Numerical predictions of the anisotropic. viscoelastic response of uni-directional fibre composites

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1 Nuerical predictions o the anisotropic viscoelastic response o uni-directional ibre coposites M.V. Pathan a, V.L. Tagarielli a,, S.Patsias b a Departent o Aeronautics, Iperial College London, South Kensington Capus, SW7 2AZ, UK b Rolls-Royce plc, PO Box 31, DE24 8BJ Derby, UK Abstract Finite Eleent (FE) siulations are conducted to predict the viscoelastic properties o unidirectional (UD) ibre coposites. The response o both periodic unit cells and rando stochastic volue eleents (SVEs) is analysed; the ibres are assued to behave as linear elastic isotropic solids while the atrix is taken as a linear viscoelastic solid. Monte Carlo analyses are conducted to deterine the probability distributions o all viscoelastic properties. Siulations are conducted on SVEs o increasing size in order to deterine the size o a representative volue eleent (RVE); or the ibre volue ractions analysed (0.3 and 0.6), we conclude that elastic properties can be eectively predicted using RVEs o size equal to 24 ties the ibre radius, whereas nuerical predictions o loss actors require saller RVEs, o size equal to 12 ties the ibre radius. The predictions o the FE siulations are copared to those o existing theories and it is ound that the Mori-Tanaka [1] and Lielens [2] odels are the ost eective in predicting the anisotropic viscoelastic response o the RVE. Keywords: Coposite, Daping, RVE, Finite Eleent 1. INTRODUCTION Fibre-reinorced polyers (FRPs) are widely used in industry due to their excellent speciic strength and stiness and also display relatively high aterial daping copared to etals o siilar stiness. Knowledge o their echanical properties is essential to achieve optial Corresponding author Eail address: v.tagarielli@iperial.ac.uk (V.L. Tagarielli) Preprint subitted to Coposites Part A April 27, 2016

2 designs with FRPs; while the anisotropic stiness and strength o FRPs have received great attention ro the research counity, less studies exist on their daping properties, which are particularly iportant in aerospace applications. The daping o FRPs is strongly anisotropic and depends on the iposed requency and teperature; experiental investigations are thereore tie-consuing and require specialist equipent. For these reasons, eective nuerical and theoretical predictions o the daping properties need to be developed and validated. Nuerous theoretical odels exist to predict the elastic response o UD coposites; these can be easily extended to the case o viscoelastic aterials via the elastic-viscoelastic correspondence principle. In addition to the upper and lower bounds given by the Voigt[3] and Reuss odel[4], respectively, Hashin[5] and Hill[6] derived narrower bounds or transversely isotropic coposites with isotropic constituents. Hashin and Rosen[7] later derived a predictive odel based on a coposite cylinder asseblage (CCA). Several predictive odels are based on ean-ield hoogenisation, in which the icroields within each constituent o an inhoogeneous aterial are approxiated by their phase averages by using Eshelby s odel[8]. Exaples include the Mori-Tanaka [1] odel, the Sel-Consistent Method (Hill [9]) and Lielens odel[2]. Other theoretical odels have ocused on predictions o viscoelastic properties via extension o previously developed elastic odels, the ost popular being such as Hashin[10 12], Christensen[13] and Saravanos and Chais [14]. Several studies attepted validation o the above analytical odels via nuerical analysis; or exaple, Chandra et al [15] and Brinson et al [16,17] considered the viscoelastic response o square or hexagonal periodic unit cells; Tsai and Chi [18] pointed out that the daping properties predicted by siulations on unit cell showed are strongly dependent on the choice 2

3 o unit cell. Such studies were either liited to a ew selected loading cases or they analysed only daping properties but not the elastic response. Since the spatial distribution o ibres in a UD coposite is closer to being rando than periodic, it is intuitive to expect that an analysis o a rando icrostructure should yield ore realistic results than the analysis o a periodic unit cell. Several authors have analysed nuerically rando icrostructures; or exaple Arnold et al. [19] analysed stiness and strength o UD ibre coposites and copared the predictions o periodic unit cells and rando icrostructures; Gusev et al. [20] analysed rando distributions o spherical particles in a continuous atrix to extract its eective elastic properties. Several researchers have ocused on the dependence o nuerical predictions upon the size o the aterial volue investigated and gave guidelines or the choice o an eective iniu size. For the case o coposites with spherical iller particles, Drugan and Willis [21] ound that the elastic properties could be eectively predicted using Representative Volue Eleents (RVEs) o size 4R, where R is the radius o the spherical particle. Trias et al. [22] exained elasticity o UD carbon/epoxy coposites and suggested an RVE size greater 50R. In the present work, we present a coprehensive nuerical analysis o the anisotropic viscoelastic response o a UD ibre coposite laina, siulating both periodic unit cells and rando icrostructures. For the case o rando icrostructures we analyse the sizedependence o the FE predictions and their scatter, deterining an eective RVE size. Predictions are also copared to existing theoretical approaches with the objective o ranking the eectiveness o dierent odels in predicting the viscoelastic properties. The outline o the paper is as ollows: in Section 2 we review linear viscoelasticity and selected analytical odels; the FE siulations are described in detail in Section 3 and the 3

4 corresponding nuerical results are presented in Section 4. In Section 5 we present and discuss a coparison o nuerical and theoretical predictions. 2. REVIEW OF THEORETICAL PREDICTIONS OF THE VISCOELASTIC PROPERTIES 2.1 Response o the constituent aterials Daping in FRPs is priarily due to the viscoelastic nature o the polyeric atrix, since the ost coonly used reinorcing ibres are inorganic (e.g. carbon, glass) and their daping properties are negligible. Accordingly, in this work we shall assue a linear elastic response o the ibres. In noral operating conditions coposites experience sall deorations; this justiies odelling the polyeric atrix as a linear viscoelastic aterial. Assuing an isotropic response o the atrix, the constitutive equations o viscoelasticity are given as t deij s ( t) 2 G( t ) d ij d (1) t d ii p ( t) 3 K( t ) d ii d (2) where s and ij p are the coponents o the deviatoric and hydrostatic stress tensor, ii respectively, e and ij ij are the corresponding deviatoric and dilatational strains, Gt () and Kt () are tie dependent shear and bulk oduli, respectively [16]. Taking a Fourier Transor o the equations (1) and (2) gives s i t G e (3) ( ) 2 ( ) ( ) ij ij p i t K (4) ( ) 3 ( ) ( ). ii ii 4

5 The above equations are analogous to those governing isotropic elasticity but are expressed in the Fourier doain; this correspondence is reerred to as the elastic-viscoelastic correspondence principle. G ( ) and can be decoposed in their real and iaginary parts K ( ) are Fourier transors o Gt () and Kt () and ' '' G ( ) G ( ) ig ( ) (5) ' '' K ( ) K ( ) ik ( ) (6) The real parts G ' ( ) and K ' ( ) are deined as storage oduli, while G '' ( ) and K '' ( ) are the corresponding loss oduli. Loss actors are deined as ratios o the loss odulus to the corresponding storage odulus, i.e. G G''/ G'; K ''/ K '. (7) K For typical polyers it is typically, due to the act that dissipative echaniss are K G ore pronounced in presence o deviatoric strains. Existing predictive odels o the eective elastic properties o ibre coposites can be extended to the case o viscoelastic coposites by using the elastic-viscoelastic correspondence principle. The viscoelastic aterials can be odelled using noralized Prony series based on the generalized Maxwell odel [23] as ollows: Gt () G 0 N i 1 ( t/ ) i 1 g (1 e ) i (8) Kt () K 0 N i 1 ( t/ ) i 1 k (1 e ) i (9) 5

6 Where G and K are instantaneous shear and bulk oduli, and g, k and 0 0 i i i are the noralized shear and bulk oduli and relaxation tie constant o the i-th ar o the generalised Maxwell odel. 2.2 Response o a transversely isotropic laina For a transversely isotropic, uni-directional coposite laina, viscoelasticity can be expressed, in ters o coplex engineering constants, as 11 1/ E11 21 / E22 21 / E / E11 1/ E22 23 / E / E11 23 / E22 1/ E / G / G / G (10) Five independent loss actors can also be deined or such transversely isotropic aterial, as E E G G (11) '' '' '' '' '' ; 22 ; ' ' 12 ; ' 23 ; ' 12 ' E11 E22 G12 G23 12 where, a prie indicates storage properties and a double prie reers to loss properties. The engineering constants in eqns. (10) and (11) have to be deterined experientally or predicted nuerically. Several analytical odels exist to predict the values o such engineering constants and loss actors. We shall copare our nuerical predictions to those o selected analytical odels, naely: direct and inverse rule o ixture [3][4], Hashin s upper and lower bounds [5][6][23][24], Saravanos and Chais odel [14][25], Coposite Cylinder Asseblage odel [7], Mori-Tanaka odel [1] and Lielens interpolative odel [2]. The details o these analytical odels and the corresponding orulae are provided in Appendix A. 6

7 3. NUMERICAL METHODS We eployed the Finite Eleent (FE) ethod to siulate the viscoelastic response o a transversely isotropic laina and to copare the nuerical predictions to those o the existing theoretical odels described in the previous section. We conducted coprehensive nuerical analyses aied at deterining a hoogenised viscoelastic tensor or a coposite laina. This was done by analysing the response o three-diensional rando arrays o cylindrical ibres, iicking the icrostructure o a unidirectional ibre coposite. Such icrostructure was subjected to our dierent cyclic loading cases, naely: uniaxial tension-copression in the ibre and transverse directions; transverse and axial shear. A steady state dynaics direct analysis step in the requency doain was perored in ABAQUS Standard; the acroscopic strains iposed on the RVEs were pure sine waves o aplitude arbitrarily set to 0.01 and varying requencies; the analysis allowed calculation o the corresponding acroscopic stress histories; such histories were interpreted as phasors and split into two coponents, in-phase and out-o-phase with respect to the iposed strain. The ratio o the in-phase stress aplitude to the corresponding strain aplitude provided the values o the storage oduli; siilarly, the ratio o the out-o-phase stress aplitude to the strain aplitude gave the loss oduli. The icrostructures siulated are rando and in general their response is aected by intrinsic scatter; ollowing the deinition in [32][a1][a2], the icrostructures analysed are statistical Volue Eleents (SVEs). Monte Carlo Siulations are perored on each SVE: ultiple realisations o the SVEs are siulated, allowing calculation o the cuulative probability distributions o all visco-elastic properties. Monte Carlo analyses are repeated at increasing SVE size in order to deterine the size o a Representative Volue Eleent (RVE). 7

8 Figure 1: Exaples o icrostructures at v =0.6 or dierent RVE sizes. Figure 2: Exaples o dierent realizations o a icrostructure with v =0.6, δ=24. Figure 3: Exaples o icrostructures o dierent volue ractions, or δ=12. Deinitions o an RVE are given, or exaple, in [33] and [34]. In brie, while SVEs provide only apparent properties o the aterial, which are in general size-dependent, a RVE is suiciently large to saple a great nuber o the icrostructural eatures o a certain aterial, and this gives a size-independent response, associated to inial scatter. For the case o a ibre coposite, the iniu size o an RVE depends on the volue raction o ibres as well as on the constitutive response o atrix and ibres. Many authors [22,35] have calculated the iniu RVE size suitable to deterine the elastic stiness constants o unidirectional coposites; on the other hand, siilar inoration is lacking or the case o daping properties and our study ais at addressing this. In our study we choose properties o the constituent aterials representative o an epoxy resin and glass ibres and we explore 8

9 the eect o the ibre volue raction ( v ) on aterial response and iniu RVE size. For the purpose o coparison, FE analyses are also conducted on square and hexagonal periodic unit cells. The siulations were run using the ABAQUS inite eleent sotware. 3.1 Details o the FE odels The icrostructures analysed consisted o rando arrays o parallel, circular cylindrical ibres o equal diaeter. In the plane perpendicular to the ibres the volue eleents analysed were squares o side length L, and had thickness t in the ibre direction. These were generated via a new algorith recently proposed by the authors [36], based on optiisation techniques. The icrostructures generated were geoetrically periodic and they were shown to be eectively rando or L/ R 7 (or the case o v 0.65 ; R is the radius o the ibres). Python scripts were used to generate autoatically ultiple realisations o each SVE, to apply appropriate boundary conditions, to esh the SVEs and autoatically peror the Monte Carlo analyses and extract the relevant outputs. The icrostructures were eshed using a cobination o hexahedral and tetrahedral inite eleents with linear shape unctions (C3D8 and C3D6). A esh sensitivity study was perored to deterine the optial eleent size in each loading case and it was ound that inite eleents o side length less than R/4 gave esh-insensitive predictions or all loading cases. 9

10 Figure 4: Exaples o periodic unit cells with v =0.3. Three selected SVE sizes were analysed, naelyδ=8,12 and 24, exaples o which are shown in Figs. 2(a)-(c) or the case v 0.6 ; the axiu value o was chosen to give L coparable to the ply thickness o typical coposites. The thickness t was shown to have negligible eect on the results, in line with the indings o other authors (e.g. Melro et al. [37]); in our analysis we assued t 4R. Exaples o ultiple realisations o the largest SVE ( 24, v 0.6 ) are presented in Fig. 3. Due to the geoetric periodicity, the SVEs can only contain an integer nuber o ibres. The SVE sizes (L) were adjusted in order to achieve the desired volue ractions. The exact SVE sizes L are , and 120μ ; in all SVEs it was R 5 μ. When conducting Monte Carlo analyses, siulations were repeated N ties where N was 60, 30 and 10 or 8,12 and 24, respectively. Exaples o SVEs o equal size and dierent volue ractions are presented in Fig. 4. Figure 5 presents exaples o the square and hexagonal unit cells analysed, or the case v 0.3. It is widely accepted that periodic boundary conditions (PBC) are the ost appropriate boundary conditions to analyse a geoetrically periodic RVE [32,38] and PBCs are iposed on the SVEs analysed here, ollowing, e.g., [39] or [37]. Loading was applied by iposing 10

11 nodal displaceents on appropriate duy nodes via the ethod o acroscopic degrees o reedo, as introduced by Michel et al. [40] and used by Tucker and Liang [30]. Periodic boundary conditions were also iposed or the analysis o periodic unit cells. The constitutive responses o ibres and atrix were chosen to be representative o a glass/epoxy coposite. The ibres were linear elastic isotropic solids while the atrix was odelled as an isotropic, linear viscoelastic solid. For siplicity a one-ter Prony series was considered to odel the visco-elasticity o the atrix and the sae noralised Prony series coeicients and relaxation tie constant were used or both deviatoric and voluetric deoration odes. Relevant aterial constants are provided in Table 1 and 2. Table 1: Mechanical properties o constituent aterials. Elastic Modulus GPa Poisson Ratio Epoxy Carbon Fibre Table 2: Prony series coeicients. g i k i i (sec) Epoxy RESULTS AND DISCUSSION FE siulations were conducted under steady state haronic conditions, iposing a acroscopic strain aplitude o 0.01 at requencies in the range to10 Hz. The response spectru or selected viscoelastic constants are given in Fig. 6 or dierent SVE sizes and or the case v 0.3 ; the results are averages o siulation outputs over the ultiple 11

12 realizations analysed, and each aterial property is noralised by the corresponding property o the neat atrix. As previously reported in literature [32], or all loading cases we detect a size dependence o the elastic constants, however predictions becoe practically insensitive to size or 24. Such size dependence is ore pronounced at high loading requencies; this can be explained as ollows: at lower requencies (and thereore low strain rates) the eective atrix stiness is very low, corresponding to low strain energy and viscous dissipation in the atrix; predictions becoe less sensitive to the detail o ibres arrangeent, giving a low sizedependence o the predictions. At any requency, the ost substantial size eect is observed or the transverse shear odulus G. For the case o daping properties (loss oduli), predictions are practically insensitive to size (in the range exained). ' 23 We now present the results o the Monte Carlo analyses, exaining the probability distributions o the predicted viscoelastic properties, or a selected loading requency o 1 2, corresponding to the peak o loss oduli or both the atrix and the coposite. 4.1 Results o the Monte Carlo analyses We begin by presenting the cuulative density unctions (CDF) or the predicted axial storage odulus E. In Fig. 7 we copare the cases o v 0.3 and v 0.6 ; or both ' 11 volue ractions predictions are size independent and associated to inial scatter, indicating that a sall SVE is suicient to obtain accurate prediction o the axial odulus. Since the ibres are odelled as purely elastic, the corresponding axial loss actors are negligible and are oitted. 12

13 Figure 5: Eect o iposed requency on the viscoelastic response o SVEs o dierent size (v =0.3). In Fig. 8 we present a siilar coparison or the case o the ajor Poisson ratio; or this elastic property, we observe that average predictions are size-independent, however they display a substantial scatter, which reduces onotonically with increasing SVE size. 13

14 We proceed to analyse the predicted viscoelastic properties o SVEs with v 0.3 ; CDFs o ' ' ' predicted values o the reaining elastic constants ( E, G, G ) and the corresponding loss actors ( 22, 12, 23 ) are presented in Fig 9. The scale on the horizontal axis is chosen to Figure 6: Cuulative distribution unctions o E 11 or v =0.3 (let) and v =0.6 (right). Figure 7: Cuulative distribution unctions o ν 12 or v =0.3 (let) and v =0.6 (right). display values ro 60% to 140% o the ean, to allow a direct coparison o the predicted scatter or dierent viscoelastic properties. ' For the case o the elastic constants E ' and G, predicted average values are sizeindependent, as observed or the case o ' E 11 and 12 ; again, as expected, the scatter in the predictions reduces with increasing SVE size. In contrast, predictions or ' G 23 display a 14

15 aterial size eect with the transverse shear odulus increasing with increasing SVE size; the size dependence is however negligible or 12. For case o the loss actors, predictions are independent o size and the corresponding scatter is inial or this volue raction. Figure 10 presents identical inoration as Fig. 9 or the case v 0.6. The trends displayed by the predicted aterial s constant are siilar to those observed at the lower volue raction: all predictions are size-independent with the exception o ' G 23, or which a size eect is detected; the agnitude o such size eect is greater than observed or v 0.3. Predictions o the loss actors at this higher volue raction are associated to a larger scatter than observed at v 0.3 ; however such scatter is substantially less than that deterined or the corresponding elastic oduli. 15

16 Figure 8: Cuulative distribution unctions o viscoelastic properties (v =0.3). For the case v 0.3 we perored a best it o the data to a Gaussian distribution and ound, as expected, a very good it, with Pearson s coeicients higher than 97% or all saples sizes. The paraeters o this best it are as given in Table 3; ro this we can deduce that the highest scatter in the predictions is associated with G ' 12, ollowed by ' G 23, or every SVE size analysed; this indicates that coparatively ore nuber o realisations ust be analysed to obtain accurate average predictions o these two shear oduli. 16

17 Figure 9: Cuulative distribution unctions o viscoelastic properties (v =0.6). 4.2 Deterination o the iniu RVE size We now analyse the ean values o the predicted viscoelastic properties and the corresponding 95% conidence interval obtained with SVEs o dierent size; the objective o this analysis is to deterine the iniu SVE size to give size-independent predictions o the ean properties as well as intrinsic scatter below a given tolerance, or a given nuber o 17

18 realisation. In other words, we seek the iniu size o a RVE suitable to deterine the viscoelastic properties o a uni-directional ibre coposite. Figure 11 presents nuerical predictions or a coposite with v 0.3 ; we oit results obtained or the axial odulus ' E and corresponding loss actor as predictions or these properties were ound to be size independent and associated to inial scatter in the range o SVE sizes analysed (see Fig. 7). The dashed horizontal lines in each graph represent a range o 5% around the average predictions o each property, obtained by analysis o the largest icrostructure. For all the aterial properties considered, predicted ean values are practically insensitive to size or 12. As expected, the scatter in the predictions reduces onotonically with increasing SVE size. For the case o elastic properties, it is ound that or icrostructures with 24 the scatter in the predictions is less than 5% ; thereore, or the case o 10 repeated siulations on dierent realisations o the icrostructure, 24 represents the iniu RVE size or prediction o the elastic properties. In contrast, the scatter in the predictions o the loss actors is below 5% or all the SVE sizes analysed; consequently i one is interested in deterining only the loss actors, the analysis o a SVE o size 8 will provide suiciently accurate predictions, or the case o 60 repeated siulations. Table 3: Curve it paraeters or v=0.3 ( Mean, Standard Deviation and R Pearson's coeicient). 18

19 Copare now with the case v 0.6, presented in Fig. 12. The trend displayed by the data is siilar to that observed or v 0.3 ; again, predicted ean values are scarcely insensitive to size or 12, and predictions o elastic properties are associated with scatter o less than 5% or 24, which thereore represents a suitable RVE size even or this higher volue raction (or 10 repeated siulations). In contrast, predictions o the loss actors are associated with higher scatter than what observed or v 0.3 and thereore the iniu RVE size necessary or accurate predictions o daping properties is 12, corresponding to analysis o 30 dierent realisations. This can be explained observing that or the case v 0.3 ibres are ore isolated within the polyeric atrix, thereore subject to a strain ield scarcely dependent upon ibre position; on the other hand or v 0.6 the strain ields around the ibres depend ore strongly on the ibre arrangeent in each realisation o the SVE, and predictions o the daping properties are expected to be ore sensitive to SVE size. In suary we ind that an SVE with 24 is suiciently large to provide sizeindependent predictions o the eective elastic properties with an accuracy o 5% (or the case o 10 repeated siulations). This is in contrast with the value 30 previously reported in the literature by Trias et al. [22] or an accuracy o 10%. These authors analysed a CFRP with v 0.5, characterised by a lower stiness contrast o the constituent aterials E / E 6 than that used in the present study / 30 E E ; it has been shown previously [41,42] that the iniu RVE size (or a given accuracy) increases with increasing stiness contrast; considering that the volue ractions in the two analyses were 19

20 siilar, the relatively larger RVE size 30 proposed by [22] ust be explained in ters o the dierent eatures o the SVEs analysed (these were non-periodic and eectively rando only at 40 ) and the type o boundary conditions used. No data is available, to the best o the authors knowledge, on the iniu RVE sizes or accurate predictions o loss actors; in our analysis, we ind that an SVE o size 12 provides size-independent predictions o accuracy 5%, or the case o 30 repeated siulations. 20

21 Figure 10: Means and 95% conidence interval o the predictions obtained with dierent SVE size (v=0.3). 4.3 Coparison o nuerical and theoretical predictions Finally, we copare theoretical and nuerical predictions o the viscoelastic properties. In Fig. 13 we present predictions o all elastic oduli and corresponding loss actors, at increasing ibre volue ractions, or 24 and 10 repeated siulations. The igure includes nuerical predictions obtained by FE analysis o square and hexagonal unit cells (Fig. 5), or the purpose o coparison. We observe initially that the theoretical predictions 21

22 o Mori-Tanaka, CCA odel and Hashin s lower bound are practically coincident or all properties considered. For case o the longitudinal odulus and corresponding loss actor, all theoretical and nuerical predictions coincide; accurate estiates o these properties can be ade by any o the nuerical or theoretical analyses considered, and these properties are independent o ibre arrangeent. For the transverse Young odulus and loss actor, the predictions obtained by analysis o rando RVEs are bounded by those obtained via nuerical analysis o periodic unit cells. The theoretical approaches which provide predictions closest to those obtained ro the analysis o rando RVEs are the odels by Lielens and Mori-Tanaka (the latter coincides with Hashin s lower bound and very close to the CCA odel). The nuerical predictions lie approxiately idway between those o Lielens and those o Mori-Tanaka, such that an average o the predictions o these odels would give very accurate estiates o both transverse odulus and loss actors. For the case o axial shear odulus and loss actors, the analysis o RVEs and unit cells provides siilar eective properties. Again, nuerical predictions obtained via RVE analysis are bounded by the theoretical predictions o Lielens and Mori-Tanaka, with the latter being ore eective than the orer. Finally, or the transverse shear odulus and loss actors, the analysis o unit cells gives predictions substantially dierent ro those o rando RVEs; in particular, analyses conducted on square unit cells appear to violate Hashin s lower bound, as observed by other authors [37,38,43]. As observed or the transverse Young s odulus and loss actors, predictions ro RVE analyses are bounded by the theoretical odels o Lielens and Mori- 22

23 Tanaka and lie approxiately idway between these bounds, such that an average o these theoretical predictions provides accurate estiates o the transverse shear properties. For all load cases considered, the odel by Saravanos and Chais [14] provides reasonably good estiates o elastic oduli but poor predictions o the loss actors, estiating loss actors which increase with ibre content. It is expected that the nuerical predictions obtained ro RVE analyses and presented in this study should be closer to easureents than those obtained ro analysing unit cells. Considering that all theoretical approaches presented rely on soe approxiations, while our RVEs ore realistically represent the details o the coposite icrostructures, we expect the RVE analyses to be ore accurate than any o the theoretical odels. This should be, however, veriied by conducting careul easureents on both coposite lainae and their constituent aterials (ibres and atrix); this is let as a subject o a uture study. 5. CONCLUSIONS The ain conclusions ro the nuerical studies presented are suarised as ollows: The predictions o viscoelastic properties obtained ro analyses o SVEs display an intrinsic scatter which decrease with increasing ibre volue raction. For an accuracy on the predictions o 5%, it was ound that the iniu RVE size or predictions o the elastic properties is L/ R 24, or 10 repeated siulations; this value is saller than previously reported in the literature. For predictions o the daping properties with accuracy 5%, the iniu RVE size was ound to be 12, or 30 repeated siulations on dierent realisations o the icrostructure; no siilar conclusions had been previously reported in the literature. 23

24 The prediction o the transverse shear odulus ' G 23 shows the slowest convergence with increasing SVE size, at any volue raction; i predictions o ' G 23 are not required, and an accuracy o 7% is suicient, an RVE o size 12 (30 repeated siulations) is adequate or volue raction v 0.6. A coparison o nuerical and theoretical predictions o the viscoelastic properties o UD coposites showed that the transverse storage oduli, E and G, and the ' ' corresponding loss actors, 22 and, can be predicted accurately by considering a 23 plain average o the theoretical predictions o Lielens [2] and Mori-Tanaka [1]. For the case o the in-plane shear odulus and corresponding loss actor, G and, the ' Mori-Tanaka odel provides the ost accurate predictions aong the theoretical odels copared in this study; however this odel under-predicts ' G 12 by approxiately 15% and over-predicts 12 by a siilar extent. 6. ACKNOWLEDGEMENTS This work is part o a collaborative R&T Project SILOET II Project 10 which is co-unded by Innovate UK and Rolls-Royce plc and carried out by Rolls-Royce plc and Iperial College. The authors are grateul to Rolls-Royce plc or giving perission to publish it. 24

25 Figure 11: Means and 95% conidence interval o the predictions obtained with dierent SVE size (v=0.6). 25

26 Figure 12: Coparison o theoretical and nuerical predictions. 26

27 APPENDIX A In this appendix we review a nuber o selected analytical odels and provide the corresponding predictions or the elastic engineering constants and the corresponding loss actors. A.1 Voigt and Reuss bounds Such bounds correspond to assuing either a unior strain (Voigt [3], providing an upper bound) or a unior stress (Reuss [4], providing a lower bound) in the dierent phases o a coposite. The eective engineering constants o a two-coponent coposite, ust be coprised between the Voigt and Reuss bounds, i.e. v P P 1 v P P v P v ij (A1) In the above equation P ij denotes a general engineering constant o the coposite, P, P are the corresponding constant or iller and atrix and, are the associated volue ractions. It is known that the direct rule-o-ixtures (ROM) corresponding to the Voigt bound provides accurate predictions o E 11 and 12 ; on the other hand, the inverse rule-oixtures (IROM) given by the Reuss odel can eectively predicts E 22 and G 12. A.2 Hashin - Hill bounds Hashin and Shtrikann [24] developed ore accurate bounds or the eective elastic constants o ultiphase aterials with an arbitrary icrostructure. Subsequently Hashin[5] and Hill[6] extended the variational orulation in [23] to the case o transversely isotropic coposites. By application o the correspondence principle, the Hashin-Hill bounds or the viscoelastic engineering laina constants are given as 27

28 v k k k v k k k G k k k G v v (A2) v v G G G ( 2 ) ( 2 ) 23 1 v k 1 G v k G G G 2 G ( k G ) G G 2 G ( k G ) (A3) v G G G 12 1 v 1 v 2 G G G G G 2G v (A4) v v v E v E v E v v 11 v 4( ) k k G k k G v v (A5) v v v v v v 12 v v 1 ( 1 )( 1 ) v v 1 k k G k k G k k (A6) Where, k, k represent the coplex bulk oduli o ibres and atrix and k 23 is the plane strain bulk odulus. Bounds on E 22 can be obtained ro anipulation o equations -. A.3 Saravanos and Chais odel Saravanos and Chais [14],[25] constructed icroechanical odels to predict stiness and daping o transversely isotropic laina. Elastic constants were obtained ro analysis o a square unit cell, while loss actors ollowed ro a strain energy ethod; naely, the authors assued that loss actors or the coposite can be calculated as a weighted average o the loss actors o the constituents, the relative weights being the ractions o strain energy stored 28

29 in the two constituents. For the case o isotropic ibres and atrix, Saravanos and Chais provide E E v E v E ; v v E E E (A7) ve E E E v E v v (1 ) ; (1 ) E E E 1 v (1 ) E (A8) vg G G G v G v v (1 ) ; (1 ) s s G G G 1 v (1 ) E (A9) E G G G ; v (1 v ) s s 2(1 v ) G G 23 (A10) 23 (1 v ) v ( ) 1 v 1 v (A11) where, and are the shear loss actors o ibres and atrix, respectively. s s A.4 Coposite Cylinder Asseblage (CCA) odel Hashin and Rosen [7] derived predictions o our engineering constants o a coposite laina, naely longitudinal odulus, longitudinal Poisson s ratio, longitudinal shear odulus and plane strain bulk odulus. The coposite is odelled as an asseblage o solid circular cylinders (representing the ibres) surrounded by annular cylindrical regions, representing the atrix. The ratio o ibre diaeter and thickness o the annular atrix region is dictated by the volue raction o the coposite. Hashin [11] also developed a siilar odel based on an asseblage o coposite spheres, to predict the properties o coposites 29

30 with spherical inclusions. The predictions were ound to be in good agreeent with easureents [10,11]. The eective properties predicted by the CCA odel are given below. 4 v v ( ) G 2 11 vg vg G G K K E v E v E (A12) G G ( )( ) vv G G K K v v vg vg 1 G G K K 3 3 (A13) k K 23 G 3 1 v G G 4G ( K K ) K 3 v 3 (A14) G (1 v ) G v G G (A15) G v G v 12 (1 ) G G 23 2 B AC B A (A16) where A, B, C are unctions o the engineering constants o the constituent aterials [11]. 30

31 Figure A1: Scheatic description o dierent hoogenization schees. A.5 Mori-Tanaka odel Mori and Tanaka [1] constructed a predictive icroechanical odel or the elastic properties o two-phase coposites, which was later sipliied by Bensvite [26] as it is presented here. Mori and Tanaka introduced inter-particle interaction by adopting an average strain in the atrix phase given by the eans o superposition o the ar ield applied strain 0 to a strain perturbation caused by the inhoogeneous icrostructure; such average strain in the atrix phase is related to the ar-ield applied strain by 0 AMT and siilarly the average strain in the inclusions is related to the ar-ield strain by 0 AMT. The odelling approach is sketched in Fig. 1(a); the coposite is considered as the assebly o an elliptical inclusion, o stiness C and volue, ebedded in a large atrix doain o volue and stiness C. The relationship between the average strain in the inclusion and in the atrix is given via Eshleby s result [8] or ebedded inhoogeneity in an ininite atrix at dilute concentrations, as A, where dil A dil is the strain concentration tensor or dilute concentrations and is given as A I J S C C 1 dil [ Eshelby ( )] (A17) 31

32 Where, I is the identity atrix, J is the Eshelby tensor, S is the copliance tensor o Eshelby the atrix aterial and C, C are the stiness tensors or atrix and inclusion aterials, respectively. The Eshelby tensor solely depends on the Poisson s ratio and geoetry o the inclusion; explicit expressions or this tensor can be ound in text by Mura et al. [27]. The Mori-Tanaka strain concentration tensors, A, A are given by MT MT A v I v A i 1 MT [(1 ) dil ] (A18) A A v I v A (A19) i i 1 MT dil[(1 ) dil ]. The eective stiness tensor or the coposite is then calculated as C v C A (A20) i MT i i MT Where, the index i reers to the various phases o the coposite. For two-phase ibre coposites, explicit Mori-Tanaka expressions or the eective aterial constants can be ound in Dvorak[28]. The Mori-Tanaka odel gives predictions close to those o the Hashin- Hill lower bound in the case o ibres stier than the atrix, while it approaches the Hashin- Hill upper bound in the case o copliant ibres in a relatively stier atrix; in this case the odel is oten reerred to as the inverse Mori-Tanaka odel C. IMT A.7 Double Inclusion Model A scheatic description o this odel, suggested by Neat-Nasser and Hori [29], is given in Fig. 1b. An inclusion o stiness C is surrounded by a shell o atrix aterial (o stiness C ) which in turn is ebedded in a reerence aterial o stiness C r. The odel reduces to the M-T ethod or the choice o Cr C, while it recovers the inverse M-T ethod or 32

33 C r C. Lielens [2] proposed an interpolation between these two liiting cases in ters o a rule-o-ixtures based on the volue raction o inclusions, as 2 2 [(1 v v ) 1 ( v v ) 1 ] 1 C C C (A21) Lielens ( MT ) ( IMT ) 2 2 It was ound that the above expression gives accurate predictions o the aterials properties over a wide range o volue ractions [30,31]. 33

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