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\DOI{10.5802/crmeca.293}
\datereceived{2024-10-31}
\daterevised{2025-03-09}
\dateaccepted{2025-03-10}
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\dateposted{2025-05-20}
\begin{document}

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\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Particle size effect on the strength of particle-reinforced
composites. Experimental analysis and comparison with the coupled
criterion}

\alttitle{Effet de la taille des particules sur la r\'esistance des
composites renforc\'es par particules : analyse exp\'erimentale et
comparaison avec le crit\`ere coupl\'e}

\author{\firstname{Israel Garc\'ia} \lastname{Garc\'ia}\CDRorcid{0000-0002-3894-5304}\IsCorresp}
\address{Departamento de Mec\'anica de Medios Continuos y Teor\'ia de
Estructuras, Escuela T\'ecnica Superior de Ingenier\'ia, Escuela
Polit\'ecnica Superior. Universidad de Sevilla. Camino de los
Descubrimienos s/n, 41092 Sevilla, Spain}
\email[I. G. Garc\'ia]{israelgarcia@us.es}

\author{\firstname{Jes\'{u}s} \lastname{Justo}\CDRorcid{0000-0003-1396-8056}}
\addressSameAs{1}{Departamento de Mec\'anica de Medios Continuos y
Teor\'ia de Estructuras, Escuela T\'ecnica Superior de Ingenier\'ia,
Escuela Polit\'ecnica Superior. Universidad de Sevilla. Camino de los
Descubrimienos s/n, 41092 Sevilla, Spain}
\email[J. Justo]{jjusto@us.es}

\author{\firstname{Alejandro} \lastname{Zurita Van-Dinter}}
\addressSameAs{1}{Departamento de Mec\'anica de Medios Continuos y
Teor\'ia de Estructuras, Escuela T\'ecnica Superior de Ingenier\'ia,
Escuela Polit\'ecnica Superior. Universidad de Sevilla. Camino de los
Descubrimienos s/n, 41092 Sevilla, Spain}
\email[A. Zurita Van-Dinter]{azurita.etsi@gmail.com}

\author{\firstname{Vladislav}\nobreakauthor\lastname{Manti\v{c}}\CDRorcid{0000-0002-7569-7442}}
\addressSameAs{1}{Departamento de Mec\'anica de Medios Continuos y
Teor\'ia de Estructuras, Escuela T\'ecnica Superior de Ingenier\'ia,
Escuela Polit\'ecnica Superior. Universidad de Sevilla. Camino de los
Descubrimienos s/n, 41092 Sevilla, Spain}
\email[V. Manti\v{c}]{mantic@us.es}

\shortrunauthors

\begin{abstract} 
Particle-reinforced composites are widely used in industry, primarily
due to their versatile fabrication methods and the ability to tailor
their properties. In many cases, extensive experimental campaigns are
required to determine the optimal characteristics of the system to
enhance specific properties. Micromechanical models can serve as a
useful alternative or initial approach during the material design
process. One of the easiest characteristics that can be modified is the
size of the reinforcement, which, according to some models and
preliminary evidence, can significantly affect the  mechanical
properties of the material.


The objective of this work is to experimentally evaluate the size
effect of reinforcement on the composite strength and to compare it 
with the predictions by the coupled criterion of finite fracture
mechanics (CC-FFM). A secondary objective is to visualize the
initiation of the failure mechanism, which starts at the
particle-matrix interface and progresses toward a crack that splits the
specimen. To achieve this, a new specimen design is proposed along with
an optimized fabrication procedure. The tests were recorded using a
high-speed camera, which allowed for the visualization of crack
initiation at the particle-matrix interface.

The experimental results show a strong size effect, where smaller
particles correspond to higher apparent strength. The results are in
relatively good agreement with the predictions of the CC-FFM.
\end{abstract}

\begin{altabstract} 
Les composites renforc\'es par particules sont largement utilis\'es
dans l'industrie, principalement en raison de la diversit\'e de leurs
m\'ethodes de fabrication et de la possibilit\'e d'adapter leurs
propri\'et\'es. Dans de nombreux cas, des campagnes exp\'erimentales
approfondies sont n\'ecessaires pour optimiser les caract\'eristiques
du syst\`eme et am\'eliorer certaines propri\'et\'es sp\'ecifiques. Les
mod\`eles microm\'ecaniques constituent une alternative utile ou une
premi\`ere approche lors du processus de conception des mat\'eriaux.
L'une des caract\'eristiques les plus faciles \`a modifier est la
taille du renfort, qui, selon certains mod\`eles et preuves
pr\'eliminaires, peut influencer significativement les propri\'et\'es
m\'ecaniques du mat\'eriau.

L'objectif de cette \'etude est d'\'evaluer exp\'erimentalement
l'effet de la taille du renfort sur la r\'esistance du composite et
de le comparer aux pr\'edictions du crit\`ere coupl\'e de la
m\'ecanique de la rupture finie (CC-MRF). Un objectif secondaire est
d'observer le m\'ecanisme d'amor\c{c}age de la rupture, qui d\'ebute \`a 
l'interface particule-matrice et \'evolue vers une fissure scindant
l'\'eprouvette. Pour cela, un nouveau design d'\'eprouvette est
propos\'e, ainsi qu'un proc\'ed\'e de fabrication optimis\'e. Les
essais ont \'et\'e enregistr\'es \`a l'aide d'une cam\'era \`a haute
vitesse, permettant de visualiser l'initiation de la fissure \`a 
l'interface particule-matrice.

Les r\'esultats exp\'erimentaux mettent en \'evidence un fort effet de
taille, o\`u des inclusions plus petites entra\^inent une r\'esistance
apparente plus \'elev\'ee. Ces r\'esultats sont en accord relativement
bon avec les pr\'edictions du CC-MRF.
\end{altabstract}

\keywords{\kwd{Size effect}
\kwd{Particle-reinforced composites}
\kwd{Spherical inhomogeneity}
\kwd{Coupled criterion}
\kwd{Finite fracture mechanics}
\kwd{Experimental fracture mechanics}}

\altkeywords{\kwd{Effet de taille}
\kwd{Composites renforc\'es par particules} 
\kwd{Inhomog\'en\'eit\'e sph\'erique} 
\kwd{Crit\`ere coupl\'e} 
\kwd{M\'ecanique de la rupture finie} 
\kwd{M\'ecanique exp\'erimentale de la rupture}}

\dedicatory{This article is  dedicated to the memory of Professor
Dominique Leguillon}

\thanks{Ministerio de Ciencia e Innovaci\'on de Espa\~na (Projects
PID2020-117001GB-I00/AEI/10.13039/{\ubreak}501100011033, PID2021-123325OB-I00), 
European Regional Development Fund (Project UNSE15-CE-3581)}

\maketitle

\end{noXML}

\section{Introduction} \label{sec-Intro}
Particle-reinforced composites are becoming more and more prevalent in
industrial applications~\cite{raja2019}. This type of composite
materials is particularly adequate for tailoring certain properties of
the unreinforced matrix without increasing significantly the complexity
of the fabrication process. The addition of particles to certain
materials is able to enhance some properties, such as electrical or
thermal conductivity~\cite{murakami2024}, stiffness~\cite{zhang2024},
tensile strength~\cite{satheeskumar2024}, or fracture
toughness~\cite{garg1988,nakamura1992}.

The mechanical properties of this type of material can be effectively
enhanced by carefully selecting the particles to be added. The
modification of these properties depends not only on the type and
volumetric fraction of the added materials but also on the particle
shape and size. This relationship has been shown in various
experiments, see~\cite{enginosoy2019} for a review. Given the influence
of micromechanics on macroscopic behavior, it is essential to fully
understand the failure mechanisms at the microscale, as highlighted
in~\cite{paris2001} in a similar material system. This understanding is
crucial for predicting and tailoring mechanical performance, ultimately
ensuring reliability in both macroscopic and mesoscopic models, which
are based primarily on phenomenological
observations~\cite{khurshudyan2019, majewski2020}.

The first stage of the failure mechanism in particle-reinforced
composites is typically associated with the particle-matrix interface.
There are two main reasons for this: (a) the stress concentration
generated by the presence of a particle, either at the poles for
particles stiffer than the matrix, see e.g.~\cite{cho2006}, or at the
equator for softer particles, and (b) the usually lower strength and
fracture toughness properties of the interface. For stiffer particles,
which is the typical case when enhancing mechanical properties is the
main objective of the added particles, the failure initiates as small
debonds at the particle-matrix interface~\cite{cho2006}. Typically, in
the most simple case of uniaxial tension, the debond appears initially
at the interface poles, progressing along the interface and finally
kinking towards the matrix~\cite{harding1997}. The first stage of the
failure mechanism is similar to the related problem of debonding at the
fiber-matrix interface, see, e.g.,~\cite{tabiai2018,uddin2024}.

In this initiation stage, two micromechanical characteristics strongly
influence the initiation and progression
of failure mechanisms: (i) the strength and fracture toughness of the
interface and (ii) the size and shape of the particle, see a review
in~\cite{fu2008}.

The influence of the interface properties has been reported both
theoretically~\cite{ye2017}  and experimentally~\cite{harding1997}.
Similar results have been reported in~\cite{tabiai2018} for a similar
system with fibrous reinforcement. The authors carried out experiments on
dog-bone specimens with a single fiber embedded. The~3D digital image
correlation technique allowed them to characterize the whole sequence
of the failure mechanism in this very related problem. 

The influence of the shape and size of the particles has been reported
by some limited experiments~\cite{cho2006}. The size effect has been
explained by using diverse models: In~\cite{liu2021}, the authors
compared different models predicting the size effect and found an
accurate correlation with experiments with a stress model combined with
the effect of nanoscale damage at the interface.  In~\cite{ye2017}, a
model was presented based on prescribing a cohesive law at the
interface and a coupled plasticity-damage model for the matrix in a
Representative Volume Element (RVE). The prediction of this model match
the experimental results in terms of the stress--strain curve.
However, the size effect was not studied,
but was expected to be inherent in the approaches used. In the same
line, in~\cite{magri2022} a computational approach was presented based
on micromechanical simulations prescribing a non-local plasticity model
in the matrix and a non-local ductile damage model based on the
well-known Gurson-Tvergaard-Needleman model, within a
Fast-Fourier-Transform approach. The results show a strong size effect,
even for systems with relatively high volumetric fractions. 

Among the diverse approaches used to predict this size effect, it is
worth highlighting the coupled criterion of  finite fracture mechanics
(CC-FFM). This is an interesting approach not only to predict, but also
to provide a physical explanation for the size effect. This criterion,
proposed in~\cite{leguillon2002,cornetti2006}, postulates that
debonding will initiate with a finite extension when both stress and
energy conditions are met simultaneously, see~\cite{doitrand2024} for a
recent review on this criterion. Size effect has been predicted by this
approach for very diverse problems, see
e.g.~\cite{mantic2009,sapora2018,doitrand2021,chao2021,ferrian2023,baldassari2023,aranda2023}, 
and even used to explain the size effect found by other
approaches~\cite{garcia2014-EFM,molnar2020}. In this sense, the present
authors~\cite{garcia2015-CSTE} proposed a model based on the CC-FFM to
predict the crack initiation at the particle-matrix interface. The
model presents, among other results, a strong size effect, predicting a
delay in crack initiation for smaller particles under monotonous
loading. A more general model was presented in~\cite{martin2020},
including residual thermal stresses and different volumetric fractions
with multi-particle models. In~\cite{gentieu2019},  a comparison of the
predictions of the CC-FFM with a computational model
using a cohesive zone model at the particle-matrix interface 
was presented. The
results show a good agreement between the two on  a qualitative level, but
a quantitative divergence in the asymptotic tendency for small
particles, in line with the results in~\cite{garcia2014-EFM} for the
problem of fiber-matrix debonding. 

Despite the fact that different models predict the size effect on this
type of material, the experimental results   validating this size
effect are limited.  It is worth highlighting the experiments
in~\cite{cho2006}, which demonstrate a strong size effect that aligns,
at least qualitatively, with the predictions of most previous models,
particularly those based on the CC-FFM. However, these experimental
results are limited because of the use of multi-particle specimens with
no control over the inter-particle distance, and thus the influence of
nearby particles.  Lauke~\cite{lauke2016} presented a specimen
specifically designed to study the effect of multiaxial loading in the
failure mechanisms around the spherical particle. However, to the best
of our knowledge, there are no experimental results in the literature
using this proposal yet. In view of these facts, it is necessary to
obtain new experimental evidence to validate the main results predicted
by these models. 

The objective of the present work is to design a single-particle
specimen to be tested under tension. The key idea is to validate the
predictions on the size effect, keeping the specimen  as simple as possible to avoid
any influence of other parameters, such as volume fraction, interface
finish, or multiaxial loading. These effects could be the objective of
further work once the model is validated for the simplest case. The
secondary objective is to observe the sequence of the failure
mechanism, the symmetry of debonding, and the manner in which the crack
migrates towards the matrix. This observation will contribute to
validating the failure sequence assumed by the theoretical models. 

The document is organized as follows: First, the CC-FFM model for this
problem will be briefly introduced in \refsec{sec-CoupledCriterion}.
The experimental setup will be described in terms of specimen design
and fabrication in \refsec{sec-SpecimenDesign}. Finally,
the experimental results are presented and discussed in
\refsec{sec-ExpResults}, where they are compared with the predictions
given by the CC-FFM model.


\section{Coupled criterion analysis} \label{sec-CoupledCriterion}
The problem under study is schematized in \reffig{fig:sph-schema},
where a particle-reinforced material is subjected to uniaxial tension.
Assuming a low volumetric fraction, a spherical particle of radius $R$
surrounded by an infinite matrix can be studied as a representative
volume element, neglecting the effect of nearby particles. The particle
material is considered to be stiffer than the matrix. This simplified
axisymmetric
model allows focusing the analysis on the sequence of events of the
failure mechanism and the size effect. In addition, this simplicity and
the CC-FFM allow us to obtain quasi-analytical expressions for the
stresses leading to the formation of the first debond. Note that the
schematic assumes that the debond will appear in only one of the poles,
in a non-symmetric manner. This result is predicted by the CC-FFM, as
proved in~\cite{garcia2015-CSTE} and discussed in detail for a related
problem in~\cite{garcia2015-EJMAS}. Thus, this result is also to be
validated by the experiments presented here. 

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:sph-schema}Schematic of the problem under study.}
\end{figure}

The CC-FFM is based on assuming that a finite-length crack onset takes
place when two conditions are met simultaneously: a stress condition
defined on the stresses before the crack onset and an energy
condition based on an energetic balance between the states before and
after the crack onset. In this case, the states before and after the
debond are represented in \reffig{fig:sph-schema}. In what follows, the
two criteria will be briefly studied in
\refsecs{subsec-StressCriterion}{subsec-EnergyCriterion} to be
subsequently combined in \refsec{subsec-CouplingTheTwoCriteria}, where
the expression that will be compared with the experiments is fully
developed.

\subsection{Stress criterion} \label{subsec-StressCriterion}
The stress criterion is based on the evaluation of the stresses before
the debond onset, i.e.~the state (I) in \reffig{fig:sph-schema}.
According to the CC-FFM, a debond onset is possible only at those
points with angle $\theta \in$ [0\textdegree, 180\textdegree] of the
interface where a certain combination of the interface normal and shear
tractions, $\sigma (\theta)$ and $\tau (\theta)$, respectively, is
greater or equal to a critical value. Since the poles are the most
stressed points at the interface, it is assumed that the debond onset
will always occupy the pole  ($\theta=0$\textdegree) and surrounding
points up to a certain angle $\Delta \theta$. In this case, the
condition given by the stress criterion for a debond onset will be:
{\begin{equation}
\sigma_{\text{eq}} (\sigma,\tau) =
\sqrt[p]{\left\langle\text{sgn}(\sigma) \left|\sigma\right|^p + 
\left( \frac{|\tau|}{\upmu} \right)^p\right\rangle_{+} } 
\geq \sigma_{\mathrm{c}} \quad \forall \theta \in [0,\Delta \theta] ,
 \label{eq-iniSC}
\end{equation}}\unskip
where $\sigma_{\mathrm{c}}$ is the interface tensile strength, and
$\mu=\tau_{\mathrm{c}}/\sigma_{\mathrm{c}}$ is the ratio of interface shear to
interface tensile strength. The exponent $p\geq 1$ controls the
coupling between normal and shear tractions, see~\cite{garcia2012-IJSS}
for more details about this expression for the stress criterion.

Assuming linear elastic behavior for the spherical particle and matrix
material and assuming a perfect interface before the debond onset, the
normal $\sigma$ and shear $\tau$ tractions can be obtained exactly
using the expressions by~\cite{goodier1933} as,
{\begin{equation}
\sigma(\theta)  = \sigma^\infty \hat{\sigma}(\theta)=
\sigma^\infty\left(k + m \cos{2 \theta}\right),\quad
\tau(\theta)  =  \sigma^\infty \hat{\tau}(\theta)=
\sigma^\infty  m \sin{2 \theta}
 \label{eq-tension}
\end{equation}}\unskip
where $\sigma^{\infty}$ is the remote tension and $k$ and $m$ depend on
the elastic properties of particle and matrix in the following form:
{\begin{subequations}
\label{eq:km}
\begin{equation}
k = \frac{1}{2} \frac{1+\alpha}{1+\beta} \frac{2+\alpha-\beta}
{1+\alpha-2 \beta},
\end{equation}
\begin{equation}
m = \frac{1+\alpha}{1+\beta},
\end{equation}
\end{subequations}}\unskip
{\begin{subequations}
\begin{equation}
 \alpha = \frac{\mu_1 (\kappa_2+1) - \mu_2 (\kappa_1+1)}{\mu_1 
 (\kappa_2+1) + \mu_2 (\kappa_1+1)},
\end{equation}
\begin{equation}
 \beta = \frac{\mu_1 (\kappa_2-1) - \mu_2 (\kappa_1-1)}
 {\mu_1 (\kappa_2+1) + \mu_2 (\kappa_1+1)},
\end{equation}
\end{subequations}}\unskip
where $\mu_i = E_i /(2(1+\nu_i))$ and $\kappa_i=3-4 \nu_i$, $E_i$ and
$\nu_i$ denoting Young's modulus and Poisson's ratio, respectively, of
particle ($i=1$) and matrix ($i=2$), and $\alpha$ and $\beta$ are 
Dundurs parameters of the bimaterial.

Introducing \refeqn{eq-tension} in \refeqn{eq-iniSC} and after some
rearrangements, the expression obtained for the stress criterion is:
{\begin{equation}
 \frac{\sigma^{\infty}}{\sigma_{\mathrm{c}}} \geq s (\Delta \theta) = 
 \max_{\theta \in [0,\Delta \theta]} \frac{1}{\sqrt[p]{\left\langle
 \text{sgn}(k+m\cos 2 \theta) \left|k+m\cos 2 
 \theta\right|^p + \left(\frac{|m \sin 2 \theta|}{\mu} 
 \right)^p\right\rangle_{+} }}.
 \label{eq-finalSC}
\end{equation}}\unskip
The dimensionless function $s (\Delta \theta)$ gives the minimum value required for
the remote tension as a function of the size of the debond at its
onset.

The expression presented here is the most simple, assuming linearity in
the material behavior and conditions, in order to keep the comparison
as simple as possible to be able to understand the basic behavior. The
extension of the stress criterion to take into account viscous,
nonlinear material effects and even higher volumetric fractions could
be obtained by using the expressions in~\cite{kammer2022}.

\subsection{Energy criterion} \label{subsec-EnergyCriterion}
The energy criterion, within the context of the finite fracture
mechanics, postulates that the debond onset is possible if the
energetic balance between the states before (I) and after the debond
onset (II) is thermodynamically admissible. This is an extension of the
classical Griffth criterion~\cite{griffith1921} to an incremental
balance between two states, rather than a differential approach. In
this sense, the incremental balance can be written as:
{\begin{equation}
 \Delta \Pi + \Delta E_{\mathrm{k}} + E_{\mathrm{d}} = 0,
 \label{eq-ECinit}
\end{equation}}\unskip
where $\Delta \Pi$ and $\Delta E_{\mathrm{k}}$ are the increments of
potential elastic and kinetic energy, respectively. The term
$E_{\mathrm{d}}$ refers to the energy dissipated during the crack
initiation process. If the initial state is assumed to be static,
$\Delta E_{\mathrm{k}} \geq 0$, the expression in \refeqn{eq-ECinit} can be
rewritten as:
{\begin{equation}
 - \Delta \Pi \geq E_{\mathrm{d}},
 \label{eq-EC-balance2}
\end{equation}}\unskip
so the application of the energy criterion reduces to evaluate the
change in potential elastic energy and the dissipated energy.

The change in potential elastic energy $\Delta \Pi$ can be obtained in
different manners, see Section~3.4.1 in~\cite{garciatesis} for a
review. In this case, in line with~\cite{garcia2015-CSTE}, this term
will be obtained by integrating the energy release rate (ERR) $G$ for a
crack slowly growing from a debond angle $\theta_{\mathrm{d}} = 0$  to an
angle $\theta_{\mathrm{d}} = \Delta \theta$ from the pole
{\begin{equation}
 - \Delta \Pi = \int_{\theta_{\mathrm{d}} = 0}^{\theta_{\mathrm{d}} = 
 \Delta \theta} G (\theta_{\mathrm{d}}) 2 \uppi R^2
 \sin(\theta_{\mathrm{d}}) \mathrm{d} \theta_{\mathrm{d}},
 \label{eq-DeltaPi-init}
\end{equation}}\unskip
where $G (\theta_{\mathrm{d}})$ can be extracted from simple and linear
elastic computational analyses, with a set of models with debonds with angle
$\theta_{\mathrm{d}}$, see~\cite{garcia2015-CSTE} for details and
computation.

Assuming linearity of the models with the remote tension
$\sigma^\infty$, and after a dimensional analysis of the problem, the
dependence of the ERR $G (\theta_{\mathrm{d}})$ can be expressed as
 follows defining a dimensionless ERR $\hat{G}$, 
{\begin{equation}
 \hat{G} \left(\theta_{\mathrm{d}}; k ,m, \frac{\mu_1}{\mu_2} \right) =  
 \frac{E^*}{(\sigma^\infty)^2 R}  G \left(\theta_{\mathrm{d}}; E_1, 
 E_2, \nu_1, \nu_2, \sigma^\infty, R \right), 
\label{eq-dimlessG}
 \end{equation}}\unskip
where $R$ is the particle radius and,
{\begin{equation}
 E^* = \frac{2}{ \frac{1-\nu_1^2}{E_1} + \frac{1-\nu_2^2}{E_2}}
\end{equation}}\unskip
is the harmonic mean of the plane-strain elastic moduli. The terms $k$
and $m$ in \refeqn{eq-dimlessG} refer to the two independent bimaterial
elastic properties defined in \refeqn{eq:km} and $\mu_1/\mu_2$ is the
ratio of particle to matrix shear moduli.

The dimensionless expression of $G (\theta_{\mathrm{d}})$ in
\refeqn{eq-dimlessG} allows to reduce the number of computational
analyses and show explicitly the dependence on the particle radius $R$,
making the resulting size effect explicit. Thus, introducing
\refeqn{eq-dimlessG} in  \refeqn{eq-DeltaPi-init}, $\Delta \Pi$ can be
obtained for any radius $R$ and remote tension $\sigma^\infty$ directly
from a set of computations for different values of the debond angle
$\theta_{\mathrm{d}}$, 
{\begin{equation}
 - \Delta \Pi = \frac{\left(\sigma^\infty \right)^2 R^3}{E^*}
 \int_{\theta_{\mathrm{d}} = 0}^{\theta_{\mathrm{d}} = 
 \Delta \theta} \hat{G} (\theta_{\mathrm{d}};k,m) \mathrm{d} \theta_{\mathrm{d}}.
 \label{eq-DeltaPi-DL}
\end{equation}}\unskip

Concerning the dissipated energy, it can be calculated following the
same strategy: integrating the fracture energy  along all the path of
the new crack,
{\begin{equation}
 E_{\mathrm{d}} = \int_{\theta = 0}^{\theta = \Delta \theta} G_{\mathrm{c}} 
 (\theta) 2 \uppi R^2 \sin(\theta) \mathrm{d} \theta,
\end{equation}}\unskip
where $G_{\mathrm{c}} (\theta)$ is the interface fracture energy, that
could depend on the point of the interface $\theta$. Assuming a uniform
interface, the main variation in $G_{\mathrm{c}} (\theta)$ could come
from the variation of the fracture mode-mixity along the interface and
the effect of this variation on $G_{\mathrm{c}} (\theta)$. In the
context of the finite fracture mechanics, this effect has been taken
into account in various ways, see~\cite{garcia2012-IJSS,talmon2017} for
a discussion. The expression can be normalized with the value of the
interface fracture energy in pure mode 1, $G_{\mathrm{1c}}$, and the
particle radius $R$,
{\begin{equation}
 E_{\mathrm{d}} = G_{\mathrm{1c}} 2\uppi R^2 \int_{\theta = 0}^{\theta = \Delta \theta} 
 \hat{G}_{\mathrm{c}} (\theta) \sin{\theta} \mathrm{d} \theta.
 \label{eq-DE-final}
\end{equation}}\unskip
In the present case, the dimensionless
value of $\hat{G}_{\mathrm{c}} (\theta)$ is
calculated using the Hutchinson and Suo phenomenological
law~\cite{hutchinson1992}, which has been shown
in~\cite{garcia2015-CSTE} to capture very accurately the fracture
energy in a similar system,
{\begin{equation}
\hat{G}_{\mathrm{c}} (\phi) = 1 + \tan^2 [ (1-\lambda) \phi],
 \label{eq-sph-Ghatc}
\end{equation}}\unskip
where $\lambda$ is a dimensionless parameter modulating the influence
of the fracture mode-mixity on the fracture toughness. The term $\phi$
is a measure of this mixity, which has been evaluated following
different alternatives in the literature, see~\cite{garcia2012-IJSS},
Section 4.2, for a discussion. In the present case, for the reasons
given in~\cite{garcia2012-IJSS}, the value of $\phi$ is based on the
stress state before the crack onset as follows:
{\begin{equation}
\phi (\theta) = \tan^{-1} \left(\frac{\tau (\theta)}{\sigma (\theta)} \right),
\end{equation}}\unskip
where $\sigma(\theta)$ and $\tau(\theta)$ are calculated using
\refeqn{eq-tension}.

Introducing the expressions for the change in potential elastic energy
\refeqn{eq-DeltaPi-DL} and dissipated energy \refeqn{eq-DE-final} in
the energy balance in \refeqn{eq-EC-balance2}, the following expression can be
obtained:
{\begin{equation}
\frac{\left(\sigma^\infty \right)^2 R}{E^*}\int_{\theta_{\mathrm{d}} = 0}^{\theta_{\mathrm{d}} = 
\Delta \theta} \hat{G} (\theta_{\mathrm{d}};k,m) \sin (\theta_{\mathrm{d}}) \mathrm{d} 
\theta_{\mathrm{d}} \geq {G}_{\mathrm{1c}}  \int_{\theta = 0}^{\theta = \Delta \theta} 
\hat{G}_{\mathrm{c}} (\theta) \sin (\theta) \mathrm{d} \theta.
\end{equation}}\unskip
To express this condition in a form similar to  the stress criterion in
\refeqn{eq-finalSC}, it is rearranged so that it is explicitly
expressed in terms of the remote tension $\sigma^\infty$   divided by
the interface tensile strength $\sigma_{\mathrm{c}}$,
{\begin{equation}
 \frac{\sigma^\infty}{\sigma_{\mathrm{c}}} \geq \gamma \sqrt{g(\Delta \theta)}=
 \underbrace{\frac{1}{\sigma_{\mathrm{c}}} \sqrt{\frac{G_{\mathrm{c}} E^*}{R}}}_{\gamma} 
 \underbrace{\sqrt{\frac{\int_{\theta = 0}^{\theta = \Delta \theta} \hat{G}_{\mathrm{c}} 
 (\theta) \sin(\theta) \mathrm{d} \theta}{\int_{\theta_{\mathrm{d}} = 0}^{\theta_{\mathrm{d}} = 
 \Delta \theta} \hat{G} (\theta_{\mathrm{d}};k,m) \sin(\theta_{\mathrm{d}}) 
 \mathrm{d}\theta_{\mathrm{d}}}}}_{\sqrt{g(\Delta \theta)}},
\end{equation}}\unskip
where $\gamma$ is a brittleness number proposed in~\cite{mantic2009}
and $g(\Delta \theta)$ is a dimensionless function, analogous to
$s(\Delta \theta)$ defined for the stress criterion. Similarly, but
modulated with $\gamma$, this function represents the value of the
remote tension $\sigma^\infty$ required for a certain debond onset with
angle $\Delta \theta$ to be admissible from the energetic point of
view.

\subsection{Combining the stress and energy criteria} \label{subsec-CouplingTheTwoCriteria}
Once the stress and energy criteria have been developed separately,
Leguillon's postulate~\cite{leguillon2002} establishes that the debond
onset will occur when the conditions for the two criteria are met
simultaneously. Assuming that the remote tension is increased
quasistatically from zero to the value leading to the debond onset, the
value of the remote tension $\sigma_{\mathrm{onset}}^\infty$ for which
it occurs is the minimum value meeting both criteria, thus given by the
next optimization problem
{\begin{equation}
 \frac{\sigma_{\mathrm{onset}}^\infty}{\sigma_{\mathrm{c}}} = 
 \min_{\Delta \theta}\left( \max \left\{ s (\Delta \theta), 
 \gamma \sqrt{g (\Delta \theta)} \right\}\right),
 \label{eq-CC-final}
\end{equation}}\unskip
where it can be observed that the debond angle at onset $\Delta \theta$
is the optimization variable and is obtained as a result of the
optimization process. The manner in which this problem is solved is
widely discussed in~\cite{garcia2015-CSTE}. 

Note that $\gamma$ depends explicitly on the particle radius with
$\gamma \propto 1/\sqrt{R}$, so it is expected that, if the energy
criterion plays a role in the optimization problem in
\refeqn{eq-CC-final}, a size effect will be predicted. 


\section{Specimen design and fabrication} \label{sec-SpecimenDesign}
This section is devoted to describe and justify the design and
fabrication of the non-standard specimens proposed and used in the
experiments. The design is based on two objectives, as enumerated in
the introductory section: (i) to evaluate the effect of the size of the
particle on the first stages of the failure mechanism and (ii) to
directly observe and characterize the failure sequence in the first
stage of the failure mechanism. These two objectives will constrain the
design.

In view of the objectives, some requirements are prescribed for the
design of the specimen, see \reffig{fig:Experiments-Requirements},
\begin{itemize}
\item The material of the matrix is required to be transparent, to
allow optical observation of the first stages of the failure
mechanism, that are expected to occur at the particle-matrix
interface. 
\item The specimen should contain only a single particle due to several
reasons: first, the presence of several particles does not allow to fix
the observation in detail in the failure, because it would be difficult
to know, a priori, which is the particle around which the failure will
start. The second reason is that in typical particle-reinforced
composites, nearby particle can affect strongly the initiation of the
failure mechanism, see e.g.~\cite{liu2018,magri2022}. This fact would
generate scatter in the experimental results, because it would add the
non-controlled influence of the nearby particles. Thus, the strategy is
to approach the real problem for nearby particles as a perturbation of
the failure mechanism for an isolated particle.
\item The particles should be spherical and with a very high quality in
terms of dimensional and geometrical tolerances. Variations from the
perfect sphere would generate scatter in the result, given the strong
influence of the shape on the failure initiation~\cite{gentieu2018}.
The particles should be available in different sizes with a high
quality in their calibration. 
\item All the specimen dimensions should keep fixed ratios between them
to reduce the influence of geometrical parameters to one: the particle
size. 
 \end{itemize}

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig:Experiments-Requirements}Requirements for the
design of the specimen used to evaluate the size effect of the particle
on the initiation of the failure mechanism and the observation of this
initiation.}
\end{figure}

Based on the above requirements, the following decisions were made in
the selection of materials and elements and in the design:

The spherical particles used for the fabrication of the specimens were
stainless steel bearing balls. They have the advantage of having a very
high quality in terms of geometrical and dimensional tolerances. In
addition, the surface finishing is very accurate, presenting an
extremely low roughness, necessary for the regular purpose of these
elements. Finally, they are available in a wide range of
well-calibrated sizes.

The matrix should be significantly softer than the particle material. A
low-viscosity room-temperature curing epoxy resin is selected:
Resoltech WWA/WWB4. The low-viscosity property is key in ensuring that
the space around the sphere is perfectly filled, avoiding trapped air
and bubbles. The room-temperature curing will allow to control
carefully the location of the spherical particle, which will require,
as shown below, a set of auxiliary elements.

The specimen will be subjected to uniaxial tension, thus, the typical
bar-type specimen is selected. In particular, a cylindrical shape is
chosen here to keep the axisymmetry inherent to the spherical
particle. Dog-bone or other kind of geometries to increase the extreme
sections are typically included in this type of specimens to promote
failure in the central section of the specimen, where the stresses are
better known. However, in this case, the presence of a stiff particle
in the central part of the specimen already introduces a stress
concentration. Thus, the specimen will be totally cylindrical, as can
be seen in \reffig{fig:Experiments-DimAna}.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig:Experiments-DimAna}Schematic of the specimen
design.}
\end{figure}

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig:sph-specimens}Details of the mould proposed for the fabrication.}
\end{figure}

In what follows, the optimized fabrication procedure for the specimen
is outlined, remarking the decisions taken to guarantee that the
failure behavior of the specimens would be representative of what was
wanted to be observed:
\begin{enumerate}[(1)]
\item External molds were prepared with different internal diameters,
corresponding to the external diameters required for the specimens. The
molds were fabricated from tubes of Polyvinyl chloride (PVC), typically
used in plumbing. After being cut and machined at the required length,
the extremes were closed with PVC pipe plugs and some adhesive, see
\reffig{fig5}a. Finally, three holes per specimen were machined, one
in the center to introduce the spherical particle, as will explained
later, and two near the extremes to allow the air to escape during the
resin filling, see \reffig{fig5}b. After trying several strategies,
this approach proved to be the optimal one. The finished mold can be
observed in \reffig{fig5}c.

\item  The steel spheres were washed with alcohol before being placed
for the fabrication. The objective was to achieve a complete and
homogeneous adhesion along the interface but limiting the adhesion
strength, since a high strength could avoid   failure initiation at the
interface. Tabiai et~al.~\cite{tabiai2018} showed the strong influence
of the interfacial bonding on the qualitative and quantitative progress
of the failure mechanism in the problem of fiber-matrix, which presents
many similarities with the present problem. Similar results were found
in~\cite{harding1997}, where the effect of   different surface
treatments on this problem was studied. They found that, in the absence
of adhesion, the failure mechanism is an equatorial crack, so this is
an aspect that will be checked in the post-mortem\break analysis.


\item Several strategies were tested to place the spherical particle in
the middle of the specimen during the curing of the epoxy. The optimum
solution was to adhere a thin fishing line to the ball in one extreme
and to a small bar in the other extreme, that helped to situate the
spherical particle in the required position during the curing,  see
\reffig{fig5}d. The fishing line could be considered an imperfection,
but it is situated in the equator of the sphere, far from the pole
where it is expected the debond to appear. Since the failure mechanism
will be recorded, it will be possible to evaluate if debond starts at
the pole without any influence of the fishing line.

\item The bi-component epoxy, Resoltech WWA, is mixed with Resoltech
WWB4 in a weight ratio of 100:40. After mixing for 10~min with a
slow stir to avoid bubbles, the epoxy resin is ready for filling and
can begin the curing process. The spherical particle should be
positioned inside the mold using its central hole prior to this,
see Figure~\ref{fig:sph-specimens}. 

\item All the molds filled with the epoxy resin and the spherical
particle adequately placed were introduced in a climate chamber with
the temperature controlled at 35~\textdegree C during 20~h, see
\reffig{fig5}e. Subsequently, a post-cured stage was carried out at 
60\textdegree~for 4~h. Once both stages were completed, the
temperature was slowly decreased to room temperature  and the molds
were extracted from the climate chamber and left to consolidate during
1 week.

\item The mold was removed by machining the specimen with a lathe and
removing the pipe plugs by cutting the specimen, see \reffig{fig5}f.
The specimens were machined up to arrive to the calibrated external
diameter planned. Finally, the region near the spherical particle was
finely polished to enhance the transparency, in order to observe the
failure mechanism during the test.

\item All the specimens were carefully inspected. A total of 4 types of
specimens were fabricated with spherical particles of radius $1.5$,
$2.0$, $3.0$ and $4.0$ mm. Absence of air bubbles, correct positioning
of the spherical particle, correct filling and dimensional tolerance
were evaluated.  The details of the measures are shown in
\reftab{tab:sph-inc-results} along with the test results. In parallel,
a set of cylindrical specimens without particle were fabricated using
exactly the same procedure with external radius of $8.25$ mm, in order
to evaluate the mechanical properties of the epoxy resin employed. One
specimen for each set is shown in \reffig{fig:sph-mould7}.
\end{enumerate}

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}Details of the fabrication procedure.}
\end{figure}

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig:sph-mould7}Examples of fabricated specimens.}
\end{figure}


\section{Experimental results} \label{sec-ExpResults} 
The specimens were tested on tensile testing machines INSTRON 8802 (for
particle radii $2$, $3$ and $4$ mm) and INSTRON 4482 (for particle
radii $1.5$ mm and specimens without particle). The length between the
grips was controlled and set to follow the same
length-to-particle-radius ratio for all the specimens. The specimens
were subjected to monotonic tensile testing at a low strain rate, which
can be considered quasistatic loading. The strain rate is the same for
all the sets of specimens, thanks to fixing the deformation
rate-to-free-length ratio for all the sets, see
\reftab{tab:sphinc-configtest} for details.  In this manner, viscous
effects, if any, are expected to affect similarly to all sets. 

%tab1
\begin{table}
\caption{\label{tab:sphinc-configtest}Details of the experimental
set-up{\vspace*{-2pt}}}
\begin{tabular}{ccc}
\thead
Particle radius (mm) &amp; Free length (mm) &amp; Deformation rate (mm/min)  \\ 
\endthead
{4.0} &amp;{80.0} &amp;{1.000} \\ 
{3.0} &amp;{60.0} &amp;{0.750} \\
{2.0} &amp;{40.0} &amp;{0.500} \\
{1.5} &amp;{30.0} &amp;{0.375}
\botline
\end{tabular}
{\vspace*{3pt}}
\end{table}

%tab2
\begin{table}
\caption{\label{tab:sph-inc-results}Details of the dimensions and
results for all the specimens{\vspace*{-2pt}}}
\begin{tabular}{ccccc}
\thead
Specimen &amp; Part.~radius (mm) &amp; Sp.~diameter (mm) &amp; Stress at failure (MPa)   &amp; Deb.~poles  \\ 
\endthead
P8-1 &amp; 4.0 &amp;  24.23 &amp;  10.78 &amp; 1\\
P8-2 &amp; 4.0 &amp;  24.26 &amp;  13.95 &amp; 1\\
P8-3 &amp; 4.0 &amp;  24.05 &amp;  13.70 &amp; 1\\
P8-5 &amp; 4.0 &amp;  24.29 &amp;  14.64 &amp; 1\vspace*{5pt}\\
P6-1 &amp; 3.0 &amp;  17.90 &amp;  14.08 &amp; 1\\
P6-3 &amp; 3.0 &amp;  18.02 &amp;  16.02 &amp; 1\\
P6-4 &amp; 3.0 &amp;  18.02 &amp;  22.66 &amp; 2\vspace*{5pt}\\
P4-1 &amp; 2.0 &amp;  11.89 &amp; 17.71 &amp; 1\\
P4-2 &amp; 2.0 &amp;  12.07 &amp; 21.70 &amp; 1\\
P4-3 &amp; 2.0 &amp;  11.89 &amp; 21.36 &amp; 1\\
P4-5 &amp; 2.0 &amp;  11.92 &amp; 18.38 &amp; 1\vspace*{5pt}\\
P3-2 &amp; 1.5 &amp; \09.32 &amp; 22.20 &amp;1\\
P3-3 &amp; 1.5 &amp; \09.23 &amp; 23.81 &amp;1
\botline
\end{tabular}
\end{table}


A high-speed visualization system was set and used to observe the first
stages of the failure mechanism. The system is composed by a Photron
camera able to record at 0.9~Mfps and a set of powerful light system.
For this test, the camera was set to 216 kfps with a resolution of
$128\times 80$ pixels. The failure in all the specimens was recorded in order
to evaluate the hypothesis on the failure mechanism, but also to check
if the fabrication procedure was correct and was not affecting the
failure in the first stages.

\reffig{fig:curvadesplazamiento} shows the stress--strain curves for all
the specimens with particles. The stress represented corresponds to the
engineering stress, i.e.~the force measured by the load cell divided
over the initial cross section. The represented homogenized strain
corresponds to the displacement of the moving clamp divided by the
initial free length. The curves are grouped for sets. As it can be
observed, the stiffness is more regular for sets with larger particles,
which could be due to two reasons: (i) for larger specimens it is easier
to reach accurate levels of fabrication standards and (ii) stress at
failure is lower for larger particles, avoiding the activation of
nonlinear phenomena at higher stress levels. This is in accordance with
the observation of a more nonlinear behavior for the specimens with
smaller particles. A possible reason of the behavior observed for the
smallest diameters is the ductility observed by~\cite{fiedler2001} when
the triaxiality is decreased. The spherical particle could have this
effect on the stress state for high level of strain. The effect of the
triaxiality could be tested using the proposal by~\cite{lauke2006}.

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig:curvadesplazamiento}Stress--strain curves for all the specimens.}
\end{figure}

According to the stress--strain curves, for all the sets and specimens,
the results show a total failure of the specimens after the first
failure event. The stress level at this failure event, along with the
real dimensions of the specimens and the post-mortem observation of the
symmetry of the failure, are presented in \reftab{tab:sph-inc-results}.
It is interesting to remark that the observed failure was non-symmetric
in all the cases except for specimen P6-4.

Given that all the stages of the failure mechanism occur dynamically,
it is necessary to observe the high-speed recording to visualize the
different stages. \reffig{fig:sph-altavelocidad} shows the key frames
to visualize the failure mechanism for the specimen P6-3. In the first
frame (a) the interface is intact. A small debond is observed in the
second frame, which progresses along the interface and migrates towards
the matrix in frame (c). This already existing matrix crack progresses
quickly along the matrix to reach the free edge in frames (d), (e) and
(f). The whole failure sequence occurs in 88~$\upmu$s, which shows the
highly dynamic character of this failure.

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig:sph-altavelocidad}Failure sequence observed with
the high-speed visualization system for specimen P6-3.}
\end{figure}

The sequence presented before is representative of all the specimens
except for the specimen P6-4, which is the only one where a symmetric
failure in the two poles was found in the post-mortem observation. To
clarify what happened in this case, \reffig{fig:sph-altavelocidadSIM}
shows its failure sequence. As it can be observed, the first stages of
the failure mechanisms are the same that for the other specimens,
leading to a non-symmetric total failure of the specimen. However, the
subsequent vibration of the two parts of the specimen generated a
debond onset at the other pole, that slowly progressed along the
interface to finally end in the total failure of the matrix dynamically
again. This result confirms the theoretical prediction by the CC-FFM,
according to which the non-symmetric debond in only one of the poles is
the preferred solution. This was predicted in~\cite{garcia2015-CSTE}
for this problem and physically explained in detail
in~\cite{garcia2015-EJMAS} for a related system.

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig:sph-altavelocidadSIM}Failure sequence observed for
the only specimen (P6-4) presenting a 2-debond fracture surface in the
post-mortem analysis.}
\end{figure}

\reffig{fig:sph-postmortem} shows the fracture surface for the specimen
P6-3. First, it is interesting to observe the debond at the spherical
particle, perfectly located around the pole. The migration toward the
matrix is also visible and occurs at a distance from the pole that is
very uniform. In addition, the migration seems to have taken place at
different points of the interface simultaneously, as evidenced by the
fact that the fracture surface at the matrix presents some steps. This
is a consequence of crack migration towards the matrix at different
points of the interface at the same time. These steps are a consequence
of the slight difference in the debond angle at which the crack
migrated. This shows an almost perfect axisymmetry in the problem,
allowing to neglect any effect of the fishing line on the failure
mechanism. Finally, the reduction of the step when the fracture surface
approaches the external edge is an evidence of the direction of crack
growth, from the spherical particle to the external boundary.

\begin{figure}
\includegraphics{fig10}
\caption{\label{fig:sph-postmortem}Post-mortem observation of the
specimen P6-3, where the spherical particle can be observed on the
right.}
\end{figure}

\reffig{fig:sizeeffect} shows the results of the stress at failure for
all the specimens as a function of the particle radius. As already
predicted~\cite{gentieu2018} and observed in some previous works in the
literature~\cite{cho2006}, the apparent strength increases for smaller
particle radii. This is qualitatively in agreement with the predictions
given by the model presented in \refsec{sec-CoupledCriterion}. The
quantitative comparison requires the value of certain properties, some
of which are represented in \reftab{tab:sph-inc-matprops}. The
properties of AISI 52100 steel used for bearing balls are nominal
values. For epoxy resin, elastic modulus and tensile strength are
extracted from the experimental tests carried out on specimens without
particles. Poisson's ratio is taken as value of reference and the
fracture toughness is taken from experiments carried out in similar
resins in the laboratory.

\begin{figure}
{\vspace*{-2pt}}
\includegraphics{fig11}
{\vspace*{-3pt}}
\caption{\label{fig:sizeeffect}Stress at failure as a function of the
particle radius. Comparison with the predictions of the CC-FFM.}
{\vspace*{-2pt}}
\end{figure}

%tab3
\begin{table}[t!]
\caption{\label{tab:sph-inc-matprops}Material properties for particle
and matrix{\vspace*{-2pt}}}
\begin{tabular}{ccc}
\thead
Material &amp; AISI 52100 Steel &amp; Resoltech WWA/WWB4 epoxy \\
\endthead
Elastic modulus (GPa)  &amp; $200$ &amp; \0$2.10$\0 \\
Poisson's ratio  &amp;  $0.3$ &amp; \0$0.3$\0\0 \\
Tensile strength (MPa)  &amp; - &amp;    $27.7$\0\0 \\
Fracture toughness (N/mm)  &amp;-&amp;  \0$0.478$
\botline
\end{tabular}
{\vspace*{-2pt}}
\end{table}

The values of the fracture toughness and strength of the interface are
very difficult to obtain directly. This fact has motivated the proposal
of indirect methods, see e.g.~\cite{harding1997,girard2023}. To ensure
the representativeness of the results provided by these methods, it is
key to mimic all the conditions involved in the interface behavior.
These conditions, such as surface finish, surface treatment,
temperature and pressure at curing in the local region of the
interface, are difficult to mimic. That is the reason why, in many
cases, it is preferential to employ indirect methods. In this case, the
fact that failure starts at the interface is evidence that fracture
toughness and strength are lower at the interface than in the bulk, as
the most stressed point is not located at the interface, as was
discussed by~\cite{qi2017}, among others. For comparison and
following~\cite{garcia2015-CSTE}, the exponent of the stress criterion
is set to $p=2$, the shear-to-tensile interface strength
$\tau_{\mathrm{c}}/\sigma_{\mathrm{c}} = 2$, and the sensitivity
parameter to the fracture mode mixity $\lambda = 0.11$.

In view of this, the comparison presented here is made for an extreme
case where the tensile strength and fracture toughness of the interface
are the same as those of the epoxy resin, and for another case where
the two properties are half of the corresponding values at the
interface. As can be observed, the results of this second hypothesis
agree well with the experiments, whereas the prediction when the epoxy
and the interface are similar in terms of strength and fracture
toughness overestimates the stress at failure, as expected.

The size effect found here has a physical interpretation from the 
point of view of the \mbox{CC-FFM,} in particular  the energy criterion: When
all dimensions are scaled for a given load level, the energy available
to be released increases with the cube of the scale factor (because it
is stored in a volume), whereas the dissipated energy scales
quadratically with the scale factor (because it is associated with a
surface). As a result, a large size requires a lower load level to meet
the energy balance of the energy criterion.

The results found here for the size effect are in agreement with the
results presented in the review in~\cite{fu2008}. A qualitatively
similar size effect was found in~\cite{gentieu2019} when the interface
failure is modeled using cohesive zone model for this region. However,
they found that the asymptotic tendency for the smaller particles is
different for CZM and CC-FFM predictions. Whereas 
\mbox{CC-FFM} predicts a critical
stress proportional to $1/\sqrt{R}$, the CZM approach gives an
asymptotic tendency proportional to $1/R$. These results were found
also by other authors, such as~\cite{doitrand2019,garcia2014-EFM}. In
fact, a physical interpretation for this mismatch was proposed
in~\cite{garcia2014-EFM}. Even for more complex models, such
as the one proposed in ~\cite{magri2022}, based on predicting ductile
damage at the bulk, the asymptotic tendency for the size effect matches
the results for CZM.

\section{Concluding remarks} \label{sec-Conclusions}

A new type of specimen has been proposed to evaluate the first stages
of the failure mechanism in particle-reinforced composites. This
specimen is relatively easy to fabricate and allows visualization of
the failure mechanism and verification of the size effect predicted by
diverse models in the literature. The experiments performed on these
specimens confirmed a strong size effect on the failure mechanism.

The experimental results were compared with the predictions of the
CC-FFM. The prediction of a non-symmetric initiation of the failure as
a debond in only one of the poles was confirmed in all the specimens.
The tendency of the size effect was correctly captured qualitatively 
with increasing apparent strength for smaller particle. However, the
quantitative comparison requires a set of interface properties which
are not available. They could be obtained in further experimental
campaigns adapting the procedure proposed in~\cite{girard2023}. The
comparison with some estimated properties showed a good agreement. In
addition, as highlighted in~\cite{leguillon2015}, the origins of the
size effect are multiple. The CC-FFM is able to account for the size
effect with an energetic origin, but not for those with a statistical
origin. Further developments should include them.

Some authors~\cite{chen2007} have highlighted the effect of the viscous
phenomena in the mechanical behavior of the particle reinforced
composites. In the experiments presented here, an attempt has been made
to keep this effect from affecting the size-effect results, by setting
the deformation rate directly proportional to the free length of the
specimen in order to keep the strain rate as similar as possible for
all the specimens. This allows comparison  with the CC-FFM model, which
does not include viscous effects at this time. However, viscous effects
could be introduced in the context of the CC-FFM.

Polymeric matrices can also present a nonlinear mechanical behavior.
That could be at the origin of the nonlinear results observed in
\reffig{fig:curvadesplazamiento}. The CC-FFM model could be extended
also to take into account this nonlinear effect, in line with the
developments in~\cite{rosendahl2019,leite2021}. The inclusion of
plasticity could also improve the predictions, see
e.g.~\cite{magri2022}.

The model described here is based on the assumption of zero-thickness
for the interface. However, the model could be extended to take into
account a finite-thickness interface, even including anisotropy with
the expressions from~\cite{pham2019}. In this case, the zero-thickness
assumption is very representative of the situation because no surface
treatment was implemented during the fabrication.

Multiaxial effects can play an important role in the initiation of this
failure mechanism in particle-reinforced composites. In this sense, the
experiments presented here could be extended to include multiaxial
loading. With this objective, a specimen shape has been proposed
by~\cite{lauke2006,lauke2016} along a detailed stress-field calculation
which could be used for an analysis based on the \mbox{CC-FFM}.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Dedication}

The manuscript was written through contributions of all authors. All
authors have given approval to the final version of the manuscript.

\section*{Acknowledgments}
The advice of Antonio Ca\~{n}as about the specimen fabrication is
greatly appreciated. IGG and VM  acknowledge  the support of 
Ministerio de Ciencia e Innovaci\'on de Espa\~na (Projects
PID2020-117001GB-I00/AEI/10.13039/501100011033 and
PID2021-123325OB-I00, respectively). IGG,~JJ and VM  acknowledge  the
support of the European Regional Development Fund (Project
UNSE15-CE-3581). 

\CDRGrant[MICIU]{PID2020-117001GB-I00/AEI/10.13039/501100011033}
\CDRGrant[MICIU]{PID2021-123325OB-I00}
\CDRGrant[ERDF]{UNSE15-CE-3581}

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