Estimation of Lamina Stiffness and Strength of Quadriaxial Non-Crimp Fabric Composites Based on Semi-Laminar Considerations

Transcription

1 appied sciences Artice Estimation of Lamina Stiffness and Strength of Quadriaxia Non-Crimp Fabric Composites Based on Semi-Laminar Considerations Yong Cao, Yunwen Feng *, Wenzhi Wang *, Danqing Wu and Zhengzheng Zhu Schoo of Aeronautics, Northwestern Poytechnica University, P.O. Box 120, Xi an , Shaanxi, China; Y.C.); D.W.); Z.Z.) * Correspondence: fengyunwen@nwpu.edu.cn Y.F.); wangwenzhi@nwpu.edu.cn W.W.); Te.: Y.F.); W.W.) Academic Editor: Peter Van Puyvede Received: 5 Juy 2016; Accepted: 14 September 2016; Pubished: 19 September 2016 Abstract: Quadriaxia non-crimp fabric QNCF) composites are increasingy being used as primary structura materias in aircraft and automotive appications. Predicting mechanica properties of QNCF amina is more compicated compared with that of unidirectiona UD) composites, because of knitting connection of different pies. In this study, to anayze stiffness and strength of QNCF composites, a nove modeing strategy for meso-scae features is presented based on semi-aminar assumption. Foowing view of mechanica properties of singe composite amina, compex QNCF ayer is decomposed into individua pies. Three different representative unit ces aong fiber direction are seected to predict mechanica performance of QNCF, incuding in-pane stiffness, damage initiation, and stiffness degradation. To vaidate deveoped modeing strategy, predictions are compared with existing experimenta resuts, where a good agreement is presented on prediction of in-pane stiffness and strength. Furrmore, effect of in-pane fiber distortion, induced by stitching yarn on mechanica properties, is studied. Keywords: non-crimp fabric NCF); fiber distortion; mechanica properties; mutiscae anaysis 1. Introduction Non-crimp fabric NCF) is constituted by a arge amount of fairy straight fiber tows that are paced side by side and bounded by warp-knitting [1]. Compared with unidirectiona UD) pre-preg composites, NCF composites have many advantages, such as ower production cost, higher out-of-pane damage toerance and fracture toughness. Thus, it is becoming more popuar in manufacture of compex and thicker parts than UD pre-preg composites [2]. The NCF has two main types, open structure and continuous pies [3]. The fiber tows of continuous pies are aid as cosey as possibe to reduce waviness of fiber tows. However, fiber distortion sti exists in pies [4]. Mechanica properties of singe amina are basic parameters for composites. Many ories and faiure criteria are deveoped based on assumptions that each amina of aminated composites is homogeneous and orthotropic [5]. Since each amina in NCF has a unique fiber direction, it can be considered as semi-aminar [6]. For mechanica anaysis of NCF composite structures, basic mechanica properties of semi-aminar composites shoud be known. To obtain mechanica properties of each amina of quadriaxia non-crimp fabric QNCF) composites materia is suppied with a ayer stacking sequence 45 /90 / 45 /0 )) through experiment, four kinds of UD specimens need to be produced one for each fiber direction, incuding through-thickness yarn) [7]. This process is very compicated and costy, for it is not very easy to adjust warp-knitting machine for a sma batch production of se UD specimens. Aternativey, mechanica properties of NCF App. Sci. 2016, 6, 267; doi: /app

2 App. Sci. 2016, 6, of 17 composites can be predicted by using a mutiscae anaysis approach. However, a compete mutiscae anaysis is aso very time-consuming [6]. An equivaent anaysis method was proposed based on semi-aminar considerations [6,8,9]. In this method, mechanica properties of NCF amina are determined by equivaent UD composite amina, which is mutipied with knock-down factors to take into consideration effect of fiber waviness. The effect of fiber waviness on mechanica properties was studied using different modeing methods in references [10 12]. The knock-down factor or stiffness reduction in previous works mainy focused on out-of-pane fiber waviness of opening structure NCF composites. However, waviness of continuous pies NCF composites mainy presents in in-pane direction of fabrics rar than out of pane direction. Thus, effect of in-pane fiber waviness shoud be considered when evauating mechanica performance of this kind of NCF composites. Severa researchers have studied mechanica properties of continuous pies NCF using eir experimenta or mutiscae modeing approach. Truong et a. [13] compared experimenta resuts of eastic moduus with that of cassica aminate ory CLT) predictions, and found that effect of stitching on stiffness of this materia was not significant. However, earier works [14,15] caimed that experimenta measurements were about 17% ower than those obtained by CLT. Mikhauk et a. [16] used acoustic emission AE) registration and X-ray imaging to detect damage initiation and evoution in quadriaxia aminates under tensie oad, and numerica simuation of faiure process was aso carried out. Ivanov et a. [17] summarized previous meso-scae mode, and set up a meso-scae mechanicay representative voume mrve), and predicted effect of stitching on mechanica properties of non-crimp fabric composite, such as, stiffness, faiure initiation, stiffness degradation and strength. The current researches were focused on performance of a whoe continuous pies QNCF ayer or aminate, but few studies has been done on amina mechanica properties of materias. Since NCF composites can be addressed as being semi-aminar [6], stiffness and strength of QNCF can be estimated from view of mechanica properties of singe QNCF amina. After evauation, QNCF amina can be equivaent to UD amina combined with knock-down factor or effective mechanica properties of amina. Then, genera method can be appied to anaysis QNCF composites, such as CLT, ayerwise ory, FE-she eement, etc. Foowing this idea, a simpified modeing strategy is presented for meso-scae anaysis to predict effective stiffness properties and py strength of QNCF. Considering imitations described above, re are two main contributions in this paper. Firsty, stiffness and strength of QNCF amina composites is predicted, and effect of in-pane fiber distortion on mechanica properties is studied at amina eve. The resuts can be used to create an equivaent continuum mode for composites at macroscae eve. Secondy, a nove modeing strategy for meso-scae features of QNCF is presented. It is feasibe for rapid modeing and meshing due to simpy description of inter-structure of QNCF. 2. Modeing Approach To perform a meso-scae anaysis based on mechanica properties of singe NCF amina, detaied interna structure of continuous pies NCF composites shoud be known. Interna geometry of a continuous pies NCF stitched by warp-knitting was investigated in Reference [4]. Based on this verified geometry data, a continuous pies QNCF which has a ayer sequence of 45 /90 / 45 /0 ) is seected for a mesomechanica mode, this quasi-isotropic ayer is widey used in engineering, and its compex interna structure contains a variety of known types of stitch yarn induced fiber distortion SYD). In this way, typica form of SYD in QNCF composites can be accounted for mutiscae anaysis. The stitching threads are ignored, foowing assumption in Reference [18]. In this modeing strategy, QNCF aminate is considered as a semi-aminar, and is assumed to be separated from ors.

3 App. Sci. 2016, 6, of App. Modeing Sci. 2016, 6, Strategy 267 Based on Mechanica Properties of Singe NCF Lamina 3 of 17 The mesomechanica unit ce of a QNCF is shown in Figure 1a. The modes incude deviations NCF, of App. fiber Sci. disturbance orientations 2016, 6, 267 caused by stitch in fiber direction is different in different pies. In in inner and outer pies, such as cracks and channes. For continuous 3 of 17 outer pies fibrous pies, stitching pus fibers aside forming ong channes in 0 amina and rhomboida NCF, disturbance caused by stitch in fiber direction is different in different pies. In outer cracks NCF, in 45 disturbance amina. In caused inner, by rhomboida stitch in fiber cracks direction are is different induced in in different fibrous pies, stitching pus fibers aside forming ong channes in 0 90 and pies. 45 In amina, outer and fibrous pies, amina and rhomboida size of se cracks in 45 inner stitching cracks pus is smaer fibers aside than forming that of ong outer channes cracks. The in direction 0 amina amina. In inner, in 90 of and rhomboida arge cracks in 45 amina. In inner, rhomboida cracks are induced in 90 and and amina, diagona amina, and and of rhomb and channe corresponds to fiber direction py. size of size se of se inner inner cracks cracks is smaer is smaer than than that that of of outer cracks. The direction of of arge arge diagona of rhomb of rhomb and and channe channe corresponds to to fiber fiber direction in in py. o 0 o 0 o 45 o 45 o 90 o o Figure 1. unit ce deveopment for singe quadriaxia non Figure The Thegeometrica mode mode of of mesomechanica mesomechanica unit unit ce ce deveopment deveopment for singe for singe quadriaxia quadriaxia noncrimp fabric non-crimp amina: a) unit ces of QNCF ayer; b) individua fibrous pies in QNCF fabric QNCF) QNCF) amina: amina: a) unit a) ces unit of ces a QNCF of a QNCF ayer; b) ayer; individua b) individua fibrous fibrous pies in pies QNCF in QNCF ayer; and c) geometrica mode of unit ces of individua QNCF amina. ayer; and ayer; c) and geometrica c) geometrica mode of mode unit of ces unit of ces individua of individua QNCF QNCF amina. amina. The different openings in individua pies in QNCF composites wi ead to a difference in The mechanica different properties openings of singe NCF individua amina. In pies property in inqncf testing composites of NCF composites, wi wi ead ead different to to a difference a noncrimped properties UD specimens of singe of are singe requires NCF NCF to amina. test amina. to determinate In property In property testing mechanica testing of NCF of properties NCF composites, composites, of singe different QNCF different non in in mechanica non-crimped amina. UD The specimens UDschematic specimens are diagrams requires are requires to of test production to to test determinate to se determinate different mechanica warp knitted mechanica properties non crimped properties of singe of UD QNCF singe QNCF amina. specimens amina. The schematic are Theshown schematic in diagrams Figure diagrams 2, and of production of ange between se se different fibers and warp knitted warp-knitted process direction non crimped non-crimped are 90, UD specimens 0 and are 45, shown respectivey. in Figure Remarkaby, 2, and ange fiber distortions between of fibers UD specimens and process induced direction by stitch are 90,, 0 and yarn 45, are, respectivey. different. Geometrica Remarkaby, mode of fiber unit ces distortions of se of different UD specimens, specimens are induced shown in by Figure stitch 3. These UD specimens aso incude different gaps and channes. After injection moding, yarn are different. Geometrica mode of unit ces of of se different specimens, are are shown in in Figure 3. composite aminates are cut out foowing in fiber direction for mechanica testing. In meso scae These 3. These UD UD specimens specimens aso incude aso incude different different gaps and gaps channes. and channes. After injection After moding, injection moding, composite anaysis, unit ces of QNCF shoud aso incude meso scae geometry features of different aminates composite non crimped are aminates cut UD out specimens foowing are cut out as infoowing shown fiber in Figure direction in 3. fiber fordirection mechanica for testing. mechanica In meso-scae testing. In anaysis, meso scae unit anaysis, ces of QNCF unit ces shoud of QNCF aso incude shoud aso meso-scae incude geometry meso scae features geometry of different features non-crimped of different UD specimens non crimped as UD shown specimens in Figure as 3. shown in Figure 3. Figure 2. Schematic diagram of production different warp knitted unidirectiona non crimp fabric UD NCF): a) production of 90 Layers; b) production of 0 ayers; and c) production of 45 ayers. 1 Process direction; 2 Fiber direction; 3 Thread; 4 Transport system; 5 Weft Figure carriage 2. Schematic system. diagram of production different warp-knitted warp knitted unidirectiona non-crimp non crimp fabric UD-NCF): UD NCF): a) productionof of90 Layers; b) b) productionof of 0 0ayers; and and c) c) production of of 45 45ayers. ayers. 1 Process direction; 2 Fiber direction; 3 Thread; 4 Transport system; 5 Weft carriage system.

4 App. Sci. 2016, 6, of 17 App. Sci. 2016, 6, of 17 App. Sci. 2016, 6, of 17 Figure Figure 3. Meso-scae 3. Meso scae geometrica geometrica mode mode of unidirectiona of UD) UD) specimens specimens with with different different ange ange between fiber direction between fiber and direction process direction: and process a) direction: 90 ; b) 0a) ; and 90 ; c) b) 45 0 ;. and 1 Process c) Process direction; 2 Fiber direction; direction; 2 Fiber direction; 3 Face channe; 4 Inner crack; 5 Stitching yarn; 6 Face crack. 3 Face Figure channe; 3. Meso scae 4 Inner geometrica crack; 5 Stitching mode of yarn; unidirectiona 6 Face crack. UD) specimens with different ange between To improve fiber direction efficiency and of process meso scae direction: anaysis, a) 90 ; b) QNCF 0 ; and ayer c) 45. is decomposed 1 Process direction; into individua 2 To pies improve Fiber as shown direction; in efficiency Figure 3 Face 1b. channe; of Considering meso-scae 4 Inner that crack; anaysis, 5 Stitching testing method QNCF yarn; 6 Face ayer of mechanica iscrack. decomposed properties into of individua singe pies QNCF as shown amina, Figure unit 1b. ces Considering are seect aong thatfiber testing direction method as shown of mechanica in Figure 1b. properties Since inter 45 of singe To improve efficiency of meso scae anaysis, QNCF ayer is decomposed into individua QNCF and amina, 90 amina unit have ces same are seect voume aong fraction fiberand direction crack size, as shown y are incassified Figure 1b. as Since one mode. inter Then 45 and 90 pies as shown in Figure 1b. Considering that testing method of mechanica properties of singe amina three types haveof unit same ces voume of individua fractionqncf and crack amina size, are y formed, are cassified which are asshown one mode. in Figure Then 1c. three QNCF amina, unit ces are seect aong fiber direction as shown in Figure 1b. Since inter 45 types Athough this modeing strategy decomposes QNCF ayer, major fiber architecture is sti and of unit 90 amina ces ofhave individua same QNCF voume amina fraction areand formed, crack size, which y are are shown cassified in Figure as one 1c. mode. Athough Then this considered in mode, incuding channe, inter sma crack, face arge crack, and oca fiber contents. modeing three strategy types of unit decomposes ces of individua QNCF QNCF ayer, amina major are formed, fiber architecture which are is shown sti considered in Figure 1c. in mode, 2.2. Athough incuding Mutiscae this channe, Modeing modeing inter Procedure strategy sma of decomposes QNCF crack, Composites face arge QNCF crack, ayer, and oca major fiber fiber contents. architecture is sti considered in mode, incuding channe, inter sma crack, face arge crack, and oca fiber contents Mutiscae The workfow Modeingof Procedure mutiscae of QNCF anaysis Composites in this paper is given in Figure 4. On microscae, 2.2. mechanica Mutiscae properties Modeing of Procedure resin of and QNCF fiber Composites are input to micro scae anaysis mode, which are n The used workfow to determine ofmechanica mutiscaeproperties anaysis of in this composites paper istows. givena in periodic Figuremicrostructure 4. On microscae, threedimensiona properties workfow of mechanica The 3D) finite mutiscae eement resin anaysis FE) andmode fiber in are this is constructed, input paper to is given micro-scae in Figure with assumption anaysis 4. On mode, microscae, that fibers which are are n mechanica uniformy used to determine properties distributed mechanica of resin and in matrix, properties fiber are as shown in of input Figure to composites micro scae 4a. In this micromechanics tows. anaysis A mode, periodic which mode, microstructure are n fibermatrix interface 3D) is modeed finite eement as perfecty FE) bonded mode with is nodes constructed, merged with in a conventiona assumption mesh. that In fibers used to determine mechanica properties of composites tows. A periodic microstructure threedimensiona 3D) finite eement FE) mode is constructed, with assumption that fibers are three-dimensiona are uniformy faiure anaysis distributed process, in faiure matrix, state as of shown interface Figure is evauated 4a. Inby this faiure micromechanics of fiber and matrix mode, uniformy distributed in matrix, as shown in Figure 4a. In this micromechanics mode, fiber fiber-matrix that is adjacent interface to is modeed interface. as Simiar perfecty modeing bonded methods with nodes are expained mergedin inreference a conventiona [7]. Henceforth, interface is modeed as perfecty bonded with nodes merged in a conventiona mesh. mesh. In In term composite tows is refers straight fiber tows impregnated with resin. 1 direction and faiure faiure anaysis anaysis process, process, faiure faiure state state of of interface is isevauated by by faiure faiure of fiber of fiber and matrix and matrix ongitudina direction are direction aong axia of fiber, 2 direction and transverse that is that adjacent is adjacent to to interface. Simiar modeing methods are areexpained in reference in [7]. Henceforth, [7]. Henceforth, direction are direction transverse to axia of fiber, and 3 direction is in thickness term term direction composite of fabrics. tows tows is is refers straight fiber tows impregnated with with resin. resin. 1 direction 1-direction and and ongitudina direction are are direction aong axia of of fiber, 2 direction 2-direction and and transverse transverse direction direction are are direction direction transverse to to axia of fiber, and and 3 direction 3-direction is in is in thickness thickness direction of fabrics. direction of fabrics. E, E, G,etc. t t t X, X, S,etc. t t t T C 12 E, E, G,etc. t t t X, X, S,etc. t t t T C 12 E1, E2, G12,etc. X, X, S,etc. T C 13 E1, E2, G12,etc. X, X, S,etc. T C 13 Figure 4. The workfow of mutiscae anaysis: a) microscae; and b) mesoscae. Figure Figure 4. The 4. The workfow of of mutiscae anaysis: a) microscae; and and b) b) mesoscae.

5 App. Sci. 2016, 6, of 17 App. Sci. 2016, 6, of 17 On mesoscae, three unit ces are used to describe fiber distortion patterns in NCF pies Figure 4b). The in-pane in pane stiffness and strength of NCF amina are obtained by numerica anaysis. These 3D FE modes of singe NCF amina are deveoped based on previous geometries Figure 1c). The meso-scae meso scae modes are shown in Figure 5, where Ce A is unit ce with a arge crack, Ce B is is unit ce with a sma crack, Ce C is is unit ce with a channe, and b and are width and ength of crack, respectivey. In this mechanica mode, oca variation of fibers orientation Area with distorted fiber direction) is is aso shown in in Figure The oca variation of fiber orientations can be ocaized near crack in py. It It is is assumed that width of area with distorted fiber direction has same size as sma crack, which is is shown in Figure 5b. Based on experiment data [4], vaue of arge crack width is is mm and ength is 7.20 mm, and vaue of sma crack width is is mm and ength is is 2.64 mm, and fiber in-pane in pane crimp ange θ = 44.. The fiber crimp ange can be modeed by rotation of of materia orientation according to to actua distorted fiber direction [11]. Figure 5. Finite eement FE) mode of meso unit ces with different fiber distortion in individua non-crimp non crimp fabric NCF) amina: a) Ce A; b) Ce B; and c) Ce C. The The voume fraction of compositetows tows is isaso aso oca ocafiber fibervoume voumefraction) is is cacuated cacuated in in Reference Reference [4]. [4]. The The averaged averaged fiber fiber voume voume fraction fractionof of amina Vf of f and voume fraction of t composite tows V t are are reated reated according to to Equation 1). 1). f f t A V Vf = Vf = V t f f t = A V f A t = A B A V f 1) A A B S Void where A is cross-section cross section of amina, A tt is cross-section cross section of tow, and SSVoid is area of a crack or a channe per one knitting neede. A, B is space of stitching oops, A is perpendicuar to machine direction, B is in machine direction see Figure 1), and assumingb B = = mm, mm, and anda A = = mm mmin in this this paper. 3. Boundary Conditions and Materia Modes 3.1. Boundary Conditions To obtain homogenized materia properties, it is necessary to appy norma and shear oads on micro and mesomechanica unit ces. A simpified periodic boundary condition is appied on se unit ces [7], as shown in Figure 6.

6 App. Sci. 2016, 6, of Boundary Conditions and Materia Modes 3.1. Boundary Conditions To obtain homogenized materia properties, it is necessary to appy norma and shear oads on micro and mesomechanica unit ces. A simpified periodic boundary condition is appied on se unit App. Sci. ces 2016, [7], 6, as 267 shown in Figure 6. 6 of 17 x x Figure 6. Two oad cases for unit ces: a) norma oad: tension in x direction; and b) shear oad: shear Figure 6. Two oad cases for unit ces: a) norma oad: tension in x direction; and b) shear oad: shear in x-y pane. in x y pane. The dispacement oad δ x is x is appied appied on on panes panes x x = aa1 1 Figure 6a), and corresponding boundary conditions can be described as boundary conditions can be described as u0, y, 0, z) y, z) = = 0, 0, ua ua 1, y,, yz), z) = δ 1 x vx, vx 0,,0, z) z) = = 0 wx, wxy,, y 0),0 = ) = 00 2) 2) vx, a vx 2, z) = va, a, z) = va 1, a, 2, a 3 ) = δ δ y = const = const y wx, y, a 3 ) = wa 1, a 2, a 3 ) = δ z = const wx, y, a ) = wa, a, a ) = δ = const z Simpe shear dispacements are appied in x-direction Figure 6b), and boundary conditions is Simpe shear dispacements are appied in x direction Figure 6b), and boundary conditions as foows, is as foows, ux, 0, z) = 0 ux, ux a,0, 2, z)=0= δ x vx, uxa 0,, z), z= )=δ vx, a 2 x 2, z) = 0 3) u0, vx y,,0, z) z)= = ua vxa 1,, y,, zz) ) = 0 2 vx, 3) u0, 0, y z), z) = vx, u a, a y 2, z z) ) 1 wx, y, 0) = wx, y, a vx,0, z)= vxa,, z) 3 ) = 0 2 The boundary conditions of orwxy direction,,0)= wxya are, simiar, )=0 to Equations 2) and 3). In most finite 3 eement anaysis FEA) commercia packages, boundary conditions can be enforced using couping constraint The boundary equations. conditions In addition, of or boundary direction condition are simiar used to in Equations stiffness prediction 2) and 3). process In most wi finite be adjusted, eement anaysis which wi FEA) be introduced commercia in packages, Section 3.2. boundary conditions can be enforced using couping constraint equations. In addition, boundary condition used in stiffness prediction 3.2. process The Stiffness wi be adjusted, Cacuation which Method wi Based be introduced on Averagein Stress Section and Strain 3.2. To evauate stiffness properties of a heterogeneous materia, it is necessary to cacuate 3.2. The Stiffness Cacuation Method Based on Average Stress and Strain average stress and strain over unit ces. The constitutive reation of average stress and strain of homogeneous To evauate composite stiffness materia properties [19] of isa shown heterogeneous in Equation materia, 4). it is necessary to cacuate average stress and strain over unit ces. The constitutive reation of average stress and strain of homogeneous composite materia [19] σ α is = shown C αβ εin β Equation 4). 4) where α, β = 1,..., 6. C αβ is stiffness tensor. where αβ=1,,..., 6. C αβ is stiffness tensor. For an orthotropic materia, tensor σ α = αβεβ C 4) C αβ can be written in form: C αβ c11 c12 c c21 c22 c c 31 c32 c = c )

7 App. Sci. 2016, 6, of 17 For an orthotropic materia, tensor C αβ can be written in form: C αβ = c 11 c 12 c c 21 c 22 c c 31 c 32 c c c c 66 5) Then, reations between effective engineering constant of composites and stiffness tensor C αβ can be written as E 1 = C 11 2C 12 C 21 / C 22 + C 23 ) E 2 = [C 11 C 22 + C 23 ) 2C 12 C 21 ]C 22 C 23 )/C 11 C 22 C 12 C 21 ) G 12 = C 66, G 23 = C 44 v 12 = C 12 / C 22 + C 23 ) v 23 = C 11 C 23 C 12 C 21 )/C 11 C 22 C 12 C 21 ) 6) where E 1 and E 2 are ongitudina and transverse Young s moduus. v 12 and v 23 are Poisson s ratios. G 12 is in-pane shear moduus. To obtain components of stiffness tensor C αβ, six components of strain ε 0 ij, i, j = 1,..., 3 are appied by enforcing foowing boundary conditions on dispacement components as shown in Figure 6. u i a 1, y, z) u i 0, y, z) = a 1 ε 0 i1 u i x, a 2, z) u i x, 0, z) = a 2 ε 0 i1 u i x, y, a 3 ) u i x, y, 0) = a 3 ε 0 i3 7) The boundary conditions Equation 7)) means that a j ε 0 ij is dispacement required to enforce a strain of ε 0 ij over a distance a j. Using Equation 7), a surface strain ε 0 β can be appied on unit ce. The reationship between ε 0 β and ε0 ij [19] can be written as ε β = ε ij = ε ji 8) where β = i, if i = j ese β = 9 i j. The voume average strain ε β in unit ces equas to appied surface strain ε 0 β, ε β = 1 ε 0 β V dv = 1 ε 0 β V dv + ε 0 β dv = ε 0 β 9) V Vf V m where V is voume of unit ces, V f is voume of fiber, and V m is voume of matrix. The corresponding voume average stress σ α is described as: where σ α is stress fied in unit ces. σ α = 1 σ α dv = 1 σ α dv + σ α dv 10) V V Vf V m V

8 App. Sci. 2016, 6, of 17 For discrete finite eements, Equation 10) can be written by σ α = n k1=1 = n+m k=1 σ k1 + σ k / i+j k=1 m k2=1 σ k2 )/ n k1=1 V k1 + m k2=1 V k2 ) V k 11) where n and m are number of eement representing fiber and matrix in FE mode, respectivey, σ k is stress at eement integration point, and V k is voume of a singe eement. The numerica homogenization method in this section can be carried out using commercia finite eement software with scripting anguage, such as Abaqus and Python statements [19]. In Abaqus, to determine components C i1, with i = 1, 2, 3, in x-direction, symmetry boundary conditions are appied on panes of x = 0, y = 0, z = 0. A uniform dispacement is appied on pane x = a 1. The y and z direction boundary are simiarity to x-direction except for oads appied on respective surfaces. To determine shear moduus, simpe shear oads are appied in three principa panes. For postprocessing, a Python script is created to extract stress of integration point. The stiffness tensor C αβ is cacuation by Equation 4) based on average stress and strain, and n effective engineering constant can be obtained by Equation 6) Mutiscae Faiure Anaysis Faiure and Softening Formuation for Fiber In this paper, fiber is treated as a transversey isotropic materia in micromechanica mode, thus some faiure criteria for aminated composites can be appied to fiber faiure anaysis. A widey used poynomia faiure criterion for composite materias proposed by Tsai and Wu [20] is used. The criterion can be expressed as: F i σ i + F ij σ i σ j + F ijk σ i σ j σ k 1 i, j, k = 1,..., 6 12) where σ i, σ j, and σ k are stress components. F i, F ij, and F ijk are components of amina strength tensors in principa materia axes. The third-order tensor F ijk is usuay ignored from a practica standpoint due to arge number of materia constants required. Then, genera poynomia criterion can be reduced to a genera quadratic criterion given by F 1 σ 1 + F 2 σ 2 + F 3 σ 3 + 2F 12 σ 1 σ 2 + 2F 13 σ 1 σ 3 + 2F 23 σ 2 σ 3 + F 11 σ 1 2 +F 22 σ F 33 σ F 44 σ F 55 σ F 66 σ ) For mechanica properties of dry fibers, fiber manufacturer ony provides tension and compressive strength parameters; refore, Equations 12) and 13) are modified as F 1 σ 1 + F 2 σ 2 + F 3 σ 3 + 2F 12 σ 1 σ 2 + 2F 31 σ 3 σ 1 + 2F 23 σ 2 σ 3 + F 11 σ 1 2 +F 22 σ F 33 σ ) where tensors in Equations 13) and 14) can be determined as foows. F 1 = X 1 C X 1 T, F 2 = Y 1 C Y 1 T, F 3 = Z 1 C Z 1 T F 11 = X 1 T X C, F 22 = Y 1 T Y C, F 33 = Z 1 T Z C 15) 1 F 12 = 2 1, F X T X C Y T Y 23 = C 2 1, F Y T Y C Z T Z 31 = C 2 Z T Z C X T X C where X T, Y T, and Z T are tensie strength in 1-direction, 2-direction, and 3-direction, respectivey. X C, Y C, and Z C are compression strength in 1-direction, 2-direction, and 3-direction, respectivey.

9 App. Sci. 2016, 6, of 17 The modified Tsai-Wu Criterion is an expression that ony considers tensie and compressive strength, and we use criterion to determine initia faiure of fiber. Post-initia faiure in fiber direction is modeed by a gradua unoading mode, where one or more of eastic materia properties of a amina are set to zero or a sma fraction of origina vaue once faiure is detected. The degradation equations are given in Reference [21]. Furrmore, in finite eement modes, materia softening aws with Tsai Wu faiure criteria have been impemented using user defined subroutines USDFLD User subroutine to redefine fied variabes at a materia point) of Abaqus Version 6.11, Dassaut systemes simuia Corp, Providence, RI, USA, 2011). For a micromechanica anaysis, properties of fibers are essentia, materia data of 12K Toray T700 50C are used in anaysis of this paper as shown in Tabe 1. The eastic data are cited from Reference [13], tensie strength is obtained from data sheet of fiber manufacturer, compressive strength are cacuated with an empirica correction k = 0.8, and empirica correction is determined according to ratio of compressive strength and tensie strength of T300 fiber, data are summarized in Word-Wide Faiure Exercise [22]. Tabe 1. The mechanica properties of fiber carbon. Parameter Vaue Tensie moduus, GPa E f1 = 230, E f2 = 28 Poisson s ratio v f12 = 0.23 Shear moduus, Gpa G f12 = 50 Tensie strength, Mpa X ft, Y ft, Z ft = 4900 Compressive strength, Mpa X fc, Y fc, Z fc = Eastic-Pastic Materia Mode for Epoxy Resin This eastic-pastic materia mode is intended to describe mechanica performance of pure resin, such as, matrix in micromechanica mode and resin-rich zone in mesomechanica mode. The eastic properties of resin Epoxy resin Epikote 828) are obtained from Reference [13]. The pastic deformation and faiure data of Epikote 828 are obtained from References [23,24]. In summary, matrix properties utiized in this paper are presented in Tabe 2. Tabe 2. Mechanica properties of Matrix. Tensie Moduus Poisson s Ratio Tensie Faiure Compressive Yied σ = 85.25MPa σ = 133MPa E m = 2.73 GPa v m = 0.4 ε = 3.62% ε = 6.5% To account different yieding behavior under uniaxia tension, uniaxia compression and shear, von Mises criterion and Drucker-Prager yied criterion are chosen. The von Mises are used to define yied and ineastic fow behavior of a meta at reativey ow temperatures. In this paper, this criterion is appied to describe tensie behaviors of epoxy resin. The Drucker-Prager criterion is appied to mode faiure behavior of pure matrix materias under compression and shear oad. Markund [2] has proved that using this criterion to predict faiure of matrix materias is feasibiity, and two criteria are readiy avaiabe in Abaqus Faiure Mode for Composites Tows A 3D progressive faiure mode [21] is used to predict fina faiure of composites tows. This faiure mode connects materia eastic properties with interna state variabes that functions of type of damage. Before oca structura faiures deveop, composite tows typicay behave are considered as inear eastic manner, and constitutive reations for undamaged are given by Equation 4).

10 App. Sci. 2016, 6, of 17 To detect onset 3D faiure incuding fiber direction faiure and transverse direction faiure, modified Hashin faiure criterion is used. In each direction, tensie and compressive faiures are handed separatey. The faiure modes are modified for case of 3D Stress as foow. Fiber faiure mode [25], σ 11 > 0 f ft = σ11 X T + τ12 S 12 + τ13 S 13 ; f ft 1 16) Matrix faiure mode, σ 11 < 0 f fc = σ11 X C ; f fc 1 17) σ 22 > 0 f mt = σ 22 < 0 f mc = σ22 Y T σ22 Y C + + τ12 S 12 τ12 S τ23 S 23 τ23 S 23 ; f mt 1 18) ; f mc 1 19) where S 12, S 13 and S 23 are strength for shear in 1 2 pane, shear in 1 3 pane and shear in 2 3 pane, respectivey. τ 12, τ 13 and τ 23 are shear stress in 1 2 pane, shear in 1 3 pane and shear in 2 3 pane, respectivey. σ 11 and σ 22 are norma stress in 1-direction and 2-direction, respectivey. f ft and f fc, are faiure indices for fiber tension and fiber compression, respectivey. f mt is faiure indices for matrix tension or shear cracking. f mc is faiure indices for matrix compression or shear cracking. The materia stiffness changes, after oca faiures within tows. The effects of damage on stiffness of tows are represented using interna state variabes. These state variabes associated with crack density under oading, a more detaied description is given in [21]. The progressive faiure mode is impemented as a user-defined materia mode using Abaqus user interface UMAT user subroutine to define a materia s mechanica behavior). In this procedure, a noninear anaysis is performed unti a converged soution is obtained. 4. Resuts and Discussion The stiffness and strength of QNCF amina are obtained using meso-scae anaysis procedure. The numerica predictions of QNCF amina are compared with experimenta resuts of non-crimped UD aminate in Reference [13]. In experiment, mechanica properties of non-crimped UD specimens with warp-knitted was reported, however, that is just one types of UD specimens mentioned in this paper. In this section, E1 t, Et 2, Et 3, Gt 12, Gt 13, vt 12, and vt 13 are eastic properties of composite tows for tensie in 1-direction, tensie in 2-direction, tensie in 3-direction, shear in 1 2 pane, shear in 1 3 pane, Poisson s ratio in 1 2 pane and Poisson s ratio in 1 3 pane, respectivey. XT t, Xt C, Yt T, Yt C, and St 12 are strength of composite tows for tension in 1-direction, compression in 1-direction, tension in 2-direction, compression in 2-direction and shear in 1 2 pane, respectivey. E1 is ongitudina Young s moduus of QNCF amina, and E2 is transverse Young s moduus of QNCF amina, and v 12 is Poisson s ratio in 1 2 pane of QNCF amina and G 12 in in-pane shear moduus of QNCF amina. XT, X C, Y T, Y C, and S 12 are strength of QNCF amina for tension in 1-direction, compression in 1-direction, tension in 2-direction, compression in 2-direction and shear in 1 2 pane, respectivey Mechanica Properties of UD Composite Tows In QNCF amina, area except for resin-rich zones can be considered as UD composites, and Vf t of outer and inner pies are different. The QNCF fabric aminate to be anayzed by mechanica modes with Vf = 42.1%. This fiber voume fraction is experimenta measurements vaues in Reference [13]. In this case, according to Equation 1), Vf t is shown in Tabe 3. As diameter of

11 App. Sci. 2016, 6, of 17 inner stitching yarn for QNCF ayer is sma, voume of resin-rich zones is about 3.34% of tota voume of NCF amina, which contributes to Vf t in inner 45 and 90 amina cose to Vf as shown in Tabe 3. In addition, stitching yarn can cause obvious fiber cracking in outer 45 and 90, which makes Vf t in outer ayer higher than that in inner ayer. Tabe 3. Vf t of each amina with V f = 42.1%. Py V t f 48.1% 43% 43% 47.3% The mechanica properties of composite tows obtained with micro-scae anaysis procedure are shown in Tabe 4. E1 t, Et 2, Gt 12, vt 12, and vt 23 are cacuated by numerica homogenization method, and E3 t and Gt 23 are determined by transversey isotropic materia assumption. The stiffness properties of UD tows estimated by an averaging technique based on rue of mixtures RM) [26] is aso given in Tabe 4. This anaytica mode can cacuate vaue of E1 t and vt 12, but has a ess accurate resut for G12 t and Et 2 because of assumption of rectange section for fiber. Therefore, anaytica vaues of E1 t, vt 12 are used in this section. According to Tabe 4, it is cear that resuts obtained using numerica modes are consistent with those obtained using RM. Among unit ces, composite tows in Ce A has highest fiber voume fraction Vf t = 48.1%), correspondingy, ongitudina Young s moduus of tows is aso higher than or two cases. This is because ongitudina Young s moduus of composites is a fiber-dominated property. For numerica predictions of E2 t and Gt 12, tows with high oca fiber voume fraction aso has a arger transverse and shear moduus compared with ow fiber voume fractions. This can be interpreted that athough transverse and shear moduus of UD composite materias are matrix-dominated, fiber voume fraction, ratio of matrix property and fiber property aso affects moduus simutaneousy [26]. Tabe 4. The engineering constants of composite tows with different fiber voume fraction obtained using micromechanica mode/gpa. Mode V t f E t E t 2 = Et 3 v t 12 v t 13 G t 12 = Gt 13 G t 23 Ref-RM 42.1% Ref-FEM 42.1% In Ce A 48.1% In Ce B 43% In Ce C 47.3% Ref-RM is reference vaue obtained with rue of mixtures, and Ref-FEM is reference vaue obtained by micromehanica mode for UD un-stitched composites, and In Ce A, In Ce B and In Ce C mean that composite tows in Ce A, Ce B and Ce C, respectivey. The numerica predicted strength properties of composite tows with different fiber voume fractions are shown in Tabe 5. The anaytica vaues of XT t and Xt C based on RM are aso presented for a comparison. The anaytica vaues are obtained by Equations 20) and 21). The existing simpe anaytica modes cannot accuratey predict transverse strength and shear strength at micromechanica eve. Therefore, anaytica resuts of YT t, Yt C, and St 12 are not presented here. X t T = X ft X t C = X fc [ V t [ V t f + E m Vt m E f1 f + E m Vt m E f1 ] ] 20) 21)

12 App. Sci. 2016, 6, of 17 t is matrix voume fraction of composite tows. where Vm Compared with reference vaues Ref-RM), numerica predictions vaue Ref-FEM) XTt and t of composite tows are decreased by 14.5% and 27%, respectivey. Athough difference is obvious, XC considering that RM is ess accurate in prediction of strength than predicting of eastic App. Sci. 2016, 6, of 17 properties, this comparison just iustrates possibe strength vaue of this materia with different voume fraction. Therefore se strength properties of UD tows obtained by micromechanica mode micromechanica mode sti can be used for cacuation of meso scae mode, and this paper sti can be used for cacuation of meso-scae mode, and this paper compares numerica compares numerica predicted and experimenta vaues on mesoscae. predicted and experimenta vaues on mesoscae. Tabe 5. Strength of composite tows obtained using micromechanica mode/mpa. Tabe 5. Strength of composite tows obtained using micromechanica mode/mpa. Mode Mode Vtf Ref RM Ref-RM Ref FEM 42.1% Ref-FEM 42.1% A In Ce AIn Ce 48.1% B In Ce B In Ce43% In Ce CIn Ce 47.3% C Vft XtT X Tt 42.1% % % % % X Ct XtC YTt YtT YCt t S12 YtC St In Pane In-Pane Stiffness Stiffness of of QNCF QNCF Lamina Lamina According According to to inter structure inter-structure of of QNCF QNCF amina, amina, three three kinds kinds of of unit unit ces ceshave havebeen beenestabished. estabished. Certainy, predicted resuts with any of three unit ces cannot represent actua vaues. As Certainy, predicted resuts with any of three unit ces cannot represent actua vaues. method of taking data average is is commony used As method of taking data average commony usedinin data dataprocessing processingof ofcomposite composite stiffness stiffness test, test, averaged averaged vaue vaue of of predicted predicted resuts resuts empoying empoying three three unit unit ces ces is is taken taken as as stiffness stiffness properties of QNCF QNCF amina. amina. properties of The representative voume The dispacement dispacement oad oad is is appied appied on on surface surface of of three three kinds kinds of of representative voume three unit ces are shown in Figure 7. eements, and deformation and stress distribution of eements, and deformation and stress distribution of three unit ces are shown in Figure 7. Figure Deformed Deformed shape shape and and contour contour pot pot of of stress stress in different dispacement oads. Figure Gare v12 The estimated, andg12 are isted Tabe estimatedin pane in-paneeffective effectiveengineering engineeringconstants constantsee1 1,, E22,, v12, and isted inin Tabe The averaged stiffness of three unit ces and numerica homogenization resuts obtained by 6. The averaged stiffness of three unit ces and numerica homogenization resuts obtained micromechanicamode modefor forud UDcomposite composite tows aso presented in Tabe a reference. micromechanica tows areare aso presented in Tabe 6 as6 aasreference. The averaged stiffness obtained with three unit ces are in a good agreement with experimenta resuts. The eastic data of UD aminate test in Reference [13] are used in this are 0.9% and 6.3% higher than corresponding comparison. The average vaue of E1 and v12 experiment resuts, respectivey, and average vaue of G12 is 5.3% ower than test vaue.

13 App. Sci. 2016, 6, of 17 The averaged stiffness obtained with three unit ces are in a good agreement with experimenta resuts. The eastic data of UD aminate test in Reference [13] are used in this comparison. The average vaue of E1 and v 12 are 0.9% and 6.3% higher than corresponding experiment resuts, respectivey, and average vaue of G12 is 5.3% ower than test vaue. Considering that, procedures for predicting stiffness is done in inear eastic range of materia, and stiffness of constituent materias attributed to numerica cacuation are experimenta data, and reasonabe boundary conditions are used in numerica homogenization method, which make it possibe to get a reativey accurate resuts. Tabe 6. In-pane stiffness of NCF non-crimp fabric) amina obtained with mesomechanica anaysis procedure GPa). Description V f V t f E 1 E 2 v 12 G 12 Ce A 42.1% 48.1% Ce B 42.1% 43% Ce C 42.1% 47.3% Average 42.1% Ref-FEM 42.1% Experiment [13] 42.8% ± 0.8% ± ± ± 0.8 When we compare average vaue with reference vaue Ref-FEM), as shown in Tabe 6, E1 is reduced by 3.34%, but this difference is not significant. The maximum difference of E 1 is about 4.8% among three unit ces. To consider same averaged fiber voume fraction, this difference in eastic properties coud be caused by fiber distortion. This resut is consistent with view of Reference [13], which concuded that absence of a significant difference in stiffness between experimenta resuts and cassica aminate ory predictions. We confirm this concusion by mutiscae anaysis. Stitch yarn induces ocaized crack in fibrous py. If width of ocaized crack is seen as ampitude of fiber waviness, ampitude of a singe crack for arge crack, b = 0.48 mm) accounts for 9.5% of unit ce width A = 5.07 mm). With a constant fiber voume fraction, se ocaized in-pane cracks are such sma ampitude and waviness ange that effect on stiffness might be not obvious. Stitching has minor effect on in-pane stiffness of continuous pies, which is different from traditiona concusions. It is generay considered that stitching can reduce in-pane stiffness by 10% 20% [15,27], se contrary concusions may ony be appicabe for stiffness of open structure NCF composite. For E2 and G 12, average vaues are amost same as reference vaues Ref-FEM in Tabe 6): y are reduced by 1.7% and 1.3% compared to reference vaues, respectivey. The transverse and shear moduus of QNCF amina are separatey cose to UD. The possibe reason for this phenomenon is that stitch yarn induces fiber distortion, which occurs in direction of fiber tows, and transverse and shear moduus of composite materias are matrix-dominated. Moreover, considering that re is no change in matrix properties, difference between predictions empoying three unit ces and reference vaue is not obvious In-Pane Strength of QNCF Lamina The in-pane strength is predicted by meso-scae faiure mode. The progress faiure mode is introduced in Section that used to predict faiure of composite tows. The von Mises criterion and Drucker Prager yied criterion are introduced to predict faiure of resin pocket under different oads. Faiure anaysis is performed by Abaqus commercia FE code combined with user subroutine UMAT, axia dispacement oad and in-pane shear oad are appied on surface of unit ces, and n predictions of in-pane strength are isted in Tabe 7. The strength data of non-crimp UD aminate that tested in Reference [13] are used in this section as a comparison. Each of se three unit ces represents a QNCF amina, and predicted strength

14 App. Sci. 2016, 6, of 17 of each unit ce represents possibe vaues of QNCF amina. For se reasons, individua prediction of each unit ce and average vaue are both compared with experimenta resuts. Simiar to that described above, averaged vaue of predicted resuts is taken as strength properties of QNCF amina. App. Sci. Tabe 2016, 7. 6, In-pane 267 strength of non-crimp fabric NCF) amina evauated with mesomechanica faiure 14 of 17 anaysis procedure/mpa. t Description V V f f X X Y T C T Y S C Description Ce A V f 42.1% V t f 48.1% X T X C Y66.7 T Y C 69.4 S 12 Ce A Ce B 42.1% 42.1% 48.1% 43% Ce B Ce C 42.1% 42.1% 43% 47.3% Ce C Average 42.1% 42.1% 47.3% Average 42.1% Ref-FEM Ref FEM 42.1% 42.1% Experiment Experiment [13] 42.8% ± 0.8% % ± 0.8% [13] Compared with experimenta resuts in reference [13], average vaue of X obtained T using Compared above with three unit experimenta ces is 11.4% resuts higher in than reference experimenta [13], average resuts. vaue Figure of X 8 T shows obtained using fiber tension above faiure three unit modes ces in is 11.4% unit higher ces and than Figure experimenta 9a presents resuts. stress strain Figure 8 shows response fiber of tension unit faiure ces subjected modes in to uni axia unit ces and tension Figure parae 9a presents to fiber. stress-strain As shown response in Figure of 9a, unit ces numericay subjected to determined uni-axia tension curves parae agree we to fiber. with As shown experiment, in Figure athough 9a, numericay a sight overestimate determined of curves tensie agree we strength with is observed. experiment, athough a sight overestimate of tensie strength is observed. Figure 8. The fiber fracture FF) faiure modes in unit ces under ongitudina tensie. Figure Typica Typica stress-strain stress strain curve curve of micromechanica of micromechanica non-crimp non crimp fabric NCF) fabric unit ces NCF) computations: unit ces a) computations: ongitudina a) tensie ongitudina oads in tensie 1-direction; oads and in 1 direction; b) shear in and 1 2 b) pane. shear in 1 2 pane. Athough fiber reinforced composite materias are often considered as britte materias, certain Athough fiber reinforced composite materias are often considered as britte materias, a certain non inear behavior is observed in Figure 9a. This non inear behavior is caused by materia stiffness non-inear behavior is observed in Figure 9a. This non-inear behavior is caused by materia stiffness degradation after oca faiures within composite tow. In fact, test data in Reference [13] aso degradation after oca faiures within composite tow. In fact, test data in Reference [13] aso have stiffness reduction. For non crimp UD aminate in test, initia Young s moduus is 94.3 have a stiffness reduction. For non-crimp UD aminate in test, initia Young s moduus is GPa, and Young s moduus is 82.8 GPa in end of test. It can be concuded that stiffness of 94.3 GPa, and Young s moduus is 82.8 GPa in end of test. It can be concuded that stiffness of aminate is reduced by 12.2% at utimate faiure. aminate is reduced by 12.2% at utimate faiure. As shown in Figure 9a, curves of Ce A and Ce B have more obvious stiffness reduction compared with that of Ce C. This resut can be expained by rhomboida cracks in aminas which are more ikey to cause stress concentration compared with channes in 0 amina. The QNCF ayer discussed in this work is composited with four aminas, and three aminas are provided with cracks, so amina with cracks is more representative of actua stress state.

15 App. Sci. 2016, 6, of 17 As shown in Figure 9a, curves of Ce A and Ce B have more obvious stiffness reduction compared with that of Ce C. This resut can be expained by rhomboida cracks in aminas which are more ikey to cause stress concentration compared with channes in 0 amina. The QNCF ayer discussed in this work is composited with four aminas, and three aminas are provided with cracks, so amina with cracks is more representative of actua stress state. Compared with Ce C, Ce A and Ce B are unit ces with cracks, thus numerica predictions empoying se two kinds of unit ces are coser to experiment vaues. From Tabe 7, YT is 15.3% higher than test vaue, and average shear strength is cose to test vaue. For composite tows under shear oading, noninear behavior of materia shoud be considered in faiure anaysis. Generay, this noninear behavior is caused by micro crack and pasticity of matrix. According to faiure mode above, interna state variabes of materia degradation are used to anayze combined infuence of two factors. The engineering anaysis method is aso used in Reference [21] to predict matrix shear cracking. Based on this method, stress-strain response of unit ces subjected to simpe shear oad are shown in Figure 9b. The stiffness decreased about ε = 1.8% due to oca damage in matrix according to Figure 9b. After eastic range, noninear curves are obtained. With progressive damage method, reasonabe shear behaviors are obtained. However, since mechanica properties of fiber/matrix interface are modeed perfecty in a conventiona mesh, a high transverse strength vaue is predicted. Considering dispersion parameters of strength of composites and reference experiment is ony for one type of non-crimped UD specimen, refore, three unit ces given here can predict in-pane strength of QNCF amina with sufficient engineering accuracy. As can be seen in Tabe 7, in fiber direction, ongitudina tensie strength X T obtained by empoying Ce B with fiber waviness is 19.8% ower than Ce C, and average vaue of X T is 23.4% ower than reference vaue Ref-FEM). For ongitudina compressive strength X C, maximum difference among three unit ces reaches to 28.0%. Generay, differences between mechanica properties obtained using ces with fiber waviness and ce with no waviness are quite obvious. The average vaue can sti be reduced by approximatey 20% reative to reference vaue. Considering same voume friction of three unit ces, this difference coud be caused by oca stress concentration, which is a resut of fiber waviness induced by stitch yarn in this materia. This difference between three unit ces or difference between average vaue and reference vaue is obvious. Moreover, this may expain resuts of Bibo [14] who found that tensie strength of NCF composite is 34.7% ower than UD composites and compress strength is 40% ower than UD, which is caused by effect of stitching in materias. The difference of YT or Y C between three unit ces and corresponding reference vaue Ref-FEM) is not obvious, S 12 is aso a simiar trendy. This indicates that fiber disturbance has no significant effect on strength in direction transverse to fibers. According to faiure mechanism, transverse strength and shear strength are aso matrix-dominated, and it has itte reation with strength of fiber itsef. Therefore, changes of transverse and shear strength of NCF amina caused by fiber disturbance are insensitivity. 5. Concusions From view of mechanica properties of singe non-crimp fabric NCF) amina, in-pane stiffness and strength of quadriaxia non-crimp fabric QNCF) composites and effect of in-pane fiber distortion on mechanica properties are estimated. A new modeing strategy for meso-scae features of QNCF is presented. The idea of this modeing approach is derives from testing method of mechanica properties of singe NCF amina. According to this idea, compex inter-structure of QNCF is decomposed into some individua py, and unit ces aong fiber direction are seected. This simpified engineering modeing approach can improve efficiency of modeing in mutiscae anaysis of QNCF. Furrmore, it can be used to create an equivaent continuum mode for continuous pies NCF at macroscae eve, such as engineering modes based on semi-aminar consideration.

16 App. Sci. 2016, 6, of 17 In accordance with this modeing strategy, stiffness and strength of composites are evauated on meso-scae, and foowing resuts are obtained. 1) The modeing strategy based on mechanica properties of singe NCF amina can be used to evauate in-pane stiffness and strength of QNCF composite amina. The E1, G 12, and S 12 are in agreement with experimenta resuts; v 12 is 5.9% higher than experimenta resuts; and X T, Y T are 11.4%, 15.3% higher than test vaue, respectivey. 2) The effects of in-pane fiber distortion, induced by stitch yarn on ongitudina eastic moduus, is not significant, and moduus of QNCF amina has a difference of 3.34% compared with same type UD un-stitched composites. This concusion on stiffness of QNCF composites is different from open structure NCF composite in which stitching may reduce in-pane eastic properties by 10% 20%. In addition, stitching induces an assignabe effect on ongitudina strength of QNCF amina, and has ony sight effect on transverse stiffness and transverse strength of materias. Acknowedgments: The first author is thankfu to Fuqiang Yang and Dongping Zhao for Engish revisons. Additionay, authors woud ike to thank three anonymous reviewers and associate editors for ir vauabe comments. Author Contributions: Y.C. conducted research and wrote paper; Y.F. and W.W. supervised research and heped in preparation of manuscript; D.W. and Z.Z. contributed data anaysis work and revision of this paper. Conficts of Interest: The authors decare no confict of interest. References 1. Lomov, S.V. Understanding and modeing effect of stitching on geometry of non-crimp fabrics. In Non-Crimp Fabric Composites: Manufacturing, Properties and Appications; Lomov, S.V., Ed.; Woodhead: Cambridge, UK, 2011; pp Markund, E.; Asp, L.E.; Osson, R. Transverse strength of unidirectiona non-crimp fabric composites: Mutiscae modeing. Compos. Part B Eng. 2014, 65, [CrossRef] 3. Lomov, S.V.; Verpoest, I.; Robitaie, F. Manufacturing and interna geometry of texties. In Design and Manufacture of Textie Composites; Long, A.C., Ed.; Woodhead: Cambridge, UK, 2005; pp Lomov, S.V.; Beov, E.B.; Bischoff, T.B.; Ghosh, S.B.; Truong Chi, T.; Verpoest, I. Carbon composites based on mutiaxia mutipy stitched preforms. Part 1. Geometry of preform. Compos. Part A App. Sci. Manuf. 2002, 33, [CrossRef] 5. Reddy, J.N. Mechanics of Laminated Composite Pates and Shes: Theory and Anaysis, 2nd ed.; CRC: Boca Raton, FL, USA, 2004; pp Markund, E.; Varna, J.; Asp, L.E. Modeing stiffness and strength of non-crimp fabric composites: Semi-aminar anaysis. In Non-Crimp Fabric Composites: Manufacturing, Properties and Appications; Lomov, S.V., Ed.; Woodhead: Cambridge, UK, 2011; pp Ernst, G.; Voger, M.; Huhne, C.; Rofes, R. Mutiscae progressive faiure anaysis of textie composites. Compos. Sci. Techno. 2010, 70, [CrossRef] 8. Zrida, H.; Markund, E.; Ayadi, Z.; Varna, J. Master curve approach to axia stiffness cacuation for non-crimp fabric biaxia composites with out-of-pane waviness. Compos. Part B Eng. 2014, 64, [CrossRef] 9. Edgren, F.; Asp, L.E. Approximate anaytica constitutive mode for non-crimp fabric composites. Compos. Part A App. Sci. Manuf. 2005, 36, [CrossRef] 10. Gonzaez, A.; Graciani, E.; Federico, P. Prediction of in-pane stiffness properties of non-crimp fabric aminates by means of 3D finite eement anaysis. Compos. Sci. Techno. 2008, 68, [CrossRef] 11. Ferreira, L.M.; Graciani, E.; Paris, F. Modeing waviness of fibres in non-crimp fabric composites using 3D finite eement modes with straight tows. Compos. Struct. 2014, 107, [CrossRef] 12. Zrida, H.; Markund, E.; Ayadi, Z.; Varna, J. Effective stiffness of curved 0 -ayers for stiffness determination of cross-py non-crimp fabric composites. J. Reinf. Past. Compos. 2014, 33, [CrossRef]

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