IM- Jne 5-9 eijing ina I-6 NUMERIL PREITION O THE MEHNIL PROPERTIES O WOVEN RI OMPOSITES V. arvelli an. Poggi epartment of Strctral Engineering Polytecnic of Milan Piazza Leonaro a Vinci Milano Italy SUMMRY: Te paper presents a nmerical procere to evalate te mecanical properties of woven fabric composites an te valiation of te meto against experiments. Te nmerical approac is fone on te main assmption of perioic istribtion of te reinforcements in te composite an on two levels of omogenization. Te first omogenization concerns te warp an weft yarns wile te secon regars a representative volme of te woven fabric laminate wit omogenize yarns. n experimental investigation an te relevant nmerical preictions were performe on varios woven fabric laminates incling for types of fibers: carbon glass arami an polyvinylalcool (PV). KEYWORS: woven fabric finite element meto experimental investigation mecanical featres. INTROUTION Te application of textile composites in engineering strctres as been riven by varios attractive aspects like ease of anling ig aaptability amage tolerance an if compare to niirectional laminates better ot-of-plane stiffness properties. Terefore an in-ept knowlege of te mecanical properties is of fnamental importance. Te most known types of fibers are se in woven fabric (W) laminates on te basis of specific nees an te geometry of te reinforcement (yarn spacing yarn tickness an sape an te weave type) is cosen to reac te reqeste mecanical properties. It is terefore important to proce an valiate micromecanical moels tat can preict bot te internal stress state of a laminate an te macroscopic beavior. Te micromecanical moels for woven fabric composites available in te literatre can be grope into two classes []. Te moels of te first class tat incles analytical metos sc as iso-stress an iso-strain provie a reasonable estimate of te mecanical properties bt te evalation of te internal stresses is not satisfactory [ ]. Te secon class is base on 'finite element' moels of an elementary cell (representative volme RV) of te laminate [ 4]. Te meto can provie aeqate estimates of te internal stresses of te local amage an can be se to evalate te ltimate failre of te composite. Tis paper presents a finite element moel sitable for te analysis of W composites sbject to any stress state. n omogenization process to pass from a micro to a macro scale an to relate te mecanical properties of te macro element to te properties of te single pases is applie at two levels. Te first level concerns te yarns consiere as niirectional fiber reinforce composites to etermine teir mecanical properties. Te secon level of omogenization is applie to te elementary cell of te W. In tis case a E moel is proce an te necessary bonary conitions to ensre te perioicity of te material are applie. In parallel an experimental campaign on W composite laminates was nertaken. Te reslts are smmarise in te first part of te paper. Te nmerical reslts are ten compare to te experimental vales of ifferent laminates in terms of elastic properties. EXPERIMENTL INVESTIGTIONS Te mecanical properties of a series of woven fabric laminates incling ifferent manfactring tecniqes geometries an types of fibers ave been recently investigate
experimentally [5]. Te final objective was te experimental evalation of te inflence of all tese parameters on te mecanical properties of te laminates. or types of fibers an tree types of epoxy resins were consiere. In tis work only te reslts of specimens mae wit plain-weave W sing te bag moling tecniqe are consiere wile te reslts of oter materials may be fone in [5]. or panels were manfactre sing te fibers an te matrix reporte in Tab. an te woven fabric liste in Tab.. Te laminates were bilt wit ifferent nmber of layers tat range from to. It sol be note tat all fabrics are balance an present te same nmber of fibers an geometry in te warp an weft irections. Tab. E [GPa] ν σ arbon. 4 ibers Glass 76. rami 8. 8 PV 9. 4 Matrix Epoxy SV.5.5 48 Yong s mols (E) Poisson s ratio (ν) an ltimate tensile strengt (σ ) of te fibers an matrices. ibers material Weigt [g/m ] onstrction [yarn/cm] (in weigt [%]) Warp Weft iber cont [tex] arbon 5 (5) 5 (5) Glass 8 6 (5) 6 (5) rami 46 6.7 (5) 6.7 (5) 6 PV 4 6 (5) 6 (5) Tab. Tecnical caracteristics of te reinforcing fabrics. Te specimens for te experimental caracterisation of te laminate were proce from flat plates ctting five nominally ientical specimens per type. Tensile tests interlaminar sear strengt (ILSS) bening an impact tests were performe accoring to te stanars STM 9 44 79M an 594 respectively. In tis work te attention is focse only on tensile tests. Te tensile properties of te laminates were evalate on two series of five specimens (approximate imensions 5x5 mm) ct from te same plate in two ifferent irections ( an 45 ). Te volme fractions V f were calclate from te weigt of te specimens an te weigt per nit area of te ry reinforcement. Te voi contents were not measre. Te tests were execte at te ambient conitions sing a tensile-torsional testing macine (capacity: kn axial Nm torsional) available at te Material Testing Laboratory of Polytecnic of Milan. Te torsional moment generate ring te tensile tests was recore to etect possible copling. Tis qantity is a significant parameter to evalate possible misalignments of te fibers an can be also aopte as a parameter for a qality control of te manfactring [5]. Te longitinal an transversal strains were measre by means of a biaxial transcer tat was remove from te specimens wen te loa reace abot 5% of te expecte ltimate loa. Te accracy of te extensometer was also verifie by means of electrical strain gages applie to some specimens an comparing te transversal an longitinal strains. Neverteless it mst be nerline tat te measrement of te transversal eformation is always a elicate matter an even te extensometer can fail becase of imperfections of te matrix in te zone of te specimen were te transcer is positione.
Te tickness of eac specimen was measre in at least tree sections to verify its reglarity. Te mean vales of te elastic mols te Poisson s ratio an te ltimate strengt were calclate as an average of five reslts from eac series of tests. Te stanar eviation an te coefficient of variation (stanar eviation over mean vale) were also calclate to ave a measre of te variability an te tests were consiere satisfactory wen te coefficient of variation was lower tan.5. Te reslts of te niaxial tensile tests for te W laminates are reporte in Tab. were te most significant properties (axial elastic mols E; Poisson s ratio ν; ltimate stress σ ) are liste. Te vales are te average of five tests. Te properties of te fabric evalate on specimens ct at are extensible to te irection 9 becase of te fll symmetry in te warp an weft irections. o /9 o (warp/weft) 45 o ibers material Layers nmber V f % E ν σ E ν σ arbon 7 4.6 959.89 64 87.79 5 Glass 5 5.6 859.6 69 585.7 6 rami 5.6 9858.9 84.8 --- PV 4 4.8 858. 5 698.788 9 Tab. Experimental tensile properties of te woven fabric laminates. THE NUMERIL HOMOGENISTION PROEURE Te propose nmerical meto for tree-imensional analyses of W composites is presente in tis section [6]. Te following ypoteses are assme: (i) reglar istribtion of te fibers in te yarns; (ii) reglar arrangement of te reinforcing yarns in te composites. Tese ypoteses allow to sty te problem witin te framework of te micromecanical teory of eterogeneos perioic materials [7] an correspon to two levels of omogenisation. Te first level concerns te yarns tat are firstly examine by means of an inepenent moel. Eac yarns is consiere as niirectional fiber reinforce composite (see ig. a) wit fiber volme fraction eqal to te packing ensity (p ). Te mecanical properties of te omogenize yarns (ig. c) are obtaine by means of nmerical analyses on a representative volme as etaile in [8]. In ig. b te representative volme of a U composite (exagonal pattern) is sowe. pplications an valiation of tis approac for niirectional laminates may be fon in [8]. x ig. (a) x (b) x (a) Woven fabric composite; (b) representative volme of te yarn consiere as niirectional fiber reinforce composites; (c) omogenize yarns. Te secon omogenization level smmarize below eals wit a fictitios woven fabric composite (W) containing te omogenize yarns (ig. a). Te reglar istribtion of te reinforcements in te material allows s to extract a representative volme (RV) (see ig. b) (c)
of te W to investigate te response of te omogenize material (ig. c) in terms of macroscopic (or global) qantities. (b) ig. (a) (a) Woven fabric composite wit omogenize yarns; (b) representative volme; (c) omogenize material. Te global constittive law links macroscopic stresses an strains wose components are collecte in vectors S E respectively (bol symbols enote matrices an vectors). Tey are efine as volmetric averages of te relevant microscopic variables σ(x) ε(x) wic are fnctions of te position vector x in te RV. Te macroscopic constittive law escribes te global response of any RV belonging to an ieal infinite meim sbjecte to niform bonary conitions eiter in stresses or isplacements. or composites consisting of nonlinear components tis law as to be etermine in incremental form. Te local fiels of stress σ(x) an strain ε(x) mst reproce te perioicity of te eterogeneos material. Sc property is satisfie if te following relation between te isplacement an te strain fiels ols [7]: ( x) = + x + E x+ ~ ( x) () were represents a rigi isplacement of te RV an Ω is te anti-symmetric tensor relate to te small rigi rotation of te RV. Te pre strain moes of te RV are escribe by te last two terms in (): a constant term (te macroscopic strain E) an a V-perioic term wit zero average vale associate wit te V-perioic part ~ of te microscopic isplacement fiel (being V te volme of te RV). Te incremental problem to efine te macroscopic constittive law can be expresse as follows: given & fin &( E& ) sc tat: iv & = in V (a) t & = & n anti - perioic on V (b) & = V & V (c) V ~ & = & E& x& V perioic () &E = V & V (e) V & = ( & ( & )) in V (f) were V an V inicate te volme an te bonary of te RV respectively an n te otwar nit-normal on V. Eqn. (f) represents te microscopic incremental constittive law. Te soltion of te incremental problem () is obtaine by means of a moel base on te isplacement formlation of te finite element meto. Te main task to implement problem () in a E coe base on te isplacement formlation is te assignment of te bonary conitions to ensre tat te isplacement fiel complies (c) 4
5 wit Eqn. (). Referring to te RV single ot from a plane woven composite in ig. a sc conitions can be written as: Y J K H = (a) P P = ˆ (b) In Eqn. (a) J an Y are one of te mipoints pairs ( ) ( ); H an K are points corresponing in te perioicity belonging to te sies x = / an x = / respectively if ( ) are consiere or belonging to te sies x = / an x = / if ( ) are consiere (see ig. b). In Eqn. (b) P is any point on te sie x = an Pˆ is te corresponing point on te sie x =. (a) (b) ig. Te RV geometry an te E moel of te reinforcement; (b) caracteristic points of te finite element moel. Te macroscopic isplacement graient are relate to some isplacement components on V by employing te perioicity conition (ab) an acconting for te general expression of te isplacement fiel () namely: = + E (4) Rigi translations of te moel are sppresse by prescribing O = (see ig. b). Rigi rotations can be avoie by imposing te skew symmetric tensor Ω to vanis setting: = + (5a) = + (5b) = + (5c) Te six components of te symmetric tensor E can be explicitly written as:
E E = ; E = ; E = ; (6abc) = ; E = ; E = (6ef) Relations (6) provie te kinematic bonary conitions tat allow te simlation of any prescribe macroscopic strain trog only some free noal isplacements ( ). Prescribe macroscopic stresses acting along te ortogonal axes x x x of te moel can also be impose trog kinematic bonary conitions sing te following sceme: = prescribe Σ; = prescribe Σ = prescribe Σ ; = prescribe Σ = prescribe Σ ; = prescribe Σ (7ab) (7c) (7ef) In te nmerical simlations a macroscopic component (stress or strain) is prescribe an te remaining ones are evalate by te volmetric averages (ce) of te microscopic conterparts in te Gass points. PPLITIONS N OMPRISONS Te procere was valiate preicting te elastic properties of te woven fabrics presente in te experimental section above. Te mecanical properties of te composite components are liste in Tab.. Te geometrical featres of te representative volmes are etaile in Tab. 4 wit reference to ig. a. Tese geometric parameters were obtaine by microscopic observations tat provie also te yarns morpology as sown in ig. 4. ig. 4 Microscopic observation of te arami W composite. Te fiber volme fraction of te yarns consiere as niirectional fiber reinforce composite in te first omogenisation step is assme eqal to te packing ensity factor p =.8 for carbon glass an PV an p =.75 for arami reinforcement. Te fiber volme fractions V f of te yarns in te W laminates in te secon omogenisation step are liste in Tab.. ibers material = w =w t =t p =p arbon 4..5.8.4.5 Glass.4.8.5.5.6 rami.4..6.48.7 PV.8.7.5.4.69 Tab. 4 Geometrical featres of te consiere RV (see ig. a) Te finite element meses employe in te first omogenisation step were mae wit 8- noes soli elements as alreay experience in [8]. Te elastic featres of te transversal isotropic omogenize yarns obtaine in te first step are etaile in Tab. 5 (x is te fiber axis wile x -x inicates te plane ortogonal to te fibers see te exagonal RV in ig. b). 6
or-noe tetraerons finite elements were se in te iscretization of te woven fabric composite RV in te secon omogenisation step. Te etails of te meses are reporte in Tab. 6. s an example te E mes of te reinforcements of te carbon woven fabric composite is rawn in ig. a. Tab. 5 arbon Glass rami PV E =E 844 645 86 5 E 84545 595 6756 77 G 644 7 6646 54 G =G 9787 6 6449 5 ν.5.8..4 ν =ν.47.4.9.76 ν =ν.8.55..8 Elastic featres of te yarns obtaine in te first omogenisation step. Tab. 6 ibers material Nmber of noes Nmber of elements arbon 6 65 Glass 448 766 rami 57 589 PV 476 787 Noes an elements nmber of te finite element meses employe. Te experimental elastic moli measre in irection o are epicte in ig. 5 wit teir stanar eviations for te for woven fabric composites consiere. Te vales of te elastic mols obtaine wit te present nmerical procere are also presente in ig. 5 for comparison. goo agreement between experimental an nmerical vales is evient. On te contrary some ifferences were etecte in terms of Poisson s ratios (ig. 6). Neverteless it mst be nerline tat te experimental measrement of te transversal eformation is always a elicate matter. Tis problem will be certainly treate in ftre evelopments of te researc. ig. 7 sows te eforme mes of te omogenize yarns in a sear test of te PV specimen. Tis permits to verify te effectiveness of te impose perioicity bonary conitions on te RV of te W composite to reproce te perioicity. ig. 5 Elastic Mols [GPa] 5 4 carbon glass arami PV ibers material Nmerical Experimental omparison of te nmerical an experimental elastic moli for traction in o irection. 7
. ig. 6 Poisson's ratio... carbon glass arami PV ibers material Nmerical Experimental omparison of te nmerical an experimental Poisson s ratio for traction in o irection. ig. 7 eforme mes of te yarns from sear test in te plane x -x of PV specimen ONLUSIONS nmerical procere to evalate te mecanical properties of woven fabric composites as been presente an valiate against experimental reslts. Te meto is base on te omogenization tecniqe for perioic meia. Tis is first applie to te warp an weft yarns by means of E moels an allows s to etermine te mecanical properties of te yarns. Te secon level of omogenization is stie sing an elementary volme of te woven fabric an imposing te proper bonary conitions to ensre te perioicity. ll te components of te stiffness matrix may be obtaine. Te moel as been valiate simlating ifferent tensile experiments of flat specimens mae wit varios types of fibers. cknowlegement: particlar tank goes to Raol avino an Stefano Mazzoleni erracini for teir elp in te nmerical comptations. Te ators wis to tank te company Tecnotessile (Prato Italy) for manfactring te specimens an for te financial spport. REERENES. T. W. o an. K. Ko Textile strctral composites Elsevier New York (989).. T. Isikawa an T. W. o Stiffness an strengt beavior of woven fabric composites J. of omposite Materials 7 (98) pp. -.. N. K. Naik an P. S. Sembekar Elastic beavior of woven fabric composites: I Lamina analysis J. of omposite Materials 6 (99) pp. 96-5. 8
4. J. Witcomb K. Woo an S. Gnapaneni Macro finite element analysis of textile composites J. of omposite Materials 7 (994) pp. 67-68. 5. V. arvelli. e ngelis. Poggi R. Poti Effects of manfactirng tecniqes on te mecanical properties of composite laminates Proc. of Experimental tecniqes an esign in omposite Materials agliari Sept.. Sbmitte to Key Engineering Materials. 6. V. arvelli an. Poggi Homogenization Procere for te Nmerical nalysis of Woven abric omposites omposites Part in print. 7. P. Sqet Elements of omogenization for inelastic soli mecanics In Homogenization tecniqes for composite meia Lectre Notes in Pysics 7 (E. by E. Sancez-Palencia an. Zaoi) Springer Wien pp. 9-78 (985). 8. V. arvelli an. Taliercio micromecanical moel for te analysis of niirectional elastoplastic composites sbjecte to stresses Mecanics Researc ommnications 6 (999) pp. 547-55. 9