Estimation of homogenized elastic coefficients of pre-impregnated composite materials

Transcription

1 Proceedngs of the nd IASME / WSEAS Internatonal Conference on Contnm Mechancs (CM'7) Portoroz Slovena Ma Estmaton of homogenzed elastc coeffcents of pre-mpregnated composte materals HORATIU TEODORESCU SORIN VLASE IOAN CANDEA LUMINITA SCUTARU Department of Mechancal Engneerng Translvana Unverst of Brasov 9 Erolor Blvd 536 Brasov ROMANIA hteodoresc@ahoo Abstract: - The am of the paper s to present an orgnal homogenzaton method for ellptc eqatons appled to pre-mpregnated composte materals nown as prepregs In ths class of prepregs can be nclded Sheetand Bl Moldng Componds Sheet Moldng Componds (SMC) are characterzed n general as mltphase heterogeneos and ansotropc composte materals wth randoml dscontnos renforcement The pper and lower lmts of the homogenzed coeffcents for a 7% fber volme fracton SMC are compted It s presented a comparson between the pper and lower lmts of the homogenzed elastc coeffcents of a SMC materal and the epermental data The comptng model sed as a homogenzaton method of these heterogeneos composte materals gave emphass to a good agreement between ths method and epermental data Ke-Words: - Sheet Moldng Componds Bl Moldng Componds Prepregs Homogenzaton Heterogenet Comptng model Ellptc eqatons Elastc coeffcents Rovng Introdcton Theoretcal researches regardng the behavor of heterogeneos materals lead to the elaboraton of some homogenzaton methods that tr to replace a heterogeneos materal wth a homogeneos one The am s to obtan a comptng model whch taes nto accont the mcrostrctre or the local heterogenet of a materal The homogenzaton theor s a comptng method to std the dfferental operators convergence wth perodc coeffcents Ths method s ndcated n the std of meda wth perodc strctre The most obvos mechancal model whch reflects ths model s a Sheet Moldng Compond (SMC) materal A SMC s a prempregnated composte materal nown as prepreg chemcall thcened manfactred as a contnos mat of chopped glass fbers resn (nown as matr) fller and addtves from whch blans can be ct and placed nto a press for hot press moldng The reslt of ths combnaton of chemcal componds s a heterogeneos ansotropc composte materal renforced wth randoml dsposed dscontnos renforcement [] [] [3] The matr- and fllers elastc coeffcents are ver dfferent bt perodcal n spatal varables Ths perodct or freqenc s stable to appl the homogenzaton theor to the std of heterogeneos materals le SMCs Problem formlaton Let s consder a doman from R 3 space n coordnates doman consdered a SMC composte materal n whch a so called sbsttte matr (resn and fller) s represented b the feld and the renforcement occpes the feld seen as a bndle of glass fbers (fg ) Let s consder the followng eqaton [4]: f ( ) a ( ) ; a a () or nder the eqvalent form: p f ; p a () In the case of SMC materals that present a perodc strctre contanng nclsons a () s a fncton of If the perod s dmensons are small n comparson wth the dmensons of the whole doman then the solton of the eqaton () can be consdered eqal wth the solton stable for a homogenzed materal where the coeffcents a are constants In the R 3 space of coordnates a parallelepped wth sdes (fg ) s consdered as well as paralleleppeds obtaned b translaton n (n nteger) n aes drectons The fnctons:

2 Proceedngs of the nd IASME / WSEAS Internatonal Conference on Contnm Mechancs (CM'7) Portoroz Slovena Ma a ( ) a (3) can be defned where s a real postve parameter Notce that the fnctons a () are -perodcal n varable (B beng the parallelepped wth sdes) F where () are -perodcal n varable The fnctons () are defned on R 3 so that the dervatves behave n the followng manner: d (7) d If the vales of are compared n two homologos ponts P and P homologos throgh perodct n neghbor perods t can be notce that the dependence n s the same and the dependence n s almost the same snce the dstance P P s small (fg ) Let s consder P 3 a pont homologos to P throgh perodct stated far from P The dependence of n s the same bt the dependence n s ver dfferent snce P and P 3 are far awa For nstance n the case of two ponts P and P 4 stated n the same perod the dependence n s almost the same snce P and P 4 are ver close bt the dependence n s ver dfferent snce P and P 4 are not homologos throgh perodct The fncton depends on the perodc coeffcents a on the fncton f() and on the bondar The development (6) s vald at the nner of the bondar where the perodc phenomena are prevalent bt near and on the bondar the non-perodc phenomena preval [5] [6] [7] [8] [9] Fg Domans- and nclsons perodct defnton of SMC composte materals [4] If the fncton f() s n defned the problem at lmt can be consdered: f ( ) a ( ) (4) Smlar wth eqaton () the vector p can be defned wth the elements: p ( ) a ( ) (5) For the fncton () an asmptotc development wll be loong for nder the form: () () () () ; (6) Fg Phscal meanng of SMCs nclsons perodct [4] Usng the development (6) the epressons and p P 4 P P P 3 can be compted as followng:

3 Proceedngs of the nd IASME / WSEAS Internatonal Conference on Contnm Mechancs (CM'7) Portoroz Slovena Ma ( ) (8) p ( ) p( ) p( ) p ( ) (9) where: p ( ) a ( ) () p ( ) a ( ) The fncton f() presented n eqaton (4) can be wrtten n the followng manner: f ( ) ( p p ) () The terms - and wll be: p () p p f ( ) (3) Eqaton (3) leads to the homogenzed- or macroscopc eqaton For ths we ntrodce the medm operator defned for an fncton G() - perodcal: ( )d (4) where represents the perodct cell volme To obtan the homogenzed eqaton the operator (4) s appled to the eqaton (3): P p I f ( ) (5) Accordng to the operator (4) the second term of the left sde of the eqaton (5) becomes: p p d p n ds (6) De to -perodct of p and the fact that n s the normal vector at the bondar of the relaton (6) s eqal wth zero So the eqaton (5) becomes: PI f ( ) (7) Wth help of relaton () the eqaton () can be wrtten as follows: a ( ) (8) therefore: a a ( ) (9) The solton () of eqaton (9) s -perodcal and to determne t s necessar to ntrodce the space U ( ) { H ( ) perodcal} The eqaton (9) s eqvalent wth the problem to fnd the solton U that verfes: a a ( ) v d vd () for vu If U s ntrodced wth that satsf: a a ( ) v d vd () for vu then from the lneart of problem () ts solton can be wrtten nder the form: ( ) ( ) c( ) () where c() s a constant as a fncton of Knowng the epresson of as a fncton of from the epressons () wth () the homogenzed coeffcents can be compted: p ( ) a ( ) a ( ) (3) a ( ) a ( ) Applng the medm operator (4) the relaton (3) can be wrtten: p ( ) a (4) a a ( ) a( ) (5) a ( ) a a Therefore the relaton (5) becomes an eqaton n wth constant coeffcents: f a (6) For a composte materal n whch the matr occpes the doman and presents the coeffcent a and the nclson occpes the doman wth the coeffcent a separated b a srface the eqaton (3) mst be seen as a dstrbton

4 Proceedngs of the nd IASME / WSEAS Internatonal Conference on Contnm Mechancs (CM'7) Portoroz Slovena Ma Problem solton for a SMC In the case of a SMC composte materal whch behaves macroscopcall as a homogeneos elastc envronment s mportant the nowledge of the elastc coeffcents Unfortnatel a precse calcls of the homogenzed coeffcents can be acheved onl n two cases: the ndmensonal one and the case n whch the matr- and nclson coeffcents are fnctons of onl one varable For a SMC materal s preferable to estmate these homogenzed coeffcents between an pper and a lower lmt Snce the fber volme fracton of common SMCs s 7% to lghten the calcls an ellpsodal nclson of area 7 stated n a sqare of sde s consdered The plane problem wll be consdered and the homogenzed coeffcents wll be n matr and n the ellpsodal nclson In fg 3 the strctre s perodct cell of a SMC composte materal s presented where the fbers bndle s seen as an ellpsodal nclson Fg 3 Strctre s perodct cell of a SMC materal wth 7% fbers volme fracton Let s consder the fncton f( ) n nclson and n matr To determne the pper and the lower lmt of the homogenzed coeffcents frst the arthmetc mean as a fncton of followed b the harmonc mean as a fncton of mst be compted The lower lmt s obtaned comptng frst the harmonc mean as a fncton of and then the arthmetc mean as a fncton of If we wrte wth -( ) the arthmetc mean aganst of the fncton f( ) t follows: ( for - 5 ) f ( )d ( 5; 45 ) ( 45; 5) 5 (7) 5 ( ) f ( )d (8) for ( 45; 45 ) The pper lmt s obtaned comptng the harmonc mean of the fncton -( ): a 5 d ( ) 5 (9) d d d To compte the lower lmt we consder ( ) the harmonc mean of the fncton f( ) aganst :! ( ) 5 d f ( ) 5 (3) for ( 5 ; 9 ) ( 9 ; 5 )!( ) d f ( ) 5 (3) for ( 9; 9 ) The lower lmt wll be gven b the arthmetc mean of the fncton ( ): 9 a! ( )d 9 5 d d d (3) 4 Reslts Snce the ellpsodal nclson of the SMC strctre ma var anglar aganst the aes center the pper and lower lmts of the homogenzed coeffcents wll var as a fncton of the ntersecton ponts coordnates of the ellpses wth the aes and of the perodct cell In table the pper and lower lmts of the homogenzed coeffcents for a SMC materal s presented and table shows the basc elastct propertes of the sotropc componds The materal s coeffcents estmaton depends both on the basc elastct propertes of the sotropc componds and the volme fracton of each compond If we wrte P M the basc elastct propert of the matr P F and P f the basc elastct

5 Proceedngs of the nd IASME / WSEAS Internatonal Conference on Contnm Mechancs (CM'7) Portoroz Slovena Ma propert of the fbers respectve of the fller K M the matr volme fracton K F and K f the fbersrespectve the fller volme fracton then the pper lmt of the homogenzed coeffcents can be estmated comptng the arthmetc mean of these basc elastct propertes tang nto accont the volme fractons of the componds also: PM M PF F Pf f A (33) 3 The lower lmt of the homogenzed elastc coeffcents can be estmated comptng the harmonc mean of the basc elastct propertes of the sotropc componds: A 3 (34) PM M PF F Pf f where P and A can be the ong modls respectve the shear modls Table : Upper and lower lmts of the homogenzed coeffcents for a SMC materals Anglar varaton of the Upper lmt a Lower lmt a_ ellpsod nclson 5 83 ± ± dstrbton s ver hgh Table : Basc elastct propertes of the sotropc componds and the volme fractons of the SMC componds Matr E-fber glass Fller ong mod E [GPa] Shear mod G [GPa] Volme fracton [%] The glass fbers represent the basc element of SMC prepreg renforcement The qantt and orentaton of the rovngs determne n a decsve manner the sbseqent profle of the SMC strctre s propertes There are dfferent grades of SMC prepregs: R- SMC (wth randoml orented renforcement) D- SMC (wth ndrectonal orentaton of the chopped fbers) C-SMC (wth ndrectonal orented contnos fbers) and a combnaton between R- SMC and C-SMC nown as C/R-SMC The followng mcrographs present the etreme heterogenet and the laered strctre of these materals as well as the glass fbers and fllers dstrbton The mcrographs show that there are areas between µm n whch the glass fbers are mssng and areas where the fbers Fg 4 Mcrographs of varos SMCs taen n-plane and perpendclar to ther thcness [8]

6 Proceedngs of the nd IASME / WSEAS Internatonal Conference on Contnm Mechancs (CM'7) Portoroz Slovena Ma Fgre 5 shows the ong modl and fgre 6 presents the shear modl of the sotropc SMC componds and the pper and lower lmts of the homogenzed elastc coeffcents E [GPa] Resn 73 Fber 478 Fller 7 E (-) E () Epermental Fg 5 The vales of ong modl of the sotropc SMC componds and the pper and lower lmts of the homogenzed elastc coeffcents G [GPa] Resn 78 Fber 8 Fller 5 G (-) 534 G () 45 Epermental Fg 6 The vales of shear modl of the sotropc SMC componds and the pper and lower lmts of the homogenzed elastc coeffcents 5 Conclsons The presented reslts sggest that the envronmental geometr gven throgh the anglar varaton of the ellpsodal domans can leads to dfferent reslts for the same fbers volme fracton Ths fact s de to the etreme heterogenet and ansotrop of these materals The pper lmts of the homogenzed elastc coeffcents are ver close to the epermental data The proposed estmaton of the homogenzed elastc coeffcents of pre-mpregnated composte materals can be etended to determne the elastc propertes of an mltphase heterogeneos and ansotropc composte materals References: [] Teodoresc H Bascs and Mechancs of Polmerc Composte Materals Translvana Unverst Pblshng Hose 7 (n Romanan) [] Ka HG Sheet Moldng Componds Scence and Technolog Hanser 993 [3] Teodoresc F Teodoresc H Sheet Moldng Compond (SMC) Materals 7 th Scentfc Conference Mltar Techncal Academ 997 pp 5-58 (n Romanan) [4] Ene HI Pasa GI Homogenzaton Method Applcatons at Composte Materals Theor Academ Pblshng Hose 987 (n Romanan) [5] Zohd TI Wrggers P Introdcton to Comptatonal Mcromechancs Lectre Notes n Appled and Comptatonal Mechancs No Sprnger 5 [6] Whtne JM McCllogh RL Delaware Composte Desgn Encclopeda Vol : Mcromechancal Materals Modelng edted b LA Carlsson Technomc Pblshng Co 99 [7] Mall PK Fber Renforced Composte Materals Manfactrng and Desgn Dept of Mech Eng Unverst of Mchgan Dearborn Mchgan Marcel Deer Inc New or Basel Hong Kong 993 [8] Teodoresc F Contrbtons Regardng the Modelng of Fber Renforced Composte Strctres PhD thess Translvana Unverst of Brasov (n Romanan) [9] Tsa SW Hahn HT Introdcton to Composte Materals Technomc Pblshng Co Westport 98