is considered: $x+y=1$). The
enthalpy of mixing is rarely known for multiconstituent metallic
liquids. Furthermore, it might not represent the situation in amorphous
systems, inasmuch as the glassy network exhibits some peculiar
structural features such as chemical segregation, clustering etc.
Nevertheless, in most cases, it seems that the enthalpy of mixture is
much smaller than the atomization enthalpy in absolute values. The
atomisation (sublimation) enthalpy is typically of several hundreds
kJ${\cdot}$mol$^{-1}$ (605, 326, 431, and 338 for Zr, Al, Ni and Cu
respectively), whereas the enthalpy of mixing is less than the enthalpy
of formation (about
$-34$~kJ${\cdot}$mol$^{-1}$ for ZrCu,
$-57$~kJ${\cdot}$mol$^{-1}$ for PdSi,
$-30$~kJ${\cdot}$mol$^{-1}$ for Ni\tsub{5}P\,\ldots) and is of few tens
kJ${\cdot}$mol$^{-1}$~\cite{7}. In the case of a disordered arrangement
of atoms, $C_{g}$ can be determined from the ratio of the effective
volume occupied by a mole of atoms ($V_{a}$), as estimated from the
atomic radii ($V_{a} = \mathcal{N} \sum _{i} ({4}/{3})\rmpi
x_{i}{r}_{i}^{3}$, where $\mathcal{N}$ is Avogadro number, $x_{i}$ and
$r_{i}$ are the atomic fraction and ionic radius of the $i${th}
element), to the corresponding volume of glass ($V_0$), calculated from
the specific mass of the glass ($\rho$) and the molar mass of the
constituents ($V_{0} =(1/\rho) \sum _{i}x_{i}m_{i}$, where $m_{i}$ is
the molar mass of the $i${th} element):
{\begin{equation}\label{eq3}
C_{g} = \dfrac{\mathcal{N} \sum _{i}(4/3)\rmpi x_{i}{r}_{i}^{3}}
{\left(\sum _{i}x_{i}m_{i}\right)/\rho}.
\end{equation}}\unskip

In multiconstituent systems where atoms are relatively free to explore
the empty space of the atomic network, large $C_{g}$ values may be
obtained. This situation is observed in {MG}s based on precious
metals, which exhibit $C_{g}$ values as large as 0.8. In the case of
silicate glasses, $C_{g}$ ranges typically between 0.45 and
0.55~\cite{8,9,10}. Most metals crystallized in dense structures
involving close-packed arrangements of atoms such as in the
face-centered cubic and hexagonal compact lattices, which both
correspond to a value of $C_{g}$ of 0.74 (pure substances). This is why
in the case of metals the elastic moduli are mainly correlated with the
dissociation energy. The situation is very different in the case of
glasses since the presence of free volume and of directional bonds
(non-metallic systems) result in relatively lose-packed atomic
arrangements for which $C_{g}$ spreads over a wide interval, typically
from 0.41 for silicon-oxycarbide glasses to 0.48 for precious metal
based {MG}s. As a matter of fact, there is no one to one
relationship between $K$ and $\langle U_{0}\rangle$
(Figure~\ref{fig1}). For example, $K$ is around 114~GPa for
Pd\tsub{80}Si\tsub{20} and Zr\tsub{55}Co\tsub{25}Al\tsub{20}
{MG}s and for a
Y\tsub{14,1}Si\tsub{18.5}Al\tsub{5.3}O\tsub{54.7}N\tsub{7.5} oxynitride
glass (horizontal dashed straight line in Figure~\ref{fig1}), although
the values for $\langle U_{0}\rangle$ are respectively of 392.8, 504
and 641 kJ${\cdot}$mol$^{-1}$. In these glasses, the decrease of the
atomic packing density counterbalances the increase in $\langle
U_{0}\rangle $ (or in $T_{g}$). The same observation can be made with
Young's modulus. For example, Tm\tsub{53}Al\tsub{25}Co\tsub{20},
amorphous silica (a-SiO\tsub{2}), a standard window glass
(soda-lime-silica) and an electric glass ($E$-glass) (about 54~mol\%
Al\tsub{2}O\tsub{3}, 22 mol\% B\tsub{2}O\tsub{3}, 14 mol\%
\textit{AE} and 10 mol\% \textit{A}) have a common Young's modulus
value close to 70~GPa, as for aluminum, despite the fact that their
glass transition temperatures spread over the 277--1190~\textdegree C
interval. Let us consider the case of vitreous silica. The bulk modulus
($K$) of a-SiO\tsub{2} is equal to 33~GPa. The addition of 30 mol\% of
a mixture of alkaline ($A$) and alkaline-earth ($AE$) oxides
(Na\tsub{2}O, K\tsub{2}O, MgO, CaO), to meet a standard composition for
window glasses, leads to a value of 44~GPa for $K$, that is a larger
value in spite of much smaller bond energies for the added
constituents: 
$U_{0(\mathrm{Na}\ndash\mathrm{O})}=73~\mathrm{kJ}{\cdot}\mathrm{mol}^{-1}$, 
$U_{0(\mathrm{K}\ndash\mathrm{O})} =49~\mathrm{kJ}{\cdot}\mathrm{mol}^{-1}$,
$U_{0(\mathrm{Mg}\ndash\mathrm{O})}=166~\mathrm{kJ}{\cdot}\mathrm{mol}^{-1}$, and
$U_{0(\mathrm{Ca}\ndash\mathrm{O})}=177~\mathrm{kJ}{\cdot}\mathrm{mol}^{-1}$, whereas
$U_{0(\mathrm{Si}\ndash\mathrm{O})}=624~\mathrm{kJ}{\cdot}\mathrm{mol}^{-1}$. However, $C_{g}$ is
increased in the presence of $A$ and $AE$ oxides, to about 0.516 for a
standard window glass, while it is as small as 0.454 for a-SiO\tsub{2}.
This is another clear evidence for the role played by $C_{g}$ on the
elastic moduli which kills off the myth that strong bonds make stiff
materials! 

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}The bulk modulus of glasses from different
chemical systems as a function of the dissociation energy per mole of
gram-atom. Data were extracted from Refs.~\cite{4,6}. The horizontal
dashed line intersects with different glass compositions exhibiting the
same $K$ value.}
\end{figure}

Let us move one step further in the interpretation of the dependence of
the elastic moduli on the glass composition by looking at the
correlation between $K$ and the volume density of energy,  $\langle
U_{0}\rangle/\langle V_{0} \rangle $ (Figure~\ref{fig2}). It turns out
that for a given chemical system, the data points are quite nicely
aligned. Nevertheless, the rates at which $K$ changes with the volume
density of energy differ from one series to the other. The slope is
relatively steep for metallic glasses. It reaches about 3.6 for those
with palladium or platinum as the major host elements, it is smaller
for Zr-, Cu-, and Ti-based alloys (about 2.5) and is at about 1.5 for
rare-earth-based ones. In the case of phosphate glasses, $K \approx
\langle U_{0}\rangle/\langle V_{0} \rangle $. The slope is equal to 3/4
in average for silicates and tends to increase with nitrogen in
silicon-oxynitride glasses. Actually, the rate at which $K$ changes
with $\langle U_{0}\rangle/\langle V_{0} \rangle $ reflects the
interatomic bonding character. For sake of simplicity, considering a
Lennard--Jones (L--J) type potential for the driving force of
elasticity at the atomic or structural network scale, with $m$ and $n$
being the exponents associated with the repulsive and attractive
branches of the potential, then  $K/(\langle U_{0}\rangle/\langle V_{0}
\rangle)$ is proportional to the $mn$ product (this is known as the
1{st} Gr\"{u}neisen's rule). Despite some fundamental difficulties,
including (i) the transposition of the approach to the dissociation
energy  $\langle U_0\rangle$, which is not an interatomic bond
strength, and (ii) the assumption that the cohesive energy can be
differentiated with respect to the atomic volume (see
Ref.~\cite{6} for details), the present investigations suggest a
steep potential for {MG}s and especially those based on precious
metals, which leaves little room for changes in the inter-unit
distance, so that shear is expected to predominate, in agreement with
the fact that Poisson's ratio ($\nu$) is relatively large for these
glasses. Here, a structural unit can be either an atom, a group of
atoms representative of the glass stoichiometry, or a cluster with a
peculiar chemical composition, depending on the way the glass behaves
under mechanical loading at the atomic scale, as illustrated in the
schematic drawings at the bottom in Figure~\ref{fig3}. On the contrary,
for a relatively small $mn$ product, the structural units might move
each with respect to the others with center-to-center distance
variations and local volume contraction or expansion. In the case of
precious-metal (Pt, Au, Pd,\,{\ldots})-based {MG}s, the atomic
packing efficiency reaches 0.8, and $\nu$ meets values as large as 0.4.
In the case of oxide glasses, it is noteworthy that whatever the
chemical system, $\nu$ and $C_{g}$ are significantly smaller than for
the {MG}s, and so is the $mn$ product. For silicate glasses,
$C_{g}$ is typically between 0.4 and 0.55, and  $K/(\langle
U_{0}\rangle/\langle V_{0} \rangle) < 1$, consistently with the fact
that oxide glasses are much more brittle than the metallic ones, and
undergo significant ``permanent'' volume changes under mechanical
loading (for instance pure SiO\tsub{2} glass, with $\nu \approx 0.15$,
can experience up to 20\% densification under hydrostatic
loading)~\cite{11}. The explanation chiefly lies in the atomic packing
density, which mostly gets better and better as the network
cross-linking degree and the bond strength are decreased. As a matter
of fact, it is easier to pack ``hard'' spheres, such as ``good'' metal
atoms, in a box, than stiff structural units such as rods, triangles or
even worse, tetrahedra!

\begin{figure}
{\vspace*{-2pt}}
\includegraphics{fig02}
\caption{\label{fig2}The bulk modulus of glasses from different
chemical systems as a function of the volume density of energy (in
J${\cdot}$m$^{-3}$ or in~GPa). Marks correspond to different chemical
systems. Data were extracted from Refs.~\cite{4,6}.}
{\vspace*{-2pt}}
\end{figure}

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Schematic drawings of the way Poisson's ratio, a
macroscopic elastic property, correlates to the fine details of the
deformation kinematics.}
\end{figure}

\section{Hardness and permanent deformation mechanism at an indentation
site}\label{sec3}

Typical hardness ($H$) values for inorganic, non-metallic glasses range
between 1 (chalcogenides) and 12 (silcon-oxynitrides) GPa~\cite{12}. It
is noteworthy that the apparent hardness, as measured by means of
indentation methods, is known to change with the applied load ($P$)---a
decrease of $H$ is mostly observed with an increase of $P$---, with the
loading rate and duration, and with the environment. For a-SiO\tsub{2},
$H$ decreases from ${\approx}11.3$ to ${\approx}6.3$~GPa as the load
increases from 10~mN to 1~N, and for a standard window glass it changes
from ${\approx}6.3$ to 4.5~GPa as the load increases from 10~mN to
10~N; this is the so-called Indentation-Size-Effect
(ISE)~\cite{13,14,15}. Besides, $H$ depends on the way the glass is
processed, and the purity of the constituents. As a matter of fact,
different values are reported for vitreous silica. For example, in the
case of fused silica glasses, hardness as obtained by means of a
Vickers indentation test at 1~N for 15~s is 8.7~GPa for a ``vitreosil''
glass (Saint-Gobain Co.)~\cite{15}, whereas in the same conditions,
$H=9.6$~GPa for a ``suprasil'' (Heraus Conamic Co.)~\cite{16}. Hardness
was found to scale with $\mu$ in many crystallized systems and is also
more or less scaling with the melting point, $T_{m}$, for crystals and
with $T_{g}$ for glasses. But these are rough approximations
(Figure~\ref{fig4}) since, on the one hand, the atomic packing factor
is ignored, on the other different mechanisms may result in the same
hardness number. For example, the dashed line in Figure~\ref{fig4}
crosses data points associated with a hardness of 7~GPa and
corresponding to glass compositions from very different systems,
including  Cu\tsub{60}Hf\tsub{10}Zr\tsub{20}Ti\tsub{10}
($T_{g}=481$~\textdegree C),
Si\tsub{0.168}Ca\tsub{0.084}Mg\tsub{0.084}Al\tsub{0.064}O\tsub{0.6}
($T_{g}=732$~\textdegree C) and a-SiO\tsub{2} ($T_{g}=1190$~\textdegree
C). As\tsub{0.2}Se\tsub{0.8},
B\tsub{0.08}Ca\tsub{0.09}P\tsub{0.18}O\tsub{0.65}, and
Ce\tsub{70}Al\tsub{10}Ni\tsub{10}Cu\tsub{10} glasses have also the same
hardness, $H=1.5$~GPa, despite very different transition temperatures
(from 97~\textdegree C for the chalcogenide composition to
611~\textdegree C for the boro-phosphate one). The hardness of glasses
containing a large free volume content, such as vitreous silica
(a-SiO\tsub{2}), is predominantly controlled by the volume density of
atomic bonds, regardless of the interatomic bond energy. This is the
reason behind a-SiO\tsub{2} ($H \approx$ 7--9~GPa) being softer than
its crystalline counterparts, quartz ($H=$ 10--12~GPa), coesite ($H
\approx 10$~GPa) or than its high-density variant, the stishovite ($H
\approx 33$~GPa)~\cite{17,18} (Figure~\ref{fig5}). With an increase in
the packing density, the bond strength becomes more important than the
packing density itself, and hardness is more and more controlled by the
resistance the glass offer to the shear flow. This is why {MG}s
can be stiffer than---and as hard as---oxide glasses, despite much
weaker interatomic bonding, thanks to a remarkable atomic packing\break
density.

\begin{figure}
{\vspace*{4pt}}
\includegraphics{fig04}
{\vspace*{4pt}}
\caption{\label{fig4}The hardness of glass (mostly measured by means of
the Vickers indentation test) as a function of the glass transition
temperature. Reprinted and adapted from Ref.~\cite{12}.}
\end{figure}

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}The hardness of glass as a function of the shear
modulus. Data on crystallized polymorphs and metamorphic minerals were
taken from Refs.~\cite{17,18}. Other data are from
Refs.~\cite{4,6,12}.}
\end{figure}

In the case of crystalline materials, and especially metals, the
residual indentation print can be attributed to the movement of
dislocation and is chiefly correlated to the shear modulus (per the
Peierls expression for the lattice friction force), and to a lesser
extent to some diffusion processes (especially at elevated temperature)
when the transport of matter plays a role in the permanent deformation,
as in the cases of dislocation climb or diffusion creep for example.
Therefore, hardness values are typically below 3~GPa for metals, and
$H$ is proportional to the macroscopic flow stress and to the shear
modulus: $H\approx 0.01\mu$ (Figure~\ref{fig4}). {MG}s, which
deform by localized shear-transformation-zone processes~\cite{19}, are
mostly hard ($H$ ranges between 2 and 18~GPa)  and exhibit a much
larger slope, $H\approx 0.145\mu$, but smaller than the one for
silicate glasses, $H\approx 0.2\mu$. As a consequence, although
{MG}s are mostly much stiffer than silicate glasses, they are
only slightly harder. The differences observed between these classes of
materials derive from the fundamental changes in the deformation
mechanism and is discussed below.

The possible and eventually concomitant mechanisms resulting in a
permanent imprint at the surface of glass in ambient conditions include
(i) densification; (ii) isochoric shear flow; and (iii) damage-based
processes (microcracking)~\cite{20}. The stress in the contact area,
which decreases toward the hardness value upon loading (as the contact
area grows), is obviously much larger than the stress that glasses can
withstand during ``ordinary'' mechanical testing (such as bending or
compression tests) or in service conditions, and are sufficient to
generate some densification in the process zone beneath the imprint.
Densification at indentation sites was deduced from changes in the
refractive index as measured by optical interferometry~\cite{21,22}, as
well as by Raman spectroscopy~\cite{23,24}, and was recognized to be a
general property of inorganic non-metallic glasses. Densification,
which is a displacive transformation, shows up through a persistent
change of the atomic network structure in the region that extends up to
several times the indentation size beneath the surface, and involves a
collapse of matter into a relatively-more close packed structure. The
extent to which a glass can densify during indentation depends on its
atomic packing density. The smaller $C_{g}$ is, and the larger the
magnitude of the volume shrinkage becomes. In the case of amorphous
silica (a-SiO\tsub{2}), densification accounts for 80\% of the
indentation volume, whereas for a Zr-based {MG}s, which has a
random close packed structure, it contributes to less than 10\% of the
deformed volume~\cite{11,25}. A striking feature of the densification
mechanism is that it is promoted and eventually triggered by
shear~\cite{26}. The temptation is thus great to use an additive
relationship for the shear and the pressure contribution, as was
suggested by independent research teams~\cite{27,28}, who proposed a
yield function $f(\tilde{\sigma})$ that ressembles the Mohr--Coulomb
law used in soil mechanics (including granular media) where the local
shear is represented by $\uptau_{\mathrm{Max}}$, or the Drucker--Prager
criterion used in structure mechanics where an equivalent shear stress
is considered, but in which both shear and pressure work in favour of
yielding (unlike in soil mechanics)
{\begin{equation}\label{eq4}
f(\tilde{\sigma}) = {-}\sigma _{h} + \zeta \tau _{e} - \sigma _{o} 
\end{equation}}\unskip
where $\tilde{\sigma}$ is the stress tensor, $\upsigma_{h}$ is the
hydrostatic stress (mostly ${<}0$ beneath the contact area),
$\uptau_{e}$ is an equivalent shear stress (for instance $\tau_{e} =
\sqrt{\tilde{s}\colon \tilde{s}/2}$ where $\tilde{s}$ is the deviatoric
stress), $\upsigma_{o}$ is the critical yield stress (pressure) in
absolute value, and $\upzeta$ determines the sensitivity to the shear
component.

However, whereas the former expression supposes that shear would itself
be able to induce irreversible yield in the absence of any pressure,
provided $\zeta \tau _{e}=\sigma _{o}$, there is no evidence that
densification, which accounts for over 80\% of the indentation print in
a-SiO\tsub{2}, can occur solely in shear. An alternate proposition in
line with the schematic drawing proposed by Mackenzie~\cite{26} is that
shear plays the role of a trigger of an instability process
(kinematics), involving local shape changes of structural units of the
glass network, and resulting in a global volume contraction further
driven by pressure. This mechanism could resemble the one by which a
folding chair is folded hence reducing the space it occupies. The
``folding chair'' yield criterion could be written
{\begin{equation}\label{eq5}
f(\tilde{\sigma}) = {-}[1 + \zeta' (\tau_{e})] \sigma _{h} - \sigma _{o} 
\end{equation}}\unskip
where $\zeta'(\tau_{e})\geq 0$ and $\zeta'(0)=0$. 

A simple form for the shear function could be $\zeta'(\tau_{e}) = 
|\tau_{e}|/\tau_{n}$ where $\tau_{n}$ is a normalizing constant.
Further investigations in this area should account for some key and
seemingly unique features of the densification of amorphous silica such
as (i) the astonishing shear yield strength dependence on
pressure~\cite{29}, which is intimately linked to structural
rearrangements (in particular the increase in the Si coordination
number) occurring at the atomic scale, and (ii) the strong non-linear
elastic behaviour that shows up in glass in the stress range of
concern, and (iii) the dramatic change in the elastic properties
occurring as densification proceeds.

Beside densification, the other major contribution to the formation of
an imprint is the isochoric and shear-driven flow of matter. The shear
deformation processes, which are reconstructive by nature, may greatly
differ depending on the material. Shear-thinning viscous flow, where
the non-linear rheological behavior stems from the high pressure that
builds at indentation site, can be invoked for non-metallic glasses. In
the case of {MG}s, the fundamental unit processes of deformation
occur via the collective shuffling of clusters of atoms to accommodate
the applied shear strain. These are termed ``shear transformation
zones'' ({STZ}s), and result in an inhomogeneous plastic flow
into narrow shear bands with typical thickness of ${\sim}10$
nm~\cite{19,30}. These shear bands, after propagating through a
characteristic distance, become shear cracks. The absence of
dislocations in {MG}s, however, makes them considerably harder
than their crystalline counterparts. 

Whereas elastic moduli, and at the first place $K$, can be seen as bulk
material properties, reflecting the energy content per unit volume in
the material, or in other words the arrangement of bonds in space,
hardness accounts for some fine heterogeneous details of the atomic
organization, such as the presence of easy shear paths (slip planes for
crystals, weak channels\break in glass).

\section{Strength, fracture surface energy and fracture toughness}
\label{sec4}

The intrinsic strength of silica and silicate glasses is of the order
of 14--20~GPa in vacuum, as shown both experimentally in
tension~\cite{31} and by {MD} simulation~\cite{32}. However, the
strength of a-SiO\tsub{2} decreases down to 6--7~GPa in moisture
environment~\cite{31} and, indeed, some more recent studies dealing
with the behavior of $\upmu$m size micropillars suggested a yield
stress (plastic flow) in compression in air of about 7~GPa~\cite{33}.
The fracture surface energy ($\gamma$) of a-SiO\tsub{2}, as obtained in
vacuum using the double-cantilever-cleavage technique~\cite{34} is
equal to 3.62~J${\cdot}$m$^{-2}$. Taking a value of 70~GPa for Young's
modulus ($E$) and 20~GPa for $\sigma_{d}$ further results in a critical
distance $r_{0}$ equal to 1.27~nm in the Orowan--Gilman equation for
the theoretical cohesive strength:
{\begin{equation}\label{eq6}
\sigma _{d} = \sqrt{\dfrac{2\gamma E}{r_{0}}}.
\end{equation}}\unskip

This value is much larger (by about one order of magnitude) than any
near-neighbor interatomic distances in glasses. For example, the Si--O
distance in a-SiO\tsub{2} is close to 1.62~\AA{}. Glasses are
characterized by some medium range ordering occurring at a mesoscale,
with a length scale in-between the interatomic distance and say a few
nanometers, and involving structural units such as chains
(chalcogenides), clusters (MGs), and rings (silicates, borates, etc.).
For example, rings consisting of SiO\tsub{4} tetrahedra connected
through Si--O--Si bridges of different sizes are found in a-SiO\tsub{2}
and silica-rich glasses. Although the average ring size is at about
6~\AA{} and corresponds to 6--8 members rings, some larger rings, with
10--12 members are observed, with a size that approaches the critical
$r_{0}$ value, and this is presumably the weakest link of the atomic
network where fracture is initiated. The value for $r_{0}$ being much
larger than any distance at the short-order arrangement scale, it is
hypothesized that the critical length it is not much affected by
compositional changes within a specific chemical system. 

Ductility can be assessed in different ways, such as by means of the
measurement of the tensile strain at fracture or using fracture
toughness measurements. In the latter case, the measured fracture
toughness value ($K_{Ic}$ in opening mode) mostly accounts for the
occurrence of dissipation mechanisms at the vicinity of the crack
front, such as confined plasticity and viscoplasticity. As~a~matter of
fact, the obtained $K_{Ic}$ values are much larger than the one that
would be predicted from the Irwin--Griffith similarity principal: 
{\begin{equation}\label{eq7}
K_{Ic} = \sqrt{2E'\gamma}
\end{equation}}\unskip
where $E'=E/(1-\nu^{2})$ and $E'=E$ respectively in plane strain and in
plane stress situations.

Consequently, $K_{Ic}$ can no longer be considered as an intrinsic
property of the material as it strongly depends on the loading-rate, on
temperature and also on the sample size in cases where the confined
deformation zone extends to a significant fraction of the specimen
thickness. In this regard, inorganic and non-metallic glasses offer a
unique opportunity to measure the intrinsic toughness, that is a
parameter directly related to the energy required to create the
fracture surface through a bond disruption process, provided the crack
is atomically sharp. In order to fulfil this requirement, a few
self-consistent methods were successfully applied to glass samples from
different chemical systems but metallic ones, namely the Controlled
Surface Flaw technique ({CSF}), the Chevron-Notched beam technique
({CN}), and the Single Edge Pre-cracked Beam technique ({SEPB}) (see
Ref.~\cite{35} for details). The experimental results are plotted as a
function of hardness in Figure~\ref{fig6}. The general trend is that
the hardest materials are also the most brittle. Nevertheless, the fine
details within a class of materials might be different, and as a matter
of fact, the toughest glasses among non-metallic inorganic systems,
namely the silicon oxynitride glasses, are also the hardest ones! 

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}The fracture toughness (intrinsic) of glasses from
different chemical systems as a function of hardness. Data were
extracted from  Refs.~\cite{36,37}.}
\end{figure}

In addition to the experimental investigations, a relatively simple
approach to predict $\gamma$ and $K_{Ic}$ in a quantitative manner
consists in assuming that a propagating crack extends following a path
disrupting the weakest links of the energy landscape, and to estimate
the surface energy from the bond strength and the bond concentration
along this fracture surface~\cite{36}. The intrinsic (or theoretical)
fracture surface energy is obtained from the surface density of
representative structural units and from the relevant bond strength.
This approach was successfully applied to relatively complex glass
systems replacing the cleavage plane by a plane of arbitrary
orientation (random) and taking the number of disrupted bonds from the
stoichiometric formula (gram-atom), assuming that a crack
preferentially cuts the weakest links of the network. Let $x_{i}$ be
the stoichiometric fraction of the species involved in the $i${th}
bonding energy $U_{oi}$ (in J${\cdot}$mol$^{-1}$), between the $i${th}
cation (${A}_{i}$) and a first neighbor oxygen anion in the case of an
oxide glass, and let $n_{i}$ be the number of such bonds supposed to be
broken as the crack front propagates to the next unit, then $\gamma$ is
expressed as:
{\begin{equation}\label{eq8}
\upgamma = \dfrac{1}{2} \left(\frac{\rho}{M_{o}}\right)^{2/3}
\mathcal{N}^{-1/3} \displaystyle \sum _{i}x_{i}n_{i}U_{oi}
\end{equation}}\unskip
where the ${1}/{2}$ pre-factor on the right-hand side member accounts
for the fact that the bond disruption process leads to the formation of
two complementary surfaces.

An equivalent expression to (\ref{eq8}) is obtained with $C_{g}$: 
{\begin{equation}\label{eq9}
\upgamma = \dfrac{1}{2} \left(\displaystyle \sum _{i} \dfrac{4}{3}
\rmpi x_{i}{r}_{i}^{3}\right)^{-2/3} \mathcal{N}^{-1}{C}_{g}^{2/3}
\langle U_{0} \rangle  
\end{equation}}\unskip
where $\langle U_{0} \rangle = \sum _{i}x_{i}n_{i}U_{oi}$ is the mean
energy content in a gram-atom of glass.

Some theoretical and experimental data were extracted from
Refs.~\cite{36,37,38,39,40,41,42,43} and are summarized in
Table~\ref{tab1}. The agreement between the theoretical (last two
columns) and experimental values of $\gamma$ for glasses from various
chemical systems corroborate the assumption that the crack front
opening displacement is within the interatomic distance, and fracture
is purely elastic.

\begin{table}
\caption{\label{tab1}$K_{Ic}$ and $\gamma$ of glasses from different
chemical systems (from Refs~\cite{36,37})}
\tabcolsep2.5pt\fontsize{9}{10.8}\selectfont
\begin{tabular}{ccccccc}
\thead
Glass & \parbox[t]{2pc}{\centering $E$ (GPa)} & $\nu$ & 
\parbox[t]{4pc}{\centering $K_{Ic}$ (MPa${\cdot}$m$^{0.5}$)} & Method & 
\parbox[t]{3pc}{\centering $\gamma$~(\ref{eq8}) (J${\cdot}$m$^{-2}$)} & 
\parbox[t]{4pc}{\centering $K_{Ic}$~(\ref{eq7}) (MPa${\cdot}$m$^{0.5}$)}
\vspace*{2pt} \\
\endthead
a-SiO\tsub{2} & 70 & 0.15\0 & 0.73 & \parbox[t]{3pc}{\centering {DCC}
(in vac.) \cite{38}} & 3.62 & 0.718\vspace*{2pt} \\ 

Si\tsub{0.25}Na\tsub{0.92}Ca\tsub{0.035}Mg\tsub{0.021}O\tsub{0.602} & 
72 & 0.224 & 0.7\0 & 3.55 \cite{36} & & 0.734 \\ 

Ti\tsub{0.013}Si\tsub{0.287}Na\tsub{0.067}O\tsub{0.633} & 65.3 &  0.215
& 0.68 & {SEPB} \cite{39} & 3.57 & 0.70\0 \\ 

Ti\tsub{0.013}Si\tsub{0.270}Na\tsub{0.1}O\tsub{0.617} & 66.1 & 0.201 &
0.6\0 & {SEPB} \cite{39} & 3.55 & 0.699 \\ 

Ti\tsub{0.013}Si\tsub{0.253}Na\tsub{0.133}O\tsub{0.6} & 63.9 & 0.232 &
0.6\0 & {SEPB} \cite{39} & 3.46 & 0.679 \\ 

Ti\tsub{0.037}Ba\tsub{0.111}Si\tsub{0.222}O\tsub{0.63} & 74.9 & 0.276 &
0.47 & {SEPB} \cite{36} & 0.86 & 0.374 \\ 

Si\tsub{0.196}Na\tsub{0.008}K\tsub{0.005}Pb\tsub{0.196}O\tsub{0.595} &
55 & 0.248 & 0.62 & {CN} \cite{40} & 2.83 & 0.576 \\ 

Pb\tsub{1/7}B\tsub{2/7}O\tsub{4/7} & 57.5 & 0.289 & 0.35 &  {SEPB}
\cite{36} & 0.88 & 0.332 \\ 

Pb\tsub{0.114}B\tsub{0.286}Cu\tsub{0.029}O\tsub{0.571} & 67.4 & 0.298 &
0.42 & {SEPB} \cite{36} & 0.86 & 0.358 \\ 

Zn\tsub{1/7}B\tsub{2/7}O\tsub{4/7} & 80.9 & 0.33\0 & 0.4\0 & {SEPB}
\cite{36} & 0.62 & 0.336 \\ 

Si\tsub{0.215}K\tsub{0.031}B\tsub{0.123}O\tsub{0.631} & 64 & 0.233 & 
0.73 & {SEPB} \cite{41} & 4.09 & 0.744 \\ 

Si\tsub{0.226}Al\tsub{0.065}B\tsub{0.065}O\tsub{0.645} & 70 & 0.208 &
0.79 & {SEPB} \cite{41} & 4.03 & 0.767 \\ 

Si\tsub{0.235}B\tsub{0.039}Pb\tsub{0.098}O\tsub{0.627} & 63 & 0.261 &
0.66 & {SEPB} \cite{41} & 3.67 & 0.704 \\ 

Si\tsub{0.245}Na\tsub{0.024}Al\tsub{0.012}B\tsub{0.079}O\tsub{0.639} &
63.7 & 0.2\0\0 & 0.68 & {CN} \cite{42} & 3.88 & 0.718 \\ 

P\tsub{0.216}Al\tsub{0.038}K\tsub{0.056}Ba\tsub{0.023}Nd\tsub{0.008}O\tsub{0.659}
& 50 & 0.256 & 0.48 & {CN} \cite{40} & 2.55 & 0.522 \\ 

Y\tsub{4.70}Mg\tsub{6.56}Si\tsub{16.78}Al\tsub{11.67}O\tsub{51.5}N\tsub{8.75} 
& 134 & 0.28\0 & 1.18 & {CN} \cite{43} & 3.73 & 1.04\0 \\ 

Ge\tsub{0.15}Se\tsub{0.85} & 13.8 & 0.295 & \00.122 & {CN} \cite{36} &
2.29 & 0.26\0 \\ 

Ge\tsub{0.25}Se\tsub{0.75} & 16.1 & 0.281 & \00.215 & {CN} \cite{36} &
1.77 & 0.25\0 \\ 

Ge\tsub{0.3}Se\tsub{0.7} & 17.9 & 0.264 & \00.210 & {CN} \cite{36} &
1.55 & 0.24\0 \\ 

SiO\tsub{0.141}C\tsub{0.3} & 96.7 & 0.18\0 &  0.73 & {SEPB} \cite{37} &
{nd} & {nd} \\ 

SiO\tsub{1.82}C\tsub{0.42}Zr\tsub{0.1} & 85.4 & 0.21\0 & 0.91 &  {SEPB}
\cite{37} & {nd} & {nd}\vspace*{2pt}
\botline
\end{tabular}
\tabnote{The theoretical value for $K_{Ic}$ is derived from $\gamma$,
$E$ and $\nu$ using the plane strain assumption. Experimental errors
are typically of $\pm$1~GPa for the elastic moduli and ${\pm}$0.05
MPa$\cdot\sqrt{\mathrm{m}}$ for fracture toughness as measured by the
{SEPB} or {CN} methods. {DCC} stands for Double Cantilever Cleavage
(also known as Double Cantilever Beam) specimen. ``{nd}'' stands for
``non-determined''.}
\end{table}

As far as non-metallic inorganic glasses are concerned, $K_{Ic}$ and
$\gamma$ range typically from 0.1 to  1.5~MPa$\cdot\sqrt{\mathrm{m}}$
and from 0.5 to 8 J${\cdot}$m$^{-2}$ respectively. The most brittle
glasses are found in the chalcogenide and lead-borate systems, which
turn out to exhibit at the same time a relatively large atomic packing
density and a relatively small interatomic bond strength. In contrast,
silicon oxynitride and oxycarbide glass forming systems provide the
toughest glasses. The situation is nevertheless different in these
latter systems. In the case of silicon oxynitride glasses, the large
$K_{Ic}$ values are attributed to the efficient packing of strongly
bonded atoms (these glasses are also the stiffest), whereas in the case
of silicon oxycarbide, the resistance toward crack initiation is
supposed to derive from the important $3D$ cross-linking of a network
consisting of strong bonds, and on possible crack tip shielding
phenomena as the crack meets the nanoscale cavities where the free
volume concentrates. It is noteworthy that the most brittle glasses,
namely the chalcogenides, are also the softest!

{\vspace*{-4pt}}

\section{Conclusion}\label{sec5}

{\vspace*{-2pt}}

Interatomic bonds are quite similar in a glass and its crystallized
polymorphs. Nevertheless, the atomic bonding arrangement in space is
different. In particular, the presence of free volume makes the volume
density of bonds, and thus the density of energy, smaller in glasses.
Therefore, elastic moduli, which are given in J${\cdot}$m$^{3}$ (Pa), are
usually smaller. Consequently, the bulk modulus of different polymorphs
depends nearly linearly on the atomic packing density. The free volume
content, or the atomic packing density has a tremendous influence on
the elastic moduli of glasses, and predominates over the bond strength
in many cases. This is why silicon oxynitride glasses are much stiffer
than their parent oxide glasses or why sodium alumino-silicate glasses
are stiffer than sodium alumino-borate glasses despite the fact that
the Si--N bond is weaker than the Si--O one and the B--O bond strength
is larger than the Si--O one.

However, as far as mechanical properties are concerned, it is important
to distinguish phenomena occurring at the global scale such as
volumetric change, and those occurring at some mesoscopic scale that
are more heterogeneous in nature, such as shear-induced processes and
fracture. These latter phenomena are mediated by the fine heterogeneous
details of the atomic network (sliding and cleavage planes in crystals
and easy shear paths in glass). 

Hardness is intimately linked to a combination of contributing
deformation mechanisms, including densification and shear processes.
Therefore, it is a composite property involving both the global and the
mesoscale at which shear localization occurs. The ease for dislocation
glide in pure metals makes them relatively soft in comparison to
metallic glasses, which are typically 15 times harder for a given shear
modulus. Ionocovalent crystals are advantaged by a more efficient
atomic packing than the glass with the same composition (when such a
material exists) and are mostly (slightly) harder than their amorphous
counterparts.

The intrinsic fracture surface energy ($\gamma$) is intimately linked
to the crack path. For ``simple'' mono-constituent glass such as
a-SiO\tsub{2}, the experimental value for $\gamma$ is in agreement with
the prediction based on a random planar fracture path through the
glass. In the case of multicomponent systems, it is necessary to
account for the weakest links of the atomic network. Fracture
toughness, as measured from a self-consistent method involving an
atomically sharp crack ranges from 0.1 to
1.5~MPa$\cdot\sqrt{\mathrm{m}}$ ($\gamma$ is between 0.5 and
8~J${\cdot}$m$^{-2}$), silicon oxynitride and oxycarbide glasses being
the toughest among non-metallic inorganic glasses. Interestingly, the
toughest glasses are also the hardest ones.

\section*{Conflicts of interest}

The author has no conflict of interest to declare.

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