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

Comptes Rendus Mécanique Israel García García, Jesús Justo, Alejandro Zurita Van-Dinter and Vladislav Mantiˇc Particle size effect on the strength of particle-reinforced composites. Experimental analysis and comparison with the coupled criterion Volume 353 (2025), p. 627-646 Online since: 20 May 2025 https://doi.org/10.5802/crmeca.293 This article is licensed under the Creative Commons Attribution 4.0International License. http://creativecommons.org/licenses/by/4.0/ CENTRE MERSENNE The Comptes Rendus. Mécanique are a member of the Mersenne Center for open scientific publishing www.centre-mersenne.org — e-ISSN : 1873-7234 Comptes Rendus. Mécanique 2025, Vol. 353, p. 627-646 https://doi.org/10.5802/crmeca.293 Research article / Article de recherche Particle size effect on the strength of particle-reinforced composites. Experimental analysis and comparison with the coupled criterion Effet de la taille des particules sur la résistance des composites renforcés par particules : analyse expérimentale et comparaison avec le critère couplé Israel García García ,∗,a, Jesús Justo ,a, Alejandro Zurita Van-Dinter aand Vladislav Mantiˇc ,a aDepartamento de Mecánica de Medios Continuos y Teoría de Estructuras, Escuela Técnica Superior de Ingeniería, Escuela Politécnica Superior. Universidad de Sevilla. Camino de los Descubrimienos s/n, 41092 Sevilla, Spain E-mails: israelgar[email protected] (I. G. García), [email protected] (J. Justo), [email protected] (A. Zurita Van-Dinter), [email protected] (V. Mantiˇc) This article is dedicated to the memory of Professor Dominique Leguillon 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 (CCFFM). A secondary objective is to visualize the initiation of the failure mechanism, which starts at the particlematrix 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 highspeed 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. Résumé. Les composites renforcés par particules sont largement utilisés dans l’industrie, principalement en raison de la diversité de leurs méthodes de fabrication et de la possibilité d’adapter leurs propriétés. Dans de nombreux cas, des campagnes expérimentales approfondies sont nécessaires pour optimiser les caractéristiques du système et améliorer certaines propriétés spécifiques. Les modèles micromécaniques constituent une alternative utile ou une première approche lors du processus de conception des matériaux. L’une des caractéristiques les plus faciles à modifier est la taille du renfort, qui, selon certains modèles et preuves préliminaires, peut influencer significativement les propriétés mécaniques du matériau. ∗Corresponding author ISSN (électronique) : 1873-7234 https://comptes-rendus.academie-sciences.fr/mecanique/ 628 Israel García García et al. L’objectif de cette étude est d’évaluer expérimentalement l’effet de la taille du renfort sur la résistance du composite et de le comparer aux prédictions du critère couplé de la mécanique de la rupture finie (CCMRF). Un objectif secondaire est d’observer le mécanisme d’amorçage de la rupture, qui débute à l’interface particule-matrice et évolue vers une fissure scindant l’éprouvette. Pour cela, un nouveau design d’éprouvette est proposé, ainsi qu’un procédé de fabrication optimisé. Les essais ont été enregistrés à l’aide d’une caméra à haute vitesse, permettant de visualiser l’initiation de la fissure à l’interface particule-matrice. Les résultats expérimentaux mettent en évidence un fort effet de taille, où des inclusions plus petites entraînent une résistance apparente plus élevée. Ces résultats sont en accord relativement bon avec les prédictions du CC-MRF. Keywords. Size effect, Particle-reinforced composites, Spherical inhomogeneity, Coupled criterion, Finite fracture mechanics, Experimental fracture mechanics. Mots-clés. Effet de taille, Composites renforcés par particules, Inhomogénéité sphérique, Critère couplé, Mécanique de la rupture finie, Mécanique expérimentale de la rupture. Funding. Ministerio de Ciencia e Innovación de España (Projects PID2020-117001GB-I00/AEI/10.13039/ 501100011033, PID2021-123325OB-I00), European Regional Development Fund (Project UNSE15-CE-3581). Manuscript received 31 October 2024, revised 9 March 2025, accepted 10 March 2025. 1. Introduction Particle-reinforced composites are becoming more and more prevalent in industrial applications [1]. 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 [2], stiffness [3], tensile strength [4], or fracture toughness [5,6]. 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 [7] 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 [8] 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 [9,10]. 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. [11], 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 [11]. 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 [12]. The first stage of the failure mechanism is similar to the related problem of debonding at the fiber-matrix interface, see, e.g., [13,14]. 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 [15]. The influence of the interface properties has been reported both theoretically [16] and experimentally [12]. Similar results have been reported in [13] for a similar system with fibrous reinforcement. The authors carried out experiments on dog-bone specimens with a single fiber Israel García García et al. 629 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 [11]. The size effect has been explained by using diverse models: In [17], 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 [16], 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 [18] 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-FourierTransform 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 [19,20], postulates that debonding will initiate with a finite extension when both stress and energy conditions are met simultaneously, see [21] for a recent review on this criterion. Size effect has been predicted by this approach for very diverse problems, see e.g. [22–28], and even used to explain the size effect found by other approaches [29,30]. In this sense, the present authors [31] 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 [32], including residual thermal stresses and different volumetric fractions with multi-particle models. In [33], 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 [29] 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 [11], 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 [34] 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. 630 Israel García García et al. Figure 1. Schematic of the problem under study. The document is organized as follows: First, the CC-FFM model for this problem will be briefly introduced in Section 2. The experimental setup will be described in terms of specimen design and fabrication in Section 3. Finally, the experimental results are presented and discussed in Section 4, where they are compared with the predictions given by the CC-FFM model. 2. Coupled criterion analysis The problem under study is schematized in Figure 1, 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 nonsymmetric manner. This result is predicted by the CC-FFM, as proved in [31] and discussed in detail for a related problem in [35]. Thus, this result is also to be validated by the experiments presented here. 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 Figure 1. In what follows, the two criteria will be briefly studied in Sections 2.1 and 2.2 to be subsequently combined in Section 2.3, where the expression that will be compared with the experiments is fully developed. 2.1. Stress criterion The stress criterion is based on the evaluation of the stresses before the debond onset, i.e. the state (I) in Figure 1. According to the CC-FFM, a debond onset is possible only at those points with angle θ∈[0°, 180°] of the interface where a certain combination of the interface normal and shear tractions, σ(θ) and τ(θ), respectively, is greater or equal to a critical value. Since the poles are the Israel García García et al. 631 most stressed points at the interface, it is assumed that the debond onset will always occupy the pole (θ=0°) and surrounding points up to a certain angle ∆θ. In this case, the condition given by the stress criterion for a debond onset will be: σeq(σ,τ)=p s¿sgn(σ)|σ|p+µ|τ| µ¶pÀ+≥σc∀θ∈[0,∆θ], (1) where σcis the interface tensile strength, and µ=τc/σcis the ratio of interface shear to interface tensile strength. The exponent p≥1 controls the coupling between normal and shear tractions, see [36] 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 σand shear τtractions can be obtained exactly using the expressions by [37] as, σ(θ)=σ∞ˆ σ(θ)=σ∞(k+mcos2θ),τ(θ)=σ∞ˆ τ(θ)=σ∞msin2θ(2) where σ∞is the remote tension and kand mdepend on the elastic properties of particle and matrix in the following form: k=1 2 1+α 1+β 2+α−β 1+α−2β, (3a) m=1+α 1+β, (3b) α=µ1(κ2+1)−µ2(κ1+1) µ1(κ2+1)+µ2(κ1+1), (4a) β=µ1(κ2−1)−µ2(κ1−1) µ1(κ2+1)+µ2(κ1+1), (4b) where µi=Ei/(2(1 +νi)) and κi=3−4νi,Eiand νidenoting Young’s modulus and Poisson’s ratio, respectively, of particle (i=1) and matrix (i=2), and αand βare Dundurs parameters of the bimaterial. Introducing (2) in (1) and after some rearrangements, the expression obtained for the stress criterion is: σ∞ σc≥s(∆θ)=max θ∈[0,∆θ] 1 p rDsgn(k+mcos2θ)|k+mcos2θ|p+³|msin2θ| µ´pE+ . (5) The dimensionless function s(∆θ) 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 [38]. 2.2. Energy criterion 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 [39] to an incremental balance between two states, rather than a differential approach. In this sense, the incremental balance can be written as: ∆Π +∆Ek+Ed=0, (6) 632 Israel García García et al. where ∆Π and ∆Ekare the increments of potential elastic and kinetic energy, respectively. The term Edrefers to the energy dissipated during the crack initiation process. If the initial state is assumed to be static, ∆Ek≥0, the expression in (6) can be rewritten as: −∆Π ≥Ed, (7) 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 ∆Π can be obtained in different manners, see Section 3.4.1 in [40] for a review. In this case, in line with [31], this term will be obtained by integrating the energy release rate (ERR) Gfor a crack slowly growing from a debond angle θd=0 to an angle θd=∆θfrom the pole −∆Π =Zθd=∆θ θd=0 G(θd)2πR2sin(θd)dθd, (8) where G(θd) can be extracted from simple and linear elastic computational analyses, with a set of models with debonds with angle θd, see [31] for details and computation. Assuming linearity of the models with the remote tension σ∞, and after a dimensional analysis of the problem, the dependence of the ERR G(θd) can be expressed as follows defining a dimensionless ERR ˆ G, ˆ Gµθd;k,m,µ1 µ2¶=E∗ (σ∞)2RG¡θd;E1,E2,ν1,ν2,σ∞,R¢, (9) where Ris the particle radius and, E∗=2 1−ν2 1 E1+1−ν2 2 E2 (10) is the harmonic mean of the plane-strain elastic moduli. The terms kand min (9) refer to the two independent bimaterial elastic properties defined in (3) and µ1/µ2is the ratio of particle to matrix shear moduli. The dimensionless expression of G(θd) in (9) 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 (9) in (8), ∆Π can be obtained for any radius Rand remote tension σ∞directly from a set of computations for different values of the debond angle θd, −∆Π =(σ∞)2R3 E∗Zθd=∆θ θd=0 ˆ G(θd;k,m)dθd. (11) Concerning the dissipated energy, it can be calculated following the same strategy: integrating the fracture energy along all the path of the new crack, Ed=Zθ=∆θ θ=0 Gc(θ)2πR2sin(θ)dθ, (12) where Gc(θ) is the interface fracture energy, that could depend on the point of the interface θ. Assuming a uniform interface, the main variation in Gc(θ) could come from the variation of the fracture mode-mixity along the interface and the effect of this variation on Gc(θ). In the context of the finite fracture mechanics, this effect has been taken into account in various ways, see [36,41] for a discussion. The expression can be normalized with the value of the interface fracture energy in pure mode 1, G1c, and the particle radius R, Ed=G1c2πR2Zθ=∆θ θ=0 ˆ Gc(θ)sinθdθ. (13) Israel García García et al. 633 In the present case, the dimensionless value of ˆ Gc(θ) is calculated using the Hutchinson and Suo phenomenological law [42], which has been shown in [31] to capture very accurately the fracture energy in a similar system, ˆ Gc(φ)=1+tan2[(1−λ)φ], (14) where λis a dimensionless parameter modulating the influence of the fracture mode-mixity on the fracture toughness. The term φis a measure of this mixity, which has been evaluated following different alternatives in the literature, see [36], Section 4.2, for a discussion. In the present case, for the reasons given in [36], the value of φis based on the stress state before the crack onset as follows: φ(θ)=tan−1µτ(θ) σ(θ)¶, (15) where σ(θ) and τ(θ) are calculated using (2). Introducing the expressions for the change in potential elastic energy (11) and dissipated energy (13) in the energy balance in (7), the following expression can be obtained: (σ∞)2R E∗Zθd=∆θ θd=0 ˆ G(θd;k,m)sin(θd)dθd≥G1c Zθ=∆θ θ=0 ˆ Gc(θ)sin(θ)dθ. (16) To express this condition in a form similar to the stress criterion in (5), it is rearranged so that it is explicitly expressed in terms of the remote tension σ∞divided by the interface tensile strength σc, σ∞ σc≥γqg(∆θ)=1 σcrGcE∗ R | {z } γ v u u u tRθ=∆θ θ=0ˆ Gc(θ)sin(θ)dθ Rθd=∆θ θd=0ˆ G(θd;k,m)sin(θd)dθd | {z } pg(∆θ) , (17) where γis a brittleness number proposed in [22] and g(∆θ) is a dimensionless function, analogous to s(∆θ) defined for the stress criterion. Similarly, but modulated with γ, this function represents the value of the remote tension σ∞required for a certain debond onset with angle ∆θto be admissible from the energetic point of view. 2.3. Combining the stress and energy criteria Once the stress and energy criteria have been developed separately, Leguillon’s postulate [19] 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 σ∞ onset for which it occurs is the minimum value meeting both criteria, thus given by the next optimization problem σ∞ onset σc=min ∆θ³maxns(∆θ),γqg(∆θ)o´, (18) where it can be observed that the debond angle at onset ∆θis 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 [31]. Note that γdepends explicitly on the particle radius with γ∝1/pR, so it is expected that, if the energy criterion plays a role in the optimization problem in (18), a size effect will be predicted. 634 Israel García García et al. Figure 2. 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. 3. Specimen design and fabrication 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 Figure 2, •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. •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. [18,43]. 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. •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 [44]. The particles should be available in different sizes with a high quality in their calibration. •All the specimen dimensions should keep fixed ratios between them to reduce the influence of geometrical parameters to one: the particle size. 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. Israel García García et al. 641 Figure 9. Failure sequence observed for the only specimen (P6-4) presenting a 2-debond fracture surface in the post-mortem analysis. Figure 11 shows the results of the stress at failure for all the specimens as a function of the particle radius. As already predicted [44] and observed in some previous works in the literature [11], the apparent strength increases for smaller particle radii. This is qualitatively in agreement with the predictions given by the model presented in Section 2. The quantitative comparison requires the value of certain properties, some of which are represented in Table 3. 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. 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. [12,47]. 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, 642 Israel García García et al. Figure 10. Post-mortem observation of the specimen P6-3, where the spherical particle can be observed on the right. Figure 11. Stress at failure as a function of the particle radius. Comparison with the predictions of the CC-FFM. Table 3. Material properties for particle and matrix Material AISI 52100 Steel Resoltech WWA/WWB4 epoxy Elastic modulus (GPa) 200 2.10 Poisson’s ratio 0.3 0.3 Tensile strength (MPa) - 27.7 Fracture toughness (N/mm) - 0.478 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 [48], among others. For comparison and following [31], the exponent of the stress criterion is set to p=2, the shear-to-tensile interface strength τc/σc=2, and the sensitivity parameter to the fracture mode mixity λ=0.11. Israel García García et al. 643 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 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 [15]. A qualitatively similar size effect was found in [33] 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 CC-FFM predicts a critical stress proportional to 1/pR, the CZM approach gives an asymptotic tendency proportional to 1/R. These results were found also by other authors, such as [29,49]. In fact, a physical interpretation for this mismatch was proposed in [29]. Even for more complex models, such as the one proposed in [18], based on predicting ductile damage at the bulk, the asymptotic tendency for the size effect matches the results for CZM. 5. Concluding remarks 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 [47]. The comparison with some estimated properties showed a good agreement. In addition, as highlighted in [50], 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 [51] 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 Figure 7. The CC-FFM model could be extended also to take into account this nonlinear effect, in line with the developments in [52,53]. The inclusion of plasticity could also improve the predictions, see e.g. [18]. 644 Israel García García et al. 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 [54]. 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 [34,46] along a detailed stress-field calculation which could be used for an analysis based on the CC-FFM. 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. Dedication The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Acknowledgments The advice of Antonio Cañas about the specimen fabrication is greatly appreciated. IGG and VM acknowledge the support of Ministerio de Ciencia e Innovación de España (Projects PID2020117001GB-I00/AEI/10.13039/501100011033 and PID2021-123325OB-I00, respectively). IGG, JJ and VM acknowledge the support of the European Regional Development Fund (Project UNSE15CE-3581). References [1] D. K. Rajak, D. D. Pagar, R. Kumar and C. I. Pruncu, “Recent progress of reinforcement materials: a comprehensive overview of composite materials”, J. Mater. Res. Technol. 8(2019), no. 6, pp. 6354–6374. [2] Y. Murakami, Y. Ichiba, T. Kawashima and N. Hozumi, “Filler particle size effect on electrically and thermally properties of thermal conductive thermoplastic polyimide/hexagonal boron nitride composite materials fabricated by the electrostatic adsorption method”, IEEJ Trans. Electr. Electron. Eng. 19 (2024), no. 10, pp. 1590–1595. [3] X. Zhang, X. Li, F. Chi, et al., “Synergistic enhancement of modulus and ductility in Mg matrix composites: A new strategy for GNPsMgOnp and SiCp hybrid reinforcement”, Compos. Part A: Appl. Sci. Manuf. 187 (2024), article no. 110110. [4] S. Sathees Kumar, S. Seenivasan, I. Isaac Premkumar, S. Vijayakumar, P. Anusha and A. Pradeep, “Analysis of wear mechanisms in AA2024/TiB2 composites under different loads”, Interactions 245 (2024), no. 1, pp. 245–257. [5] A. C. Garg and Y.-W. Mai, “Failure mechanisms in toughened epoxy resins-A review”, Compos. Sci. Technol. 31 (1988), no. 3, pp. 179–223. [6] N. Yoshinobu, M. Yamaguchi, M. Okubo and T. Matsumoto, “Effects of particle size on mechanical and impact properties of epoxy resin filled with spherical silica”, J. Appl. Polym. Sci. 45 (1992), no. 7, pp. 1281–1289. [7] H. M. Enginsoy, F. Gatamorta, E. Bayraktar, M. H. Robert and I. Miskioglu, “Experimental and numerical study of Al-Nb 2 Al composites via associated procedure of powder metallurgy and thixoforming”, Compos. Part B: Eng. 162 (2019), pp. 397–410. [8] F. París, A Study of Failure Criteria of Fibrous Composite Materials, Technical Report, National Aeronautics and Space Administration (NASA), 2001. Israel García García et al. 645 [9] A. Z. Khurshudyan, “Derivation of a mesoscopic model for nonlinear particle-reinforced composites from a fully microscopic model”, Acta Mech. 230 (2019), no. 10, pp. 3543–3554. [10] M. Majewski, P. Holobut, M. Kursa and K. Kowalczyk-Gajewska, “Packing and size effects in elastic-plastic particulate composites: Micromechanical modelling and numerical verification”, Int. J. Eng. Sci. 151 (2020), article no. 103271. [11] J. Cho, M. S. Joshi and C. T. Sun, “Effect of inclusion size on mechanical properties of polymeric composites with micro and nano particles”, Compos. Sci. Technol. 66 (2006), no. 13, pp. 1941–1952. [12] P. H. Harding and J. C. Berg, “The characterization of interfacial strength using single-particle composites”, J. Adhes. Sci. Technol. 11 (1997), no. 8, pp. 1063–1076. [13] I. Tabiai, R. Delorme, D. Therriault and M. Levesque, “In-situ full field measurements during inter-facial debonding in single fiber composite under transverse load”, Exp. Mech. 58 (2018), no. 9, pp. 1451–1467. [14] K. Uddin, H. Girard, N. Mennie, A. Doitrand and B. Koohbor, “Simultaneous measurement of fiber-matrix interface debonding and tunneling using a dual-vision experimental setup”, Exp. Mech. 64 (2024), no. 9, pp. 1497– 1511. [15] S.-F. Fu, X.-Q. Feng, B. Lauke and Y.-W. Mai, “Effects of particle size, particle/matrix interface adhesion and particle loading on mechanical properties of particulate–polymer composites”, Compos. Part B: Eng. 39 (2008), no. 6, pp. 933–961. [16] J. Ye, C. Chu, Z. Zhai, Y. Wang, B. Shi and Y. Qiu, “The interphase influences on the particle-reinforced composites with periodic particle configuration”, Appl. Sci. 7(2017), no. 1, pp. 1–13. [17] Y. Liu, J. Chen and X. Dong, “A particle debonding model considering interface nanoscale damage effect”, Model. Simul. Mat. Sci. Eng. 29 (2021), no. 1, article no. 015007. [18] M. Magri, L. Adam and J. Segurado, “Particle size effects in ductile composites: An FFT homogenization study”, J. Mech. Phys. Solids 160 (2021), article no. 104759. [19] D. Leguillon, “Strength or toughness? A criterion for crack onset at a notch”, Eur. J. Mech. Solids 21 (2002), no. 1, pp. 61–72. [20] P. Cornetti, N. Pugno, A. Carpinteri and D. Taylor, “Finite fracture mechanics: A coupled stress and energy failure criterion”, Eng. Fract. Mech. 73 (2006), no. 14, pp. 2021–2033. [21] A. Doitrand, T. Duminy, H. Girard and X. Chen, “A review of the coupled criterion”, J. Theor. Comput. Appl. Mech. (2024), pp. 1–37. [22] V. Mantiˇc, “Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion”, Int. J. Solids Struct. 46 (2009), no. 6, pp. 1287–1304. [23] A. Sapora and P. Cornetti, “Crack onset and propagation stability from a circular hole under biaxial loading”, Int. J. Fract. 214 (2018), no. 1, pp. 97–104. [24] A. Doitrand, R. Henry, T. Lube and S. Meille, “Size effect assessment by Weibull’s approach and the coupled criterion”, Eng. Fract. Mech. 256 (2021), article no. 107979. [25] A. Chao Correas, M. Corrado, A. Sapora and P. Cornetti, “Size-effect on the apparent tensile strength of brittle materials with spherical cavities”, Theor. Appl. Fract. Mech. 116 (2021), article no. 103120. [26] F. Ferrian, A. Chao Correas, P. Cornetti and A. Sapora, “Size effects on spheroidal voids by Finite Fracture Mechanics and application to corrosion pits”, Fatigue Fract. Eng. Mater. Struct. 46 (2023), no. 3, pp. 875–885. [27] M. Baldassari, A. Monaco, A. Sapora and P. Cornetti, “Size effect on flexural strength of notched and unnotched concrete and rock specimens by Finite Fracture Mechanics”, Theor. Appl. Fract. Mech. 125 (2023), article no. 103787. [28] M. T. Aranda, I. G. García, A. Quintanas-Corominas and J. Reinoso, “Crack impinging on a curved weak interface: Penetration or deflection?”, J. Mech. Phys. Solids 178 (2023), article no. 105326. [29] I. G. García, M. Paggi and V. Mantiˇc, “Fiber-size effects on the onset of fiber-matrix debonding under transverse tension: A comparison between cohesive zone and finite fracture mechanics models”, Eng. Fract. Mech. 115 (2014), pp. 96–110. [30] G. Molnár, A. Doitrand, R. Estevez and A. Gravouil, “Toughness or strength? Regularization in phase-field fracture explained by the coupled criterion”, Theor. Appl. Fract. Mech. 109 (2020), article no. 102736. [31] I. G. García, V. Mantiˇc and E. Graciani, “A model for the prediction of debond onset in spherical-particlereinforced composites under tension: Application of a coupled stress and energy criterion”, Compos. Sci. Technol. 106 (2015), pp. 60–67. [32] E. Martin, D. Leguillon, A. Catapano and N. Carrère, “Prediction of interfacial debonding between stiffspherical particles and a soft matrix with the coupled criterion”, Theor. Appl. Fract. Mech. 109 (2020), article no. 102749. [33] T. Gentieu, J. Jumel, A. Catapano and J. Broughton, “Size effect in particle debonding: Comparisons between finite fracture mechanics and cohesive zone model”, J. Compos. Mater. 53 (2019), no. 14, pp. 1941–1954. [34] B. Lauke, “Stress field calculation around a particle in elastic-plastic polymer matrix under multiaxial loading as basis for the determination of adhesion strength”, Compos. Interfaces 23 (2016), no. 1, pp. 1–14. 646 Israel García García et al. [35] I. G. García, V. Mantiˇc and E. Graciani, “Debonding at the fibre-matrix interface under remote transverse tension. One debond or two symmetric debonds?”, Eur. J. Mech. A/Solids 53 (2015), pp. 75–88. [36] I. G. García and D. Leguillon, “Mixed-mode crack initiation at a V-notch in presence of an adhesive joint”, Int. J. Solids Struct. 49 (2012), no. 15–16, pp. 2138–2149. [37] J. N. Goodier, “Concentration of stress around spherical and cylindrical inclusions and flaws”, J. Appl. Mech. 55 (1933), pp. 39–44. [38] C. Kammer and P. P. Castañeda, “Variational estimates for the effective properties and field statistics of composites with variable particle interaction strengths”, J. Mech. Phys. Solids 167 (2022), article no. 104996. [39] A. A. Griffith, “The phenomena of rupture and flow in solids”, Phil. Trans. R. Soc. Lond. A. 221 (1921), pp. 163–198. [40] I. G. García, Crack initiation in composites at micro and meso scales, Development and applications of finite fracture mechanics, PhD thesis, School of Engineering, University of Seville, 2014. [41] A. Talmon l’Armée, S. Hell, P. L. Rosendahl, J. Felger and W. Becker, “Nonlinear crack opening integral: Mode mixity for finite cracks”, Eng. Fract. Mech. 186 (2017), pp. 283–299. [42] J. W. Hutchinson and Z. Suo, “Mixed mode cracking in layered materials”, Adv. Appl. Mech. 29 (1992), pp. 63–191. [43] Q. Liu, F. gong Qi, H. min Ding, X. liang Fan and Y. Tue, “Distribution of stress and strain between adjacent particles in particulate reinforced metal matrix composites”, Trans. Nonferrous Met. Soc. China 28 (2018), no. 11, pp. 2314– 2323. [44] T. Gentieu, J. Jumel, A. Catapano and J. Broughton, “Size effect in particle debonding: Comparisons between finite fracture mechanics and cohesive zone model”, J. Compos. Mater. 53 (2018), no. 14, pp. 1941–1954. [45] B. Fiedler, M. Hojo, S. Ochiai, K. Schulte and M. Ochi, “Finite-element modeling of initial matrix failure in CFRP under static transverse tensile load”, Compos. Sci. Technol. 61 (2001), no. 14, pp. 95–105. [46] B. Lauke, “Determination of adhesion strength between a coated particle and polymer matrix”, Compos. Sci. Technol. 66 (2006), no. 16, pp. 3153–3160. [47] H. Girard, A. Doitrand, B. Koohbor, R. Rinaldi, N. Godin, D. Long, J. Bikard and L. Trouillet-Fonti, “Numerical simulation of fiber–matrix debonding: Inverse identification of interface properties”, Eng. Fract. Mech. 286 (2023), article no. 109254. [48] F. Qi, H. Ding, X. Wang, Q. Liu, B. Du and X. Zhang, “The stress and strain field distribution around the reinforced particles in Al/TiC composites: A finite element modeling study”, Comput. Mater. Sci. 137 (2017), pp. 297–305. [49] A. Doitrand, R. Estevez and D. Leguillon, “Comparison between cohesive zone and coupled criterion modeling of crack initiation in rhombus hole specimens under quasi-static compression”, Theor. Appl. Fract. Mech. 99 (2019), pp. 51–59. [50] D. Leguillon, E. Martin and M.-C. Lafarie-Frenot, “Flexural vs. tensile strength in brittle materials”, C. R. Méc. 343 (2015), no. 4, pp. 275–281. [51] J.-K. Chen and M. Yuan, “Decoupling of viscous dissipation and damage dissipation in particulate-reinforced polymeric materials”, Comput. Mater. Sci. 40 (2007), no. 2, pp. 267–274. [52] P. L. Rosendahl, Y. Staudt, A. P. Schneider, J. Schneider and W. Becker, “Nonlinear elastic finite fracture mechanics: Modeling mixed-mode crack nucleation in structural glazing silicone sealants”, Mater. Des. 182 (2019), article no. 108057. [53] A. Leite, V. Mantiˇc and F. París, “Crack onset in stretched open hole PMMA plates considering linear and nonlinear elastic behaviours”, Theor. Appl. Fract. Mech. 114 (2021), article no. 102931. [54] D. C. Pham, T. K. Nguyen and B. V. Tran, “Macroscopic elastic moduli of spherically-symmetric-inclusion composites and the microscopic stress-strain fields”, Int. J. Solids Struct. 169 (2019), pp. 141–165.