COMPOSITES

Composites Science and Technology 66 (2006) 1030–1037

www.elsevier.com/locate/compscitech

SCIENCE ANDTECHNOLOGY

Modeling of ceramic particles filled polymer–matrix nanocomposites

V. Cannillo a,*, F. Bondioli a, L. Lusvarghi a, M. Montorsi a, M. Avella b,M.E. Errico b, M. Malinconico b

a Dipartimento di Ingegneria dei Materiali e dell�Ambiente, University of Modena and Reggio Emilia, Via Vignolese 905, 41100, Modena, Italyb Istituto di Chimica e Tecnologia dei Polimeri (ICTP)-CNR, Pozzuoli, Napoli, Italy

Received 29 April 2005; received in revised form 26 July 2005; accepted 30 July 2005Available online 19 September 2005

Abstract

In this work, the mechanical properties of polymer matrix–ceramic fillers nanocomposites were investigated. A PCL (poly-cap-rolactone) matrix was reinforced with increasing amount of nano-sized silica particles in the range 1–2.5% by weight, and the result-ing properties were determined as a function of reinforcement characteristics and volume fraction. In order to gain a deeper insightinto the mechanical behaviour of such nanocomposites, a numerical model able to reproduce the peculiar composite features was setup. The study focussed on the effect of particles size and amount on the achieved increment in the overall stiffness. The computa-tional approach revealed that a third phase, namely the interphase, has to be taken into account in the model in order to accuratelyreproduce the experimental results.� 2005 Elsevier Ltd. All rights reserved.

Keywords: A. Particle-reinforced composites; B. Mechanical properties; B. Modelling; C. Finite element analysis (FEA); Nanocomposite

1. Introduction

Poly-caprolactone is a biodegradable polymer whichhas been investigated for medical devices, drug deliverysystems, and recently it has been considered as a candi-date material for tissue engineering [1,2]. Moreover PCLis suitable for preparing degradable packaging [3]. How-ever, the main disadvantages associated with PCL are itslow melting temperature and, mainly, its low elasticmodulus and abrasion resistance.

A traditional approach to improve properties of poly-mers is to add a second phase either in the form of fibresor particulate. In the past twenty years, several compos-ites have been produced by adding to the polymer ma-trix micro-sized reinforcements. Recently, the scientificand industrial interest has been focused on the so-callednanocomposites, i.e., materials in which the secondary

0266-3538/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compscitech.2005.07.030

* Corresponding author. Tel.: +39 59 2056240; fax: +39 59 2056243.E-mail address: valeria@unimore.it (V. Cannillo).

phase is nano-sized. In fact, such composites show en-hanced performance compared to traditional ones. Inparticular, a great deal of attention has been devotedto bi-phase or multi-phase systems where inorganicnanometric fillers are added to the polymer. Such rein-forcement, if well dispersed in the matrix, offers a largerspecific surface area compared to usual fillers; thus, thepotential of these system is that the interfacial interac-tions between the matrix and the particles can be en-hanced, leading to an improvement of the propertiesof the material [4]. Some authors reported that even atvery low filler volume content such as 1–5% a consider-able improvement of the mechanical and tribologicalproperties can be achieved [5]. In particular, severalauthors proved that ceramic and silica nanoparticlescan effectively reinforce bulk polymers [e.g. [5,6]].

The aim of the present research is to improve poly-caprolactone mechanical performance by reinforcingwith silica spherical nanoparticles. In fact, the additionof silica nanoparticles is beneficial with respect to the

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demanded properties without preventing the usage inthe biomedical field [7]. In a previous paper, PCL basednanocomposites reinforced with spherical SiO2 nano-particles were prepared and experimentally character-ized [8]. A significant increase in the elastic propertieswas obtained if the inclusions were functionalised in or-der to promote the interfacial adhesion between the twoconstituents and to achieve a fine nanoparticlesdispersion.

In this work, in order to gain a deeper understandinginto the mechanical performance of such nanocompos-ites and into phenomena occurring at the nanoscale, amodelling approach closely related to the compositemorphology is proposed.

In fact several authors pointed out the need for ana-lytical and computational models able to reproduce thebehaviour of complex and heterogeneous materials. Thismethodology can be very helpful for a complete investi-gation of nanocomposites: computational simulationswhich can predict the overall mechanical behaviour ofthe system are an effective tool for a reliable design ofsuch materials. Even if there are several examples ofanalytical and numerical modelling for traditional mi-cro-sized composites, in the field of nanocomposites sev-eral issues still need to be developed and addressed.However, the approaches reported in literature devotedto the investigation of nanocomposite systems cover dif-ferent length scales, ranging from atomistic to contin-uum mechanics; which approach could be the mostsuitable one is still a discussed topic.

Atomistic simulations such as Molecular Dynamics(MD), although very accurate, cannot deal with verylarge time and length scales, which are typically involvedin structural nanocomposites. On the other hand, at thenanoscale, analytical models are difficult to establish ortoo complicated to solve [9]; in fact, the analytical equa-tions which are based on micromechanics and are widelyutilized for traditional composites with micro-sizedinclusions should be tested in order to validate theirapplicability at the nanoscale.

Some recent papers proved the applicability of the Fi-nite Element Method (FEM) for modelling compositeswith a nanometric secondary phase. For example, Liuand Chen demonstrated the feasibility of the applicationof the FEM approach to carbon nanotube-based com-posites, by using a representative volume element(RVE) [9]. Other authors used finite element models inorder to gain a deeper insight into the mechanical prop-erties of nanostructured systems [e.g. [10–12]].

It is worth noting that these FEM-based models areoften applied to the so called representative volume ele-ment, thus assuming that the microstructure of the com-posite can be reproduced by assembling a large numberof such elements. However, this can be a serious limita-tion when dealing with complex and highly heteroge-neous composites microstructures, such as randomly

dispersed particulate systems. Therefore, an approachable to consider the actual microstructure morphologyof the nanocomposite is useful in order to accuratelypredict the overall properties.

Thus, in this work, a computational tool able to mapa composite microstructure onto a finite element mesh isadopted in order to construct models which accuratelyreproduce the composite morphology and characteris-tics. Since it is well known that the shape, the spatialarrangement, the volume fraction and above all the sizeof the nanoparticles play a major role in the resultingperformance, this numerical approach enables to havea deeper insight into the mechanical behaviour of suchmaterials. In particular, the microstructural featureshaving a significant impact on the overall propertiescan be identified.

Thus, the aim of the present paper is to thoroughlyinvestigate mechanical properties of the PCL-silicananocomposites. In particular, the computational modelwill be used to interpret the available experimental re-sults [8], focussing on the fundamental mechanisms aris-ing in such systems.

2. Materials and methods

PCL-SiO2 nanocomposites were prepared by addingnanoparticles in the amount of 1% and 2.5% by weight,respectively.

The details of nanocomposites preparation are re-ported elsewhere [8] and here are only briefly summa-rized. The silica particles, with a spherical shape and asize in the range 100–200 nm, were functionalised bygrafting a hydroxyl end-capped PCL in order to achievea fine particulate dispersion and to improve the poly-meric matrix–inorganic nanofiller interfacial adhesion.

Moreover, PCL based nanocomposites filled withneat silica nanopowders were prepared for comparison,to effectively evaluate the beneficial effect of thefunctionalisation.

SEM observation of fracture surfaces allowed theinvestigation of the SiO2 nanoparticles dispersion andthe PCL/nanoparticles interfacial adhesion. An exampleof the nanocomposite microstructure is reported inFig. 1. As can be seen from the image, the silica fillersare evenly distributed into the matrix and the bond be-tween the two phases appears strong.

The mechanical properties of the so obtained nano-composites were evaluated by tensile tests using an In-stron machine.

Starting from nanocomposites microstructures suchas Fig. 1, computational models for the different nano-composites were prepared by using the computationalcode OOF [13,14]. OOF is an innovative finite elementtool: in fact, by using its pre-processor, digitised imagessuch as SEM micrographs can be easily mapped onto

Table 1Mechanical properties of the composite constituents [8,32]

PCL matrix Silica particles

Young modulus 265 MPa 80 GPaa

Poisson�s ratio 0.35 0.18Density q (g/cm3) 1.15 1.9

a From the literature data [33].

Fig. 1. Nanocomposite microstructure. The filler weight fraction is 2.5%.

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finite elements grids. Thus the mesh reproduces exactlythe original microstructure, namely the inclusions size,morphology, spatial distribution, and the respective vol-ume fraction of the different constituents. Such methodcan be used to prepare computational models of multi-phase systems, i.e., to each phase can be attributed thecorresponding material properties. OOF has been exten-sively used in the literature in order to correlate themicrostructure of the material to the resulting proper-ties. Several authors investigated for example the resid-ual stress distribution in polycrystallines, compositesand coatings; moreover, fracture propagation and otherrelated mechanisms such as damage accumulation andtoughening were assessed as a function of microstruc-tural features in heterogeneous materials [15–31].

In this work, this computational tool was used to pre-pare models of the nanocomposites for each fraction ofsecond phase, namely 1% and 2.5% by weight. SEMimages, such as that in Fig. 1, were acquired, the constit-uent phases were identified and the intrinsic propertiesof each phase were assigned according to Table 1[8,32,33]. The finite element mesh was created, thus dis-

Fig. 2. An example of a microstructure-based finite element grid: the microgrgrid (detail, on the right).

playing all microstructural details, i.e., the inclusionswith their actual size, shape, spatial positioning and inthe exact amount, as illustrated in Fig. 2 for the micro-graph of Fig. 1. Moreover, an adaptive meshing strategyhas been exploited to refine the grid at the matrix–nano-filler interfaces, where the stresses are expected to pres-ent the highest gradients due to the elastic modulusmismatch between the two phases and therefore a highdegree of refinement in the mesh is required to capturevariations in the stress field.

The so constructed models were subjected to tensileload in order to simulate a tensile test. In the computa-tions, the plane stress hypothesis was selected. Both the

aph of Fig. 1 (detail, on the left) has been mapped on a computational

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polymeric matrix and the silica particles were consideredto behave as linear elastic materials: since the focus ofthe present approach is the obtainment of the elasticproperties of the nanocomposites, this hypothesis canbe considered reasonable. Moreover, it was assumed aperfect bonding between the two constituents, havingthe silica fillers been functionalised as described in theprevious section in order to improve matrix-reinforce-ment interfacial adhesion; in fact, SEM micrographs,such as the one reported in Fig. 1, confirmed the goodbond of the two phases.

The stress state was calculated by the finite elementsolver; from stress–strain curves the overall elastic mod-ulus of the nanocomposites was estimated. The numeri-cal results were critically analyzed by comparison withavailable experimental data [8].

Fig. 3. Comparison of the modelling results (two-phase model) withexperimental data and analytical equations. The error bars refer toexperimental data. LN, Lewis–Nielsen equation; HT, Halpin–Tsaiequation.

3. Results and discussion

Experimental tensile tests performed on the availablesamples revealed that a considerable improvement in theelastic modulus could be obtained as a function ofthe reinforcement content in the nanocomposites withthe modified particles, whilst for unmodified fillers nosignificant enhancement was observed [8]. Thus thesedata prove that in fact bulk PCL can be effectively rein-forced by preparing nanocomposites systems, in partic-ular if the nanoparticles are treated to improveinterfacial characteristics between the two phases.

The elastic moduli of the nanocomposites were thenevaluated by means of the computational model, start-ing from the microstructural information obtained bySEM and the intrinsic constituents properties (matrixand particles), as described in the previous section.The obtained results are reported in Fig. 3, along withthe experimental data for comparison. As can be noticedfrom the graph, the modelling results are significantlylower that the experimental ones for the functionalisedparticles nanocomposites; as regards the compositeswith unmodified fillers, it can be observed that for suchcomposites the predicted values are close, even if thecomputational results are slightly greater.

In order to better understand such results, the elasticmoduli of the nanocomposites were also estimated byanalytical equations widely adopted for traditional com-posites, namely the Halpin–Tsai equation e.g. [34] andthe Lewis–Nielsen equation [35]. The comparison ofall these data (reported in Fig. 3 as well) is useful to bet-ter understand the nanocomposite behaviour. The val-ues obtained by means of the analytical equations areclose to the modelling ones. This means that modelscommonly employed for the description of compositeswith micro-sized inclusions are not suitable for theinvestigation of nanocomposites. Nanocomposites showenhanced properties with respect to traditional compos-

ites with the same volume fraction of reinforcement. Infact the values predicted both by the numerical modeland the analytical equations would be the elastic moduliof the composites with silica inclusions of micrometricdimension.

Thus, new modelling strategies should be set up in or-der to accurately reproduce the behaviour of nanocom-posites. It has been often reported in literature that aninterphase region may form between the matrix andthe nanoparticles. The interphase polymer layer nearthe inorganic surface dramatically differs from the bulkpolymer [4]. Because of this interphase layer, compositesshow enhanced properties with respect to constituentsbulk properties: in fact, due to the large surface areaof nanoreinforcements, the interphase polymer is ex-pected to significantly affect the performance ofnanocomposites.

Therefore, an accurate modelling approach shouldaccount for a new phase, namely the interphase. In thecomputational grids, the interphase should be insertedas a third phase, i.e., a layer surrounding the silicananofillers, and with this new model all the elastic prop-erties should be recalculated.

It is worth noting that this hypothesis of an inter-phase considered as a third phase around the particlesis reasonable for the nanocomposites with functional-ised nanoparticles. As a matter of fact, fracture surfacesof this nanocomposites typology (see for example Fig. 1)showed that the polymer adhered well to the particlesand demonstrated the good bond between the twophases; analogous observations were pointed out alsoby other authors for similar systems [5,6]. Thus the sur-face chemical modification of the nanoparticles pro-moted the bonding with the polymer matrix and

Fig. 4. Microstructure with an interphase layer (black, particles; lightgrey, matrix; dark grey, interphase).

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consequently the formation of an interfacial layer.Moreover, the even distribution of particles favoursthe formation of a strong interface [5].

On the other hand, unmodified particles did not showa remarkable improvement in elastic moduli. Thismeans that the functionalisation of silica particles isessential to improve the filler-matrix coupling quality.Moreover, SEM observation of these composites re-vealed that not-treated particles were not homoge-neously dispersed into the polymer matrix. Asmentioned, it is well known that the dispersion state ofthe fillers favours the mechanical properties [5]. If theparticles are not evenly distributed, stress concentrationsmay arise due to particles clusters and this may have adetrimental effect on the overall response. This explainswhy unmodified particles nanocomposites displayproperties which are even lower than the predictedvalues.

In fact, the need to include the interphase layer as athird phase in the model is demanding in functionalisednanocomposites, due to the ultra high interfacial areawhich is present when the characteristic size of the inclu-sion is a few hundred nanometers.

Thus, new models which included the interphaselayer were prepared. In particular, the interphase wasmodelled as a region with uniform properties through-out which were intermediate between those of the matrixand the fillers, as assumed by a previous investigation[36]. So, the elastic modulus of the interfacial layerwas initially set equal to 40 GPa (which is the averagebetween those of the composite constituents) and withan extension of about 0.5 of the particles averagediameter, i.e., 75 nm. The microstructures for thesecalculations were randomly generated by means of acomputer algorithm written ad hoc, which allowed toset the particles dimension, the characteristic size ofthe interphase and the volume fraction of the fillersand thus to be representative of the real sample. Anexample of a so created microstructure is illustrated inFig. 4. With these assumptions, the computationalresults were very close to the experimental data: forthe nanocomposite with 2.5% by weight silica fillersthe estimated modulus was equal to 321.9 MPa, whichis in very good agreement with the experimental valueof 325.0 MPa (see also Fig. 3 and consider error bars).

Since it is well-known that the constituents character-istics greatly affect the overall composite behaviour, asystematic study was performed to investigate thedependence of the composite elastic properties on theinterphase features.

Firstly, a parametric study was carried out as a func-tion of the interphase elastic modulus, varying such va-lue between that of the matrix and that of the particles.The results are reported in Fig. 5, for a weight fractionof particles equal to 2.5% and for an interphase sizeequal to half particle average diameter.

As can be observed from the picture, while the resultsare greatly affected by the interphase modulus up to5 GPa, the variation is smooth for an interphase modu-lus greater than 7.5 GPa. Thus, it can be concluded thatinterphase values in the range 7.5–80 GPa provide anoverall elastic modulus of the nanocomposite which isconsistent with the experimental one. It is worth notingthat also values below that of the matrix were consid-ered, which represent a weak bonding between the twophases: in this case, the overall modulus of the compos-ite can be even lower than that of the unreinforced ma-trix because a load transfer mechanism is not effective.

Then, the effect of the interphase dimension was eval-uated, varying its size up to half the particles averagediameter, while keeping its elastic modulus equal to40 GPa. The results, displayed in Fig. 6, demonstratethat the layer characteristic dimension has a significanteffect on the resulting composite modulus. In fact, theexperimental results are accurately reproduced if the sizeis at least 0.45 times the average diameter. Moreover,the same study was repeated (and reported in Fig. 6 aswell) for an interphase elastic modulus equal to 7.5and 2.5 GPa, respectively. It should be noted that valuesin the range 7.5–40 GPa for the interphase elastic mod-ulus are able to reproduce the experimental results (for aproper interphase dimension), thus confirming the re-sults previously reported in Fig. 5.

In conclusion, the applicability of the finite elementmethod even at the nanoscale has been proven to beeffective to evaluate the overall properties as a functionof microstructural characteristics, if the model are prop-erly prepared, i.e., including an interphase layer. In thepresent paper, the interphase properties which give esti-mates consistent with experimental data are intermedi-

Fig. 5. (a) Modulus of the nanocomposite as a function of theinterphase elastic modulus. The horizontal line represents the value forthe unreinforced matrix. Weight fraction of particles equal to 2.5%.(b) Zoom.

Fig. 6. Effect of the interphase characteristic size on the overallmodulus of the nanocomposite. The x-axis is referred to the particlesaverage diameter (thus the scale ranges from 0 to half diameter). Ei,elastic modulus of the interphase.

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ate between the matrix and the particles. Even if thisassumption is reasonable, it would be advisable to setup experimental techniques able to determine the inter-phase characteristics in terms of stiffness; moreover, theextension of the interphase layer should be assessed.

Therefore, the focus of future research for PCL-silicananocomposites should provide for an accurate estimateof the nanocomposite interphase properties, in order toincorporate these data in the numerical model. In partic-ular, the use of a nanoindentation equipment eventuallycoupled with AFM would contribute to the determina-tion of the interphase mechanical characteristics[6,36,37] and the average width; the latter could be esti-mated assessing its dependence on the filler diameter andsurface chemical treatment [36].

With such data, it would be possible to set up a reli-able model, which could be used as a predictive tool,e.g., for estimating properties of similar systems and alsofor other contents of second phase. In fact, appropriate

computational approaches can play a significant role inthe development of organic–inorganic nanocompositesby providing indications for the analysis and design ofsuch materials.

Moreover, future developments could include a morerefined model for the description of the interphase: in-stead of a homogeneous region with uniform propertiesthroughout, it can be treated as a layer with graded prop-erties between those of the matrix and those of the parti-cles. Since the interphase is a dominant factor governingthe overall composite behaviour [38], detailed informa-tion regarding this region would lead to a more realisticsimulation and thus to more accurate results. In addition,computational models could also be used for interphaseformation and thus properties prediction [38]. A multi-scale model could be set up, integrating information fromdifferent length scales: for example, the interphase forma-tion model or other simulations at the atomic scale couldbe integrated with FEM computations.

Finally, the present study should be extended to ana-lyse also crack propagation, as already performed formicro-sized inclusion particulate systems [e.g. [26,27]].In particular, the correlation between strength and fillersize and amount has to be assessed in order to under-stand fracture mechanisms. In fact, phenomena suchas crack propagation and damage accumulation are af-fected by the nanocomposite nature, i.e., the crack inter-acts with many nanoparticles instead of crossing only afew microparticles.

4. Conclusions

In this work, modelling strategies for the descriptionof poly-caprolactone/silica nanocomposites have been

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proposed and discussed. As a matter of fact, in recentyears, nanocomposites have attracted a great deal ofattention due to the new tremendous opportunities thatsuch materials offer; the ultra-high portion of interfacialarea between the matrix and the nanofillers confers un-ique properties. The need for numerical and analyticalmodels able to predict the behaviour of these new sys-tems is thus demanding.

In this paper, the effective elastic properties have beenevaluated by means of FEM based computational mod-els which could accurately reproduce the nanocompositemicrostructure, namely the fillers size, amount and spa-tial arrangement. The modelling results clearly pointedout the need to include an interphase layer around theparticles as a third constituent material. In fact, bothanalytical equations widely employed for traditionalcomposites and two-phase numerical models failed inreproducing the nanocomposites performance. On theother hand, a three-phase model having interphase char-acteristics intermediate between the two constituentswas able to capture the peculiar properties.

Even if the assumptions of the present model needexperimental verification in terms of a more detailedinterphase characterization, the results give importantindications regarding nanocomposites modelling.

Acknowledgements

The OOF authors [13,14] are gratefully acknowl-edged. Ms. M. Franchi is acknowledged for her helpwith part of the computations.

References

[1] Cohn D, Hotovely Salomon A. Designing biodegradable multi-block PCL/PLA thermoplastic elastomers. Biomaterials2005;26:2297–305.

[2] Pitt C. Poly e-caprolactone and its copolymers. In: Chasin M,Langer R, editors. Biodegradable polymers as drug deliverysystems. New York: Marcel-Dekker; 1990. p. 71–120.

[3] Averous L. Biodegradable multiphase systems based on plasti-cized starch: a review. J Macromol Sci- Pol R 2004;C44:231–74.

[4] Schmidt D, Shah D, Giannelis EP. New advances in polymer/layered silicate nanocomposites. Curr Opin Solid State Mater Sci2002;6:205–12.

[5] Wetzel B, Haupert F, Zhang MQ. Epoxy nanocomposites withhigh mechanical and tribological performance. Compos SciTechnol 2003;63:2055–67.

[6] Wang H, Bai Y, Liu S, Wu J, Wong CP. Combined effects of silicafiller and its interface in epoxy resin. Acta Mater 2002;50:4369–77.

[7] Tian D, Dubois P, Grandfils C, Jerome R. A novel biodegradableand biocompatible ceramer prepared by the sol–gel process. ChemMat 1997;9:871.

[8] Avella M, Bondioli F, Cannillo V, Errico ME, Ferrari AM,Focher B, et al. Preparation, characterisation and computationalstudy of poly(e-caprolactone) based nanocomposites. Mater SciTechnol 2004;20:1340–4.

[9] Liu YJ, Chen XL. Evaluations of the effective material propertiesof carbon nanotube-based composites using a nanoscale repre-sentative volume element. Mech Mater 2003;35:69–81.

[10] Katti DR, Katti KS, Sopp JM, Sarikaya M. 3D finite elementmodelling of mechanical response in nacre-based hybrid nano-composites. Comput Theoret Polymer Sci 2001;11:397–404.

[11] Wang XY, Wang X. Numerical simulation for bending modulusof carbon nanotubes and some explanations for experiment.Composites Part B 2004;35:79–86.

[12] Wang Y, Sun C, Sun X, Hinkley J, Odegard G, Gates TS. 2-Dnano-scale finite element analysis of a polymer field. Compos SciTechnol 2003;63:1581–90.

[13] Carter WC, Langer SA, Fuller ER. The OOF manual: version 1.0;1998.

[14] Langer SA, Fuller ER, Carter WC. OOF: an image-based finiteelement analysis of material microstructures. Comput Sci Eng2001;3:15–23.

[15] Hsueh CH, Fuller ER, Langer SA, Carter WC. Analytical andnumerical analyses for two-dimensional stress transfer. Mater SciEng A 1999;268:1–7.

[16] Hsueh CH, Haynes JA, Lance MJ, Becher PF, Ferber MK, FullerER, et al. Effects of interface roughness on residual stresses inthermal barrier coatings. J Am Ceram Soc 1999;82:1073–5.

[17] Hsueh C-H, Fuller Jr ER. Residual stresses in thermal barriercoatings: effects of interface asperity curvature/height and oxidethickness. Mater Sci Eng A 2000;283:46–55.

[18] Zimmermann A, Fuller ER, Rodel J. Residual stress distributionsin ceramics. J Am Ceram Soc 1999;82(11):3155–60.

[19] Saigal A, Fuller Jr ER. Analysis of stresses in aluminum–siliconalloys. Comput Mater Sci 2001;21:149–58.

[20] Zimmermann MH, Baskin DM, Faber KT, Fuller Jr ER, AllenAJ, Keane DT. Fracture of a textured anisotropic ceramic. ActaMater 2001;49:3231–42.

[21] Zimmermann A, Carter WC, Fuller ER. Damage evolutionduring microcracking of brittle solids. Acta Mater2001;49:127–37.

[22] Vedula VR, Glass SJ, Saylor DM, Rohrer GS, Carter WC,Langer SA, et al. Residual-stress prediction in polycrystallinealumina. J Am Ceram Soc 2001;84(12):2947–54.

[23] Saigal A, Fuller Jr ER, Langer SA, Carter WC, ZimmermannMH, Faber KT. Effect of interface properties on microcracking ofiron titanate. Scripta Mater 1998;38:1449–53.

[24] Cannillo V, Carter WC. Computation and simulation of reliabil-ity parameters and their variations in heterogeneous materials.Acta Mater 2000;48:3593–605.

[25] Cannillo V, Leonelli C, Boccaccini AR. Numerical models forthermal residual stresses in Al2O3 platelets/borosilicate glassmatrix composites. Mater Sci Eng A 2002;323:246–50.

[26] Cannillo V, Pellacani GC, Leonelli C, Boccaccini AR. Numericalmodeling of the fracture behavior of a glass matrix compositereinforced with alumina platelets. Composites Part A2003;34:43–51.

[27] Cannillo V, Leonelli C, Manfredini T, Montorsi M, BoccacciniAR. Computational simulations for the assessment of themechanical properties of glass with controlled porosity. J PorousMater 2003;10:189–200.

[28] Cannillo V, Manfredini T, Montorsi M, Boccaccini AR. Use ofnumerical approaches to predict mechanical properties of brittlebodies containing controlled porosity. J Mater Sci2004;39:4335–7.

[29] Cannillo V, Manfredini T, Montorsi M, Boccaccini AR. Inves-tigation of the mechanical properties of Mo-reinforced glass-matrix composites. J Non-Crystal Solids 2004;344:88–93.

[30] Minay E, Boccaccini AR, Veronesi P, Cannillo V, Leonelli C.Processing of novel glass matrix composites by microwaveheating. J Mater Process Technol 2004;155–156:1749–55.

V. Cannillo et al. / Composites Science and Technology 66 (2006) 1030–1037 1037

[31] Chawla N, Patel BV, Koopman M, Chawla KK, Saha R,Patterson BR, et al. Microstructure-based simulation of ther-momechanical behavior of composite materials by object-oriented finite element analysis. Mater Charact2002;49:395–407.

[32] Avella M, Bondioli F, Cannillo V, Cosco S, Errico ME, FerrariAM, et al. Properties/structure relationships in innovative PCL-SiO2 nanocomposites. Macromol Symp 2004;218:201–10.

[33] Odegard GM, Clancy TC, Gates TS. Modeling of the mechanicalproperties of nanoparticle/polymer composites. Polymer2005;46:553–62.

[34] Barbero EJ. Introduction to composite materials design. NewYork: Taylor & Francis Inc.; 1998.

[35] Nielsen LE, Landel RF. Mechanical properties of polymer andcomposites. New York: Marcel Dekker Inc.; 1994.

[36] Gao S-L, Mader E. Characterisation of interphase nanoscaleproperty variations in glass fibre reinforced polypropilene andepoxy resin composites. Composites Part A 2002;33: 559–76.

[37] Bogetti TA, Wang T, VanLandingham MR, Gillespie Jr JW.Characterization of nanoscale property variations in polymercomposite systems: 2. Numerical modelling. Composites Part A1999;30:85–94.

[38] Yang F, Pitchumani R. Effects of interphase formation on themodulus and stress concentration factor of fiber-reinforcedthermosetting-matrix composites. Compos Sci Technol2004;64:145–1437.