AN ANALYTICAL STUDY ON THE ELASTIC-PLASTIC BEHAVIOR OF METAL MATRIX COMPOSITES UNDER TENSILE LOADING

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1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 5,, pp , Wrsw 014 AN ANALYTICAL STUDY ON THE ELASTIC-PLASTIC BEHAVIOR OF METAL MATRIX COMPOSITES UNDER TENSILE LOADING Sdf Khosoussi Aerospce Engineering Deprtment, Shrif University of Technology, Tehrn, Irn Mehdi Mondli Deprtment of Mechnicl nd Aerospce Engineering, Islmic Azd University, Science nd Reserch Brnch, Tehrn, Irn Ali Abedin Aerospce Engineering Deprtment, Shrif University of Technology, Tehrn, Irn; e-mil: An nlyticl pproch is proposed for studying the elstic-plstic behvior of short fiber reinforced metl mtrix composites under tensile loding. In the proposed reserch, micromechnicl pproch is employed considering n xi-symmetric unit cell including one fiber nd the surrounding mtrix. First, the governing equtions nd the boundry conditions re derived nd the elstic solution is obtined bsed on some sher lg type methods. Since under norml loding conditions nd ccording to the fiber mteril chrcteristics the metl mtrix undergoes plstic deformtion, while the fiber remins within the elstic region, plstic deformtion is obtined for the mtrix under ech smll tensile loding step. Then, the elstic-plstic stress trnsfer behvior of the composite is studied considering this plstic deformtion. The results re finlly compred with the numericl results obtined from the FE nlysis of the considered micromechnicl model. Keywords: metl mtrix composites, plsticity in mtrix, sher lg 1. Introduction The use of metl mtrix composite mterils(mmc) in erospce structures nd engine prts hsbecomemorefrequentrecently.stresstrnsferfromthemtrixtothefiberisknowns the most importnt mechnism governing the deformtion nd the frcture response of such MMCs. The stress trnsfer chrcteristic of fiber reinforced composite mterils under vrious mechnicl nd therml lodings hs been studied frequently. Since the nlyticl study of the behvior of MMCs nd the ssumptions involved re complex, the mjority of the investigtions is limited to the numericl finite element or experimentl methods. Recent efforts for nlyticl study of such mterils hve led to development of vrious micromechnicl models including the fiber nd the surrounding mtrix, which considering severl simplifying ssumptions, hve tried to interpret the behvior of MMCs. Due to the complexity of the model nd vrious ssumptions involved, most of the performed studies hve been limitedtotheelsticbehviornlysisofthefiberndthemtrix.though,sincethemtrixis metllic nd thus the plsticity will occur t low strins, unpredicted filure my hppen in the prts mde from MMCs. The solution to the concerned problem consists of two prts, first the elstic nlysis of the MMCs nd then the plstic solution for the metl mtrix, which is bsed on the stress/strin fields obtined from the former elstic nlysis. In generl, numerous methods hve been developedforelsticnlysisofsuchmmcs.oneoftheminpprochestosuchproblemisthe Sher Lg Method(Cox, 195; Kelly, 1966; Piggott, 1980; Fukud nd Chou, 1981; Nrdone nd Prewo, 1986; Clyne, 1989; Krbhri nd Wilkins, 1991; Strink nd Syngellkis, 1999; Go

2 34 S. Khosoussi et l. ndli,005),whichduetothegooddescriptionofthelodtrnsfermechnismfromthefiber tothemtrixisofmjorimportncendppliction.but,itshouldbenotedthtthevrious simplifying ssumptions involved re considered s the min disdvntge of this model. Since duetosuchssumptions,themodelisnotthtmuchccurte,ithsbeenmodifiedbyothers uthors, see Hsueh( ), leding to incresed efficiency, enbling the clcultion of the stressesttheendofthefiberndthesherstresstthefiber/mtrixinterfce.inthismethod, clled the Imginry Fiber Technique, the problem is initilly solved for continuous long fiber, nd by mens of pplying the boundry conditions, the consistency condition, nd considering fiberwiththesmemtrixmterilttheendsection,thesolutiontotheshortfiberproblemis obtined ccordingly. The next effort for modeling the concerned elstic problem ws performed byjingetl.(1998,004).intheinitilsteps,helsoppliedsomeoftheprevioussimplifying ssumptions involved, but eliminting some of these ssumptions in the finl stges, he chieved better comptibility with the existing FE results. After this brief review of the existing solutions for the elstic nlysis of the current problem, some of the efforts for solving the plstic problem will be discussed shortly. The Sher Lg Method first introduced by Cox(195) ws vstly used due to its mthemticl simplicity nd its good prediction of this mechnism. After the elstic nlysis of the existing problem, Jing et l.(004) studied the plstic behvior of short fiber reinforced MMCs, pplying the Sher Lg Method. In this study, the effect of the mtrix plstic deformtion on the lod trnsfer mechnism hs been defined vi introduction of plstic strin. The results show tht the efficiency of the lod trnsfer in the elstic region decreses due to locl plstic deformtions inthemtrix.though,inthisstudy,verysimplepprochhsbeenusedforplsticnlysis ofthemtrix.sincethegenerlsherlgmodelppliedisnotcpbleofpredictingthestress distribution in the mtrix, the problem hs been simplified by mens of ssuming n verge xil plstic deformtion in the mtrix, defined by some liner distribution ssumptions mde lter. Furthermore, the plstic strin term considered hs been limited to the xil strin only, nd the effect of the sher stress hs been totlly neglected by pplying n pproximte reltion between the xil stress nd the verge xil plstic deformtion in the mtrix. In the present study, n nlyticl pproch is proposed for studying the elstic-plstic behvior of short fiber reinforced metl mtrix composites under smll tensile loding steps. In the proposed reserch, employing micromechnicl pproch, n xi-symmetric unit cell including one fiber nd the surrounding mtrix is considered. Using sher-lg bsed formultion, first, the governing equtions nd the boundry conditions re derived nd the elstic solution is obtined for both the fiber nd mtrix. Since under norml loding conditions nd ccording to the fiber mteril chrcteristics, the metl mtrix undergoes plstic deformtion first, plstic deformtion is considered in the mtrix. The governing reltions re then obtined, considering both xil nd sher stress terms, e.g. the equivlent stress nd the plstic strin components. Finlly, n pproximte estimte for plstic strin distribution in the mtrix nd the effects of this plstic deformtion on the stress trnsfer mechnism of the composite re resulted. Some numericl results obtined by FE nlysis of the model re shown s well.. Mthemticlformultion In order to study the elstic-plstic behvior of MMCs under simple tensile loding, cylindricl xi-symmetric unit cell consisting of fiber nd the surrounding mtrix hs been considered s showninfigs.1nd1b.thecellissubjectedtouniformtensilestressσ 0. According to FEM results nd the governing equtions, which cn be dmitted by common sense lso, the plstic deformtion in the mtrix will strt t somewhere in the djcent region tothefiber,regionishowninfig.1b,duetothelowmtrixyieldstressndhighstress

3 An nlyticl study on the elstic-plstic behvior of metl mtrix composites Fig. 1.() Micromechnicl unit cell.(b) Axi-symmetric model concentrtions in tht region. Considering the occurrence of plstic deformtion in the mtrix ndfocusingonregioniofmtrixfromnowon,thestress-strinreltionsforthemtrixinthis region cn be re-written s(mendelson, 1986) ε m r = 1 E m[σm r ν m (σ m z +σ m θ)]+ε p r ε m θ = 1 E m[σm θ ν m (σ m r +σ m z)]+ε p θ ε m z = 1 E m[σm z νm (σ m θ +σm r )]+εp z ε m rz =1+νm E m τm rz +εp rz where E m nd ν m reyoung smodulusndpoisson srtioofthemtrix,respectively,nd ε p termsdefinetheplsticstrincomponents.itisclerthoughthtsthelodingisincresed, the plstic region expnds within the mtrix, while the points for which the yield criterion is notstisfiedyet,stillreminintheelsticregion,withε p termsbeingzeroineqs.(.1).to summrize,techlodingstep,theyieldcriterionmustbecheckedforthemtrixinorderto determine the plstic region s will be shown lter. Considering the xi-symmetry of the model, nd thus neglecting the derivtives with respect toθ,thegenerlequilibriumequtionsforboththefiberndmtrixcnbewrittens σ z z + τ rz r +τ rz r =0 σ r r + τ rz z +σ r σ θ r (.1) =0 (.) Assuming σ m z / z=g(z)ndusingeq.(.1) 1,itcnbeshowntht(Go,005) τ m rz (z)= b ( b r r ) τ i (z) (.3) whereτ i (z)isthesherstresstthefiber/mtrixinterfce.accordingtothelstofthestrin- -displcement reltions ε r = u r ε θ = u r ε z = w z ε rz = 1 ( u ) z + w r withubeingtherdilndwthexildisplcement,ndneglectingtheterm u m / zformtrix,ccordingtothessumptiontht u m / z w m / r,whichduetothetensileloding condition nd the symmetry of the model is resonble ssumption, the rdil displcement ofthemtrixw m isresultedsfollows w m (r,z)=w m + wb m wm b b ln( b ε p rz(r,z)dr ) b [ b ln ( r ) r ] r + (.4) ε p rz(r,z)dr (.5) where w m isthemtrixxildisplcementttheinterfce r=,nd wm b isthexil displcementt r=b.fromtheotherhnd,ndssumingthttherdilndtngentil stressesinthemtrixremuchsmllercompredtothexilstressterm, σ m r +σ m θ σm z,

4 36 S. Khosoussi et l. ndthusneglectingtheterm(σ m r +σm θ )ineq.(.1) 3,differentitingEq.(.5)withrespecttoz nd finlly considering the strin-displcement reltions, Eqs.(.4), it cn be shown tht σ m z (r,z)=σm +Em ε p z (,z)+em z r ε p rz (r,z)dr σb m+em ε p z(b,z) σ m E m ε p z(,z) E m z + ) b ln( b b [ ( r b ln ) r ] E m ε p z(r,z) b ε p rz(r,z)dr whereσ m ndσm b rethemtrixxilstressestr=ndr=b,respectively.writingthe equilibriumequtionofcrosssectionofthecell,itcnbeconcludedtht b σf z(z)+ (.6) rσ m z(r,z)dr= b σ 0 (.7) inwhichσ f z (z)isthevergexilstressinfiber.substitutingσm z (r,z)fromeq.(.6)into(.7) ndre-writingeq.(.6)withtheresultedσ m b,xilstressinthemtrixcnbefounds σ m z(r,z)=σ m +E m ε p z(,z) E m ε p z(r,z)+e m z b ln( r + b ln( 4 b 4E m z ) 1 (r ) ) 1 4 (b )(3b ) b r r { b ε p rz (r,z)drdr+em r ε p rz(r,z)dr σ f z+b σ 0 (b )[σ m +E m ε p z(,z)] ε p z (r,z)rdr } Ontheotherhnd,fromthefirstofequilibriumequtions(.1) 1 forfiberusingthevergexil stress introduced in Eq.(.7), the following reltion known s sher lg eqution cn be derived dσ f z(z) dz (.8) = τ i(z) (.9) Combining Eq.(.9), Eq.(.3), stress-strin reltions, Eqs.(.1) nd strin-displcement equtionseqs.(.4)ndfinllysubstitutingforσ m z(r,z)fromeq.(.8),thefollowingreltionwill bederivedforthefibervergexilstressσ f z (z) d σ f z (z) dz = { [ (1+ν m ) b ln( 4 b b ) ] 1 4 (b )(3b ) σ f z+b σ 0 (b )[σ m +E m ε p z(,z)] (.10) 4E m z b r r b ε p rz (r,z)drdr+em rε p z (r,z)dr } Ascnbeseen,theonlyunknownterminEq.(.10)isthemtrixxilstressttheinterfceσ m, which cn be derived s follows. The ssumption of perfect bond between the fiber nd mtrix

5 An nlyticl study on the elstic-plstic behvior of metl mtrix composites t the interfce implies the boundry condition of equlity of xil strins long the interfce. ReferringtoEq.(.1) 3 ndrememberingthepreviouslymdessumptionccordingtowhich the summtion of the rdil nd tngentil stresses were neglected compred to the xil term ofstress,theequlityofxilstrinsttheboundrywillledtothefollowing σ m =E m( σ f z (z) ) E f ε p z(,z) (.11) wheree f isyoung smodulusofthefiber.itshouldbenotedthtthestress-strinreltionused forthefiberisintheelsticstte,whichisobviousccordingtothebsicssumptionofthe problem, previously stted in the introduction. Therefore, substituting Eq.(.11) into Eq.(.10) nd solving the resulted ordinry differentil eqution, one cn esily derive the reltion for fiber verge xil stress, including the effect of plstic deformtion in the mtrix { σ f z (z)=exp( ABz) A B C 1 1 +exp( { ABz) C + 1 A B exp( } ABz)[C+F(z)]dz exp( } ABz)[C+F(z)]dz (.1) in which A= b [ ) ] (1+ν m ) b ln( 4 b 1 4 (b )(3b ) B= +(b ) Em E f C= b σ 0 F(z)=4E m z b r r b ε p rz(r,z)drdr E m rε p z(r,z)dr Using Eq.(.9), fiber sher stress cn be derived ccordingly τ i (z)= { ABexp( [ ABz) + ABexp( [ ABz) C + 1 A B C 1 1 A B exp( ABz)[C+F(z)]dz exp( } ABz)[C+F(z)]dz (.13) (.14) Ascleredtthebeginningoftheformultion,llthebovediscussionsuptothispointrevlid forregioniofthemtrixshowninfig.1bndthefiberwithin0 z lrnge.considering thel z l re,thesmepprochstheoneproposedbyhsueh(1995),hsuehndbecher (1996), known s the Imginry Fiber Technique hs been pplied. In other words, the fiber isconsideredtobecontinuouslongthewholecelllength,withthe l z l regionbeing knownstheimginryfiber.thesmetretmentstheonediscussedboveisppliedtothis fiber, except tht t the end of the process, the mteril properties of this imginry section will be replced by the mtrix properties. Considering the locl plstic deformtion in region Iofthemtrixsdiscussed,thesmergumentwithJingetl.(004)hsbeenuseds follows. As stted in tht study, when the mtrix locl plstic deformtion occurs in region I, itwilllsooccurintheregionnerthefiberendfcewiththesmemgnitude.therefore,the stresstrnsferinthefiberendregionwillnotbeffectedbytheplsticdeformtioninthefiber region. Following such n rgument, the governing differentil eqution for the imginry fiber xilstresscnbewrittens

6 38 S. Khosoussi et l. d σ f z(z) dz = b [ ) ]( b (1+ν m ) b ln( σ f 4 b 1 4 (b )(3b z +b σ 0 ) (.15) ) Therefore, the xil nd sher stress for imginry fiber cn be derived s follows σ f z (z)=c 3exp( AB z)+c 4 exp( AB z) C B τ i(z)= AB [ C 3 exp( AB z)+c 4 exp( ] (.16) AB z) whereconstntsandcrepreviouslydefinedineq.(.13) 1 nd(.13) 3,whileB is B =b (.17) AscnbeseeninEqs.(.1),(.14)nd(.16),fourunknownconstnts C 1,C,C 3 ndc 4 re still to be clculted. For this reson, the following boundry conditions re pplied to the xil nd sher stresses previously derived σ f z = { σ0 t z=l σ f z t z=l τ i= { τi t z=l 0 t z=0 (.18) In order to clculte the plstic deformtion nd its effect on stress trnsfer in the composite, firstofll,theyieldcriterionshouldbedeterminedinordertoindentifytheregioninwhichthe plstic deformtion occurs under ech loding step. Considering the von-mises yield criterion (Mendelson, 1986) s follows σ e =σ y (.19) theyieldingwilloccurssoonstheequivlentstressσ e incertinpointrechestheyield stressσ y inwhichtheequivlentstressisdefineds σ e = 1 (σ r σ θ ) +(σ θ σ z ) +(σ z σ r ) +6τ rz (.0) Accordingtothepreviouslysttedssumption, σ m r +σ m θ σm z,theyieldcriterionforthe current problem cn be restted s σ m z +3τ m rz =σ y (.1) Asdescribedbeforendwillbeshowninthenumericlresults,itisevidentthttheyielding inthemtrixwillstrtttheinterfcesomewherettheendofthefiber.thus,substituting for σ m z nd τ m rzt (r,z)=(,l)intoeq.(.1),thecriticlstress σ 0 forwhichtheyielding will strt for the first time in the composite, will be clculted. Therefter, for stress vlues bove this criticl threshold, the region in the mtrix in which the plstic deformtion occurs willbeexpnded.inordertoobtinnestimteforthebehviorofthemodelconsidering plstic deformtions in the mtrix, the reltions obtined up to this point will be simplified s follows. The plstic strin increments re relted to the stresses through the yield criterion nd the ssocited flow rule. According to the von-mises yield criterion, Eq.(.19) previously pplied nd the Prndtl-Reuss reltions considered here, the z nd rz strin increments cn be obtined s follows(mendelson, 1986) ε p z= εe p (σ z σ r σ θ ) ε p σ rz= 3 ε e p τ rz (.) e σ e

7 An nlyticl study on the elstic-plstic behvior of metl mtrix composites in which ε e p = 3 ( ε p r ε p θ ) +( ε p θ εp z) +( ε p z ε p r) +6 ε p rz (.3) Assuming proportionl loding, i.e. considering loding level for which the yielded re is reltivelysmllcompredtothefiberdimeter,itcnbessumedthtthestressstteinthe yieldedregioninthemtrixisthesmestheelsticstte,e.g. σz me (r,z) τrz me (r,z) =σm z P (r,z) τrz mp (r,z) (.4) where E nd P superscripts refer to elstic nd plstic stress fields, respectively. With such n rgument nd rewriting Eqs.(.) for the proportionl loding cse nd thus omitting the increment nottion ccording to the deformtion or totl theory of plsticity nd remembering tht the rdil nd xil stresses were neglected compred to the xil term, the following reltion cnbederivedbetweenzndrzplsticstrins ε p z ε p rz = σ z (r,z) 3τ rz (r,z) (.5) Therefter, s previously stted, considering smll loding steps, nd thus smll plstic region, liner distribution for plstic deformtions cn be ssumed over the plstic region for certin loding step ε p z (r,z)=εp z1 ( r) d ε p rz (r,z)=εp rz1 ( r) d εp z1 (l z) dz εp rz1 (l z) dz +ε p z1 +ε p rz1 (.6) inwhich ε p z1 nd εp rz1rethemximumvluesoftheplsticdeformtiont (r,z)=(,l), nd d nd dz re rdil nd xil boundries of the plstic region, respectively. The nlysis ppliedwillbethensfollows.forlodingstep σ 01 greterthntheinitilcriticlstress, theyieldboundries,e.g. dnd dzineqs.(.6),inthemtrixredetermined.usingeq. (.1),(.14) nd(.16) nd rewriting the reltions, substituting for one of the plstic strins in termsoftheotherccordingtoeq.(.5)t(r,z)=(,l),thexilstressinthefiberndthe interfce sher stress will be clculted s functions of the considered plstic strin. Therefter, substitutingtheresultsintoeq.(.8)nd(.3),thexilndsherstressinthemtrixwillbe determined ccordingly. Clculting the equivlent stress for the mtrix, the equivlent plstic strin will be determined from the uni-xil stress-strin curve s follows σ e =σ y + EEp E E pεp e (.7) inwhiche p istheplsticmodulusintheuni-xilstress-strincurvendε p eistheequivlent plstic strin, previously defined incrementlly in Eq.(.3). It should be noted tht this reltion is vlid for perfectly plstic mteril. Also, since the rdil nd the tngentil plstic strins re considered to be ε p r =εp θ = 1 εp z (.8) the equivlent strin term, Eq.(.3), cn be simplified to 9 ε e p= 3 εp z +6ε p rz (.9) Finlly,solvingEq.(.7)t (r,z)=(,l)bymaplecodedeveloped,theplsticstrin terms, nd therefore the stresses, cn be derived.

8 330 S. Khosoussi et l. 3. Results nd discussion In order to understnd the effects of the plstic deformtion in the mtrix on stress trnsfer behviorofthecomposite,theeffectivestressescontributinginthismechnism, σ f z nd τ i, will be studied. The clcultions re done for Al6061/SiC0% composite, with the following specifictions Tble 1. Geometricl specifictions of Al6061/SiC0% composite s k f f Tble. Mechnicl properties of Al6061/SiC0% composite Mteril ρ[g/cm 3 ] E[GP] E p [GP] υ Al SiC Figure shows the equivlent stress distribution in the mtrix for n rbitrry elsticloding.itisclerthtthisstressismximumttheinterfce,tthefiberendpoint (r,z)=(,l),sclimedbefore.asshowninthisfigure,fornppliedlodingofσ 0 =74MP, theequivlentstresstthecriticlpointof(r,z)=(,l)rechestheyieldvlueσ y =76MP for the first time. Therefter, compring this equivlent stress with the yield stress ccording to the von-mises criterion, the expnsion of plstic region in the mtrix for different loding stepsσ 0 isshowninfig.b.ascnbeseen,theboundriesofthisplsticregiondnddz in Eqs.(.6) expnd with the increse of the loding pplied. As described before, the reltion between the two plstic strin components cn be estimted from Eq.(.5). The clculted vluet(r,z)=(,l)hsbeenroundedto1forsimplicity,i.e. ε p z1 =εp rz1 ndthezndrz termsoftheplsticstrinhvebeenconsideredtobeequlε p z(r,z)=ε p rz(r,z).thevrition of the plstic strin components hve been previously defined by liner functions in Eqs.(.6). With this ssumption, the fiber verge xil stress, the interfcil sher stress, nd the z/rz terms of the mtrix plstic strin hve been clculted for ech loding step s shown. The plsticstrinfunctionsreillustrtedinfigs.3nd3btz=lndr=,respectively,for different loding steps. Fig..()Mtrixequivlentstressforσ 0 =74MP,(b)expnsionoftheplsticregioninthemtrix model Generlbehviorofthesherstresst r=ndfibervergexilstressreshownin Figs.4nd4b,respectively.AcloserviewofthesestresscurvesreshowninFigs.5nd5b

9 An nlyticl study on the elstic-plstic behvior of metl mtrix composites Fig.3.ε p z(r,l)/ε p rz(r,l),plsticstrinst:()z=l,(b)r= Fig.4.Sherstresstr=()ndfibervergexilstress(b)forσ 0 =90MP Fig.5.Sherstresstr=()ndfibervergexilstress(b)forσ 0 =90MP forsmllregionboutthefiberendz=l.thestresseshvebeenclcultedforσ 0 =90MP both for the elstic nd elstic-plstic conditions. As cn be seen, the results comply with the initil expecttions tht the occurrence of plsticity in the mtrix will reduce the efficiency of the lod trnsfer mechnism in the composite. Since the sher stress t the interfce nd the xil stressinthefiberreofmjorimportnceintheprocessofthelodtrnsferfromthemtrixto the fiber, this decrese in their vlues will ffect the initilly expected lod bering chrcteristics ofthemteril.therefter,figs.6nd6bshowthevritionofthesherstressttheinterfce nd the fiber verge xil stress for different loding steps. It cn be noted tht incresing the loding, the stresses in presence of the plstic strins will increse more slowly compred to the elstic condition. This is nother verifiction of the dverse effect of the plsticity in the mtrix over the mechnicl chrcteristics of the MMCs. Finlly, some numericl FEM results using ANSYS10cnbefoundinFig.7toFig.9forgenerlcomprison.Tkingglnceoverthese results,firstofllitcnbenoticedthtthessumptionoftheequlityoftheplsticstrin terms,ε p z (r,z)=εp rz (r,z),iscceptblewithinthelodingregionpplied.furthermore,scn be seen, the liner distribution considered for plstic strin distribution over the plstic region

10 33 S. Khosoussi et l. is lso resonble ssumption for the considered loding steps. Therefter, the vlues derived numericllyforthesmplelodingofσ 0 =90MPshowreltivelygoodcomptibilitywith the theoreticl results obtined. Fig.6.Sherstresstr=()ndfibervergexilstress(b),differentlodingsσ 0 Fig.7.Numericlresults ε p z (r,l)t:()z=lnd(b)z=l Fig.8.Numericlresults ε p z (,z)t:()r=nd(b)r= Fig.9.Numericlresults sherstresstr=()ndfibervergexilstress(b)forσ 0 =90MP

11 An nlyticl study on the elstic-plstic behvior of metl mtrix composites Conclusion nd remrks Inthepresentstudy,nnlyticlsherlgbsedmodelwsproposedtostudytheeffectsof plstic deformtions in the mtrix on overll stress trnsfer behvior of fiber reinforced metl mtrix composite. For this reson, cylindricl unit cell consisting of fiber nd the surrounding metl mtrix ws considered under tensile loding. Writing the sher lg bsed reltions for this model, the stress terms for both the mtrix nd fiber were clculted considering the occurrence ofplsticityinthemtrixduetotheductilityofthemetlmtrixcompredtothebrittle chrcteristics of the hrd fiber. Unlike reltively similr study performed by Jing et l. (004) bsed on the sher lg pproch, the present work hs tken into ccount the stress distribution in the mtrix nd hs derived the governing reltions for tht region. Also, the effect of the equivlent stress, i.e. both sher nd xil stresses, hve been considered rther thn the mere xil stress pplied in tht work. Moreover, both of the plstic strin terms, i.e. xil nd sher strin components hve been tken into ccount lso vi the equivlent plstic strin term in the plsticity reltions. Furthermore, the performed study is ble to predict the plstic region boundriesinthemtrixforgivenlodingndiscpbleofproposingnestimteforthe growth of the yielded region with n increse in the loding. To summrize, hving obtined the sher lg bsed reltions for the problem, the plstic strin distribution in the mtrix hs been derived for given loding provided tht the yield re is smll compred to the fiber dimeter. Therefter, the effects of these plstic strins on the interfce sher nd the fiber verge xil stress hve been obtined. It ws lso verified tht the occurrence of locl plsticity in the mtrix hs n dverse effect over the stress trnsfer efficiency of the composite vi reduction in these two criticl lod trnsfer mechnisms compred to the stress distribution in the bsence of such deformtions. Moreover, it ws shown tht s the locl plstic deformtion occurs in the mtrix, the increse in stresses hppens more slowly compredtotheelsticcse,whichisnotherevidencefortheefficiencylossinthepresenceof the mtrix plsticity. References 1. Clyne T.W., 1989, A simple development of the sher lg theory pproprite for composites with reltively smll modulus mismtch, Mterils Science nd Engineering, A1, Cox H.L., 195, The elsticity nd strength of pper nd other fibrous mterils, British Journl of Applied Physics, 3, Fukud H., Chou T.W., 1981, An dvnced sher-lg model pplicble to discontinuous fiber composites, Journl of Composite Mterils, 15, Go X.L., Li K., 005, A sher-lg for crbon nnotube-reinforced polymer composites, Interntionl Journl of Solids nd Structures, 4, Hsueh C.H., 1988, Anlyticl evlution of interfcil sher strength for fiber-reinforced cermic composites, Journl of the Americn Cermic Society, 71, 6, Hsueh C.H., 1988b, Elstic lod trnsfer from prtilly embedded xilly loded fiber to mtrix, Journl of Mterils Science Letters, 7, 5, Hsueh C.H., 1990, Interfcil debonding nd fiber pullout stresses of fiber-reinforced composites, Mterils Science nd Engineering, A13, 1, Hsueh C.H., 199, Interfcil debonding nd fiber pull-out stresses of fiber-reinforced composites VII: improved nlyses for bonded interfces, Mterils Science nd Engineering, A154, Hsueh C.H., 1995, A modified nlysis for stress trnsfer in fiber-reinforced composites with bonded fiber ends, Journl of Mterils Science, 30, 19-4

12 334 S. Khosoussi et l. 10. Hsueh C.H., 1994, A two-dimensionl stress trnsfer model for pltelet reinforcement, Compos. Eng. 4(10) Hsueh C.H., Becher P.F., 1988, Therml expnsion coefficients of unidirectionl fiber reinforced cermics, Journl of the Americn Cermic Society, 71, 10, Hsueh C.H., Becher P.F., 1996, Residul therml stresses in cermic composites, Prt II: with short fibers, Mterils Science nd Engineering, A1, Hsueh C.H., Fuller E.R., Lnger S.A., Crter W.C., 1999, Anlyticl nd numericl nlyses for two-dimensionl stress trnsfer, Mterils Science nd Engineering, A68, Hsueh C.H., Young R.J., Yng X., Becher P.F., 1997, Stress trnsfer in model composite contining single embedded fiber, Act Mterili, 45, 4, JingZ.,LiG.,LinJ.,DingX.,SunJ.,004,Elstic-plsticstresstrnsferinshortfibrereinforced metl mtrix composites, Composites Science nd Technology, 64, JingZ.,LinJ.,YngD.,DongS.,1998,Annlyticlstudyoftheinfluenceoftherml residul stresses on the elstic nd yield behviors of short fiber-reinforced metl mtrix composites, Mterils Science nd Engineering, A48, Jing Z., Liu X., Li G., Lin J., 004b, A new nlyticl model for three-dimensionl elstic stress field distribution in short fiber composite, Mterils Science nd Engineering, A366, Krbhri V.M., Wilkins D.J., 1991, An engineering modifiction to the sherlg model s pplied to whisker nd prticulte reinforced composites, Script Metllurgic, 5, Kelly A., 1966, Strong Solids, Clrendon Press, Oxford, pp Mendelson A., 1986, Plsticity: Theory nd Appliction, R.E. Krieger 1. Nrdone V.C., Prewo K.M., 1986, On the strength of discontinuous silicon crbide reinforced luminum composites, Script Metllurgic, 0, Piggott M.R., 1980, Lod Bering Fiber Composites, Pergmon Press, New York, pp Strink M.J., Syngellkis S., 1999, Sher lg models for discontinuous composites: fiber end stresses nd wek interfce lyers, Mterils Science nd Engineering, A70, Mnuscript received My 19, 013; ccepted for print September 5, 013