Elastic properties of multi-cracked composite materials

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1 Eur. Phys. J. B 76, 6 69 () DOI:./epjb/e-7-6 Elast propertes o mult-raked omposte materals S. Gordano, M.I. Saba and L. Colombo

2 Eur. Phys. J. B 76, 6 69 () DOI:./epjb/e-7-6 Regular Artle THE EUROPEAN PHYSICAL JOURNAL B Elast propertes o mult-raked omposte materals S. Gordano a, M.I. Saba, and L. Colombo Department o Physs o the Unversty o Caglar and Isttuto Ona de Materal del CNR (CNR-IOM) Untà SLACS, Cttadella Unverstara, 9 Monserrato (Ca), Italy Reeved Deember 9 / Reeved n nal orm Marh Publshed onlne June EDP Senes, Soetà Italana d Fsa, Sprnger-Verlag Abstrat. The problem o predtng the eetve elast propertes o mult-raked and/or omposte materals s both undamental n materals mehans and o large tehnologal mpat. In ths paper we develop a ontnuum elastty model, based on the Eshelby theory and on the derental homogenzaton tehnque, or the eetve elast modul o a bro-renored system and we address t to elaborate an estmaton o the average alure ondton o suh ompostes. Introduton Ths work s amed at evaluatng the eetve elast propertes o materals ontanng a gven dstrbuton o raks and nlusons. A paradgmat example o suh mult-raked materals wth large tehnologal mpat s oered by nanoompostes (.e. nanobro-renored materals): here a talored texture o bers s nserted nto a matrx, so generatng a struture wth overall mproved mehanal propertes (e.g., hgher rature toughness) [,]. In many real applatons arbon nanotubes and nanobers (also hemally untonalzed) are used n renored ompostes [,]. Moreover, slon arbde ompostes renored by arbon bers (C/SC) are eram materals, haraterzed by thermo-mehanal stablty, low densty and hgh rature toughness [5,6]. Whle the overall eet o a rak populaton embedded n a homogeneous matrx has been evaluated by the method o tratons (or pseudotratons) [7,8], through the sel-onsstent tehnque [9] and by the derental methodology [ ], the nterplay among raks and renorng nlusons (lke, e.g., bers) s stll largely unexplored. In the present paper we ntrodue a methodology or homogenzng a mult-raked and omposte materal, takng nto aount the nteratons between raks and renorng partles. Ths work, thereore, belongs to the vast eld o homogenzaton tehnques [,]. As or the elast haratersaton o dspersons, several works have been developed: an exat result exsts or a materal omposed by a very dlute onentraton o spheral or ylndral nlusons dspersed n a sold matrx. Ths result, attrbuted to several authors [5,6], has been generalzed to the ase o any nte volume raton by teratve [7,8] or derental [9 ] methods both or spheral or ylndral nlusons [] and or ellpsodal partles []. a e-mal: steano.gordano@ds.una.t From the hstoral pont o vew, one o the rst attempt to obtan the eetve behavor o an elast dsperson has been perormed through a sel-onsstent method (SCM) leadng to the so-alled Mor-Tanaka sheme []. The range o valdty o suh a method has been dsussed or a polydsperse (sze) suspenson o spheral nlusons n a ontnuous matrx phase [5]. Moreover, spe results have been obtaned or ompostes renored by algned or randomly orented, transversely sotrop bers or platelets by usng the above Mor-Tanaka sheme [6]. A urther generalzaton has been perormed through a method to obtan the transversely-sotrop eetve thermomehanal propertes o undretonal ompostes renored wth arbtrarly oated ylndral bers [7]. Some other methodologes have been developed as well to ope wth derent phases ormng the renorng system: or example, the double-nluson model (ormed by an ellpsodal nluson wth an ellpsodal heterogenety, embedded n an nntely extended homogeneous doman) has been ntrodued and t has been generalzed to the mult-nluson model [8]. These approahes have been appled to a omposte ontanng nlusons wth multlayer oatngs and to a omposte onsstng o several dstnt materals: n eah ase ther overall modul are analytally estmated [8]. Fnally, a generalzed selonsstent method (GSCM) based on the energy equvalene and a matrx-omposte model have been proposed and broadly appled to ompostes wth three or more phases [9]. The above theoretal methods have a ommon eature, namely: they have been appled to sngle- or multple-phase renorng nlusons, but do not take nto onsderaton the nterplay among nlusons and raks. Thereore, we devote the rst part o the present paper to the development o a derental method whh s able to onsder (arbtrarly large) volume ratons o an arbtrary number o nteratng populatons o derent nhomogenetes (ether renorng nlusons or raks)

3 6 The European Physal Journal B Ĉ Ĉ Ĉ b b e a Fg.. (Color onlne) Sheme o a multraked and brorenored materal. Ĉ, Ĉ and Ĉ are, respetvely, the stness tensors o the matrx, the raks and the nlusons. On the rght we sketh the geometral parameters or the ellpt raks. embedded n the hostng matrx. In partular, we derve a set o ordnary derental equatons desrbng the eetve elast modul (the stness tensor) o the omposte struture wth an arbtrary stohometr omposton. In the seond part o the present paper we nvestgate the paradgmat stuaton o a bro-renored and multraked omposte materal wth parallel ylndral bers, muh ster than the host matrx. In ths ase bres and raks represent two derent embedded populatons. A shemat representaton o suh a struture an be ound n Fgure, where the geometr detals are shown. Ths stuaton deally meets the C/SC ase above dsussed. The present derental method s addressed to obtan the eetve elast behavor o ths system through a par o losed-orm relatonshps (or the eetve Young modulus and the Posson rato, respetvely). The resultng elast model s dsussed wth the support o numeral results, whh desrbe the behavor o the system wth derent propertes o ts onsttuents. Fnally, we desrbe an approxmate applaton o the Grth stablty rteron or a omposte materal, degraded by a gven assembly o raks. We obtan an estmaton o the average alure stress n terms o the volume raton o renorng bres and the rak densty. It s thereore possble to predt approxmately the atual volume raton o bers to embed n order to get the desred value o alure stress, when t s dened the largest rak densty admtted n the sample. The struture o the paper s the ollowng: n Seton we outlne the multomponent derental proedure. In Seton we desrbe the applaton o suh a method to the mult-raked and bro-renored struture. Fnally, n Seton we desrbe the estmaton o the alure stress or the heterogeneous system. Multomponent derental method In general, we dene a omposte as a matrx (wth stness tensor Ĉ) embeddng N populatons o derent nlusons, haraterzed by volume ratons v and stness tensors Ĉ ( =...N). For sotrop phases eah stness tensors Ĉ an be wrtten n terms o the Young modulus E and the Posson rato ν, takng nto aount the elast response o the -th materal [ ]. As ustomary n mromehans [ 5], we assume that eah nluson s an ellpsod, sne suh a shape s the mother geometry o a all the strutures o nterest or ths work (.e., ylndral bers and slt raks). For sake o smplty, all the ellpsods o the -th populaton are taken equal. Wthn ths ramework, we make use o the Eshelby theory [6,7] or desrbng ther elast behavor upon loadng. Ths theory provdes the relatonshp between the unorm stran ˆɛ wthn eah nluson and the remotely appled stran )] ˆɛ,namely:ˆɛ = [Î Ŝ (Î Ĉ Ĉ ˆɛ,whereŜ s the Eshelby tensor o the -th populaton []. In the ase o randomly orented nlusons, we are nterested n the average value ˆɛ = ˆɛ o the nternal stran over all the possble orentatons o the ellpsods, where we have )] dened  = [Î Ŝ (Î Ĉ Ĉ []. The applaton o the Eshelby theory leads to a rst homogenzaton step vald only or dluted dspersons. Under the assumpton v, we an evaluate the average stran n the system as ( ˆɛ = ) v ˆɛ + v ˆɛ. () On the other hand, the average value o the stress tensor s gven by ˆT = ˆTdv V V = ˆTdv+ ˆTdv V V V V = V Ĉ ˆɛdv + Ĉ ˆɛdv V V V + V Ĉ ˆɛdv V V Ĉ ˆɛdv V = V Ĉ ˆɛdv + V V V (Ĉ Ĉ) ˆɛdv V V = Ĉ ˆɛ + v (Ĉ Ĉ) ˆɛ () where V s the volume o the matrx, V s the volume o the -th populaton o nlusons and V s the total volume o the omposte materal. By drawng a omparson betweenthe expressonsor ˆɛ and ˆT, we eventually obtan the eetve stness tensor Ĉ e = Ĉ + v (Ĉ Ĉ) [( ) v Î + ] v Â. () In the present ase o dluted dspersons wth v equaton () an be smpled by negletng the last nverse tensor [8]. Thereore, the nal expresson or the eetve stness tensor s Ĉ e = Ĉ + v (Ĉ Ĉ)Â. ()

4 S. Gordano et al.: Elast propertes o mult-raked omposte materals 6 For brevty, equaton () an be summarzed n the orm Ĉ e = (Ĉ, {Ĉ}, {v }), where the unton ully desrbes the eetve behavor o a dlute dsperson. The next step s to onsder hgher values o the volume ratons. We dene V as the volume o the -th populaton o nlusons and V as the total volume o the omposte materal (so that v = V V ). An teratve sheme an be mplemented by usng a sngle salar varable t dened by v = α t,wth α =and t. We observe that t represents the total volume raton o nlusons sne v = α t = t. We onsder the ntal onguraton wth v = V V = α t and V = α tv,oraverysmallt. The ntal eetve elast tensor s alulated as Ĉe = (Ĉ, {Ĉ},v ), aordng to equaton (). Next, we slghtly nrease the nlusons volume by ΔV = α ΔtV and we get the orrespondng eetve stness tensor Ĉ e = (Ĉe, {Ĉ},v ). We stress that v ΔV = V + k ΔV k = Δt +Δt α (5) are the volume ratons o the nlusons embedded n a vrtual matrx o stness tensor Ĉe. SneΔt s small by onstruton, t s possble to elaborate a rst order expanson o Ĉ e Ĉ e = Ĉe + v Δt +Δt α, (6) where the symbol means that v must be alulated or v =andĉ = Ĉe. Through the above proedure we an denty the ntal volume ratons v n = α t and the atual nal volume ratons v n = V + ΔV V + k ΔV k = α t + Δt +Δt. (7) Thereore, the orrespondng nrements are gven by Δv = v n v n We an reast equaton (6) n the orm ( ) Ĉ e {v n } Ĉe ( {v n } ) = = α Δt t +Δt. (8) v Δt +Δt α. (9) Thenrementothevarablet, when we nrease the volume ratons v n up to the values v n,sgvenby Δv α = Δt t +Δt. So, by dvdng both sdes o equaton (9) by the quantty Δv α = Δt t +Δt, we obtan the derental quotent o the eetve elast tensor Ĉe. Fnally, n the lmt o Δt, the teratve sheme onverges to a derental equaton dĉe dt = α () t v wth ntal ondton Ĉe(t =)=Ĉ. One solved the derental equaton, we must restore t and α through the denttes t = v and α = v / k v k. Eetve behavor o the mult-raked omposte We now address the above ormal deve to the paradgmat heterogeneous system we are gong to nvestgate. We onsder an sotrop matrx wth an elast tensor Ĉ embeddng just two knds o nlusons, namely: an assembly o rgd parallel ylndral bers (stness Ĉ wth Young modulus E ) and an assembly o slt raks (stness Ĉ wth E ). The orrespondng volume ratons are v and v, respetvely. All the nlusons are orented along the same dreton perpendular to a gven plane, as shown n Fgure. We mpose the plane stran ondton, thus translatng our problem nto a twodmensonal one. We remark that the slt raks are randomly orented wthn the plane and, thereore, we wll observe an overall sotrop elast behavor o the twodmensonal system []. Provsonally, a slt rak s treated as a ylnder wth a strongly oblate ellpt base (orrespondng to a vanshng aspet rato e, as dened n Fg. ). Ths approah s very onvenent sne we wll develop our arguments by takng prot rom general results holdng or ellpsodal nlusons [,]. Sne the area o the ellpt base s gven by πab = πea (see Fg. ), we an easly obtan the volume raton o the assembly o raks as v = Nπea /A = e, where N s the number o raks dspersed over the area A and = Nπa /A s the rak densty. From now on, the two varables v = (volume raton o bers) and (rak densty) wll be used to desrbe the omposton o the system. We an ombne the general result gven n equaton () wth the spe ase o dlute populatons o bers and raks. In both ases, we benet o the explt expressons or the Eshelby tensors, hereater reerred to as Ŝ and Ŝ, respetvely []. In the present ase equaton (), takng nto aount the populaton o random orented ellpt ylnders (vod) and the populaton o renorng rular ylnders (rgd), assumes the orm Ĉ e = Ĉ + v (Ĉ Ĉ) v Ĉ  () where or the renorng bres we have v = and  = [Î Ŝ (Î Ĉ Ĉ )] () and or the random orented raks v = e and  = (Î Ŝ). () Ater some algebra onernng the lmtng value o Ĉ e or E, the homogenzaton proedure eventually provdes the eetve Young modulus and the eetve Posson rato = E ( + G + eg ) = + H + eh ()

5 6 The European Physal Journal B wherewehaventroduedtheourquanttesg, G, H and H as ollows G = ( )(8ν +5) ( )( +) rg = ( )( e + + e + e + e +) e( +) H = ( )( )( ) H = e e +ν e + ν e + e ν. (5) e The slt rak geometry s nally reovered through the lmt b or, equvalently, e. We get = E + E ( )(8ν +5) ( )( +) + E ( )( +) + ν e = + ( )( )( ) + ( ). (6) Whle equaton (6) s vald n the dlute lmt, we am at applyng the above derental sheme n order to ope wth arbtrarly large volume ratons o bers and raks. Ths an be done by usng equaton (), where v may be easly obtaned rom equatons (5) and (6). The resultng system o derental equatons or the eetve elast modul s d dt dν e dt = t (α G + α G ) = t (α H + α H ), (7) where the untons G, G, H and H must be evaluated or = ν e. O ourse, we onsder the ntal ondtons (t =)=E and ν e (t =)=. The problem an be solved through a ouple o ntegratons as ollows νe α G (ν e)+α G (ν e) α H (ν e )+α H (ν e ) dν e = νe dν e α H (ν e )+α H (ν e ) = Ee E t d dt t. (8) One perormed the ntegratons, the relatons t = + e, α = +e and α = e +e are used n order to elmnate the temporary parameters t, α and α and to ntrodue the real quanttes and. Fnally, by takng the lmt o e, we obtan the key result o the present nvestgaton or a mult-raked omposte, namely an mplt equaton or the eetve Posson rato ln ν [ ] e +ln ( ) + + s + + r q = (9) and the explt orm o the eetve Young modulus ln = E + s ln + +ν e 8 + r q, () where we have dened the parameters +9 q = + [ ] q + 8ν q +8ν e r =ln q + 8ν e q +8 [ ] + ν ( ) ( + ) s =ln. () + ν e (ν e ) ( + ) We remark that the general result provded by equatons (9) and () reovers the more spe ase obtaned n reerene [] or a mult-raked homogeneous materal (.e., or = ). The same also holds or the partular ase o a purely brous materal ( = ): present ndngs are onsstent wth prevous lterature []. More speally, we onsder = we obtan the explt expressons ν e = +( )e () +( )e = E [ +( )e ] ( + ). () In ths ase the relaton gvng the eetve Posson rato has been expltly solved as shown n equaton (). Then,tsusedntherelatonor by obtanng equaton (). On the other hand, when = (purely brous materal) we eventually obtan ( ) / ( ) / νe ν e ν ( ) = () ν e = E +ν e + ( ) ( ) νe ν. (5) ν e In ths ase equaton () gvng the eetve Posson rato an not be solved expltly and t remans n mplt orm. Wth the am o analyzng the behavor o and ν e desrbed by equatons (9) and () we show a seres o numeral results obtaned or derent values o the matrx Posson rato. The results are organzed as ollows: =.8 nfgure; =.5 nfgure; =.5 n Fgure and, nally, =. n Fgure 5. The rst two ases onern a negatve Posson rato o the matrx, orrespondng to the realst ases reported or oams [9]. The other two ases show the results or postve values o the matrx Posson rato, typal o brttle and eram materals. In Fgure (rght) we note that when and we obtan ν e, as expeted or a matrx wthout any nluson (ber or rak). Moreover, Fgure (rght) shows that, or =and (.e. wthout raks), we obtan ν e /. Ths property an, n at, be easly vered by equaton (), and t s satsed or any value o.

6 S. Gordano et al.: Elast propertes o mult-raked omposte materals 65 /E 6 5 =.8 ν e... = Fg.. (Color onlne) Eetve Young modulus (let) and eetve Posson rato ν e (rght) n terms the rak densty alulated or a matrx wth =.8. The arrow marked by ndates an nreasng volume raton o bers n the range <<. s normalzed to the value E o the matrx Young modulus. /E 6 5 =.5 5 ν e = Fg.. (Color onlne) Eetve Young modulus (let) and eetve Posson rato ν e (rght) n terms the rak densty alulated or a matrx wth =.5. The arrow marked by ndates an nreasng volume raton o bers n the range <<. s normalzed to the value E o the matrx Young modulus =.5. =.5 /E ν e Fg.. (Color onlne) Eetve Young modulus (let) and eetve Posson rato ν e (rght) n terms the rak densty alulated or a matrx wth =.5. The arrow marked by ndates an nreasng volume raton o bers n the range <<. s normalzed to the value E o the matrx Young modulus.

7 66 The European Physal Journal B 6. 5 =.. =. /E ν e Fg. 5. (Color onlne) Eetve Young modulus (let) and eetve Posson rato ν e (rght) n terms the rak densty alulated or a matrx wth =.. The arrow marked by ndates an nreasng volume raton o bers n the range <<. s normalzed to the value E o the matrx Young modulus. We also observe that the value o the Posson rato tends to beome postve or both nreasng values o and. Ths phenomenon an be explaned by observng that the extreme ases o vod materal and rgd materal have an elast behavor whh s n opposton to the elast response dened by a negatve Posson rato (where a gven traton leads to a dlataton n the transverse dretons). As or the eetve Young modulus shown n Fgure (let), we obtan, or low values o and, avaluegreater than the Young modulus o the orgnal elast matrx. Ths s an nterestng unonventonal behavor o the eetve Young modulus o the multraked omposte sold, exhbted or negatve Posson rato o the matrx. It an be explaned as ollows: to begn, we onsder the smpler ase wth = (wthout bres). The dervatve o equaton () wth respet to (evaluated or =)anbe smply obtaned as d d ( )( + ) = = E. (6) + Thereore, the urve /E versus shows a maxmum or < < /. Aordngly, equaton (6) s postve or < < / provng that the eetve Young modulus s an nreasng unton o around the pont =. Ths phenomenon remans observable or low values o the bre volume raton and dsappears or hgher values o. Evdently, or values o ater the maxmum, the eetve Young modulus s dentvely dereasng, desrbng the progressve degradaton o the mult-raked materal. Moreover, the urves or /E n Fgure (let) show the renorng eets o the bres embedded n the omposte materal: the exponental nrease o /E wth s n at evdent. In the ase wth = /, the urves o ν e shown n Fgure (rght) do not exhbt mportant varaton wth respet to the prevous ase wth =.8. On the ontrary, as or the eetve Young modulus, t s mportant to note that the value = / represents a transton n the elast behavor. As a matter o at, as shown n equaton (6), or = / wehave d d = =.It means that, or and =,theurveo /E approahes the value wth horzontal tangent lne, as learly shown n Fgure (let). We onsder now the rst ase wth a postve Posson rato =.5. Fgure (rght) shows that ν e =/ or = and or any value o the volume raton o bres. Ths behavor s n peret agreement wth the above stated property: wthout raks ( = )and wth an hgh onentraton o bres ( ) we have ν e /. In other words, we an say that the value =/ assumes the role o a xed pont or the Posson rato o the brous materal. As or the eetve Young modulus, we observe n Fgure (let) a qualtatve behavor smlar to the prevous results. Interestngly enough, we remark that /E dereases wth startng wth an ntal negatve slope (sne > /; see Eq. (6)). Fnally, we onsder the ase wth =.. We observe n Fgure 5 (rght) a dereasng trend o ν e (rom ν e =. toν e =.5) or =when vares rom to. Ths behavor s agan n peret agreement wth the property above dsussed. The behavor o the eetve Young modulus shown n Fgure 5 (let) onrms the renorng eet o the dstrbuton o ylndral bres n the materal. Falure propertes The ornerstone o lnear elast rature mehans or homogeneous materals s represented by the renowed Grth rteron or brttle alure, namely []: upon loadng, a sngle (solated) slt rak propagates (.e. the materal als) provded that the tensle load s greater than the Grth alure stress σ G. The same model provdes a smple and very useul relaton between σ G and the Young modulus E and the Posson rato o the materal, namely σ G = γ s E πa( ν ), (7)

8 S. Gordano et al.: Elast propertes o mult-raked omposte materals 67 /E Fg. 6. (Color onlne) Eetve Young modulus and eetve Posson rato ν e n terms o the volume raton o bers and the rak densty alulated or a SC matrx wth =.8. ν e where we have mposed plane-stran ondtons and we have ndated by a and γ s the rak hal-length and the materal spe surae energy, respetvely [,]. The Grth rteron was orgnally stated n terms o an energy balane: the growth o a rak wthn a materal under loadng ores requres the reaton o two new suraes. Thereore, n order to make a materal to al (.e. n order to make the rak to nrease ts length), the mehanal work o the appled external ores must exeed the work neessary to reate new (nternal) surae. The Grth rteron an be also stated n terms o the Stress Intensty Fator (SIF) KI G : a key quantty n lnear elast rature mehans desrbng the onentraton o the stress near the rak tps, aused by a remote load. For an solated slt rak remotely loaded n mode I by a stress σ, the asymptot orm o the tensle stress near the rak tp s KI G/ πr (where r s the dstane rom the rak tp measured on the rak plane). The lnear elast theory predts the value o KI G n ths ase [,] K G I = σ πa. (8) The Grth rteron an be also stated by armng that the rak propagates KI G KI, G where the quantty KI, G = γ s E /( ν ) s the so-alled rature toughness, whh a materal onstant. From a oneptual pont o vew the energy balane underlyng the Grth rteron an be also appled to a populaton o raks dspersed n a gven body. Nevertheless, n suh a ase, the surae energy assoated wth the populaton o raks an be smply evaluated whle the elast energy o the system under stress s not avalable n losed orm (sne the elast elds n ths omplex ase an not be evaluated analytally). Thereore, although the energy balane stll remans the key oneptual tem, t s very hard to explot t n order to determne the alure ondton. So, the Grth theory an be expltly appled only to an solated rak, embedded n a homogeneous materal: a stuaton qute ar o the ommon prate. As a matter o at, real materals (both manuatured and natural) ontan ull populatons o raks (possbly takng a large densty) and dsplay ompostonal lutuatons (beause o elast nlusons, deets, preptates or bers). Whle the eetve Youngs modulus and the eetve Posson rato are volume averaged quanttes and they are almost nsenstve to lusterng o the nhomogenetes or spe spatal dstrbutons o the nlusons, the alure stress and the stress ntensty ator are very senstve to the mutual postons o nhomogenetes. Ths s true sne ther dentons depend on the worst ase o spatal arrangement o raks and bres. From ths pont o vew, these quanttes an not be determned by an homogenzaton tehnque, as the present one. Nevertheless, we an obtan a rough estmate o the alure stress and the stress ntensty ator by supposng a unorm or regular dstrbuton o objets wthn the matrx. It means that we are evaluatng the average value o a random varable whh s, however, haraterzed by a large value o the standard devaton. We an use equatons (9) and () or analyzng the rature mehans o a mult-raked and bro-renored materal. An applaton o these results s gven n Fgure 6 where and ν e are shown or a SC matrx ( =.8). By onsderng a random dstrbuton o bers and raks as shown n Fgure, the eetve average alure stress σ e n suh a omposte system an be straghtorwardly obtaned rom equaton (7) by replang and E wth ν e and.itmeansthatwe are alulatng the average value o alure stress over all the possble onguratons wth gven volume raton o bres and raks. Thereore, σ e s smply provded by the ollowng relaton σ e = σ G ν E νe. (9) The estmate n equaton (9) relets how alure propertes are aeted by the presene o heterogenetes. Thereore, t an be used also as rough estmate o the hanges o the ntrns toughness ndued by the presene o the heterogenetes. It s mportant to observe that there are two spe ases where the estmate gven n equaton (9) should be assumed as a good approxmaton o the real alure stress: () when the raks and the renorng bres are unormly spaed (n average sense) n the spae (n ths ase all the eets ndued by the proxmty o the heterogenetes are avoded); () when we are

9 68 The European Physal Journal B σ e /σ G Fg. 7. (Color onlne) Eetve alure stress σ e (let) and stress ntensty ator K e I (rght) or a mult-raked omposte as unton o ber and rak densty. Both quanttes are normalzed to the orrespondng value or an homogeneous matrx ontanng just a rak. K I e /KI G * e G σ /σ =. e G σ /σ = e G σ /σ =.8 e G σ /σ =x* value o the sutable volume raton o bers that must be embedded n order to get the desred value o alure stress rato x. Ths s shown n Fgure 8 where a omposte SC system ontanng a onentraton o rgd arbon bers s onsdered. I we need that suh a omposte ressts up to a maxmum appled stress rato as large as x, we should embed as many bers as orrespondng toavolumeraton.. *.5.5 Fg. 8. (Color onlne) Contour plot o σ e /σg n terms o the volume raton o bers and the rak densty. dealng wth raks wth the length muh greater than the radus o the bres and small (n ths ase the populaton o raks an be onsdered qute exatly as embedded n an eetve homogeneous matrx). In Fgure 7 (let) we show the σ e /σg rato versus the ber () andrak() denstes or the atual ase o a SC matrx. I = we obtaned the exat exponental law σ e /σg =exp( /), whh explans and quantes the degradaton phenomenon wth an nreasng rak densty. Moreover, the σ e /σg trend versus quanttatvely predts the renorng eatures due to the bers. In Fgure 7 (rght) we have also shown the rato between the Stress Intensty Fator (SIF, n mode I) o a rak n the omposte materal and an solated rak o the same sze [,]. Ths plot has been obtaned by observng that the ollowng relaton s vered K e I K G I = σg σ e () wth the same degree o approxmaton o equaton (9). It means that the SIF rato shows a behavor whh s smply the reverse o that o the alure stress. Fnally, n Fgure 8 we onsder the ontour plot o the rato σ e /σg : by seletng a gven rak densty admtted n a sample, t s ndeed possble to dretly predt the 5 Conlusons In onluson, a rened teratve/derental sheme has been ntrodued n order to evaluate the eetve elast propertes o a mult-raked and brous materal. The nteratons among renorng nlusons and raks have been taken nto aount n order to obtan a homogenzaton sheme whh s vald or arbtrary values o volume raton o bres and rak densty. The results have been shown or both negatve and postve values o the matrx Posson rato. The ase o the C/SC omposte struture has been onsdered rom the pont o vew o the alure stablty. We have shown that ths approah allows or an approxmate predton and desgn o the bro-renorng system, as or ts alure ondton. Ths work s ounded by Regone Autonoma della Sardegna under projet Modellazone Multsala della Meana de Materal Compost (MC). One o us (S. G.) aknowledges nanal support by CYBERSAR (Caglar, Italy). Reerenes. Handbook o Materals Modelng, edted by G.B. Olson (Sprnger, Amsterdam, 5). Ceram Armour Materals By Desgn, edtedbyj.w. MCauley et al. (The Ameran Ceram Soety, Westervlle, Oho, ). J. L, C. Papadopoulos, J.M. Xu, M. Moskovts, Appl. Phys. Lett. 75, 67 (999). S. Wang, Z. Lang, T. Lu, B. Wang, C. Zhang, Nanotehnology 7, 55 (6)

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