TECHNISCHE MECHANIK, 3, -5, (1), 484-56 Subitted: Noveber 1, 11 Multi-Scale Statistical Approach o the Elastic and Theral Behavior o a Theroplastic Polyaid-Glass Fiber Coposite M. M. Ouarou, D. Jeulin, J. Renard, P. Castaing The strong heterogeneity and the anisotropy o coposite aterials require a rigorous and precise analysis as a result o their ipact on local properties. First, echanical tests are perored to deterine the acroscopical behavior o a polyaid glass iber coposite. Then we ocus on the inluence o the heterogeneities o the icrostructure on theral and echanical properties ro inite eleent calculations on the real icrostructure, ater plane strain assuptions. 1 iages in 1 dierent sizes (5, 1, 15,, 5, 3, 35, 4, 45, 6 pixels) are analysed. The inluence o the area raction and the spatial arrangeent o ibers is then established. For the theral conductivity and the bulk odulus the iber area raction is the ost iportant actor. These properties are iproved by increasing the area raction. On the other hand, or the shear odulus, the ibers spatial arrangeent plays the paraount role i the size o the icrostructure is saller than the RVE. Thereore, to ake a good prediction ro a ulti-scale approach the knowledge o the RVE is undaental. By a statistical approach and a nuerical hoogenization ethod, we deterine the RVE o the coposite or the elastic behavior (shear and bulk oduli), the theral behavior (theral conductivity), and or the area raction. There is a relatively good agreeent between the eective properties o this RVE and the experiental acroscopical behavior. These eective properties are estiated by the Hashin-Shtrikan lower bound. 1 Introduction This work is carried out on a coposite with a theroplastic atrix reinorced by continuous glass ibers. Tests at acroscopical scale give relatively hoogeneous results with a reduced standard deviation. However, the icrostructure analysis o the sae saples shows several heterogeneities related to the anuacturing process o the aterial and the geoetry o their coponents. The study o the icrostructure o heterogeneous aterials becae an iportant branch o research during the ive last decades (Hill, 1963; Hashin and Shtrikan, 1963; Matheron, 1971; Willis, 1981; Serra, 198; Jeulin, 1991; Drugan and Willis, 1996). To have a better prediction o these aterials, dierent scales have to be considered: icroscopical scale (size o ibers), esoscopical scale (interediate size) and acroscopical scale (size o saples). This is coonly called a ulti-scale approach (Jeulin, 1991; Jeulin and Ostoja-Starzewski, 1). The ai o this approach is to deterine the sallest volue which allows or a good estiation o the acroscopical properties o the aterial, and which will be large enough to take into account all heterogeneities at a icroscopical scale. It is the Representative Volue Eleent (RVE) (Sun and Vaidya, 1996; Andrei, 1997; Ostoja-Starzewski, ; Shan and Gokhale, ; Kanit et al., 3; Xiangdong and Ostoja-Starzewski, 6; Ostoja-Starzewski, 6; Trias et al., 6; Gitan et al., 7; Gruan and Ellyin, 7; Zean and Sejnoha, 7; Thoas et al., 8; Frank Xu and Chen, 9). The RVE becae a scientiic concept with any interpretations according to various authors. Soe authors choose the geoetrical criteria o the coponents (Willis, 1981; Ostoja-Starzewski, 1993, 1998,, 6, 7; Jiang et al., 1). Moreover, it is shown that the RVE is related to the ratio o the size o the icrostructure on the diaeter o ibers. This ratio equals 5 (Trias et al., 6). Others showed the inluence o the contour o the icrostructures, as well as the volue raction o the clusters (Bhattacharyya and Lagoudas, ; Jiang et al., 1; Segurado and LIorca, 6; Jan et al., 6). In other works, the RVE is deterined according to the ibers spatial arrangeent and the distance between ibers (Jiang et al., 1; Knight et al., 3). Thoas et al. (8), ollowing Kanit et al. (3), ixed a relative error and the sallest RVE is the irst size or which the precision reaches this value. On the other hand, they stipulate that the RVE is reached only when (the standard deviation) is iniu and tends to a rather constant value. They applied this approach to the iber area raction and the theral conductivity o a carbon-epoxy coposite. Each criterion used is supposed to be the best ethod to deterine the RVE, according to its author. 484
In our case, the RVE cannot only depend on the size o ibers and icrostructure. The diaeter o ibers varies uch with a coeicient o variation o approxiately 13%. When we consider only one size o the diaeter, we ove away ro the real icrostructure o the studied coposite. In addition, in Trias et al. (6), it is stipulated that the inhoogeneities in the particle spatial distribution had a negligible inluence on the eective properties o the coposite in the elastic and plastic regies. This is true, in our case, only i the volue is larger than the sallest RVE. On the contrary, there is a property (shear odulus), or which this inhoogeneity (spatial arrangeent) can play a ore iportant role than the area raction. To deterine a statistical RVE objectively, Kanit et al. (3) have developed the use o statistical tools able to take into account all these dispersions (area raction, size o ibers, contour, ibers spatial arrangeent, size o icrostructures, precision, standard deviation, etc). We adopt this ethod to deterine the RVE or the area raction, the theral conductivity, and the elastic behavior o the coposite. This RVE depends on the easured property. The inal RVE will be the largest o all RVEs. It will be shown that it is equal to 854 854 (or N = 1 iages), and contains 131 ibers in average. The eective properties o this RVE are copared to the acroscopical behavior o the coposite. These properties can be locally estiated by Hashin-Shtrikan lower bounds. Macroscopic Approach.1 Material and Method The aterial is a coposite ade o a polyaid atrix (PA6) and continuous glass ibers reinorceent (Figure1-a). Its abrication is based on a hoogeneous ixture o ibers and atrix. This innovative product brings several advantages such as the recyclability, the weldability, a good ipregnation o glass ibers by the resin, and the absence o volatile organic copounds. In the acroscopical approach, the echanical behavior (bulk and shear oduli) and the volue raction o the coposite are studied. a) b) Figure 1. a) Plate o coposite ( 3 3. 4 ); b) Saples or echanical tests.1.1 Mechanical Behaviour Plates o a unidirectional coposite are provided by CETIM (Centre Technique des Industries Mécaniques), Figure 1-a. A unidirectional coposite (with long ibers reinorceent) is characterized by a transversely isotropic behavior (Berthelot, 1999; Bunsel and Renard, 5) which is deined by the knowledge o 5 coeicients in the elastic regie: ( EL, ET, GLT, n LT, n TT ' ), where E L is the longitudinal odulus, E T the transverse odulus, G LT the longitudinal shear odulus, n LT the longitudinal Poisson s ratio and n TT ' the transverse Poisson s ratio. These coeicients are obtained by the generalized Hooke s law, in which a linear relation is established between the strain and the stress ields, s ij = Cijkle kl or e ij = Sijkls kl (1) Where C ijkl is the stiness atrix and S ijkl the coplience atrix. Saples o diensions 5 are cut, Figure 1-b. This size is so large that all local dispersions are reoved. Standards into epoxy with 6 o length are stuck to the edges o the saples (Figure 1-b) to avoid 485
daage starting there, as a result o the claping orce. The working length is 8, suicient to place extensoeters, and to satisy the principle o Saint Venant. The oduli are deterined by quasi static tensile tests carried out on an Instron s dynaoeter, equipped with a loading cell o 1 kn. Displaceent control is used. Two extensoeters record longitudinal and transverse displaceent, and enable us to deterine the Poisson s ratios. These tensile tests are perored in the direction parallel to ibers (or E L and n LT ), perpendicular to ibers (or E T, and n TT ' ), and in 45 copared to the direction o ibers (or G LT ) (Figure 1-b). The stiness atrix is deterined by reversing the coplience atrix given bellow S ijkl È 1/ EL Í Í -n LT / E Í-n LT / E = Í Í Í Í ÍÎ L L -n -n TL 1/ E TT ' / E T T / E T -n -n TL TT ' 1/ E / E T T / E T 1/ G TT ' 1/ G LT 1/ G LT () The other coeicients ( n TL, G TT ' ) o the stiness atrix are deterined by the relations due to the syetries o the transversly isotropic behavior, as given in equaitons bellow ET n TL = n LT (3) E G TT ' L ET = (4) ( 1 n ) TT '.1. Volue Fraction Saples o diensions 5 are cut out in the plates and weighed. Then they are placed in an oven at 5 C during 1 hour or pyrolysis, to deterine the ass raction o the ibers (Figure ). This enables us to deterine the acroscopical ibers volue raction.. Experiental Results..1 Mechanical Behavior Figure. Glass ibers ater pyrolysis Ater tensile tests, the echanical acroscopic behavior is given in Table 1. Then we obtain the stiness atrix o the coposite. Table 1. Experiental result E L E T G LT n LT n TT ' 35 GPa 6.3 GPa 1.66 GPa.34.5 486
C acro È38.39 Í Í 4.84 Í 4.84 = Í Í Í Í ÍÎ 4.84 9. 4.81 4.84 4.81 9..1 1.66 GPa 1.66 For sipliications, we consider the bulk odulus ( K = ( C 3 ) C / ) and the transverse shear odulus ( = G TT ' ), which can represent this elastic behavior, and which are given by: K acro = 6. 9 GPa and 3 acro =. 1GPa... Volue Fraction Ater pyrolysis, the density o the coposite is deterined (1.616 g.c - 3 ). The rate o porosity (which is the ratio between the theorical and easured density) is o.7%. The ibers ass raction (Figure ) is deterined as 59%. Thus, the acroscopical ibers volue raction is 4%. 3 Microscopic Approach In this approach, the coposite is not regarded as one indissociable aterial. It is rather regarded as a ield ade up o separable physical entities: ibers and atrix, and, possibly, inclusions and deects in the atrix (voids). In this procedure, only the ibers and the atrix are taken into account. The purpose o this section is to deterine ro a ultiscale approach the RVE o studied properties (area raction, theral conductivity, elastic oduli) according to the relative error, by using orphological and statistical tools. Then by a nuerical hoogenization the eective properties o this RVE will be deterined and copared to the acroscopical behavior characterized above. First, an iage analysis is perored to characterize the icrostructure o the coposite using the sotware MICROMORPH, developed by Centre de Morphologie athéatique o Ecole des Mines de Paris. The analysed iages are ro SEM (Scanning Electron Microscop) ater polishing. 3.1 Iage Analysis The characterization o a rando ediu is based on orphological easuring criteria. These criteria are based on the stereology (surace and volue raction easureent) size distribution (D, 3D), distribution in space (clusters, anisotropy, etc) and connectivity. The principle o this easureent is based on two steps: orphological transoration applied to the icrostructure, and easureent o objects contained in the transored icrostructure (Serra, 198). The realization o these steps requires well adapted orphological tools which take into account all dispersions (local dispersions (area raction, size o ibers), and ibers spatial arrangeent). These dispersions ust be taken into account in the deterination o the RVE. Consequently it is not exact to consider a unit cell as representative volue eleent or this type o aterial. The icrostructures resulting ro the SEM are on grey level (luinosity between and 55, Figure 3-a-b) with a depth o 8 bits per pixel. The irst transoration consists o aking the binary (Figure 4-a). 487
a) b) Figure 3. a) Fibers spatial arrangeent (scale = ); b) Local luctuations o icrostructures a) b) Figure 4. Iage segentation The separation o ibers was ade by an iage segentation procedure, so that the detected ibers reain surrounded by the atrix in the urther steps o the calculations (Figure 4-b). These stages enable one to apply the orphological tools such as the covariance and the integral range which are briely described bellow. 3.1.1 The Covariance The covariance provides several inoration o the icrostructure, such as the spatial distribution o the coponents and their size, the anisotropy (or not), the periodicity and the volue raction (area raction in D). Morphologically it is deined or a subset K ade up o the couple o points { x } and { x h} by the probability that these two points belong to the sae set A. 488
C ( x x h) = P{ x Œ A, x h Œ A} 3.1. The Integral Range, (5) The integral range is a very iportant concept or the statistical processing o the heterogeneous icrostructures (Matheron, 1971; Serra, 198; Jeulin, 1991; Kanit et al., 3). For a given icrostructure, it deterines a doaine size or which the paraeters easured in this volue have a good statistical representativity. n In a space R, it can be written as An = C 1 ( ) - C( ) ( C( h) C( ) ) Ú - n R dh (6) Where A is the integral range and C (h) the covariance. n For a volue V (S in D), the variance o an average property Z in V can be expressed according to the integral range and the local variance i V >> then D ( V) An D Z being the local variance. 3. Deterination o the RVE Z An = DZ (7) V As seen previously, there are any deinitions or the RVE o heterogeneous aterials. The ost iportant ones are: Deinition 1: The RVE is (a) a saple that is structuraly entirely typical o the whole ixture on average, and (b) contains suicient nuber o inclusions or the apparent overal oduli to be eectively independent o the surace values o traction and displaceent, so long as these values are acroscopically unior (Hill, 1963). Deinition : The RVE is the sallest aterial volue eleent o the coposite or which the usual spatially constant overal odulus acoscopic constitutive representation is a suiciently accurate odel to represent ean constitutive response (Drugan and Willis, 1996). The RVE deines the esoscopical scale (Hill, 1963; Drugan and Willis, 1996; Jiang et al., 1; Kanit et al., 3; Ostoja-Starzewski,, 6; Trias et al., 6). To be practical and applicable, the Deinition 1 requires the use o statistical tools on icrostructures. And it enables one to obtain the evolution o the RVE according to the relative statistical error resulting ro luctuations o the icrostructure and o the ields (Kanit et al., 3). In act the accuracy o the estiation o the average property Z o a rando variable is given by the absolute error ( S ) D e Z abs = (8) n The relative error is then deduced ( S) abs DZ e rel = e = (9) Z Z n The variance o the rando variable Z can be written according to the local variance and the integral range, or very large area S (Matheron, 1971; Serra, 198; Kanit et al., 3): D Z ( S) A = DZ (1) S 489
By replacing the variance by its value in equation (1), the evolution o the statistical RVE is obtained according to the integral range, the average value Z, the nuber o observations (iages o the icrostructure), and the chosen relative error V ER Z r 4D A = (11) ne Z In practice, we can work with a single large iage (using n = 1 in equation (11)), or use n saller iages with an area equal to the RVE. In what ollows we will take n = 1. The Deinition provides the average o the ields o the easured properties (hoogenization), and akes it possible to choose that particular RVE which will be the sallest o RVEs which does not depend on boundary conditions (Hill, 1963; Kanit et al., 3). Both ethods appear copleentary rather than opposing. Still, the eleentary surace S o iages in equation (1) should be larger than the sallest size o the icrostructure, in order to insure that no bias is generated by boundary conditions in the estiation o the eective properties. Moreover, a heterogeneous aterial has a RVE or each property, and the inal RVE will be the larger one. 3.3 Hoogenization The hoogenization o a heterogeneous ediu (Figure 5) consists o deterining the equivalent hoogeneous ediu (EHM), ater reoving the local luctuations, (Hill, 1963; Willis, 1981; Torquato, 1991; Lukkassen et al, 1995; Neat-Nasser and Hori, 1999; Jeulin and Ostoja-Starzewski, 1). a) b) c) d) Figure 5. Hoogenization process. a) acroscopical scale; b) icroscopical scale; c) esoscopical scale; d) EHM In the next subsections, inite eleent ethod (FEM) hoogenization is studied and copared by the bounds estiation. The inluence o the boundary conditions on the easured properties is highlighted. 3.3.1 FEM Hoogenization The developent o new sotwares and reliable tools or icrostructures analysis akes the hoogenization by the inite eleent ethod (FEM) possible. By integrating all heterogeneities, the FEM is very eicient to estiate the eective properties. The eshing process is carried out with the sotware AVIZO, while Zebulon (developed by Ecole des Mines de Paris and ONERA) is used or inite eleent calculations. The icrostructures are prepared and eshed by the process in Figure 6. 49
Figure 6. Iage preparing and eshing process The plane strain assuption enables one to work in the plane perpendicular to the cross section o the ibers. The Cd3 type o eleent is used. For each property 1 iages o the icrostructure are considered or 1 dierent sizes. Speciic boundary conditions are needed to deterine the eective properties o the coposite (see the subsections bellow). The eective value is reached when the easured property does not depend on the type o boundary conditions any ore. The physical properties which will be studied by the FEM are given in Table. Table. Physical properties to be studied Proble Flux Measured ield Measured properties Heat conduction Heat lux Teperature gradient Theral conductivity Elasticity Stress/Strain Strain/Stress Elastic oduli a) FEM hoogenization o the theral conductivity In this part, a theral steady state proble is studied, by iposing a unior teperature gradient on the contours o iages o the icrostructures (see equation (1)). The theral conductivity is deterined by studying the heat lux o the coposite using Fourier s law (see equation (13)). T = Gx " x Œ V (1) q = -l T (13) ij ij Where q is the heat lux, T the teperature gradient, and l the theral conductivity tensor. The average heat lux is obtained by averaging the local ields in the icrostructure. Ú Q = 1 qdv (14) V V To highlight the inluence o the anisotropy o the icrostructure on the theral behavior, the gradient o teperature is iposed in two directions (horizontal and vertical). b) FEM hoogenization o the elastic properties For the echanical proble, we will study the bulk odulus and the transverse shear odulus o iages by assuing an isotropic elastic behavior o the ibers and o the atrix. 491
On each iage, kineatic unior boundary conditions called KUBC are iposed (see equation (15)). These conditions result in a unior displaceent o all points o the contour u = E x " x Œ V (15) ~ Thus, it is possible to deterine the average stress in the icrostructure by averaging the local stress ields (see equation 16) S = ~ 1 s = dv ~ V Ú s (16) V ~ It should be noted that there exist other boundary conditions such as SUBC (Static Unior Boundary Conditions which lead to unior stress conditions on the contours) and the periodic boundary conditions. However it is shown in several works that KUBC and SUBC conditions are two dierent ways to obtain asyptotically the sae result, provided the iages are large enough and thus statistically representative (Jeulin and Ostoja-Starzewski, 1; Jiang et al., ; Kanit et al., 3; Trias et al., 6; Zean and Sejnoha, 7). 3.3. Bounds Method Hoogenization The variational bounds and estiations have been used to deterine hoogenized properties o a rando ediu or several decades (Willis, 1981; Neat-Nasser and Hori, 1999; Jeulin and Ostoja-Starzewski, 1; Ostoja-Starzewski, 1998, 7). Soe o the are ore accurate than others. For exaple, the irst order bounds o Voigt and Reuss (Swan and Kosaka, 1997) only take into account the volue raction o the coponents. The Hashin-Shtrikan second order bounds take into account the volue raction and the isotropy o the icrostructure (Willis, 1981; Berthelot, 1999; Kanit et al, 3). In addition, the estiations o Mori Tanaka take into account the spherical particles in an isotropic elastic atrix (Mori and Tanaka, 1973). In this work, only Hashin-Shtrikan bounds are studied. For a transversely isotropic coposite with rando arrangeent o ibers, the D Hashin-Shtrikan upper and lower bounds or shear odulus (see equation (17) and (18)), and bulk odulus (see equation (19) and ()) are given by (Berthelot, 1999) H 3 K K - H 3 H - H = = = K = K 1- Fv 1 K Fv - K K 1-1 - F 1 - K ( 1 - F ) v ( K ) F v v K ( K ) F v ( K ) ( K ) Fv 1 1 - Fv () - K (17) (18) (19) where Ei Ei Ki = = i =, 3 1 i ( - n ) ( 1 n ) i i or atrix and iber, Fv is the volue raction For the theral conductivy, the upper and lower bounds o Hashin-Shtrikan in D are given by l HS = l l F 1 - l V V F l (1) 49
l HS- = l l F 1 - l V V F l () where l and l are (respectively) the theral conductivity o ibers and atrix. 3. 4 Nuerical Siulation and Iage Analysis Results 3.4.1 Fibers Spatial Arrangeent For a stationary rando set as will be assued here, the covariance does not depend on the point x. By taking h = in equation 3, one reaches the area raction (equivalent to the volue raction) o ibers, 4% (Figure 7-c). The changing o the covariance according to the direction (horizontal or vertical) shows the anisotropy o the icrostructure on scale o ply (Figure 7-a, and 7-b), and highlights its periodicity in the horizontal direction (Figure 7-a and 7-c). 5 45 4 35 3 5 15 1 5 1 3 4 5 6 7 8 Distance (u) a) 5 45 4 35 3 5 15 1 5 1 3 4 5 6 7 8 Distance (u) b) c) Figure 7. a) Horizontal covariance; b) Vertical covariance; c) Horizontal average covariance (or 1 icrostructures) Three inlexions appear on the covariance curve (Figure 7-c). The x coordinate o the irst inlexion point indicates the average value o the diaeter o the ibers ( 16 ). Siilarly the average size o plys ( 3 ), and the average distance between centers o plys ( 439 ) and thus the average size o the poor zone o the ibers ( 7 ) are obtained. 493
3.4. Fluctuations o Fibers Area Fraction To deterine the area raction by iage analysis, 1 iages o dierent sizes ( 5 5,1 1,15 15,,5 5,3 3,35 35,4 4,45 45,6 6 pixels) are randoly taken in the icrostructure. It can be observed that the area raction does not depend on the size o the studied iage (Figure 8-a), and that the variance and the interval o conidence decrease when the size o the iage increases (Figure 8-b), (Drugan and Willis, 1996; Kanit et al., 3). The average area raction is 4% (Figure 8-a). AF,75,65,55,45,35,5,15 DFV (S),16,14,1,1,8,6,4,5 1 3 4 5 6 7 sqrt (S) (Pixels) a) b) Figure 8. a) Area raction according to the iage size; b) Variance o the area raction 3.4.3 Fibers Size Fluctuations, 15 35 55 75 95 115 135 155 sqrt (S) (µ) To analyse the dispersions o the size o the ibers, three iages o 6 6 pixels ( 1464 1464 ) were randoly taken. Each one o these iages contains ore than 3 ibers. This akes a total o ore than 11. ibers to be analyzed. The results are given in Table 3. Table 3. Size o ibers and dispersions (or 1137 ibers) Average Standard in deviation average diaeter o ibers ax Coeicient o variation Interval o conidence at 95% 16 1.9 1 1.37 % [15.96 16.4] 3.4.4 Inluence o Fibers Spatial Arrangeent on the Local Theral Conductivity - For the calculations, the input theral conductivities o the ibers and o the atrix are respectively ( ) 1 - and. ( K ) 1 W. 1. W K The FEM calculations provide the ields o the heat lux (Figure 9). We thus deterine the theral conductivity o the coposite in two directions (orthogonal and parallel to the beas o ibers) or iages o dierent sizes. 494
a) T b) T 11 Figure 9. Fields o heat lux ater FEM siulation I the teperature gradient is applied orthogonally to the beas o ibers, the theral conductivity is lower. However, it reains quasi hoogeneous in the icrostructure (Figure 9-a). It is higher in the case o a gradient parallel to the beas, with the risk o local overheating in zones with strong density o iber (Figure 9-b). A relationship between the local area raction and the theral conductivity o 1 dierent iages o the icrostructures ro dierent sizes is exained. It can be observed in both cases that the theral conductivity depends ainly on the teperature gradient and especially the iber area raction. This is shown by the quasi linear relation in Figure 1-a and 1-b.,6,6,5 11 ג,5 ג ) -1.K (W. ג.,4,3, ) -1.K (W. ג.,4,3,,1,1 1 3 4 5 6 Area raction (%) a) 1 3 4 5 6 Area raction (%) b) Area raction (%) c) Figure 1. a) and b): local theral conductivity as a unction o the area raction; c) Local Theral conductivity copared to the bounds o Hashin- Shtrikan 495
Furtherore, or the ajority o the iages the local theral conductivity can be directly estiated by the lower Hashin-Shtrikan bounds, which corresponds to the Hashin coated disks icrostructure, not very ar ro the non overlapping discs generated by the iber section (Figure 1-c). Nevertheless dispersions are observed around bounds. 3.4.5 Inluence o Fibers Spatial Arrangeent on Local Elastic Moduli As has been seen previously in the acroscopical approach, the echanical behavior will be represented by the bulk and shear oduli. Thus, or nuerical investigations, we only copute these oduli which will be copared to the acroscopic values. The ibers and the atrix are assued to have a linear isotropic behavior. The input data or the FEM calculations are given in Table 4. Plane strain FE calculations are perored. Table 4. Input data or nuerical coputations o elastic oduli Matrix(PA6) Glass iber Young s odulus (MPa) 7 Poisson s ratio.39. K (MPa) 9 8 7 6 5 4 3 1 1 3 4 5 6 G3 (MPa) 35 3 5 15 1 5 V = 6 % G3 =,4 GPa V = 4 % G3 =,3 GPa 1 3 4 5 6 Area raction (%) Area raction (%) a) b) Area raction (%) Area raction (%) c) d) Figure 11. a) and b) Dependence o the local shear and bulk oduli to the local area raction and ibers arrangeent; c) and d) coparison o the local oduli to the Hashin-Shtrikan bounds derived ro the local iber area raction. Figure 11 (11-a, and 11-b) shows the relation between the local elastic oduli and the local iber area raction (relecting the ibers spatial arrangeent). It is observed that the bulk odulus is strongly dependent on the local area raction, and not on the ibers arrangeent. This is showed by the quasi linear relation in Figure 11-a. All that is necessary to iprove this property is to increase the iber area raction. 496
The shear odulus depends uch ore on the arrangeent o the ibers than on the area raction, or soe icrostructures. One can notice on Figure 11-b that the point with 4% o area raction has the lower shear odulus than the other with 6% o area raction. Thus, the ibers spatial arrangeent plays a role ore prevalent than the area raction, as long as the icrostructure does not reach a certain size. A coparison between local estiations and Hashin-Shtrikan bounds calculated ro local area ractions is ade in Figure 11-c and 11-d. It is shown that one cannot exactly estiate the elastic oduli or all sizes o the icrostructure with the bounds. Dispersion is observed around the, ainly or the saller iages which are ore sensitive to the boundary conditions eect. 3.4.6 Deterination o RVE 3.4.6.1 The RVE o the Area Fraction The integral range is deduced ro the slope o the curve which relates the variance and the inverse o the area (see equation (1) and Figure 1-a). D Z(S)/(Fv(1-Fv)),18,16,14,1,1,8,6,4, 16 14 A Fv,5,1,15,,5,3,35 1/S (µ - ) a) Relative error (%) 1 1 8 6 4 1 3 4 5 6 7 Size o RVE (µ) b) Figure 1. a) The integral range o the area raction; b) RVE according to the relative error Table 5. The RVE o the area raction according to the relative error or 1 iages Relative error (%) 1 3 4 5 8 1 15 Size o RVE ( ) 1311 656 437 38 63 164 13 88 497
The integral range is F A V = 55 55. It is saller than the sallest size o iages used in this study ( 5 5 pixels = 1 1 ), which validates the use o the asyptotic relation (see equation (5)) to copute the variance o estiation. One can observe that the choice o the RVE depends on the desired accuracy (Figure 1- b). In Table 5 soe values o RVE are given or various relative precisions and or 1 iages. It is instructive to copare our results to those obtained or a carbon-epoxy coposite (Thoas et al., 8): i we consider a single iage ( n = 1) or a % and 3% precision, the size are presently 6.56, and 4.37 in our case, instead o 611 and 47 given in Thoas et al. (8). This actor ten results ro a uch higher heterogeneity o the distribution o glass ibers in the present case, due to the presence o layers o resin as shown in Figure 3, resulting in a uch higher correlation length deduced ro the covariance (typically 6 as seen on Figure 7-c), instead o 1 in Thoas et al. (8). 3.4.6. The RVE o the Theral Conductivity Relative error (%) D Z(S),8,7,6,5,4,3,,1 1 3 4 5 6 7 Sqrt (S) (pixels) a) 16 14 1 1 8 6 4 1 3 4 5 6 7 RVE (µ) c) D z(s)/d Z ( app(w.-1kג.,,18,16,14 A,1,1,8,6,4, E-6 4E-6 6E-6 8E-6 1E-5 1E-5 1E-5 E-5 E-5 b) 1/S (µ - ),4,39,38,37 UGT 1 UGT,36,35,34,33,3,31,3 4 6 8 1 1 14 16 Sqrt (S) (µ) d) 498
Area raction (%) e) Figure 13. a) Variance; b) Integral range; c) RVEs according to the relative conductivities; e) Bounds o Hashin- Shtrikan or S > RVE (or n = 1 ) precision; d) Hoogenised Table 6. The RVE o the theral conductivity according to the relative error, or n = 1 iages Relative error (%) 1 3 4 5 8 1 1 Size o RVE ( ) 969 485 33 43 194 1 97 81 By a local post processing, ater FEM calculations, the local point variances o dierent size o iages o the icrostructure are deterined. This enables one to deterine the integral range which is equal to l A = 9 9. The assuption or the validity o the orula (5) is then satisied. To draw the evolution o the RVE (see equation (11)), the local point variance o the large icrostructure - ) is deterined ( ( ) 1 ( 1464 1464 - D = 3.5973 1 l W K ). The size o the RVE o the theral conductivity (given or n = 1 iages) associated to a chosen relative error is given in Table 6. For a single iage ( n = 1) and or 1% o accuracy this size is very large ( 9.69 ), as copared to the RVE o the theral conductivity o a carbon epoxy coposite ( 8 ) in Thoas et al (8) or the sae contrast o conductivities between the atrix and the ibers. Again this is a consequence o a uch larger correlation length in the present case. The FEM hoogenization akes it possible to choose the RVE, which is the sallest size reed ro the eects o the boundary conditions (Figure 13-d), and which is equal to 73 73. Its relative error is 1.3% when using 1 iages o this size, and 13.% or only one iage. For iages with a larger area the behavior is alost constant and is called eective behavior (Figure 13-d). The ibers, being ive ties ore heat conducting than the atrix, control the eective theral conductivity. The eective conductivity tensor is given by l e 3.36-1 - È = Í 1 W K Î 3.44 ( ) 1 One can notice incidently that the eective theral conductivity shows a slightly anisotropic behavior in the two ain directions as a result o a low contrast between the conductivities o the ibers and o the atrix. This dierence is o %. In addition, ro iages larger than the RVE, the theral conductivity is correctly estiated by the Hashin- Shtrikan lower bound, (Figure 13-e), with less dispersion than in Figure 1 c. A ore precise estiation o the theral conductivity is provided by the 3D Hashin-Shtrikan lower bound (Figure 14). 499
Figure 14. Bounds o Hashin & Shtrikan or Area raction (%) S > RVE (or n = 1 and 3D analytical orula) 3.4.6.3 RVE o the Elastic Moduli a) RVE o the Bulk odulus 7,E5,45 D Z(S) 6,E5 5,E5 4,E5 3,E5,E5 1,E5 K D Z(S)/D Z,4,35,3,5,,15,1,5 A k,e 1 1 1 3 4 5 6 7 Sqrt (S) (µ) a) 8,E 3,E-6 6,E-6 9,E-6 1,E-5,E-5,E-5 b) 1/S (µ - ) Relative error (%) 8 6 4 1 3 4 5 6 7 Size o RVE (µ) c) K (MPa) 7 6 5 4 5 5 75 1 15 15 Sqrt (S) (µ) d) 5
Area raction (%) e) Figure 15. a) Variance; b) Integral range, c) RVEs according to the relative precision, d) Hoogenised bulk odulus, e) local bulk odulus estiation or volue larger than the RVE Table 7. The RVE o the Bulk odulus according to the relative error or n = 1 iages Relative error (%) 1 3 4 5 8 1 5 Size o RVE 11 K (µ) 158 53 353 65 1 133 16 43 The bulk odulus local variance is D K = 581685MPa. The integral range is A K = 43 43. The size o the RVE o the bulk odulus associated to a chosen relative error is given in Table 7 or 1 iages. By the FEM, as previously seen, we deterine the sallest RVE which is not inluenced by the eect o boundary conditions (Figure 15-d). It is the RVE o the bulk odulus, and it is equals to 488 488, with an associated relative error o.17 % (or n = 1 iages). While considering only one iage o this size, the relative error o this RVE is 1.7%. Its eective value is 5.94GPa. In Figure 15-e it is observed that the Hashin-Shtrikan bounds estiation is ore accurate with iage size larger than the RVE or heterogeneous aterial (see or coparison the Figures 11-c and 11-d). This shows again the paraount iportance o the RVE o heterogenous aterial in the predicting o the acroscopical behavior. This is very useul or echanical design. b) RVE o Shear odulus 9,E5, 8,E5,16 D Z(S) 6,E5 5,E5 3,E5 µ D Z(S)/D Z,1,8 A µ,e5,4,e 3 6 9 1 15 Sqrt (S) (µ) a),35,7,15,14,175 1/S (µ - ) b) 51
1,5 9 3,4E3 Relative error (%) 7,5 6 4,5 3 µ (MPa),8E3,E3 1,6E3 1,5 1,E3 4 6 8 1 Size o RVE (µ) c) 4,E 5 5 75 1 15 15 Sqrt (S) (µ) d) Area raction (%) e) Figure 16. a) Variance; b) Integral range, c) RVEs according to the precision, d) Hoogenised shear odulus, e) local shear odulus estiation or volue larger than the RVE The local point variance or the transverse shear odulus is Table 8. The RVE o the shear odulus according to the relative error or n = 1 iages Relative error (%) 1 1.5 3 4 5 1 11 Size 11 RVE ( ) D = 53355 MPa. The integral range is A = 73 73. The size o the RVE o the transverse shear odulus associated to a choosen relative error or 1 iages is given in Table 8. As or the previous steps, the RVE reed ro boundary conditions is deterined by FEM hoogenization, and 1594 163 797 53 399 319 16 145 equals 854 854, with an associated relative error o 1.87% (or 1 iages) and 18.7% or only one iage. The eective transverse shear odulus o the RVE is.1gpa (Figure 16-d). The shear odulus has the larger RVE, as a result o its strong dependence o the ibers spatial arrangeent. Thereore, this RVE will be the inal RVE o the studied properties (area raction, theral conductivity, bulk odulus, transverse shear odulus). The relative errors induced on every property with this size o RVE or 1 iages are given in Table 9. 5
Table 9. The induced relative error on properties with an RVE size o 854 Relative error (%) Volue Fraction Theral conductivity Bulk odulus, n = 1 Shear odulus 1.19 1.134 1.4 1.87 4 Coparison between Experiental and Nuerical Results Table 1 provides the coparison between the experiental and the nuerical calculations results. Table 1. Coparison between experiental and nuerical results with an RVE size o 854 Volue Fraction Theral conductivity Bulk odulus (GPa) Shear odulus (GPa) Experiental 4% - 6.9.1 Nuerical 4%.336 /.344 5.94.1 The volue raction deterined by pyrolysis is uch closed to the area raction (D) provided by iage analysis (4 %). This volue raction is deterined beore iages segentation or nuerical siulations. The underestiation o the bulk and the shear oduli (Table (1)) by the eective behavior o the RVE can be explained: On one hand, the segentation has probably reduced the size and the nuber o ibers, and thus the coposite strength in nuerical coputations. On the other hand, the ibers and atrix input data we used or nuerical coputations could be underestiated. However, a coparision between the calculated and the acroscopical theral conductivity o the coposite is not ade, since no experiental easureents are available in the present study. 5 Conclusion Firstly, echanical tests were perored and the acroscopical elastic behavior is ully characterized by a transversely isotropic stiness atrix. This is the acroscopical approach. Then we carried out a ulti-scale iage analysis (icroscopical and esoscopical approach), with a large nuber o iages (1 iages with 1 sizes). This step enabled us to highlight the inluence o the various luctuations (local area raction and ibers spatial arrangeent) on the echanical and theral behavior o the coposite. It showed in addition the need to use a representative volue eleent (RVE) which takes into account all dispersions (local area raction, ibers spatial arrangeent, etc) or heterogeneous aterials. Several statistical tools such as the integral range and the covariance enabled us to study the evolution o the RVE according to the relative error. However, at this step the eect o boundary conditions on easured property is not known. The FEM hoogenization akes it possible to choose the irst volue (area in our case) which is reed ro boundary conditions eects and which is the RVE. Furtherore it is shown that the property which depends strongly on the iber spatial arrangeent requires the larger RVE. This was the case o the shear odulus ( 854 854, with 131 ibers). This RVE involves a relative error o 18.7% or one studied iage. The bulk odulus is the least sensitive property to the ibers distribution, with a size o RVE o 488 488. It involves a relative error o 1.7% or one studied iage. The RVE o the theral conductivity is 73 73 with 13.% o relative error or one studied iage. The inal RVE or these properties is the higher value o dierent RVEs ( 854 854 ). The eective properties o the RVE are copared to the acroscopical behavior o the coposite provided by experiental tests. A relatively good agreeent is then established. These results show that the acroscopical behavior o a heterogeneous aterial can be estiated by the calculation o iages o the icrostructure. 53
Due to the heterogeneity o the icrostructure o the studied aterial, a total area o around 5 ust be scanned in the aterial to estiate its orphological, theral and elastic properties with a 1% statistical property. Besides, it appeared that the local conductivity and elastic oduli could be estiated by the lower Hashin- Shtrikan bound, provided the size o iages is larger than the RVE. The slight dierence between the local oduli and the Hashin-Shtrikan lower bound could result ro the geoetry o ibers, which are not exactly circular (Figure 4) as a consequence o the polishing and segentation process. The originality o these bound estiation, is that one could predict the local luctuations o the properties o the coposite at dierent scales ro the local luctuations o the iber area raction that can be accessed ro iage analysis or ro non destructive testing, such as ultrasonic easureents ade on a uch larger scale in Guilleinot et al. (8). Acknowledgeents: This work was done as a part o the Probadur project. The authors are grateul to the Institute Carnot M.I.N.E.S or supporting this research and CETIM or providing the aterial. Reerences Andrei A A: Representative volue eleent size or elastic coposites: a nuerical study. Journal o the Mechanics and Physics o Solids, 45, (1997), 1449-1459. Berggren S A; Lukkassen D; Meidell A; Siula L; Narvik: Soe ethods or calculating stiness properties o periodic structures. Applications o atheatics,, (3), 97-11. Berthelot J M : Matériaux coposites. Coporteent écanique et analyse des structures. Editions Tec & doc, (1999), p. 64. Bhattacharyya A; Lagoudas D C: Eective elastic oduli o two-phase transversely isotropic coposites with aligned clustered ibers. Acta Mechanica, 145, (), 65-95. Bunsel A R; Renard J: Fundaentals o ibre reinorcced coposite aterials. Series in Materials Science and Engineering IoP, (5) p. 398. Drugan W J; Willis J R: A icroechanics-based nonlocal constitutive equation and estiates o representative volue eleent size or elastic coposites. Journal o the Mechanics and Physics o Solids, 44, (1996), 497-54. Frank Xu X; Chen X: Stochastic hoogenization o rando elastic ulti-phase coposites and size quantiication o representative volue eleent. Mechanics o Materials, 41, (9), 174-186. Gitan I M; Askes H; Sluys L J: Representative volue: existence and size deterination. Engineering Fracture Mechanics, 74, (7), 518-534. Gruan C; Ellyin F: Deterining a representative volue eleent capturing the orphology o ibre reinorced polyer coposites. Coposite Science and Technology, 67, (7), 766-775. Guilleinot J; Soize C; Kondo D; Binetruy C: Theoretical raework and experiental procedure or odeling esoscopic volue raction stochastic luctuations in iber reinorced coposites. International Journal o Solids and Structures, 45, (8), 5567 5583 Hashin Z; Shtrikan S: A variational approach to the theory o the elastic behaviour o ultiphase aterials. J. Mech. Phys. Solids, 11, (1963), 17-14. Hill R: Elastic properties o reinorced solids: soe theoretical principles. J. Mech. Phys. Solids, 11, (1963), 357-37. Jan G; Jan Z; Michal S: Quantitative analysis o iber coposite icrostructure: Inluence o boundary conditions. Probabilistic Engineering Mechanics, 1, (6), 317-39. 54
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Thoas M; Boyard N; Perez L; Jarny Y; Delaunay D: Representative volue eleent o anisotropic unidirectional carbon-epoxy coposite with high-ibre volue raction. Coposites Science and Technology, 68, (8), 3184-319. Torquato S: Rando heterogeneous edia: icrostructure and iproved bounds on eective properties. Applied Mechanics Reviews, 44, (1991), 37 76. Trias D; Costa J; Turon A; Hurtado J E: Deterination o the critical size o a statistical representative volue eleent (SRVE) or carbon reinorced polyers. Acta aterialia, 54, (6), 3471-3484. Willis J R: Variational and related ethods or the overall properties o coposites. Advances in Applied Mechanics, 1, (1981), 1 78 Xiangdong D; Ostoja-Starzewski M: On the scaling ro statistical to representative volue eleent in theroelasticity o rando aterials. Networks and Heterogeneous Media, 1, (6), 59-74. Zean J; Sejnoha M: Fro rando icrostructures to representative volue eleents. Modelling and Siulation in Materials Science Engineering, 15, (7), S35-S335. Adresses: Maane M. Ouarou Ecole des Mines de Paris, Centre des atériaux, UMR CNRS 7633, BP 87, 913 Evry Cedex, France aane.ouarou@ines-paristech.r Doinique Jeulin, Proessor Mines Paristech, Centre de Morphologie Mathéatique, 35 rue St Honoré, 773 Fontainebleau, France doinique.jeulin@ines-paristech.r Jacques Renard, Proessor Ecole des Mines de Paris, Centre des atériaux, UMR CNRS 7633, BP 87, 913 Evry Cedex, France jacques.renard@ines-paristech.r Philippe Castaing, PhD CETIM, Centre Technique des Industries Mécaniques, 74 Route Joneliere, 44 Nantes, France philippe.castaing@ceti.r 56