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1 Copyrght c 2004 Tech Scence Press CMES, vol.5, no.2, pp , 2004 Adaptve Mult-Scale Computatonal Modelng of Composte Materals P. Raghavan 1 and S. Ghosh 2 Abstract: Ths paper presents an adaptve mult-level computatonal model that combnes a conventonal dsplacement based fnte element model wth a mcrostructural Vorono cell fnte element model for mult-scale analyss of composte structures wth non-unform mcrostructural heterogenetes as obtaned from optcal or scannng electron mcrographs. Three levels of herarchy, wth dfferent resolutons, are ntroduced n ths model to overcome shortcomngs posed by modelng and dscretzaton errors. Among the three levels are: (a) level-0 of pure macroscopc analyss; (b) level-1 of macro-mcro coupled modelng, used for sgnalng the swtch over from macroscopc analyses to pure mcroscopc analyses; and (c) level-2 regons of pure mcroscopc modelng. The adaptve Vorono cell fnte element model s utlzed effectvely for analyss of extended mcrostructural regons wth hgh effcency and accuracy. Identfcaton of statstcally equvalent RVE (SERVE) for evaluatng the effectve propertes are made through the use of correlaton functons for dfferent varables. Upon determnaton of SERVE s for actual mcrostructures, numercal examples of a composte plate and a composte lamnate are solved to demonstrate the ablty of the mult-scale computatonal model n analyzng complex heterogeneous structures. keyword: Vorono cell FEM, mult-scale analyses, non-unform mcrostructures 1 Introducton The commercal use of renforced compostes n varous structural components has ncreased consderably n the last few decades. Based on desgn requrements, they are engneered to yeld superor thermo-mechancal propertes lke hgh strength or stffness to weght ratos, resultng n a tremendous advantage over monolthc ma- 1 Graduate Research Assocate 2 Professor, Department of Mechancal Engneerng, The Oho State Unversty, Columbus, Oho, USA terals. Despte property enhancements, the presence of second phase fbers or partcles n compostes often has adverse effects on ther falure propertes lke fracture toughness and stran to falure. Structural components, e.g. lamnates of many composte materals, exhbt strong non-unformtes at the mcrostructural level. The non-unformtes are n mcro-scale morphology ncludng varable fber/partcle spacng, sze, shape, volume fracton and dsperson, n meso-scale clusterng or drectonalty or n varyng consttuent materal and nterface propertes. The materal response and especally mcrostructural damage mechansms, ncludng ncluson and matrx crackng, nterfacal decoheson etc. can be very senstve to these local varatons n morphologcal and consttutve parameters. Robust analyss methods to desgn optmal composte mcrostructures are necessary for enhanced utlzaton of composte materals n load bearng hgh performance applcatons. Heterogeneous structures are conventonally analyzed wth propertes obtaned from homogenzaton of response at smaller (meso-, mcro-) length scales. [Ghonem and Cho (2002)] provdes an overvew of current and a vson for future developements n multscale smulatons for nano- and mcro-mechancs of materals. Analyss of composte materals s often performed by the method of homogenzaton wheren the macroscopc propertes are obtaned by averagng stresses and strans over a perodc representatve volume element (RVE). Commonly used methods of homogenzaton, e.g. the asymptotc expanson homogenzaton [Benssousan, Lons and Papancoulau (1978); Sanchez-Palenca (1980)], assume spatal perodcty of mcrostructural representatve volume elements or RVE s and unformty of macroscopc varables. Multple scale analyses of lnear elastc renforced compostes have been conducted by Fsh et. al. [Fsh and Wagman (1993)], Guedes and Kkuch [Guedes and Kkuch (1991)], Ghosh et. al. [Ghosh, Lee and Raghavan (2001); Raghavan, Moorthy, Ghosh and Pagano (2001)]. In ths ssue, the homogenzaton method has

2 152 Copyrght c 2004 Tech Scence Press CMES, vol.5, no.2, pp , 2004 been used n conjucton wth the boundary element method for three-dmensonal partcle renforced compostes n [Okada, Fuku and Kumazawa (2003)]. Mcromechancal tracton-dsplacement laws are embedded n the contnuum macromechancal formulaton by a varatonal multscale method n [Garkpat (2002)]. The applcaton of homogenzaton methods suffer from some shortcomngs wth respect to accuracy n specfc problems wth respect to lmtatons n the assumptons of macroscopc unformty and RVE perodcty. The unformty assumpton s not approprate n crtcal regons of hgh gradents lke free edges [Pagano and Rybck (1974); Rybck and Pagano (1976); Raghavan, Moorthy, Ghosh and Pagano (2001)], nterfaces, materal dscontnutes and most mportantly n regons of evolvng damage. Perodcty of smple unt cells s also unrealstc for non-unform mcrostructures, partcularly n the presence of clusterng. Even wth unform dstrbuton of mcrostructures, evolvng localzed stresses or strans can volate the perodc assumptons. Problems lke ths have been tackled effectvely by global-local technques ntroduced by Fsh et. al. [Fsh and Wagman (1993)], Ghosh et. al. [Ghosh, Lee and Raghavan (2001); Raghavan, Moorthy, Ghosh and Pagano (2001)] and Oden et. al. [Oden and Zohd (1997)]. Sub-structurng n these multple-scale analyss methods dfferentate between regons requrng dfferent resolutons, and enable global analyss n some parts of the doman and zoom n for complete mcroscopc modelng at regon of hgh gradents. Adaptvty s a desrable ngredent of these multple scale modelng methods, for automatcally selectng approprate regons to mnmze dscretzaton and modelng errors. Wthout adaptvty, herarchcal modelng may not be optmally effcent. Adaptve multplelevel methods have been proposed by Oden et. al. [Oden and Zohd (1997); Oden, Vemagant and Moes (1999)], Ghosh et. al. [Ghosh, Lee and Raghavan (2001); Raghavan, Moorthy, Ghosh and Pagano (2001)] to address dscretzaton and modelng error for mult-scale analyss of compostes. A second shortcomng of composte analyss usng the asymptotc homogenzaton methods s related to effcency n the concurrent executon of fnte element analyses at the macroscopc and mcroscopc scales. Enormous computatonal efforts can result from havng to solve boundary value problems of the mcrostructural RVE n each macroscopc element of a fnte element model, n addton to the macroscopc soluton. To economze computatons, many studes have assumed smple unt cells models of the RVE, conssts of a rectangular doman wth one or two fbers. These smplfed RVE s often mply unform or hexagonal closed-pack dstrbutons n the mcrostructure bearng lttle resemblance wth the actual stereographc features of the actual mcrostructure. Multple scale models ncorporatng the Vorono cell mcrostructural models have proved to possess sgnfcant edge n ths regard. The mcrostructural Vorono Cell Fnte Element Model (VCFEM) has been developed by Ghosh et. al. [Moorthy and Ghosh (1996, 1998, 2000); Ghosh, Lng, Majumdar and Km (2000)] to overcome lmtatons of unt cell models and effectvely analyze large mcrostructural regons wth arbtrary dspersons, shapes and szes of heterogenetes. By combnng assumed stress hybrd fnte element formulatons wth essental characterstcs of mcromechancs, a hgh level of computatonal effcency wth good accuracy and resoluton has been acheved wth ths method. In ths ssue a 3D model for stress and damage analyss n mult-ncluson dscontnuously renforced compostes has been proposed by Böhm et. al. [Böhm, Han and Eckschlager (2003)] and a model for woven fabrc compostes has been proposed by Kwon and Roach [Kwon and Roach (2003)]. Whle VCFEM offers a soluton to effcent analyss of complex mcrostructures, dentfyng statstcally equvalent representatve volume elements or SERVE for non-unform mcrostructures s a challenge. Underrepresentaton of SERVE s can lead to consderable errors n the values of effectve propertes and should be avoded. Varous authors have used statstcal analyses to determne the sze scale of RVE and the number of fbers contaned n t [Pyrz (1994a,b), Bulsara, Talreja and Qu (1999)]. Pyrz et. al. [Pyrz (1994a,b)] has used statstcal correlaton functons to obtan characterstc nformaton about the mcrostructure. For example, geometrc descrptors lke the second order ntensty functon and par dstrbuton functons have been used to dstngush between dfferent patterns or dstrbutons. To account for the nteracton between fbers, they have used the marked correlaton functon as an nformatve descrptor for characterzng the sze of the mcrostructural SERVE. The marked correlaton functons combnes both the geometrc descrptors as well as dstrbutons of response varables lke stresses and strans n the mcrostructure.

3 Manuscrpt Preparaton for CMES 153 In ths paper, a systematc mult-scale analyss method s establshed for fber renforced composte structures consstng of non-unformly dspersed mcrostructures. The mult-level computatonal model ntroduced n [Ghosh, Lee and Raghavan (2001); Raghavan, Moorthy, Ghosh and Pagano (2001); Lee, Moorthy and Ghosh (1999)] for smple mcrostructures s extended n ths work. The model encompasses three levels n the computatonal doman. The level-0 and level-1 subdomans use effectve propertes obtaned by homogenzaton of the statstcally equvalent RVE s, obtaned usng marked correlaton functons n the mcrostructure. Unque methods of applyng perodc boundary condtons on non-unform RVE s are developed. Level-2 sub-domans emerge wth loss of perodcty or unformty n macroscopc regons, where the model swtches to complete mcroscopc calculatons usng precse mcrostructures. Detals of the multple scale computatonal model s provded n Secton 4. Numercal examples demonstratng the effectveness of the model are provded n Secton 7. 2 The Asymptotc Expanson Homogenzaton Method Boundary value problems n a heterogeneous doman Ω ε are assumed to satsfy the equatons of lnear elastcty, gven as Equlbrum : σ ε j, j = f Knematcs : e ε kl = 1 ( u ε ) k 2 x ε + uε l l x ε k Consttutve Relatons : σ ε j = Ejkl ε eε kl n Ω ε (1) where σ ε j, eε j and uε are stress, stran and dsplacement felds respectvely. The scale parameter ε = l y l x (l y,l x correspond to length scales n the mcroscopc and macroscopc domans respectvely) s typcally an nfntesmally small number. Snce computatonal analyss of ths problem wll be prohbtvely expensve due to the presence of large number of heterogenetes, most analysts solve an equvalent homogenzed verson of the problem usng macroscopc effectve propertes obtaned by averagng mcroscopc varables. A powerful method that has been developed n conjuncton wth computatonal analyss of heterogeneous materals s the asymptotc expanson homogenzaton method [Benssousan, Lons and Papancoulau (1978); Sanchez-Palenca (1980)]. In ths method varables (stresses, strans and dsplacements) are assumed to exhbt dependence on the macroscopc as well as mcroscopc length scales. Furthermore, the mcroscopc dependence s assumed to be Y perodc, where Y s the perod of RVE. In ths method, the dsplacement feld n a heterogeneous doman s expanded asymptotcally about ts values at a macroscopc pont x, n terms of the mcroscopc coordnates y as u ε (x)=u 0 (x,y)+εu 1 (x,y)+ε 2 u 2 (x,y)+, y = x ε (2) The spatal dervatve of any mult-scale functon s gven as x ε (Φ(x,y)) = Φ + 1 Φ (3) x ε y Usng Eq. 2 and Eq. 3 n knematcs and consttutve relatons of Eq. 1 yelds the stress tensor σ ε j as σ ε j = 1 ε σ0 j+σ 1 j+εσ 2 j+ε 2 σ 3 j (4) Usng Eq. 3 and Eq. 4 n the equlbrum relaton of Eq. 1, equatng varous powers of ε and averagng over the perodc RVE, t can be shown that the followng relatons hold. u 0 = u 0 (x) σ 0 j = 0 u 1 = χ kl (y) u0 k x l σ 1 j = ˆσ kl j(y) u0 k x l (5) where ˆσ kl j, χkl p are the mcroscopc stresses and characterstc deformaton modes. Furthermore the volume average of mcroscopc stresses yelds the homogenzed stffness tensor Ejkl H for use n macroscopc calculatons. Ejkl H =< ˆσ kl j > Y = 1 Y = 1 Y Y Y ˆσ kl jdy E ε jpm(δ kp δ lm + χkl p y m )dy The macroscopc stress-stran relaton then takes the form Σ j (x)=< Ejkl ε (δ kmδ ln + χmn k ) u0 m > Y = E y l x jmne H mn (x) n (6)

4 154 Copyrght c 2004 Tech Scence Press CMES, vol.5, no.2, pp , 2004 (7) where the homogenzed varables are Σ(x) =< σ ε (x,y) > Y and e(x) =< e ε (x,y) > Y. The components of the homogenzed stffness matrx E jkl H are calculated by detaled soluton of separate boundary value problems of the entre RVE. The loadng n each of these problems s n the form of mposed unt macroscopc strans. Addtonally, the perodcty on RVE boundares mples that ponts on the boundary are constraned to dsplace perodcally. For nodes on the boundary whch are separated by the perods Y 1,Y 2,Y 3 along one or more coordnate drectons, the dsplacement constrants can be expressed as u (x 1,x 2,x 3 )=u (x 1 ±k 1 Y 1,x 2 ±k 2 Y 2,x 3 ±k 3 Y 3 ), = 1,2,3 (8) where k 1,k 2,k 3 may assume the values 0 or 1, dependng on the node locatons. 3 Estmatng Statstcally Equvalent RVE s for Non-unform Mcrostructures Macroscopc analyss of a composte structure, requres that an approprate RVE be dentfed for each macroscopc pont. RVE s can be readly dentfed for a regular arrangement of fbers lke rectangular or hexagonal dstrbutons. However, mcrostructures from real composte materals hardly possess regular dstrbuton as shown n Fg. 1(b). In Fg. 1 a composte plate wth a crcular hole and the correspondng mcrostructure from a optcal mcrograph are shown. The fbers are assumed to be algned perpendcular to the plane of paper. Snce ths mcrostructure s random, a RVE can be obtaned only n a statstcal sense and attenton s focused on dentfyng a Statstcally Equvalent RVE (SERVE), whch would exhbt a macroscopc behavor that s equvalent to the average behavor of the correspondng mcrostructure. The mcrostructure shown n Fg. 1(b) corresponds to macroscopc pont A only. Snce the constructon of SERVE s for the entre plate would requre a large number of mcrographs from varous representatve ponts n the plate, an assumpton s made that, ths s a representatve mcrostructure for the entre plate. Ths may be justfed from two consderatons from statstcal contnuum theores that have been descrbed n [Beran (1968)], vz. (a) an ensemble assumpton, n whch dfferent experments wth dfferent mcrostructural arrangements exhbt smlar macroscopc behavor, and (b) an ergodc L y 2a W σ σ (a) A x 100 µm (b) Fgure 1 : (a) Composte plate wth a hole (b) Optcal mcrograph of the mcrostructure at pont A hypothess, whch demands that all states avalable to ensemble of systems be avalable to each system of the ensemble. The justfcaton of the assumptons have been provded n [Zeman and Sejnoha (2001)]. Consequently a sngle SERVE s assumed to represent every macroscopc pont n the plate. The representatve sze of the SERVE s dentfed by the use of statstcal functons, e.g. correlaton functons. Pyrz [Pyrz (1994b)] has ntroduced marked correlaton functons for characterzng the length scales defned as the regon of nfluence n a heterogeneous neghborhood on pre-dsposed response felds lke stresses and strans. The marked correlaton functon for a heterogeneous doman of area A contanng N fbers may be expressed as M(r)= d H(r) dr g(r) where H(r)= 1 m 2 A N 2 N k k=1 (9) m m k (r) (10) In the above equaton m s a mark assocated wth th fber, k s the number of fbers whch have ther centers wthn a crcle of radus r around the th fber, m k are the marks of those fbers and m s the mean of all the marks. Marks n the marked correlaton functon can be any feld varable for example, the maxmum prncpal stress, Von Mses stress etc. assocated wth each fber.

5 Manuscrpt Preparaton for CMES 155 H(r) s called the mark ntensty functon and g(r) s the par dstrbuton functon defned as g(r)= 1 dk(r) 2πr dr (11) where K(r) s a second order ntensty functon whch s explaned n [Ghosh, Nowak and Lee (1997a,b)]. Whle K(r) can dstngush between dfferent patterns, the par dstrbuton functon g(r) characterzes the ntensty of nter fber dstances. From the defnton t can be seen that the marked correlaton functon assocates feld varables wth morphology of the mcrostructure. The radus of nfluence R n f may be nferred from a plot of M(r) vs r, n whch M(r) stablzes as r approaches R n f. Once the radus of nfluence R n f has been determned from M(r), SERVE may be constructed by usng square wndows of sze R n f R n f at varous ponts of the mcrograph. The use of square wndows to carve out the RVE for a random mcrostructure results n the ntersecton of many fbers wth the edges yeldng cut fbers near the boundary, as shown n Fg. 2(a). Whle some authors have used cut fbers n ther RVE constructon [Zeman and Sejnoha (2001)], the applcaton of perodc boundary condton s mproper wth these RVE s. Increasng the wndow sze to nclude the totalty of cut fbers, as done n [Bulsara, Talreja and Qu (1999)] (Fg. 2(b)), results n a decreased volume fracton. As a remedy, to the above dscrepances, a method of constructng the SERVE boundary by repeatng the group of fbers perodcally s adopted n ths work as shown n Fg. 2(c). The local mcrostructure s frst constructed by repeatng the randomly dstrbuted fbers obtaned from statstcal analyss n both the y 1 and y 2 drectons for several perod lengths. Perodc repettve fbers are placed at (y 1,y 2 ), (y 1 ± k 1 Y 1,y 2 ), (y 1,y 2 ± k 2 Y 2 ) and (y 1 ± k 1 Y 1,y 2 ± k 2 Y 2 ), where k 1,k 2 are ntegers. The perod lengths Y 1 and Y 2 are selected such that the volume fracton of the RVE matches wth that of the orgnal mcrostructure. The mult-fber doman s then tessellated nto a network of Vorono cells [Ghosh and Mukhopadhyay (1991)] as shown n Fg. 2(c). The boundary of the RVE, shown n bold lnes, s generated as the aggregate of the outsde edges of Vorono cells assocated wth the prmary fbers (dark colored). The consequent SERVE wll have non-straght lne edges correspondng to nonunform fber arrangements. However, nodes on the RVE (a) G H J KLM NG F A A C B C B D D E E F F GH J K LM N (c) (b) Fgure 2 : (a) Partcles cut durng the RVE generaton (b) Boundary of the RVE adjusted to accommodate the complete partcle (c) Constructon of perodc RVE wth non-straght edges boundary, created by ths procedure are perodc,.e. for every boundary node a perodc par can be dentfed on the boundary at a dstance of one perod along one or both of the coordnate drectons. In Fg. 2(c), the node pars are dentfed as AA, BB etc. The perodcty constrant condtons on nodal dsplacements can then be easly mposed. 3.1 Convergence n the Sze of SERVE A numercal example s used to demonstrate the effect of sze of the constructed SERVE on the macroscopc propertes as well as the mcroscopc stresses. The par dstrbuton functon g(r) for the entre mcrostructure (Fg. 1) s evaluated and depcted n Fg. 3, where r o s the fber

6 156 Copyrght c 2004 Tech Scence Press CMES, vol.5, no.2, pp , 2004 radus. Instead of usng a correcton factor for evaluatng K(r), perodcty of the mcrograph s used. A comparson wth a pure random Posson dstrbuton (g(r)=1) shows that the mcrostructure consdered exhbts sgnfcant devaton from randomness through clusterng etc. Hgh levels of clusterng are seen for lower levels of r (< 8r o ), and the clusterng ntensty decreases wth ncreasng radal dstance. M(r) Von Mses stress n matrx Prncpal stress n ncluson Constant stress Mcrograph Posson dstrbuton R nf r/r ο (a) g(r) Von Mses stress n matrx Prncpal stress n ncluson Constant stress r/r ο M(r) 1.2 Fgure 3 : g(r) dstrbuton for the mcrograph 1 R nf Maxmum prncpal stress n the fber and maxmum Von Mses stress n matrx n each Vorono cell are consdered as marks snce they are good ndcators of falure ntaton n the mcrostructure. These varables are obtaned by detaled computatonal analyss of the entre mcrograph under tensle loadng by the Vorono Cell fnte element method (VCFEM), descrbed n Secton 4.3, [Moorthy and Ghosh (1996)]. The VCFEM mesh s generated by tessellatng the entre mcrograph nto a network of Vorono cells [Ghosh and Mukhopadhyay (1991)]. Plots of M(r) for dfferent marks are shown n Fg. 4(a). It can be seen that M(r) s hgh at dstances less than 8r o but stablzes to a unt value at dstances greater than approxmately 8r o. Ths ndcates that the nfluence on the stress perssts for fbers wthn an approxmate radus of 8r o. It can also be seen that M(r) for both Von Mses stress and prncpal stress have a smlar behavor and stablze approxmately n the same radal range. A very smlar behavor of M(r) s also observed when the mcrograph s loaded under baxal tenson as shown n Fg. 4(b). Ths suggests that a length of around 8r o can characterze the sze scale of the statstcally equvalent r/r ο (b) Fgure 4 : Marked correlaton functons M(r) for (a) unaxal loadng and (b) baxal loadng representatve volume element. Convergence n effectve macroscopc modul and mcroscopc stress dstrbutons wth respect to the RVE sze s now studed. 5 dfferent RVE s, shown n Fg. 5, are consdered wth perodc boundary condtons. They consst of 1, 8, 18, 35 and 55 fbers respectvely wth correspondng RVE szes of r o,3r o,6r o,9r o and 12r o. The RVE s are chosen from any arbtrary regon (Regon A) n the mcrograph. The matrx materal s assumed to be epoxy wth propertes E m = 3.8 GPA and ν m = 0.34, whle the fbers are of graphte wth propertes E f = GPA and ν f = 0.2. The effectve propertes are calculated by the homogenzaton method usng Eq. 6. A Frobenus

7 Manuscrpt Preparaton for CMES Regon #1 Regon #2 E r (µm) Fgure 5 : RVE s wth 1, 8, 18, 35 and 55 fbers Fgure 6 : Convergence of E wth ncreasng RVE szes at dfferent regons norm of the effectve elastc modulus s gven as E = N N =1 j=1 E 2 j (12) where N = 3 for plane stress and plane stran. Fg. 6 shows the convergence of E wth the ncreasng RVE szes. The norm converges wth ncreasng RVE szes. The dfference n the norm between the sngle fber and 55 fbers s around 2%, whle the dfference between 18 fbers (correspondng to R n f = 10µm n Fg. 4) and 55 fbers s found to be less than 0.5%. Smlar responses are observed for other regons (Regon B) n the mcrostructure as shown wth the dotted lne n Fg. 6. Consequently, a sze scale of r = 6r o s deemed adequate for the SERVE to be used n all subsequent analyses. In the study of the effect of RVE sze on mcroscopc stresses, unt macroscopc strans e xx = 1, e yy = 0, e xy = 0, e zz = 0 are mposed on the RVE s wth perodc boundary condtons. Fg. 7 shows the macroscopc stress (Σ xx ), the maxmum Von Mses stress n the matrx and the maxmum prncpal stress n the fber, as functons of ncreasng RVE sze. Whle the macroscopc stress s almost nsenstve to the RVE sze, the maxmum stresses n the mcrostructure change consderably. The dfference n maxmum Von Mses stress n the matrx for the sngle fber RVE and 55 fber RVE s almost 60% whereas, the correspondng dfference for the 18 fber RVE and the 55 fber RVE s less than 4%. Ths study reasserts the effectveness of the marked correlaton functon n determnng the SERVE sze. Stress (GPa) Maxmum mcroscopc prncpal stress n ncluson Maxmum mcroscopc Von Mses stress n matrx Macroscopc Σ xx stress Number of fbers n RVE Fgure 7 : Convergence of macroscopc and mcroscopc stress for unt stran loadng for dfferent RVE szes 4 The Adaptve Mult-Scale Computatonal Model Even f a RVE wth a large number of heterogenetes s used, the mcroscopc stresses or strans at crtcal locatons may be grossly msrepresented n purely macroscopc studes wth effectve modul establshed through mcroscopc homogenzaton, due to the assumptons of perodcty and macroscopc unformty assocated wth the defnton of a RVE. It has been notced by many

8 158 Copyrght c 2004 Tech Scence Press CMES, vol.5, no.2, pp , 2004 authors [Pyrz (1994a); Danel and Anastassopoulos (1995)] that local morphology of fbers have strong effect on falure ntaton and propagaton. The mult-scale computatonal model has been developed n [Ghosh, Lee and Raghavan (2001); Raghavan, Moorthy, Ghosh and Pagano (2001)] to overcome the lmtatons of pure homogenzaton based analyses of heterogeneous materals. Ths model s adaptve n nature and automatcally dstngushes between crtcal and non-crtcal regons to ntroduce levels of herarchy n the computatonal model. In ths paper, the model s extended to mcrostructures wth non-unform dstrbuton of fbers. The man computatonal subdomans n the herarchcal model are shown n Fg. 8 and dscussed next. level-1 element computatons are Settng up the macroscopc stffness by solvng RVE level boundary value problems wth unt strans and perodc boundary condtons Post-processng to compute mcroscopc stresses n the RVE at each ntegraton pont of the macroscopc element 4.1 Computatonal Subdoman Level-0 Level-0 subdomans encompass regons of macroscopc analyss usng effectve propertes obtaned by homogenzaton of SERVE. Ths level s vald n regons where macroscopc gradents n varables lke stresses or strans are relatvely small. For each element n the level-0 subdoman, a SERVE s dentfed and the asymptotc homogenzaton method s then used for obtanng effectve materal propertes. Conventonal dsplacement based elements are used for the level-0 element formulaton. Each element stffness matrx and load vector s evaluated and stored for global assembly for ths subdoman. 4.2 Computatonal Subdoman Level-1 Level-1 subdomans are ntended as transton regons, where mcroscopc nformaton n the SERVE s used to decde whether mcroscopc computatons are necessary for these regons. They are seeded n regons of locally ncreasng gradents of macroscopc varables n the pure level-0 smulatons. These gradents may be caused by mcroscopc non-homogenety n the form of large localzed stresses and strans, or when the mcrostructure faces possble damage ntaton or localzaton. Computatons n ths regon are stll based on assumptons of macroscopc unformty and perodcty of the RVE. Concurrent wth macroscopc smulatons, computatons are executed n the mcrostructure to montor varables n the RVE. Computatonal requrements for mcrostructural analyss of elements n ths level are consderably hgher than that for level-0. It s therefore mportant to desgn robust crtera to avod redundant element transton from level-0 to level-1. Major steps nvolved n Fgure 8 : Mult-level mesh showng dfferent levels for the mult-scale model Adaptve Level-0 and Level-1 Mesh Enrchment The dscretzaton error n level-0 and level-1 are reduced by performng adaptve refnement to ncrease resoluton n requred regons of the model. Dfferent types of adaptatons are possble for these models. Three popular refnements vz., the h-verson, p-verson and the hp-verson have been proposed n lterature. The smple and most common s the h-verson, where refnement s accomplshed by subdvdng the element whle keepng the same polynomal order of the element. In the p-verson, the sze of an element s kept constant, but the order of nterpolatng polynomal s ncreased. In the hp-verson both of the above refnements are ncluded. When hgh accuracy s requred, the use of p-verson or the hp-verson s necessary. Mathematcal and numercal work by Babuska and co-workers [Babuska and Szabo (1991)], have shown that wth the hp-refnement, t s possble to acheve an exponental rate of convergence to the exact soluton for many problems, ncludng those wth sngulartes. The rate of convergence n the hp

9 Manuscrpt Preparaton for CMES 159 adaptve method has been estmated [Guo and Babuska (1986a,b)] as u u hp fe Chµ p (m 1) u (13) where µ= mn(p,m 1), p = polynomal order, m = regularty of the soluton and C = constant ndependent of h and p. The estmate shows that the rate of convergence wll be slow f m 1 s less than p. On the other hand f m s large, whch s the case when the soluton s very smooth, the rate of convergence wll be lmted only by the order of polynomal p. A method for estmatng the value of m s gven n [Answorth and Senor (1997)]. For problems wth sngularty, the value of m wll be low near sngular regons. The hp adaptvty procedure automatcally performs h refnement near a sngularty and p refnement outsde of the sngular regons. Ω l1 k (a) Ω k^ Ω ε 4.3 Computatonal Subdoman Level-2 Level-2 regons are characterzed as those wth sgnfcant mcrostructural non-unformtes n the form of hgh local stresses or strans that would occur e.g. near a crack tp or free edge. Hgh gradents n macroscopc varables and loss of RVE perodcty are expected n those regons. Scale effects are mportant n these regons, resultng n mesh-dependence of pure macroscopc computatons. Adaptvty s used to swtch from level-1 to level-2 elements for performng extended mcroscopc analyss. The mcroscopc model n level-2 elements s requred to encompass consderable portons of the mcrostructure wth large number of heterogenetes as shown n Fg. 9. Level-2 elements are constructed by fllng macro level-1 elements wth the exact mcrostructure at that locaton. The regon Ω k consttutng the k th level-2 element, s obtaned as the ntersecton of the entre mcrostructural regon Ω ε wth the boundary of the k th level-1 element Ω k l1,.e. Ω k = Ωε Ω k l1 (14) Steps n creatng a level-2 element are temzed below Use adaptaton crtera descrbed n Secton 6.2, to determne f a level-1 element needs to swtch to level-2 element. Identfy a regon Ωˆk Ωε that s located n the same regon n Ω k and that extends beyond Ωk by approxmately two fber lengths. (b) Fgure 9 : (a) Level-1 element boundary superposed over the actual mcrostructure (b) Level-2 element formed by carvng out the mcrostructure Tessellate the regon Ω ˆk to generate a mesh of Vorono cell elements as shown n Fg. 9(a). Carve out the regon Ω k by superposng the boundary of Ω k l1 on Ωˆk. Ths procedure wll result n dssectng some of the fbers on the boundary of Ω k. When ths happens, addtonal nodes are generated on the Vorono cell boundary at locatons where the fber surface and Vorono cell edges ntersects the boundary of Ω k. The dssected peces of a fber belong to two contguous level-2 elements are joned together when the two contguous elements share a common edge. The hgh resoluton model for level-2 elements wth many heterogenetes entals prohbtvely large computatons wth conventonal fnte element models. Consequently, the mcrostructure based Vorono Cell Fnte Element Model (VCFEM), whch has been developed by the authors [Moorthy and Ghosh (1996, 1998, 2000)] n

10 160 Copyrght c 2004 Tech Scence Press CMES, vol.5, no.2, pp , 2004 modelng non-unform heterogeneous materals s used for analyzng the level-2 elements. Extensve mcrostructural regons, obtaned from mcrographs, are effcently modeled by ths approach. In VCFEM, the computatonal mesh conssts of mult-sded Vorono polygons that naturally evolve by tessellaton of the mcrostructure [Ghosh and Mukhopadhyay (1991)]. Each element n VCFEM conssts of a heterogenety (ncluson or vod) wth ts mmedate surroundng matrx. Accuracy of analyss s mantaned for these relatvely large multphase elements by ncorporatng observed behavor of stress felds from mcromechancs n an assumed stress hybrd fnte element formulaton [Moorthy and Ghosh (1996, 1998)]. Ths method has shown consderable success n modelng elastc-plastc problems [Moorthy and Ghosh (1996)] and problems wth damage by partcle crackng and debondng [Moorthy and Ghosh (1998); Ghosh, Lng, Majumdar and Km (2000)]. VCFEM has been successful n sgnfcantly reducng computatonal degrees of freedom and current VCFEM computng effort s estmated to be tmes lower than most commercal FEM packages for modelng complex mcrostructures Transton Elements between Level-0/1 and Level-2 Elements The nterface between the macroscopc dsplacement based level-0 or level-1 elements, and level-2 elements requres a layer of transton elements for creatng a smooth transton of varables as shown n Fg. 8. These elements (tr) are essentally level-2 elements wth a dsplacement constrant mposed at the nterface wth level- 0/1 elements. In [Ghosh, Lee and Raghavan (2001)], a drect constrant has been mposed on Vorono cell FEM nodes n the transton elements to conform wth the dsplacement nterpolaton of the adjacent level-0/level-1 element boundares. Such a drect constranng process may however lead to local sngulartes n the transton element due to nduced dsplacement dscontnutes at the nterface. To avert these spurous sngulartes, a specal nterfacal layer s ntroduced between the transton elements and the level-0/level-1 elements. As shown n Fg. 10, the left sde corresponds to level-0/level-1 elements wth lower order dsplacement polynomals on the boundary. The rght sde corresponds to the boundary of transton elements whch conssts of multple edges, of Vorono cell X Y Y Y Y Y Level 0/1 Element Transton element Level 2 element Interface layer Level 0/1 nodes outsde transton layer Level 0/1 nodes on transton layer VCFEM nodes on level 2/tr boundary VCFEM nternal nodes Transton element nodes on transton layer X X X X X X X X X Fgure 10 : Interface constrant at the level-0/1 element and transton element elements dependng on the number of fbers. The ntermedate boundary segment n generally chosen to have hgher order nterpolaton than the adjacent level-0/level- 1 element boundares. As suggested n [Amnpour, Ransom and McCleary (1995)], Lagrange multplers are used to satsfy the nterfacal dsplacement contnuty constrant n a weak sense. Comprehensvely speakng, the total potental energy of the mult-level element mesh may be expressed as Π = Π Ωl0 +Π Ωl1 +Π Ω +Π Ωtr + λ l0/l1 (v u l0/l1 ) dγ Γ nt + λ tr (v u tr ) dγ Γ nt (15) where Π Ωl0, Π Ωl1, Π Ω and Π Ωtr are the total potental energes for elements n level-0, level-1, level-2 subdomans and transton regons respectvely. Γ nt corresponds to the nterfacal layer. λ l0/l1 and λ tr are Lagrange multplers on Γ nt, correspondng to boundares of Ω l0/l1 and Ω tr respectvely. The nterfacal dsplacements on the boundares of Ω l0/l1 and Ω tr elements at the nterface are desgnated as u l0/l1 and u tr. The Euler s equatons, obtaned by the varaton of Eq. 15 wth respect v, λ l0/l1 and λ tr, are λ l0/l1 =(σ j n j ) l0/l1 = λ tr = (σ j n j ) tr and (16) u l0/l1 = u tr = v

11 Manuscrpt Preparaton for CMES 161 where n s the unt outward normal vector, and λ l0/l1 and λ tr correspond to the nterface tractons on the boundares of Ω l0/l1 and Ω tr respectvely. 5 Couplng of all Levels The global stffness matrx and load vectors are derved for the complete mult-scale model consstng of level-0, level-1, level-2 and transton elements. The computatonal doman (Ω) can be descrbed as Ω = {Ω l0 Ω l1 Ω Ω tr } (17) Ω l0 = N l0 k=1 E l0 - Doman comprsed of N l0 level-0 elements wth boundary Ω l0 Ω l1 = N l1 k=1 E l1 - Doman comprsed of N l1 level-1 elements wth boundary Ω l1 Ω = N k=1 E - Doman comprsed of N level-2 elements wth boundary Ω Ω tr = N tr k=1 E tr - Doman comprsed of N tr transton elements wth boundary Ω tr The boundary of the complete doman Γ can be wrtten as Γ = {Γ l0 Γ l1 Γ Γ tr } (18) where, Γ l0, Γ l1, Γ, Γ tr are defned as Γ l0 = Ω l0 Γ, Γ l1 = Ω l1 Γ, Γ = Ω Γ, Γ tr = Ω tr Γ respectvely. The equaton for the prncple of vrtual work for the entre doman can be expressed as +δ λ l0/l1 Γ nt + δu Σ j Ωl0 lo x j δu Σ j Ωl1 l1 x j dω dω δu + σ j Ω dω x j δu + σ j Ωtr tr x j (v u l0/l1 )dγ+δ t δu l0 Γ l0 t δu Γ dω dγ dγ f δu l0 Ω l0 f δu l1 Ω l1 f δu Ω dω dω dω f δu tr dω Ω tr t δu l1 Γ l1 dγ Γ tr t δu tr dγ λ tr (v u tr )dγ = 0 Γ nt (19) An mplct assumpton s made n ths equaton that the tracton contnuty between level-0 and level-1, and level- 2 and transton elements, are satsfed n a weak sense. The terms n the box of Eq. 19 for Ω and Ω tr are analyzed usng the Vorono cell fnte element method and should be ntegrated wth the other terms obtaned by conventonal dsplacement based fnte element analyss. The VCFEM employs the assumed stress hybrd formulaton wth ndependent assumptons on equlbrated stress felds (σ) n the matrx (Ω m ) and fber phases (Ω c ) of each element, and compatble dsplacement felds u e on the element boundary Ω e and u c on the matrxncluson nterface Ω c as shown n Fg. 11. The element complmentary energy functonal s gven by Π C e (σ,u e,u c )= σ : S : σ dω σ n e u e d Ω Ω e Ω e + t u e dγ+ (σ m σ c ) n c u c d Ω Γ tm Ω c (20) where S s the elastc complance tensor, n e and n c are the outward normals on Ω e and Ω c respectvely and t s the prescrbed tracton on the boundary Γ tm. The total energy functonal for a level-2 element contanng Vorono

12 162 Copyrght c 2004 Tech Scence Press CMES, vol.5, no.2, pp , 2004 Ωnt Ω ext Ωc Ω c Ωm e Ω Ω e Γ tm Fgure 11 : A Level-2 element wth Vorono cell fnte elements cell elements can be wrtten as Π C = Π C e = + Ω e σ : S : σdω Ω e σ n e u e d Ω + Ω c (σ m σ c ) n c u c d Ω Γ tm t u e dγ (21) The varaton of the above equaton wth respect to the element boundary dsplacements u e results n δπ C = Ω e σ n e δu e d Ω+ Γ tm t δu e dγ (22) The boundares of all Vorono cells can be splt as Ω e = Ω nt Ω ext (23) where Ω ext corresponds to external Vorono cell element boundares that concde wth level-2/transton element boundary and Ω nt corresponds to all the other nternal boundares of Vorono cell elements. The external element boundares Ω ext are dentfed by thck lnes n Fg. 11. Eq. 22 can be re-wrtten as δπ C = Ω nt σ n e δu e d Ω Ω ext σ n e δu e d Ω + Γ tm t δu e dγ (24) In the vrtual work equaton Eq. 19, the boxed terms correspondng to energy n level-2 and transton elements can be re-wrtten usng dvergence theorem, n the absence of body forces as σ u e dω = σn e δu e dγ σ u e dω Ω e /tr Ω e /tr Ω e /tr (25) The frst term n Eq. 25 s obtaned as the contrbuton to the stffness from all Vorono cells and can be calculated usng Eq. 24 as σ n e δu e dγ = σ n e δu e d Ω Ω e /tr Ω vc e (26) + t δu e dγ+ σ n e δu e d Ω Γ tm Ω nt The last term n Eq. 25 drops out snce the analyss s performed usng VCFEM, whch uses an equlbrated stress feld. The contrbuton to the stffness matrx from level- 2 elements may therefore be calculated by assemblng the stffness contrbutons from all VCFEM elements usng Eq. 22 and condensng out the nternal degrees of freedom on Ω nt. To acheve ths, the dsplacement feld along the edges of VCFEM elements are nterpolated by {u e } =[L vc ]{U vc } (27) The degrees of freedom U vc can be separated nto U ext and U nt dependng on whether they belong to Ω ext or Ω nt respectvely. The stffness matrx and the load vector of the ensemble of all Vorono cell elements belongng to a level-2 element can be parttoned as [ K ext,ext K ext,nt ]{ } { } U ext F ext U nt = F nt (28) K nt,ext K nt,nt Statc condensaton of the nternal degrees of freedom leads to [ [K ext,ext] ][ ] 1 [ ] [K ] ext,nt K nt,nt K nt,ext { } U ext = { F ext } ][ [K ext,nt K mnt,nt ] 1 {F } (29) nt

13 Manuscrpt Preparaton for CMES 163 The above equaton s then used n global assembly. The dsplacements u l0 and u l1 n level-0 and level-1 elements are nterpolated by the standard or herarchcal shape functons based on Legendre polynomals as [Answorth and Senor (1997)] {u} l0/l1 =[N l0/l1 ]{U l0/l1 } =[N I l0/l1 NO l0/l1 ]{ U I l0/l1 U O l0/l1 } (30) where Ul0/l1 I corresponds to the nodal degrees of freedom at the nterface wth transton elements and U O l0/l1 are all others. A smlar separaton can also done for transton elements nto dsplacements on ths nterface, U I tr and otherwse, U O /tr. The dsplacements and the Lagrange multplers on the ntermedate boundary segment are nterpolated from nodal values usng sutably assumed shape functons as {v} =[L nt ]{U nt }, {λ lo/l1 } =[L λ l0/l1]{λ lo/l1 } {λ tr } =[L λ tr]{λ tr } (31) Substtutng nterpolatons from Eq. 29,Eq. 30 and Eq. 31 n Eq. 19 results n a coupled set of matrx equatons for the mult-level doman. The global assembly leads to the followng coupled set of equatons K I,I l0/l1 K O,I l0/l1 P T K I,O l0/l P l0/l1 0 l0/l K I,O /tr 0 0 P tr K O,O 0 0 K I,I tr 0 0 K O,I /tr K O,O /tr Q l0/l1 Q tr l0/l Q T l0/l Ptr T 0 Qtr T 0 0 Ul0/l1 I F I Ul0/l1 O l0/l1 Fl0/l1 O U I tr U O /tr U nt Λ l0/l1 Λ tr = F I tr F O /tr (32) The notaton correspondng to the superscrpt I represents quanttes on the nterface whereas those wth the superscrpt O are on the regons other than nterface. The submatrces K l0/l1, K and K tr and the vectors F l0/l1, F and F tr correspond to stffness matrces and load vectors from the respectve subdomans gven as [K l0/l1 ]= [B] T [E][B]dΩ Ω l0/l1 { } Fl0/l1 = [N l0/l1 ] T { f }dω + [N l0/l1 ] T {t}dγ Ω l0/l1 Γ l0/l1 (33) where [B] s the stran-dsplacement matrx. The stffness [K /tr ] and the load vectors { } F /tr for level-2 and transton elements are obtaned by solvng VCFEM usng the procedure descrbed n [Moorthy and Ghosh (1996)]. The couplng between the level-0/1 and transton elements s acheved through the [P] and [Q] matrces, whch are [P l0/l1 ]= [N l0/l1 ] T [L λ l0/l1] dγ Γ nt [P tr ]= [L tr ] T [L λ tr] dγ Γ nt (34) [Q l0/l1 ]= [L nt ] T [L λ l0/l1] dγ Γ nt [Q tr ]= [L nt ] T [L λ tr] dγ Γ nt The system of equatons s solved usng an teratve solver wth Lanczos method. 6 Dscretzaton and Modelng Error Indcators The errors assocated wth the mult-level model are classfed nto two groups, vz., dscretzaton errors and modelng errors. 6.1 Dscretzaton Error Dscretzaton error n level-0/1 elements s a result of nsuffcent orders of nterpolaton n the fnte element model. The hp adaptve mesh refnement suggested n [Answorth and Senor (1997)] s adopted n ths paper to reduce the dscretzaton error. The steps nvolved n ths process are as follows Evaluate the energy norm of the local error φ k for element k, by solvng the resdual n the prncpal of vrtual work as Σ j (φ k )e j (v)dω = f v dω Ω k Ω k l0/l1 l0/l1 (35) Σ j (u l0/l1 )e j (v)dω+ (g k ) v dγ Ω k l0/l1 Γ k

14 164 Copyrght c 2004 Tech Scence Press CMES, vol.5, no.2, pp , 2004 where (g k ) s the tracton dscontnuty on the element boundary Γ k. Identfy elements for hp adaptvty from the condton φ k C 1 (φ k ) max, where (φ k ) max s the maxmum elemental local error. If an element s a canddate for adaptaton, an exponent m s evaluated to determne the type of adaptvty.e f p + 2 m then p refnement else h refnement s performed. The sngularty ndcator m can be obtaned by solvng φ 2 k = φ p+q 2 k +C 2 k (p +q) 2(m 1) (36) for three dfferent values of q as outlned n [Answorth and Senor (1997)]. 6.2 Modelng Error Transton from level-0 to level-1 : Modelng errors for level-0 and level-1 elements evolve at regons where macroscopc unformty and mcroscopc perodcty become nvald. Varous condtons, dependng on the physcs of the problem beng are proposed. Two examples of such crtera consdered n ths paper are gven below. 1. If a macroscopc stress Σ j s mportant, crtera based on the gradents of ths stress can be proposed. Transton from level-0 to level-1 for an element k s performed f E k C 2 E avg (37) where E avg =( NE k=1 E2 NE )1/2 and Ek 2 = Γ k [[Σ j ]] 2 dγ Γ k dγ. NE s the total number of elements and [[ ]] s the jump operator. 2. When tracton gradent s crtcal [Raghavan, Moorthy, Ghosh and Pagano (2001)], level-0 to level-1 transton for element k wll be made f E k C 3 E avg (38) where E avg =( NE k=1 E2 NE Ek 2 = Γ k([[t x]] 2 +[[T y ]] 2 )dγ and T Γ k dγ x and T y are tractons n x and y drectons respectvely. )1/2 and Transton from level-1 to level-2 : Crtera for transton from level-1 to level-2, s based on observaton of varables n the RVE and predcton of departure from perodcty. In ths work, large local stresses n the matrx/fber or nterfaces are assumed to ndcate such departures. Two alternatve crtera are used n ths paper. 1. Level-1 to level-2 transton takes place f the local 3 mcroscopc equvalent stress σ eqv = 2 σ j σ j exceeds the average. The equvalent stress s a good ndcator for damage, especally n plastcty domnated problems. Level-1 to level-2 transton s made f (σ m eqv ) max C 4 (σ m eqv ) avg or (σ c eqv) max C 4 (σ c eqv) avg (39) at more than 1% of all ntegraton ponts n the RVE. (σ m eqv) max and (σ m eqv) avg, and (σ c eqv) max and (σ c eqv) avg represent the maxmum and average equvalent stresses n the matrx and n the fber. 2. Transton from level-1 to level-2 takes place f T C 5 T avg (40) where T represents the local nterfacal tracton ( Tn 2 +Tt 2 ) evaluated on the fber/matrx nterface. T avg s the average tracton on the fber/matrx boundary and s gven by NI =1 T NI, where NI s the total number of ntegraton ponts on the fber/matrx nterface n the RVE. Snce debondng s an mportant falure mechansm n fber renforced compostes, ths crtera s expected to warrant sgnfcant amount of debondng n the mcrostructure. All constants C 1 to C 5 are chosen from tral numercal experments. 7 Numercal Examples - Composte Plate wth a Hole Numercal experments are conducted to demonstrate the effectveness of the mult-scale model. Two examples, one of a composte plate and another of a composte lamnate are consdered.

15 Manuscrpt Preparaton for CMES Composte Plate wth a Hole A fber renforced composte plate wth a hole as shown n Fg. 1 s analyzed. The dmensons of the composte plate are W = mm, L = mm and r = mm. The representatve mcrostructure of the plate at the pont A s shown n Fg. 1 wth dmensons 100 µm µm. All fbers are assumed to be posses the same radus value of 1.75 µm. Average fber spacng s around 3.4 µm. The estmated total number of fbers for ths quarter plate s approxmately 8 mllon. The matrx materal s epoxy wth propertes E m = 3.8 GPA and ν m = The graphte fber propertes are E f = GPA and ν f = The plate s subjected to a load of unty along the y drecton on the top face. A statstcally equvalent RVE for ths plate s evaluated as descrbed n Secton 3 and s found to contan 18 fbers. The effectve modulus (n GPa) calculated by homogenzaton for ths SERVE s gven by E 1111 = 10.51, E 1122 = 4.49, E 1133 = 4.23, E 2222 = 11.19, E 1212 = 2.85, E 3333 = Usng ths modulus the macroscopc model s constructed wth 300 level-0 elements. Generally no sngulartes are expected n the soluton of the problem. Consequently, dscretzaton error would result n p adaptvty beng predomnant. Whle a p adaptaton would suffce for the problem, the h refnement facltates for sgnfcantly smaller regons of localzed modelng error dentfcaton and hence realzaton of fewer elements makng level transtons. The h adaptatons are executed to a mnmum macroscopc element sze of 25 µm and no more. Level-0 to level-1 transton takes place accordng to Eq. 37 as specfed n Secton 6.2 wth the value of C 2 taken to be 1.5. Level-1 to level-2 transton takes place accordng to Eq. 39 as specfed n Secton 6.2 wth the value of C 4 taken to be 4.0. The adapted mult-scale mesh, shown n Fg. 12, conssts of 1091 level-0 elements, 2 level-1 elements, 3 transton elements and 2 level-2 elements. The mesh n the crtcal regon s crcled and shown n Fg. 12(b). The mcrostructure for the level-2 and transton elements s shown n Fg. 12(c). The macroscopc contour plot of Σ yy stress and mcroscopc contour plot of σ yy stress for the element close to the crtcal regon A s shown n Fg. 13. It can be seen that the maxmum mcroscopc stresses are at least one order hgher than the macroscopc values. The maxmum mcroscopc stresses s near the pont where two or more fbers are located close to each other. TOTAL ELEMENTS = 1098 LEVEL 0 = 1091 LEVEL 1 = 2 TRANSITION ELEMENTS = 3 LEVEL 2 = 2 y (a) x (b) Level-2/transton element boundary (c) Fgure 12 Fgure 12 : (a) h adapted mult-level mesh (b) Mesh around local crcled area (c) Mcrostructure of level- 2/transton elements 7.2 Free Edge Composte Lamnate Subjected to Extensonal Loadng The performance of the mult-scale model n the presence of sngularty s demonstrated wth ths example. The composte lamnate conssts of randomly dstrbuted fbers on the top and bottom whereas the mddle portons conssts only of matrx materal. Wth effectve modulus theory ths archtecture corresponds to a homogenzed

16 166 c 2004 Tech Scence Press Copyrght Max. CMES, vol.5, no.2, pp , E-03 y 6.386E E-03 A h 3.165E E-03 A Mn. x 2h E-05 (a) Max E E-03 (a) (b) Fgure 14 : (a) Composte lamnate subjected to exte Fgure 14 : (a) Composte lamnate subjected to extensonal loadng (b) Optcal mcrograph of the mcrostructure near pont A 6.619E-03 of the lamnate s modeled wth effectve propertes obtaned by homogenzaton. The ntal mesh n the multlevel model conssts of 200 level-0 4-noded blnear or 9.745E-04 QUAD4 elements. The dscretzaton error n the homogenzed model s reduced by performng hp adaptatons E-03 Mn. The presence of free edge at the composte-monolthc (b) materal nterface causes a sngularty n σ yy stress. Due Fgure 13 : (a) Contour plot of macroscopc s to the sngularty, h adaptvty s domnant near the free Fgure 13 : (a) Contour plot of macroscopc Σ yy stress edge, whle regons far away from the free edge are p for the plate wth a hole (b) Contour plot of mcroscopc adapted. σyy stress for the macroscopc element close to pont A The hp adapted mesh s shown n Fg. 15(a) wth a blow-up of the mesh near the free edge s shown n Fg. 15 (b) E-03 materal sandwched between two composte ples. The lamnate s subjected to extensonal out-of-plane loadng along the fber drecton as shown n Fg. 14(a). The mcrostructure of the lamnate around the free edge pont A s assumed to be same as n the prevous problem. The dmenson of the cross secton s 64 mm 32 mm. The out-of-plane loadng, s smulated usng a generalzed plane stran condton wth prescrbed ε zz = 1. The dstrbuton of the mcrostructure s assumed to symmetrc about the x and y axes. Due to symmetry n the xz and yz planes only one quarter of the lamnate s modeled. Symmetrc boundary condtons are employed on the surfaces x = 0 and y = 0, and the top and rght surfaces are assumed to be tracton free. The top porton Due to the requrement of tracton contnuty at the materal nterface level-0 to level-1 transton s made accordng to Eq. 38. For the level-1 to level-2 transton the crtera n Eq. 40 s used. The parameter C3 and C5 are chosen to be 2.5 and 1.25 respectvely. The evolved multlevel model conssts of 513 level-0 elements, 7 level-1 elements, 3 transton elements and 1 level-2 element. The mcrostructure of the macroscopc element close to the free edge s shown n Fg. 15(c). The macroscopc Σ yy stress sngularty at the materal nterface y = h4 s shown n Fg. 16. The fgure also shows mcroscopc σ yy stress near the free edge obtaned from the mult-scale soluton. It can be observed stress sngularty observed at the free edge n the effectve modulus soluton s not present n the mult-scale soluton. The macroscopc contour plot