Journal of Solid Mechanics and Materials Engineering

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1 Journal o Solid Mechanics and Materials Engineering First Order Perturbation-based Stochastic oogeniation Analysis or Short Fiber Reinorced Coposite Materials* Sei-ichiro SAKATA**, Fuihiro ASIDA** and Ryuta TANAKA*** **Shiane University, 060 Nishikawatsu-cho, Matsue, Shiane, Japan E-ail: sakata@ecs.shiane-u.ac.jp ***Graduate School o Shiane University, 060 Nishikawatsu-cho, Matsue, Shiane, Japan Abstract This paper describes stochastic hoogeniation analysis o a uni-directionally aligned short iber reinorced coposite aterial. In case o a short iber reinorced coposite aterial, length or orientation o the short iber will be a rando variable in addition to elastic properties o coponent aterials, volue raction or cross-sectional shape o iber. Especially, a stochastic hoogeniation proble considering the length or iber orientation is analyed in this paper. For the purpose o this analysis, the irst-order perturbation-based stochastic hoogeniation ethod is eployed. Since the perturbation ter with respect to the length or the orientation cannot be explicitly obtained in using the hoogeniation ethod, a inite dierential technique is used or approxiation o the perturbation ter. Fro the nuerical results, validity, eectiveness and proble o the perturbation-based analysis are discussed. Also, inluence o the length on a stochastic characteristic o a hoogenied elastic property o a short iber reinorced coposite aterial is discussed. Key words: Stochastic oogeniation Analysis, Perturbation, oogeniation Method, Short Fiber Reinorced Coposites. Introduction *Received Oct., 2009 (No ) [DOI: 0.299/jp.4.446] Copyright 200 by JSME Since a icroscopic rando will have an inluence on a hoogenied aterial property o a coposite aterial, a stochastic hoogeniation proble has been analyed in recent. Kainski ()-(3), Ostojya, (4) Sakata (5)-(7) or Xu (8) reported several ethods or the stochastic hoogeniation analysis. This kind o a proble has been noticed or other objectives, or instance, Lobardo (9) reported a stochastic odeling or a asonry structure. For a coposite aterial, a unidirectional continuous iber reinorced coposite aterial is soeties selected as a target o the analysis, and in this case a two diensional proble is soeties assued. It ay be eective or sipliying the proble, but a three diensional analysis should be perored or a ore coplex proble observed in several kinds o coposite aterials. For instance, a short iber reinorced coposite aterial has a three diensional icrostructure, and the stochastic hoogeniation proble or the short iber reinorced coposites will be ore coplex than that o a unidirectional iber 446

2 Journal o Solid Mechanics reinorced coposites even i an assuption as the short iber is uni-directionally aligned can be acceptable. Fro this viewpoint, a stochastic hoogeniation analysis o a short iber reinorced coposite aterial is attepted in this paper. About a short iber reinorced coposite aterial, length or orientation o iber will be a rando variable in addition to elastic properties o coponent aterials, volue raction or cross-sectional shape o iber. Especially, a stochastic hoogeniation proble considering the length or iber orientation is analyed in this paper. For the purpose o the analysis, the irst-order perturbation-based stochastic hoogeniation ethod is eployed. Since the perturbation ter with respect to the length or the orientation cannot be explicitly obtained in using the inite eleent-based analysis, a inite dierential technique is used or approxiation o the perturbation ter. At irst, outline o the stochastic hoogeniation analysis is introduced. Next, the proble setting and nuerical results are described. With the nuerical results, applicability o the perturbation-based approach to the proble and inluence o the icroscopic uncertainty on the stochastic characteristics o the hoogenied elastic properties are discussed. 2. Analysis ethod 2. oogeniation Method A hoogenied elastic tensor o a coposite aterial can be coputed using the hoogeniation ethod (0)() as χ E = E I dy (2.) y where E is a hoogenied elastic tensor, I is a unit tensor, is volue o a unit cell is a elastic tensor o a icrostructure, y is a variable with respect to icroscopic coordinates and χ is a characteristic displaceent. χ is obtained as a solution o the ollowing characteristic equation. χ E dy = d (2.2) y y y Equation (2.2) can be discretied using the inite eleent ethod as [ K ][ χ] = [ F ] (2.3) where [ K ] is a coeicient atrix o the characteristic equation or a unit cell, [ F ] shows a set o orce vectors caused by inhoogeneity in aterial properties o a icrostructure. 2.2 First-Order Perturbation-based Stochastic oogeniation Method (2),(5),(2) I a rando o a icroscopic aterial property is taken into account, each ter o equation (2.3) will have a rando, then it can be rewritten as [ K *][ χ *] = [ F *] (2.4) where the superscript * shows an observed value o a probabilistic response unction. With using the irst order perturbation theory, each ter o Eq. (2.4) can be expressed by the ollowing asyptotic expansion ors 0 [ K* ] K + K α (2.5) 0 [ χ* ] χ + χ α (2.6) 0 [ F* ] F + F α (2.7) where the subscript 0 or shows the 0th or irst order perturbation ter and α is a rando variable. Coparing with the coeicients with respect to α and the constant 447

3 Journal o Solid Mechanics ter, we can obtain a siultaneous equation as K { χ } F = 0 0 ( K { χ } + K { χ }) = F Also, an asyptotic or o a hoogenied elastic tensor can be expressed as * 0 E E + E α 0 = ( E + E α ) dy ( E + E α)( B + B α) ({ } + { } α) dy χ χ (2.8) (2.9) where [ B ] is a displaceent-strain atrix. Coparing with the each coeicient, each perturbation ter o the hoogenied elastic tensor can be obtained using the ollowing equations E = E E B { χ } dy (2.0) E = E ( E B { χ } + E B { χ } + E B { χ }) dy In case o an orthotropic aterial, a stochastic or o hoogenied equivalent elastic constants can be coputed as (5) * * * Ex = * y = * = * C C22 C 33 * * * * * * * * * ν y = Ey C23, νx = E C3, νxy = Ex C2 (2.) * * * Gy =, G, * x = G * xy = * C44 C55 C 66 where C is a hoogenied copliance atrix, which can be coputed as the inverse o * the hoogenied elastic tensor E. C can be also approxiated with the asyptotic expansion as Eq.(2.2), then the irst order perturbation ter o the equivalent elastic constant can be coputed as Eq.(2.3) 0 Q ij + Q ijα + :, i j = 3 0 * R + Rα + Cij (2.2) :, i j = * ( 4,4,5,5,6,6 ) ( ) ( ) Eij 0 0 RQ Q R 0 0 Ex =, v 0 xy = ( C2 Ex + C2 Ex ) (2.3) Q where Q and R are constants coputed ro the perturbation ter o the hoogenied elastic tensor. Using a result o the perturbation-based analysis, the stochastic characteristics such as the expectation and variance o a hoogenied elastic tensor or hoogenied equivalent elastic constants can be estiated using the irst order second oent ethod (3) as 0 E E E = (2.4) Var E E ie j = cov αi, α j i j where cov α i, α j is the covariance o α i and α j. 2.3 Perturbation-based analysis or a short iber reinorced coposite aterial In case o a short iber reinorced coposite aterial, soe geoetrical paraeters such 448

4 Journal o Solid Mechanics as the length or orientation o iber will be taken into account as a rando variable in addition to the elastic properties, volue raction or cross-sectional shape o iber, which is treated as a rando variable in case o a unidirectional iber reinorced coposite aterial. In this paper, as a irst step o this study, a uni-directionally aligned short iber reinorced coposite aterial, which is illustrated in Fig. (a), is assued (4) or sipliication. Shape o a short iber is assued as ellipsoidal as shown in Fig. (b). The length or orientation o the iber can be randoly varied as Fig.(c). θ* x a 2 a l 0 l* x x y (a)short iber reinorced coposite (b)shape o a short iber (c)rando in length or orientation Fig. Schee o a unidirectionally aligned short iber reinorced coposite aterial For evaluation o a stochastic characteristic o a hoogenied elastic property o the coposite aterial caused by a rando in an elastic property o a coponent aterial, the perturbation technique described in the previous section can be usable. owever, in case o a geoetrical, an additional nuerical eort will be needed or the stochastic hoogeniation analysis. For a geoetrical rando, each perturbation ter o the hoogenied elastic tensor can be expressed as ( χ ) E = [ E] [ I] B dy (2.5) 0 0 E = { [ E]( B + B ) } dy χ χ Fro this equation, it is recognied that the irst order perturbation ter o the displaceent-strain atrix B is needed. I each nodal coordinate is a rando variable, B can be obtained analytically. owever, or a typical geoetrical rando such as the length or orientation, it cannot be explicitly expressed as a unction o a rando variable in case o using the inite eleent ethod based approach. Fro this reason, the irst order perturbation ter o the hoogenied elastic tensor with respect to the length or orientation is approxiated using the inite dierential ethod. The approxiated irst order perturbation ter o the hoogenied elastic tensor can be written as [ E] [ ] [ ] E ( 0 0 E B B ) dy + = dα χ χ (2.6) dα For the iber length or iber orientation, the perturbation ter can be coputed as dl l E E, l dα dl dθ E θ E (2.7) θ dα dθ E can be coputed ro a orward, backward or central dierence. 449

5 Journal o Solid Mechanics 2.4 Stochastic Analysis using the Monte-Carlo siulation For accuracy evaluation o the perturbation-based approach, the stochastic characteristics o the hoogenied equivalent elastic constants with respect to the icroscopic rando are also coputed using the Monte-Carlo siulation. Fro the Monte-Carlo siulation, the stochastic characteristics such as the expectation or variance o the hoogenied elastic properties can be coputed as * * E E E i (2.8) N Var i ( E ) 2 i E E E (2.9) * * * N i where N is the nuber o trials. Also the coeicient o variance (CV) can be coputed as EI * Var * E CV E = (2.20) EI * E E The Box-Mullar randoiation (5) is used or generating a set o noral distributed rando values or the Monte-Carlo siulation. In this paper, 0 4 trials are perored or the Monte-Carlo siulation. 3. Proble Settings In this paper, the stochastic characteristics o a hoogenied equivalent elastic constant o a unidirectionally aligned short iber reinorced coposite caused by a rando o a aterial property or geoetry o a icrostructure are estiated. Four elastic properties as oung s odulus o the iber ( E ) and resin ( E ), and Poisson s ratios o the iber ( ν ) and resin ( ν ) are taken into account as a rando variables related to a aterial property. Also, length ( l ) and orientation ( θ ) o iber are taken into account as geoetrical rando variables. The expected values o the elastic properties on iber and atrix are listed in Table. The properties o iber and atrix are eployed correspond to E-glass and Epoxy resin. In this case, the expected value o volue raction o iber ( V ) is It is assued that the rando variables are independently distributed according to the noral distribution. Those observed values can be expressed as ( ) 0 X* X α = + (3.) and stochastic characteristics o a rando variable α are assued as E 2 [ α ] 0,Var[ α] σ where E[ α ] is an expected value and [ ] this case, it is assued that σ = = = (3.2) Var α is variance o a stochastic variable α. In Table Expected values o elastic properties or iber and atrix Fiber ( E-glass ) Matrix ( Epoxy ) oung s odulus ( GPa ) Poisson s ratio

6 Journal o Solid Mechanics 4. Nuerical Results 4. Accuracy o the irst order perturbation-based stochastic hoogeniation ethod or the length At irst, accuracy o the irst order perturbation-based stochastic ethod or the stochastic hoogeniation analysis o the short iber reinorced coposite aterial is investigated. Figure 2 shows the relative estiation error between the results o the irst order perturbation ethod and the Monte-Carlo siulation. The expected value o is 0.0 in this case. a and a 2 are ixed at.0. Only the length o iber is a randoly changed and the orientation is ixed in this case. Fro Fig.2, it is recognied that the estiation error or the ν is larger than the others, and those are less than %. This results show that an estiated CV will be reliable or E andν. The estiated CV or ν will include a larger error than the others. The estiation errors or the length are larger than that or E andν, but those are less than about 2%. Since a result CV estiation using the Monte-Carlo siulation with 0 4 trials will include about 3-5% error, these results indicate that the perturbation-ethod will be usable or the stochastic hoogeniation analysis. owever, it should be also noted the estiation error in CV[ ] estiation reaches 6%, and the error ay becoe beyond 0% in the worst case since the result o MC includes 3-5% bias. This result shows that the proposed perturbation ethod should be careully used or CV estiation considering a large ν. Relative estiation error (%) E ν E ν l Fig. 2. Estiation error in CV o the hoogenied elastic constants or each aterial property In order to investigate inluence o the iber length on the estiation accuracy o the perturbation-based technique, the relative estiation errors are coputed or dierent iber lengths. In this case, the volue raction is ixed, and length o iber is changed. As an exaple, the relationships between a 3 and the relative error in CV estiation or E, ν and the length are shown in Fig.3. Fro this igure, it is recognied that the estiation error or CV o soe kinds o equivalent elastic properties increases according to the iber length increases, but the axiu estiation error is about % or = 20 or E, or the axiu error is about 5% or the length. The estiation error in CV o the equivalent oung's odulus or iber direction in case o ν is rapidly increases as the iber length increases, and the axiu value is about 9% or a 3 = 20. Fro these results, it can be conired that the perturbation based technique is usable or the analysis, but it should be noticed that the estiation error in CV o E will be large when a 3 is large in case o ν 45

7 Journal o Solid Mechanics Relative estiation error (%) G x (a) Relative estiation error or each length o iber in case o E Relative estiation error (%) G x (b) Relative estiation error or each length o iber in case o ν Relative estiation error (%) G x (c) Relative estiation error or each length o iber in case o the length Fig. 3. Estiation error in CV o the hoogenied elastic constants or each aterial property 452

8 Journal o Solid Mechanics 4.2 Inluence o the iber length on the stochastic characteristics o a hoogenied elastic property Next, inluence o the iber length on the stochastic characteristics o the hoogenied elastic properties is investigated. In particular, relationships between the iber length and CV o each hoogenied elastic constants caused by a icroscopic rando in aterial properties o coponent aterials are investigated. The iber length is changed under assuption on constant volue raction o iber. In this section, CV is estiated with using the perturbation-based stochastic hoogeniation ethod. Figures 4-7 show the relationship or E, ν and ν. Fro Fig.4, it is recognied that CV o E and rapidly increase as the iber length increases in the case o E. On the other hand, in case o E, CV o decreases as the length increases though CV o ν y increases. Fro these results, it is considered that inluence o the iber length will be dierent according to a kind o rando. As or the Poisson's ratio, it is recognied that the CV o the hoogenied Poisson's ratio or ν rapidly increases as the length increases. On the other hand, CV o E or ν decreases when the length is larger than 5. Fro this result and Fig.4, it can be considered that the ost CV caused by a aterial property o the iber will increase as the length increase. Also, it is recognied that the inluence o the length on the CV caused by o an elastic property o the atrix will be ore coplex than that o the iber; soe o the will increase when the length increases, but others, or instance CV o E or E or ν, will decrease i the length increases. CV G x Fig.4 Relationship between the length o iber and the CV o the hoogenied elastic constants caused by E 453

9 Journal o Solid Mechanics CV G x Fig.5 Relationship between the length o iber and the CV o the hoogenied elastic constants caused by E CV G x Fig.6 Relationship between the length o iber and the CV o the hoogenied elastic constants caused by ν CV G x Fig.7 Relationship between the length o iber and the CV o the hoogenied elastic constants caused by ν 454

10 Journal o Solid Mechanics Additionally, inluence o the iber length on the CV caused by the iber length is investigated. A relationship between the CV and the iber length is shown in Fig.8. In this igure, CV o the E and decreases as the length increases when the length is less than 2, then the CV increases when the length is less than 7.5 and then the CV decreases again. The other relationships between CV and the length are also non-linear, but the inluence o the length on the CV caused by the length is not larger than that o the elastic property s. Since the CVs o the hoogenied oung's odulus or ν y are siilar to or larger than that caused by the aterial property especially in the case that a 3 is relatively sall, the iber length should be taken into account in solving a stochastic hoogeniation proble in case o a short iber reinorced coposite aterial. CV G x Fig.8 Relationship between the length o iber and the CV o the hoogenied elastic constants caused by the length 4.3 Inluence o a rando in the length o iber on the stochastic characteristics o a hoogenied elastic property As an exaple o a stochastic hoogeniation analysis o a short iber reinorced coposite aterial, a stochastic characteristic o the hoogenied elastic constants or siultaneous rando s o the all elastic properties o coponent aterials and the iber length is coputed. In order to evaluate inluence o the length o iber and its rando, relationship between the iber length and estiated CV o the hoogenied elastic constants are investigated. Figure 9(a) shows a relationship between the CV caused by E, ν and ν. Figure 9(b) shows a relationship between the CV caused by the length in addition to those our rando s. Fro these igures, it is recognied the relationship between the CVs and the length is very siilar to each other except to the CV o E, and the CV o E x y, G y, G x and with considering the length are about 0% or ore larger than the case o considering only elastic property s. 455

11 Journal o Solid Mechanics CV G x (a) CV or all aterial properties s CV (length) G x (b) CV or all aterial properties and length o iber s Fig. 9 CV o the hoogenied elastic constants caused by all aterial properties and the iber length s In order to evaluate inluence o the iber length on CV o the hoogenied elastic constants, increasing ratio o the CV or each iber length is suaried in Fig.0. CV is the increasing ratio, which can be coputed as CVall CVMP CV= 00 (4.) CVMP where subscript "all" indicates the CV caused by E, ν, ν and the length, and subscript "MP" indicates the CV caused by E, ν and ν. Fro this igure, it can be recognied that the inluence o the iber length on the CV o E is particularly large. CV is ore than 50%, and it reaches about 80% when the length is 20. Also, it is recognied that CV o E x y, G y, G x and are ore than 0% when is sall, and it depends on the length o iber. This result shows iportance o a stochastic hoogeniation analysis considering the iber length rando, and it should be taken into account in analying a stochastic characteristic o the hoogenied elastic property o a short iber coposite aterial considering a icroscopic rando in an elastic property o a coponent aterial. 456

12 Journal o Solid Mechanics CV (%) G x Fig. 0 Inluence o the length on CV o the hoogenied elastic constants 4.4 Stochastic hoogeniation analysis considering a rando in iber-orientation. As another geoetrical, stochastic characteristics o the hoogenied elastic constants caused by a rando o iber orientation are investigated. In this case, σ = 0.04 is assued. Table 2 shows the estiated results o CV or the iber orientation with the perturbation ethod and the Monte-Carlo siulation. In this case, three types o the inite dierences as orward, backward and central dierences are used or coputation. Fro table 2, however, it is recognied that the irst order perturbation based approach cannot give an appropriate estiation o the CV o the hoogenied elastic constants or the rando o the iber orientation. One o the reasons why the irst order perturbation-based approach cannot estiate the CV o the iber orientation is a nonlinear relationship between the equivalent elastic constants and the iber orientation. As an exaple, the relationship between soe o the equivalent elastic constants and the iber orientation is illustrated in Fig.. "Exact" in the igure indicates the response curves obtained ro the hoogeniation analysis with each observed value o the iber orientation, and "Linear regression" indicates the linear approxiation o the curves obtained ro the perturbation-based approach using the central inite dierence. Fro this igure, especially in the case o E[ θ ] = 0, the irst order derivative is approxiately 0, and a linear regression using this irst order derivative cannot approxiate the exact relationship. This act shows another approach should be developed or estiation o the stochastic characteristics o the hoogenied elastic constants or a rando o iber orientation. Table 2 Estiated CVs o hoogenied elastic constants or iber orientation MC FA(orward) FA(backward) FA (central)

13 Journal o Solid Mechanics E Linear regression (E ) Exact (E ) Exact (ν ) x Linear regression (ν ) x θ Fig. A relationship between the equivalent elastic constants and iber orientation 5. Conclusion This paper discusses a stochastic hoogeniation analysis o a unidirectionally aligned short iber reinorced coposite aterial. Rando variables are elastic properties o coponent aterials, length or orientation o iber. The stochastic properties o the hoogenied elastic properties such as the expectation, variance or coeicient o variance are coputed with using the irst-order perturbation-based stochastic hoogeniation ethod. Since it is diicult to copute a perturbation ter or the iber length analytically in case o using the inite eleent ethod, the inite dierence-based approach is eployed in this paper. At irst, accuracy o the perturbation-based hoogeniation ethod or analysis o the stochastic hoogeniation proble o a short iber reinorced coposite aterial and its dependency on the length o iber are investigated. Fro the nuerical exaples, it is conired that the irst order perturbation-based hoogeniation ethod will be usable or the stochastic hoogeniation analysis considering iber length. Next, inluence o the iber length on the stochastic characteristics o the hoogenied elastic properties is investigated. Also, inluence o the iber length on the stochastic characteristics caused by a rando o elastic properties o coponent aterials is investigated. Dependency and inluence o the iber length and its are illustrated with the nuerical results. Especially, iportance o considering the iber length in stochastic hoogeniation analysis o a short iber coposite aterial is conired. Additionally, the perturbation-based hoogeniation ethod is applied to stochastic hoogeniation analysis or a iber orientation. Fro the nuerical results, however, it is recognied that the irst order perturbation ethod is not appropriate to estiate a stochastic characteristic o the hoogenied elastic property or the iber orientation. An iproved approach should be developed or the analysis in a next step o this study. Acknowledgeent The irst author is pleased to acknowledge support in part by Grants-in-Aid or oung Scientists (B) (No ) ro the Ministry o Education, Culture, Sports Science and Technology. 458

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