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1 Journal of Computational Physics 228 (2009) Contents lists available at ScienceDirect Journal of Computational Physics journal homepage: Elastic property of multiphase composites with ranom microstructures Moran Wang *,1, Ning Pan Nanomaterials in the Environment, Agriculture an Technology (NEAT), University of California at Davis, CA 95616, USA article info abstract Article history: Receive 30 August 2007 Receive in revise form 19 January 2009 Accepte 9 May 2009 Available online 15 May 2009 Keywors: Elastic property Young s moulus Multiphase composites Ranom structure Lattice Boltzmann metho We propose a computational metho with no a hoc empirical parameters to etermine the elastic properties of multiphase composites of complex geometries by numerically solving the stress strain relationships in heterogeneous materials. First the ranom microstructure of the multiphase composites is reprouce in our moel by the ranom generation-growth metho. Then a high-efficiency lattice Boltzmann metho is employe to solve the governing equation on the multiphase microstructures. After valiate against a few stanar solutions for simple geometries, the present metho is use to preict the effective elastic properties of real multiphase composites. The comparisons between the preictions an the existing experimental ata have shown that the effects of pores/ vois in composites are not negligible espite their seemingly tiny amounts. Ignorance of such effects will lea to over-preictions of the effective elastic properties compare with the experimental measurements. When the pores are taken into account an treate as a separate phase, the preicte Young s moulus, shear moulus an Poisson s ratio agree well with the available experimental ata. The present metho provies an alternative tool for analysis, esign an optimization of multiphase composite materials. Publishe by Elsevier Inc. 1. Introuction The problem of etermining the effective linear elastic properties of multiphase composites with complex microstructures is a classical, important an yet challenging issue, with applications in almost every area of material sciences [1 5]. The effective elastic moulus is one of the most important properties that characterize the mechanical performance of materials. As is well known now, the effective elastic moulus of a multiphase composite epens not only on the corresponing elastic properties an the volume fraction of each constitute component, but on the microstructures (i.e., the spatial istribution of the components) of the composite as well [4]. The theoretical approaches foun in the literature for preicting the effective elastic moulus are mostly base on the effective meium theories which in essence provie approximate estimations of the effective moulus by homogenizing the complex meium [6]. Inevitably empirical parameters have been introuce into the moels to account for the influence of the structural variations on the results [3,7]. Another alternative group of analytical approaches is to provie upper an lower bouns, which were rigorously erive theoretically an then valiate through experimental ata [1,8,9]. In terms of experimental techniques, there have been numbers of measurement schemes evelope over the years. However more often than not, it is ifficult to compare the theoretical preictions with the experimental ata, for many important features an mechanisms in a complex material, such as the shapes an spatial istribution of inclusions, an connections an interaction between ifferent phases, are ifficult to be accounte for in a theoretical moel, other than making a few rough assumptions to either ignore or over-simplify them. On the experimental * Corresponing author. Tel.: aresses: mmwang@ucavis.eu, moralwang@gmail.com (M. Wang). 1 Currently working at Los Alamos National Laboratory as an Oppenheimer Fellow /$ - see front matter Publishe by Elsevier Inc. oi: /j.jcp

2 M. Wang, N. Pan / Journal of Computational Physics 228 (2009) sie, the current technology is still not avance enough to etect many local, ranom an often irregular factors that collectively impact the material behaviors in a significant way. Furthermore, the existing theories fall even short when coming to esign non-existent novel materials, for instance the meta-materials, where initial esign an optimization have to rely on theories [3]. Computer numerical simulation has provie a promising new way in this regar, owing to the rapi evelopments of computers an computational techniques in the past few ecaes. A useful numerical scheme in ealing with complex materials generally shoul inclue two major components: reproucing the microstructures of the materials so as to bring the inherent structural complicities an internal interactions into the computer as the first an critical step, an then solving the relevant set of governing equations on the given structure. In terms of generating computerize material structures, several methos have been propose focusing on multiphase composites. The ranom location of obstacles propose by Zhang et al. in 2006 [10] is the simplest one in constructing an artificial ranom microstructure, however, the approach is too primitive to catch some of the important structural etails. Snyer et al. [3] propose a more avance metho to generate ranomly istribute circular inclusions on hexagonal pixels. Digital micro-tomographic information an statistical correlation functions have been aopte in reconstruction of the structures more accurately [9,11,12]. Inspire by the spirit of the cluster growing theory [13], Wang et al. [14] have recently evelope a simpler metho, the ranom generation-growth (RGG) metho, to reprouce the ranom microstructures of multiphase porous meia. The generate microstructure is conveniently controlle by a few statistical parameters each of which has a clear physical significance. The RGG metho has shown success in preicting thermal conuctivities of various multiphase meia, as valiate by the experimental ata [14,15]. After the reconstruction of materials via the algorithms, the relevant governing equations have to be solve. However such traitional PDE solvers as the finite-ifference scheme [4] or the popular finite-element technique [16,17] require vast gri refinements an thus eman huge computational resources when the geometries of the microstructure are complex, especially for multiphase conjugate cases. A high-efficiency lattice Boltzmann metho (LBM) has recently been evelope to tackle various PDEs with conservation an conjugation effects uring evolution of thermal an electrical transport problems [18,19]. Since complex geometry bounary conitions can be easy implemente in LBM [20 24], application of this approach to solving the governing equations of elastic mechanics in multiphase composites becomes a logical evelopment. This contribution aims to evelop a numerical metho in calculating or preicting the effective elastic properties of multiphase composites. We first evise the ranom generation-growth metho for reproucing the microstructures of threephase (3 P) composites an then solve the governing equations by the LBM metho. The calculate elastic mouli will be compare with existing experimental ata an the physical mechanisms involve will be iscusse. 2. The numerical scheme This section will escribe the etails of the numerical moel, its basic assumptions an the governing equations. The new approach reprouces the multiphase microstructure by the ranom generation-growth metho an solves the governing equations by the LBM algorithm Basic hypothesis an equations Consier a simple pure elastic compression test on a multiphase composite as shown in Fig. 1. The bottom surface of the omain is fixe an the top surface is uner a uniform normal compressive force, F. We suppose only small elastic strain occurs in the irection of the force without any multi-imensional effects. After reaching a steay state, the governing equation for escribing the isplacement fiel can be expresse by a simple Poisson equation [25], i.e. r feðrþr½aðrþuðrþšg ¼ 0 ð1þ where E is the local Young s moulus, r the position, A the cross-sectional area, an U the isplacement in the irection of force. Since no slippage is consiere, Eq. (1) is subjecte to the stress an strain continuities at each interface between two phases ði; jþ: Uj int;i ¼ Uj int;j E i ruj int;i ¼ E j ruj int;j ð2þ ð3þ The force F on the top surface leas to a secon-type (Neumann) bounary conition. However since the application of F, uner the pure elastic assumption, oes not alter the effective property noticeably, we use the Dirichlet bounary conition by assigning a known isplacement, U 1, at the top surface an U 0 at the bottom: Uj top ¼ U 1 Uj bottom ¼ U 0 ð4þ ð5þ While the perioic bounary conitions are implemente on the both sies. Once the isplacement fiel is solve, the effective Young s moulus of the composite is obtaine as

3 5980 M. Wang, N. Pan / Journal of Computational Physics 228 (2009) Fig. 1. Schematic iagram of stress an strain bounaries for a multiphase composite. E eff ¼ stress strain ¼ F=A DU=H where the external force F is calculate as F ¼ E ArU Eqs. (1) (7) escribe a simple yet novel way to etermine the effective Young s moulus of a heterogeneous multiphase composites by solving a Poisson equation at given conitions. Although is the case shown here is one-imensional, it coul be extene to three-imensional problems [26,27]. Another interesting thing is that if the external force on the top surface is a shear force, i.e. tractions on top an bottom surfaces, an the shear isplacement U only occurs in the shear irection, i.e., simple shear, the current metho can be extene to calculating the effective shear mouli of the system Structure reprouction We have propose a general ranom generation-growth algorithm in our previous work [14]. Here we aopt it for reproucing the microstructure of a composite of two soli components, one isperse into the other continuous matrix, an containing small amount of vois (or air), thus actually a three-phase system [28,29]. Before the generation process, we select the continuous soli phase as the non-growing phase, the isperse soli as the first growing phase an the vois (or pores) the secon growing phase. We use a superscript number below to inicate the corresponing growing phase. The growing process is then conucte as follows. (i) Ranomly locate cores/sees of the first growing phase in a gri system base on a core istribution probability, c 1, whose value is no greater than the volume fraction of the phase. Each cell in the gri will be assigne a ranom number of a uniform istribution function within (0,1). Each cell whose ranom number is no greater than c 1 will be chosen as a core/see; (ii) Expan every element of the growing phase to its neighboring cells in each irection base on the given irectional growth probability, D 1 i, where i represents the irection. Again for each growing element, new ranom numbers will be assigne to its neighboring cells. The neighboring cell in irection i will become part of the growing phase if its ranom number is no greater than D 1 i ; (iii) Repeat the growing process of (ii) until the volume fraction of the first growing phase reaches its given value / 1 ; (iv) As to the secon growing phase (pores), we treat the pores as another iscrete growing phase. Therefore it grows from separate sees, very similar to the process for the first growing phase escribe in (i) (iii), yet controlle by c 2 an D2 i corresponingly. (v) Stop the pores growth once its volume fraction reaches the given value / 2 (or more often expresse as the porosity e). Thus the generate microstructure of multiphase composites can be controlle by three parameters ðc ; D i ; /Þ for each growing phase. Every parameter in the generation process has a physical significance an can be etermine through experimental observation an measurements. The core istribution probability c is efine as the probability of a cell/gri to ð6þ ð7þ

4 M. Wang, N. Pan / Journal of Computational Physics 228 (2009) become to a core of the growing phase an it value epens on the number ensity of the growing units. For a growing phase with a given volume fraction v f, the value of c coul be etermine by c ¼ / V=ðN V p Þ ð8þ with V representing the total volume of the system, N the total gri number an V p the average volume of the growing particles/pores. The irectional growth probability D i is efine as the probability for a yet-to-be-occupie cell to merge into a neighboring cell in the ith irection so as to become part of the growing phase. An appropriate arrangement of the irectional growth probabilities may lea to an isotropic structure or any other irectional formations. For two-imensional cases, each gri cell has eight growing irections to its neighbors, as seen in Fig. 2. There are four main irections (1,2,3,4) an four iagonal irections (5, 6, 7, 8). To obtain an isotropic structure in such systems, we set both the main irectional growth probabilities D 1 4 an the iagonal irectional growth probabilities D 5 8 into a constant in each respective group, an the both constants in a fixe ratio. It is the relative value not the absolute value of D i that controls the anisotropy of structure. For instance by esignating the probabilities ratio, D 1 4 : D 5 8 ¼ 4, we get the irectional growth probability consistent with the equilibrium ensity istribution function for isotropic materials [30,31] The lattice Boltzmann algorithm The lattice Boltzmann metho (LBM) is intrinsically a mesoscopic approach base on the evolution of statistical istribution of particles on lattices, an has achieve consierable success in solving various PDEs [32 34,18,19]. The most important avantages of the LBM are the easy implementations of interparticle interactions an the complex geometry bounary conitions [20,21,28,29], an in general the conservation laws of mass or/an energy can hol automatically without aitional computational efforts [33]. Aitional scheme ealing with the multiphase conjugate bounary conitions has been evelope very recently using the LBM metho as well an has shown high computation efficiency [19]. For the governing equation, Eq. (1), we employ the evolution equation on iscrete lattices for each phase as g a ðr þ e a t ; t þ t Þ g a ðr; tþ ¼ 1 s ½g aðr; tþ g eq n a ðr; tþš; ð9þ The equilibrium istribution of the evolution variable, g a, for the two-imensional nine-spee (D2Q9) moel is 8 >< 0 a ¼ 0 g eq a ¼ U=6 a ¼ 1; 2; 3; 4 >: U=12 a ¼ 5; 6; 7; 8 the microscopic evolution velocity 8 >< ð0; 0Þ a ¼ 0 e a ¼ ðcos h a ; sin h a Þc; h a ¼ða 1Þp=2 a ¼ 1; 2; 3; 4 >: pffiffiffi 2 ðcos ha ; sin h a Þc; h a ¼ða 5Þp=2þp=4 a ¼ 5; 6; 7; 8 an the imensionless relaxation time s n ¼ 3 2 E n c 2 t þ 0:5 ð10þ ð11þ ð12þ Fig. 2. Eight growth irections of each point for 2D systems.

5 5982 M. Wang, N. Pan / Journal of Computational Physics 228 (2009) where the superscript n still represents the nth phase, t the time step, e r the relative ielectric constant, an c a pseuo soun spee whose value can theoretically take any positive value to insure s values within (0.5, 2) [18]. The isplacement an the force on each lattice are then calculate by U ¼ X a g a ð13þ F ¼ X a e a g a! s n 0:5 s n ð14þ For the Dirichlet bounary, we employ the bounce-back rule of the non-equilibrium istribution propose by Zou an He [35]. After the isplacement fiel is solve, the effective Young s moulus can then be etermine base on Eq. (6) as: E eff ¼ H R F L DU R L ð15þ 3. The existing theoretical moels To valiate our numerical metho initially, we will compare our results with those of the basic moels for materials with simple structures. The simplest theoretical approaches to preict the mouli of two-phase composites are the classical averaging schemes: the Voigt moel an the Reuss moel [36,37]. The Voigt moel assumes the constituents of a composite to be in parallel arrangement subjecte to the same strain (isostrain). The effective moulus of the composite is thus given by E eff ¼ E 1 / 1 þ E 2 ð1 / 1 Þ ð16þ where E 1 an E 2 are the Young s mouli of the phase 1 an phase 2, respectively, an / 1 is the volume fraction of the phase 1. In the Reuss moel, the constituents of the composite are uner the same stress (isostress) an the effective moulus is given by " # E eff ¼ /1 þ ð1 1 /1 Þ ð17þ E 1 E 2 Hashin an Shtrikman [1] evelope moels with assume macroscopical isotropy an quasi-homogeneity of composites, where the reinforcing particles are uniformly isperse into the matrix. Depening on whether the stiffness of the matrix material is larger or smaller than that of the reinforcement, the upper an lower bouns of the effective mechanical properties are calculate as: K u c ¼ K p þ K l c ¼ K m þ G u c ¼ G p þ G l c ¼ G m þ 1 / 1 K m K p þ 3/ 3K pþ4g p / 1 K p K m þ 3ð1 /Þ 3K mþ4g m 1 / 1 G m G p þ 6/ðKpþ2GpÞ 5G pð3k pþ4g pþ / 1 G p G m þ 6ð1 /ÞðKmþ2GmÞ 5G mð3k mþ4g mþ ð18þ ð19þ ð20þ ð21þ where / refers to the volume fraction of the reinforcing particles, K p ; K m an K c are the bulk mouli of the particles, matrix an composite, an G p ; G m an G c their shear mouli respectively. The superscripts u an l correspon to the upper an lower bouns, respectively. The upper an lower Hashin Shtrikman (HS) bouns for the Young s moulus of the composite can then be calculate by [8]: E HS ¼ 9K cg c 3K c þ G c where the upper an lower K c an G c values are use to etermine the respective E HS bouns. ð22þ 4. Results an iscussion The present methos will first be valiate by comparisons with the basic moels for composites of simple structures. Then they will be use to eal with multiphase composites with much more complex internal structures an the results will be verifie by the experimental ata.

6 M. Wang, N. Pan / Journal of Computational Physics 228 (2009) Reprouction of complex multiphase microstructures Using the present generation metho, we have reprouce microstructures for some real multiphase composites base on the experimental information. For instance, Tilbrook et al. [7] provie a series of microstructural images of alumina epoxy composite samples. Pores are clearly observable in the composites, although the porosity etails are not reporte. Their results show roughly that the porosity ecreases with the alumina volume fraction / Al 2O 3 when / Al 2O 3 P 50%, but increase with / Al 2O 3 otherwise. Since it is reporte that the porosity e was at most 5% in the Al 2 O 3 Epoxy composites, we estimate here a bi-linear relationship between the porosity e an the alumina volume fraction / Al 2O 3 8 < e ¼ 0:1 /Al 2O 3 ðv Al 2O 3 f < 50%Þ ð23þ : 0:1 ð1 / Al 2O 3 Þ ðv Al 2O 3 f P 50%Þ Fig. 3 shows six generate ranom microstructures of the Al 2 O 3 Epoxy Pore composites using our approach on a gri system, where the light phase is the alumina, the grey area is the epoxy an the ark spots are the pores. Compare with the microstructure images of the actual composites, see Fig. 1 in Ref. [7], the regenerate ones here have capture the structure etails an the stochastic characteristics. Fig. 3. Reprouce three-phase microstructures of alumina epoxy composites by the ranom generation-growth algorithm. c Al 2O 3 an c pore ¼ð1 / Al 2O 3 Þc Al 2O 3 for / Al 2O 3 P 50% or c pore ¼ / Al 2O 3 c Al 2O 3 for / Al 2O 3 < 50%. ¼ 0:1 / Al 2O 3 ð1 / Al 2O 3 Þ

7 5984 M. Wang, N. Pan / Journal of Computational Physics 228 (2009) Fig. 4. Two basic structures for valiations The benchmarks valiations To valiate our methos, we first compare our preicte effective elastic mouli for two simple structures, the parallel an series structures in Fig. 4, with those by the basic moels corresponingly. The Voigt moel an the Reuss moel give exact analytical solutions for these two simple structures in Eqs. (15) (16), an Fig. 5 compares the preicte effective mouli versus the volume fraction of phase 1, where E 1 ¼ 1 GPa an E 2 ¼ 10 GPa with our preictions. It is clear that the present methos agree very well with both basic moels. To further emonstrate the robustness of our approach, we allow the moulus ratio E 1 :E 2 to range from 1:2 to 1:10000 by keeping E 1 ¼ 1:0 GPa as a constant. Table 1 lists the calculate effective mouli comparing with the basic solutions for / ¼ 0:5. Such a large contrast between E 1 an E 2 leas to a long computational time for our algorithms to converge to a steay result, an yet provies a useful test on our moel performance. The eviations between the preictions are no greater than 0.006% for the parallel structure an 0.765% for the series structure, showing a goo accuracy of our approach The Young s moulus of complex materials After the valiations of our metho for simple structures, we apply it to investigating the effective elastic mouli of multiphase composites with complex microstructures. We use the ranom generation-growth metho to reprouce microstructures of the composites as introuce above an then solve the governing equations by the lattice Boltzmann metho to obtain the preictions. Then we compare in Fig. 6 our preictions with the measurements as well as the preictions by other theoretical moels in [38] for the effective Young s moulus of Al Al 2 O 3 composites. The properties use in the simulations are liste in Table 2. The present two-phase preictions (soli line with stars) in general seem to have over Fig. 5. Comparisons between our preicte effective moulus (symbols) with theoretical moels for isostrain an isostress moes (lines) at E 1 ¼ 1 GPa an E 2 ¼ 10 GPa.

8 M. Wang, N. Pan / Journal of Computational Physics 228 (2009) Table 1 Comparisons between preicte results an theoretical solutions where E 1 ¼ 1:0 ðgpaþ. E c Isostrain moe Isostress moe E 1 :E 2 Voigt moel (GPa) Present preictions (GPa) Relative eviations (%) Reuss moel (GPa) Present preictions (GPa) 1: : : : : : Relative eviations (%) Fig. 6. Comparisons of our two-phase preictions of effective Young s Moulus, as a function of alumina volume fraction, for Al Al 2 O 3 composites with experimental ata [38] an various theoretical moeling results. estimate composite property when the alumina volume fraction is lower than We also notice that the preicte HS lower boun are sometime even greater than the experimental ata, another sign of overestimation. A careful examination of the optical micrographs of the Al Al 2 O 3 composites in Fig. 1 by Moon et al. [38] shows some visible pores in the samples. We suspect such uncounte material efects may be responsible for the overestimation. Thus we moify our preictions by incluing some pores as another phase resiing in the composite. We estimate the porosity of such composite roughly at 5% when the alumina volume fraction is no greater than 90%, an at 2.5% otherwise. The elastic moulus of the air is assigne a small value E a ¼ 10 3 GPa in our simulations. Fig. 7 shows our preicte effective Young s mouli for such three-phase composites along with the same experimental ata. The agreement becomes much better in general over all the alumina volume fraction range. This inicates that the pores influence on the effect elastic properties of soli composites is not negligible even if the porosity is as small as 2.5%. Another case we calculate is the alumina epoxy composite iscusse in Section 4.1, an a bi-linear relationship between the porosity an the alumina volume fraction is alreay given in Eq. (23). We employ such a relationship as in Eq. (23) for our preictions. The properties of alumina an epoxy are liste in Table 2. Fig. 8 shows a goo agreement between the preicte effective Young s moulus an the experimental ata by Tilbrook et al. [7]. At a high moulus ratio between the two soli components ðe Al 2O 3 : E epoxy ¼ 390 : 3:4Þ, the HS bouns fail to provie acceptable estimations whereas the preictions by the present metho again exhibit high agreement with the experimental ata. If there were more etails on the porosity, the preiction accuracy coul be better. Table 2 Elastic properties of composite-materials. Property Al 2 O 3 Al Epoxy E (GPa) K (GPa) G (GPa)

9 5986 M. Wang, N. Pan / Journal of Computational Physics 228 (2009) Fig. 7. The three-phase (Al;Al 2O 3 an air) preictions of effective Young s Moulus, as a function of alumina volume fraction, for Al Al 2O 3 composites compare with the experimental ata [38]. The air volume fraction, e, is set at 5% when the alumina volume fraction / Al2O3 6 90% an e ¼ 2:5% when / Al2O3 > 90%. Fig. 8. Comparisons of three-phase (Epoxy, Al 2O 3, air) preictions of effective Young s Moulus, as a function of alumina volume fraction, for Epoxy Al 2 O 3 composites with experimental ata [7] an various theoretical moeling results. Linear relationships between porosity an volume fraction of alumina are assume: for / Al P 50%; e ¼ 0:1ð1 / Al Þ; for / Al < 50%; e ¼ 0:1/ Al The shear moulus an the Poisson s ratio As mentione before, when the irection of external force F varies from normal to parallel to the top surface, a very similar governing equation as Eq. (1) can be obtaine to escribe the shear isplacement with varying local shear moulus of composite-materials. The moeling process is quite similar to that for effective Young s moulus. Fig. 9 shows the preicte effective shear moulus of alumina epoxy composites versus the alumina volume fraction for the same three-phase microstructures in Fig. 8. The given shear mouli of Al 2 O 3 an epoxy are liste in Table 2. We assign the shear moulus of pores a tiny value ðg a ¼ 10 5 GPaÞ which actually has little effect on the final preictions. Again with the isperse pores/vois consiere, the preicte effective shear moulus of Al 2 O 3 Epoxy composites agree well with the experimental measurements [7]. After both the Young s moulus an the shear moulus are preicte, by assuming isotropy of the composite at the macro-level, the effective Poisson s ratio is subsequently calculate by [7] m ¼ E 1; 2G ð24þ

10 M. Wang, N. Pan / Journal of Computational Physics 228 (2009) Fig. 9. Preicte effective shear Moulus as a function of alumina volume fraction for Al 2 O 3 Epoxy air (three-phase) composites compare with the experimental ata [7] an HS bouns. Fig. 10. Comparisons between preicte effective Poisson s ratio as a function of alumina volume fraction for Epoxy Al 2 O 3 composites an the measurements [7]. an the results are compare with the available experimental ata in Fig. 10. The theoretical moels preict that the Poisson ratio shoul fall within the region of (0.25,0.35) if the Al 2 O 3 Epoxy composite is treate as two-phase only, again an overestimation of the lower boun. This further confirms that the effects of pores/vois on the effective elastic properties of composites are not negligible. Note that, in all the cases shown above, the present methos have emonstrate reliable an robust performance for accurate preictions without resorting to any empirical parameters. 5. Conclusions A new numerical approach for elastic properties preiction of multiphase composites has been evelope in this stuy. A ranom generation-growth metho is propose for reproucing the complex microstructures of multi-soli composite with isperse pores/vois insie. A high-efficiency lattice Boltzmann metho is then employe to solve the governing equations through multiphase microstructures with complex geometries by varying the local relaxation time corresponing to the local material property. After valiate by a few basic moels for simple geometries, the present metho is use to moel the effective elastic properties of several actual composites an the results are compare well with the existing experimental

11 5988 M. Wang, N. Pan / Journal of Computational Physics 228 (2009) ata. It is emonstrate that espite their tiny amounts, the pores/vois isperse in composites if ignore likely lea to non-negligible eviations in the preicte effective elastic properties, compare with the experimental measurements. When the pores are taken into account an treate as another separate phase, the preicte Young s moulus, shear moulus an Poisson s ratio agree well with the available experimental ata. Although the present governing equations are base on a one-imensional assumption, they seem to work well for composites of multi-imensional microstructures. The present metho thus provies a robust tool for analysis, esign an optimization of multiphase composite-materials, without resorting to empirical parameters Acknowlegments The present work is supporte by the grants from the NTC-M04-CD01. The authors woul like to thank Dr. J. Wang, Prof. S. Chen, Prof. Q. Zheng an Dr. Q. Kang for helpful iscussions. 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