Science and Technology on Scramjet Laboratory, National University of Defense Technology, Hunan, Changsha , China.

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1 st Internatonal Conference on Composte Materals X an, 0-5 th August 07 EFFECT OF FIBER VOLUME FRACTION ON THE VARIATION OF STOCHASTIC ELASTIC CONSTANTS OF PLAIN-WEAVE COMPOSITE Hao Wang, Xu-we Yu and Zhong-we Wang 3 Scence and Technology on Scramjet Laboratory, Natonal Unversty of Defense Technology, Hunan, Changsha 40073, Chna. E-mal: gfkdwh@63.com Scence and Technology on Scramjet Laboratory, Natonal Unversty of Defense Technology, Hunan, Changsha 40073, Chna. E-mal: yuxuwe3@nudt.edu.cn 3 Scence and Technology on Scramjet Laboratory, Natonal Unversty of Defense Technology, Hunan, Changsha 40073, Chna. E-mal: wang_zhwe00@63.com Keywords: Textle composte, Fber volume fracton, Mechancal propertes, Computatonal modelng ABSTRACT Fber volume fracton of yarn has obvous nfluence on the elastc constants of fber yarn. In order to nvestgate the effect of fber volume fracton of yarn on elastc constants of plan-weave composte, a new method to calculate the elastc constants of fber yarn s establshed based on Local Stochastc Varable Metrc Bass. A real yarn n the fabrc composte has stochastc wavness, scalng and twst varatons. Dfferent values of fber volume fracton are appled nto three cases wth the new method, and the effects of fber volume fracton of yarn on elastc constants of plan-weave compostes n three cases are acqured, respectvely. As the fber volume fracton of yarn ncreases, the mpact bounds of elastc modul of plan-weave compostes decrease whle the mpact bounds of shear modul ncrease. The varaton trends of the mpact bounds of Posson s ratos are dfferent n the three cases. INTRODUCTION Because of the hgh strength ratos and stffness ratos, fber renforced compostes are wdely used n the area of aeronautcs and astronautcs, whch decreases the weght of structures. Compare wth the lamnates, fabrc compostes have hgher strength and stffness and are replacng the use of lamnates. Durng the producng process of compostes, aleatorc uncertantes always exts n the mcrostructures of compostes, whch results n the scatter of mechancal propertes of compostes[]. Many research teams have studed the methods to calculate the elastc constants of a real fber yarn n textle composte[, 3]. Gommer et al [4] used a method based on Hungary algorthm to reconstruct the path of fbers n yarn and ths method can utlze tradtonal optcal magng technology to analyss the dstrbuton of fber n the yarn. The study suggests that the fbers of rregular drectons n the yarn have mpact on the mechancal propertes (complance and stffness) of the yarn. For C/Fber renforced compostes, the predcton of strength has 5% dfference to the true value. Vanaerschot et al. [5-7] defned 6 parameters to fully descrbe each yarn path and cross-secton shape n compostes. Each parameter was decomposed nto systematc value and a zero-mean stochastc devaton. Wang and Wang [8-0] establshed a local varable metrc stochastc theory to quantfy the nfluence of stochastc feature parameters of yarn on elastc propertes of plan-weave compostes. Taylor expansons of complance of yarn were executed n global mean to estmate the local mean and standard devaton of elastc constants. Volumetrc averagng method was employed to obtan the elastc constants of plan-weave compostes. Fber yarns ext n the fabrc compostes, and the fber volume fracton (FVF) of yarn has obvous effect on the elastc constants of yarn and composte[]. Lee et al [] proposed a geometrc and elastc models based on the unt cell to predct the geometrc characterstcs and the elastc constants of plan-weave carbon fabrc renforced metal matrx compostes. Increase of FVF entals the hgher angle of yarn crmp, whch n turn ncreases off-axs angle of yarns, resultng n the reduced modul of woven compostes. Tsa et al [3] proposed a parallelogram sprng model to nvestgate the effects of fabrc parameters on the elastc modul of braded composte plates. The smulaton results of FEM show that all the Young s modul ncrease almost lnearly wth the ncrease of FVF.

2 H. Wang, X.-w. Yu and Z.-w. Wang The focus of ths paper s to study the effect of yarn FVF on elastc constants of plan-weave compostes. Frstly, local stochastc varable metrc theory s establshed to transform a real yarn nto a deal yarn, and estmate the elastc constants of real yarns. Then, the volumetrc averagng method s performed to acqure the elastc constants of plan-weave composte. At last, three values of yarn FVF are appled nto the estmaton of elastc constants of plan-weave compostes. The nfluence of yarn FVF on elastc constants of plan-weave compostes s acqured by comparng the smulaton results. FEATURE PARAMETERS OF YARN A global coordnate system wth orthogonal axes --3 s defned to descrbe the yarn of planweave compostes, where -axs s along the longtudnal drecton of yarn. A real yarn n the fabrc composte can be fully characterzed by 6 feature parameters, namely path vector r (x, x, x 3) and geometrc parameters (a, b, ) (see Fg. ). " Fg. The feature parameters of yarn (.e. path vector r,scalng matrx ˆ and twst angle ), and the global e, e, e 3, local e ' ' ', e, e 3 and varable metrc basc e " " ", e, e 3 [9] Ellpse s used to descrbe the cross-secton shape of yarn and the parameters a and b are the sem-major and sem-mnor axes of ellptc cross-secton. s the orentaton of yarn s cross-secton, whch descrbes the twst extent of yarn around the tangent vector of the yarn path. The path vector descrbes the yarn wavness and deflecton, whle the geometrc parameters represent the shape and orentaton of the yarn cross-secton. For arbtrary poston of yarn center path n Fg., r s the vector from orgnal pont to pont C n. C n r e () x where e are the bass of global system, and x,,3. A local coordnate system - -3 at arbtrary poston of yarn path s defned n Fg., where - axs s n the drecton of tangent vector of yarn path, - and 3 -axes are n the drecton of sem-major and sem-mnor axes of yarn cross-secton, respectvely. A varable metrc coordnate system - -3 overlapped wth the defned local coordnate system s defned, but the metrc between them s dfferent along each of the same axs drecton. Correspondngly, three trads of unt bases e, e, e, e, e, e and,, 3 created. 3 3 e e e correspondng to the global, local and varable metrc coordnate system are

3 st Internatonal Conference on Composte Materals X an, 0-5 th August 07 Fg. The orthogonal decomposton of path vector n and 3 planes. Obvously, the relatonshp between global and local coordnate system s expressed by a rotaton transformaton, thus the followng tensor expresson s obtaned. v v e () where e v, v j j j j j are the components of vector n global and local coordnate system, respectvely, and e e s rotaton transformaton matrx where symbol denotes scalar (dot) product. Because of dfferent metrcs between local and varable metrc coordnate system, the components of vector n the two coordnate systems are not the same. Let denotes the components of vector n varable metrc coordnate system. The transformaton relatonshp between the two coordnate systems s defned as below. v v v (3) j j j j j j where are second order symmetrc tensors, named scalng tensor and j 0 when j. Due to that s a dagonal matrx, j j. It s usually represented varable metrc transformaton Eq. (3) n entty form by v ˆ v v ˆ (4) Substtute Eq. (3) nto Eq. (), one can obtan the followng transformaton rules between global and varable metrc coordnate system. v ve v e v (5) j j k kj j k k v v ˆ eˆ v ˆ (6) where ˆ ˆ eˆ s varable metrc coordnate transformaton matrx. Each feature parameter can be decomposed as, x, x, a, b, (7) 3 v where s the systematc value and s zero-mean devaton from the systematc value. The global values of a and b are obtaned by ntegratng along the yarn path. a a d L, b b dl L r Lr where dl rr d. In the varable metrc coordnate, ˆ can be defned as (8)

4 H. Wang, X.-w. Yu and Z.-w. Wang where ˆ a b dag,, 33 dag,, a b a a b b dag,, a b dag,, represents the scalng factor n -axs. 3 EXPANSION OF LOCAL STOCHASTIC VARIABLE METRIC BASIS(LSVMB) 3. CALCULATION OF LOCAL BASIS The tangent vector of the yarn path s defned as In Fg., the yarn cross-secton at path pont a has the followng relaton wth constant vector a e r r C n and plane. e 3 e3 e e3 r e e e r 3 3 (9) (0) A n ntersect n a unt vector a, and Wth the ntroducton of Rodrgues rotaton formula nto about, the unt vectors can be evaluated as e3 e cos e 3 e e3 e sn e e e 3 a cos e e e e e e e3 e e e sn 3. COVARIANCES OF FEATURE PARAMETERS Correspondng to the components of feature parameters, the covarances of feature parameters are decomposed nto three parts, namely tangent vectors, scalng matrx and twst angle. Snce the covarance functon s nd centered moment, the covarance functon s closely related to the standard devaton of each feature parameter () In stochastc wavness case, the covarance functon of tangent vector s defned as (warp genus) Kˆ dag 0, K, K K e e (3) where K x x tan, x x rr xx x3x3 xx j j e and () e3 () s standard devaton of the deflecton angle whch s defned as the angle between the drecton of the perfect yarn path and local tangent to the realzaton of the stochastc yarn path, and the symbol denotes outer tensor product. () In stochastc twst case, the covarance functon of twst angle s defned as, K (4) wth the standard devaton. (3) In stochastc scalng case, the covarance of scalng matrx [9] s defned as K ˆ ˆ ˆ ˆ K e e e e j mn j m n (5)

5 st Internatonal Conference on Composte Materals X an, 0-5 th August 07 aa a where K c a a 3.3 EXPANSION OF LSVMB The LSVMB of the path are defned as ˆ ˆ r e r v, K a bb b c b b ˆ ˆ ˆ ˆ e3 e cos e3 sn e e3 e sn e e e 3 ˆ ˆ ˆ ˆ e3 cos e e3 e cos e 3 e sn 3 e3 e e Because e, e, e 3 are mutually orthogonal and scalng tensor ˆ does not affect the drecton of local bass, the LSVMB,, 3 are mutually orthogonal. Eq.(6) shows the LSVMB are the functon of st dervatve of path vector, scalng matrx and twst angle. Expanson of the LSVMB nto Taylor s seres about the mean parameters,, yelds:,, 3 3 r r ˆ ˆ ( ) ( ) ( ) ( ) ˆ ˆ ˆ( ) ( ) ˆ a b r c : d : f r r t (7) () where a, ˆ () ) () b, cˆ(, d, ˆ () f r ˆ, rr t (), r, ˆ,. In Eq. (7), () a vb. represents the elastc constants of deal yarn, ˆ b, f ˆ () cˆ (6) ( ) ( ) represents the stochastc wavness of yarn path, represents the stochastc scalng of yarn crosssecton, and d, t represents the twst of yarn. ( ) ( ) The nd approxmaton of the mean of LSVMB s ( ) ˆ ( ) ˆ ( ) a : K f Krr t (8) The wavness of relatve to s ˆ ( ) ( ) ˆ ( ) b r cˆ : d (9) Thus, the st order approxmaton of the covarance of LSVMB s obtaned as followng T ˆ ˆ ˆ j ( ) ( j) o ( ) ( j) b K ˆ : ˆ ˆ : ˆ j rr b c c Kd d K (0) where the superscrpts T represents the transposton of 3rd order tensor, and superscrpts o ( j) j represents the transposton rule ĉ c e e e. jk j k For the purpose of convenence, the nne-component vector of varable metrc drectonal cosnes are ntroduced as follows,,,,,,,,,, () where j re rj. Then, the covarance ˆK of the LSVMB vectors can be wrtten n the matrx form

6 H. Wang, X.-w. Yu and Z.-w. Wang ˆK 3 3 symm 3 3 The local varable metrc complance or stffness tensor complance tensor CS, () are related to the global through the followng tensor transformaton law. e e e e (3) jkl mnop mr r ns sj ot tk pu ul mnop m nj ok pl The mean complance or stffness of yarn [8, 9] can be obtaned by ˆ : ˆ u K (4) 4 ELASTIC CONSTATNS OF PLAIN-WEAVE COMPOSITE The volumes of warp and weft genuses are dentcal, hence the volume fracton of each yarn genus s 0.5. The formulatons of of plan-weave composte are acqured by applyng volumetrc averagng method to the complance or stffness of all genuses of materals [8]. ˆ aver Vy I II ˆ + ˆ d ˆ m L Vm L (5) ˆ where the superscrpts (I), (II) and (m) represent warp, weft and matrx genuses, respectvely. L represents the length of a yarn perod. V V denotes the volume fracton of matrx n composte and V y m y s the volume fracton of yarn n composte. Isostress method s conducted when complance tensor, whle the sostran method s conducted when relatons between elastc constants and complance tensor[] are S, S, S3 E E E 5 RESULTS AND DISCUSSIONS S, S, S E E E33 S, S, S G3 G3 G ˆ 3 ˆ s s stffness tensor[4]. The The mpact bounds of elastc constants of plan-weave composte changes when the FVF of yarn changes. In order to study the effect of FVF on the elastc constants of plan-weave compostes, three values of FVF are appled nto the three cases of stochastc wavness, scalng and twst. The parameters of mechancal propertes and FVF of yarn n plan-weave compostes are lsted n Table. The sostress and sostran results corresponds to the upper and lower mpact bounds of elastc constants. The results of elastc characterstcs are expressed as the ratos of the mean elastc constants to the deal ones (called mpact bounds), whch reflect the relatve changes between the mean elastc constants and deal ones wth consderng stochastc fluctuatons n the feature parameters of yarn. Table Parameters of mechancal propertes and fber volume fracton of yarn n plan-weave composte Parameters of compostes Parameter values E GPa f 00 Elastc constants of E composte consttutes GPa f 00 G GPa f 50 (6)

7 st Internatonal Conference on Composte Materals X an, 0-5 th August 07 Fber volume fracton of composte consttutes 5. STOCHASTIC WAVINESS G f E m G m 3 GPa f GPa GPa m 0.3 V f V y 0.5,0.7,0.9 The nfluence of FVF on elastc constants of plan-weave compostes n stochastc wavness case s shown n Fg. 3, and the sold and dashed lne surfaces represent the results of sostress and sostran methods, respectvely. The upper and lower mpact bounds of elastc modul exchange as 0.7 y ncrease when When FVF ncrease from 0.5 to 0.9, the both mpact bounds of n-plane elastc modul E show decreasng trends. The sostress mpact bounds of out-of-plane elastc modulus shows a slght ncreasng trend when 5 As for n-plane shear modulus exchange as z s small. Smlarly, the upper and lower mpact bounds of outof-plane shear modulus ncreases whle G 3 y y G z and a decreasng trend n the other sde. z E 5. and E 33, the mpact bounds of sostress and sostran methods and G 3 exchange as y ncreases whle z s small. When FVF ncreases from 0.5 to 0.9, the both mpact bounds of shear modul ncrease. Smlar to shear modul, as for n-plane Posson s rato, the mpact bounds of sostress and sostran methods exchange as Posson s ratos 3 and 3 z ncreases when exchange as y y s small. The mpact bounds of out-of-plane ncreases when from 0.5 to 0.9, the both mpact bounds of n-plane Posson s ratos z s small. When FVF ncreases decrease when 6 and ncrease n the other sde. Meanwhle, the sostress mpact bounds of out-of-plane Posson s ratos and change lttle when 0.6 and ncrease slghtly n the other sde. The sostran 3 3 mpact bounds decrease when z z y and ncrease n the other sde. y z y

8 H. Wang, X.-w. Yu and Z.-w. Wang Fg. 3 The nfluence of V f 5. STOCHASTIC SCALING on elastc constants of plan-weave compostes n stochastc wavness cases The nfluence of FVF on elastc constants of plan-weave compostes n stochastc scalng case s shown n Fg. 4, and the sold and dashed lne surfaces represent the results of sostress and sostran methods, respectvely. When FVF ncreases from 0.5 to 0.9, the mpact bounds of n-plane elastc modulus and E decrease. Comparng wth the upper mpact bounds, the lower mpact bounds are less senstve to the varaton of FVF. Meanwhle, the varaton of FVF has no nfluence on the both mpact bounds of. The nfluence of FVF on the both mpact bounds of shear modul s gnorable. E 33 As FVF ncreases from 0.5 to 0.9, the nfluence of FVF on mpact bounds of becomes more and more sgnfcant as c v ncreases. The both mpact bounds of a 3 and 3 exchange when cv cv. The effects of FVF on mpact bounds of 3 and 3 are not obvous. On the contrary, a b the mpacts of FVF on the mpact bounds of E, E and of plan-weave compostes n stochastc scalng case are obvous. E

9 st Internatonal Conference on Composte Materals X an, 0-5 th August 07 Fg. 4 The nfluence of V f on elastc constants of plan-weave compostes n stochastc scalng cases 5.3 STOCHASTIC TWIST The nfluence of FVF on elastc constants of plan-weave compostes n stochastc twst case s shown n Fg. 5, and the sold and dashed lne surfaces represent the results of sostress and sostran methods, respectvely. The mpact bounds of n-plane elastc modulus E and E expand as FVF ncrease from 0.5 to 0.9. Isostress mpact bounds of E and E are more senstve to the varaton of FVF than

10 H. Wang, X.-w. Yu and Z.-w. Wang sostran mpact bounds. Meanwhle, FVF has almost no mpacts on the both mpact bounds of out-ofplane elastc modul. E 33 The mpact of FVF on the both mpact bounds of shear modul s not obvous. The sostress and sostran mpact bounds of shear modul of plan-weave compostes exchanges when 8. When FVF ncreases from 0.5 to 0.9, the sostress mpact bound of n-plane Posson s rato decreases whle those of out-of-plane Posson s ratos mpact bounds of Posson s ratos are almost not affected by the varaton of FVF. 3 and 3 ncrease. Besdes, the sostran

11 st Internatonal Conference on Composte Materals X an, 0-5 th August 07 Fg. 5 The nfluence of 6 CONCLUSIONS V f on elastc constants of plan-weave compostes n stochastc twst cases A real yarn can be transformed nto a deal yarn by LSVMB. Ths paper presents a new method to estmate the elastc constants of fber yarn, whch s based on the LSVMB. Volume averagng method s appled to compute the global complance or stffness of plan-weave composte. The lower and upper mpact bounds of the elastc constants of the composte are obtaned from sostress and sostran methods, respectvely. The nfluence of FVF on the mpact bounds of elastc constants of plan-weave compostes s obtaned by applyng three values of yarn FVF nto the estmaton of elastc constants of plan-weave compostes n stochastc wavness, scalng and twst cases, respectvely. From the results of numercal smulaton, the mpact bounds of elastc constants of plan-weave compostes are affected by the varaton of FVF of yarn. FVF nfluences the varaton degrees of the mpact bounds of elastc constants of plan-weave compostes whle the varaton trends keep constant. When FVF ncreases from 0.5 to 0.9, the both mpact bounds of elastc modul decrease or are nsenstve to the varatons FVF n three cases. The both mpact bounds of shear modul ncrease or are nsenstve to the varatons FVF n three cases. The mpact bounds of Posson s ratos show dfferent varaton trends n three cases. REFERENCE [] M. Olave, A. Vanaerschot, S.V. Lomov, D. Vandeptte, Internal geometry varablty of two woven compostes and related varablty of the stffness, Polymer Compostes 33(8) (0) [] R. Rnald, M. Blacklock, H. Bale, M. Begley, B. Cox, Generatng vrtual textle composte specmens usng statstcal data from mcro-computed tomography: 3D tow representatons, J Mech Phys Solds 60 (0) [3] H. Bale, M. Blacklock, M.R. Begley, D.B. Marshall, B.N. Cox, Characterzng Three-Dmensonal Textle Ceramc Compostes Usng Synchrotron X-Ray Mcro-Computed-Tomography, J. Am. Ceram. Soc. 95() (0) [4] F. Gommer, K. Wedgwood, L. Brown, Stochastc reconstructon of flament paths n fbre bundles based on two-dmensonal nput data, Compostes: Part A 76 (05) 6-7. [5] A. Vanaerschot, B.N. Cox, S.V. Lomov, D. Vandeptte, Stochastc framework for quantfyng the geometrcal varablty of lamnated textle compostes usng mcro-computed tomography, Compostes Part A 44 (03) -3. [6] A. Vanaerschot, B.N. Cox, S.V. Lomov, D. Vandeptte, Stochastc mult-scale modellng of textle compostes based on nternal geometry varablty, Computers & Structures (03) [7] A. Vanaerschot, B.N. Cox, S.V. Lomov, D. Vandeptte, Expermentally valdated stochastc geometry descrpton for textle composte renforcements, Compostes Scence and Technology (06) -9. [8] H. Wang, Z.-w. Wang, Quantfcaton of effects of stochastc feature parameters of yarn on elastc propertes of plan-weave composte. Part : Theoretcal modelng, Compostes Part A 78 (05)

12 H. Wang, X.-w. Yu and Z.-w. Wang [9] H. Wang, Z.-w. Wang, A varable metrc stochastc theory of textle compostes wth random geometrc parameters of yarn cross-secton, Composte Structures 6 (05) [0] H. Wang, Z.-w. Wang, Quantfcaton of effects of stochastc feature parameters of yarn on elastc propertes of plan-weave composte Part : Statstcal predctons vs. mechancal experments, Compostes Part A 84 (06) [] C. Herakovch, Mechancs of fbrous compostes, Wley, New York, 998. [] S.-K.B. Lee, Joon-Hyung; Hong, Soon Hyung, Effect of fber geometry on the elastc constants of the plan woven fabrc renforced alumnum matrx compostes, Materals Scence and Engneerng A 347 (003) 3. [3] K.-H.H. Tsa, Chung-L; Chen, Wen-Lang; Chu, Chang-Hsuan, A parallelogram sprng model for predctng the effectve elastc propertes of D braded compostes, Composte Structures 83 (008). [4] B. Cox, G. Flanagan, Handbook of Analytcal Methods for Textle Compostes, NASA Langley Techncal Report Server, 997.