EFFECT OF FILLER AGGREGATES STRUCTURE ON COMPOSITE ELASTIC PROPERTIES.

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1 EFFECT OF FILLER AGGREGATES STRUCTURE ON COMPOSITE ELASTIC PROPERTIES. V. G. Oshmya, S. A. Patlazha, S. A. Tima Departmet of Polymer ad Composite Materials, N.N. Semeov Istitute of Chemical Physics, 4 Kosygi St., Moscow , Russia. SUMMARY: Elastic scalig behavior of a cotiuous aisotropic fractal, 2D Sierpiski carpet, is the subject of the study. I the case of porous i elastic matrix, P. Sheg ad R. Tao [4] ad S. A. Patlazha [5] have foud that axial ad shear moduli of the carpet exhibit distict scalig with the size of the system. However, it is widely accepted that differet stiffess of isotropic fractals scale with equal expoets. The ature of such discrepacy has remaied uclear. Usig umerical positio-space reormalizatio group techique, we show that differet stiffess of the carpet also scale with equal expoets. I particular, it meas that both i the cases of porous ad rigid iclusios fractal Poisso ratio has o zero values idepedet of matrix moduli. Differece i the values of scalig expoets obtaied i previous study is caused by aalysis of the iitial fractal geeratios. KEYWORDS: elasticity, fractal, scalig, reormalizatio, disordered composite. INTRODUCTION. The elastic properties of isotropic fractal structures (regular or radom) have bee studied sice 1984 [1-5]. I particular, it was show [2,3] umerically i the framework of discrete sprig-based models that stiffess C of rigid percolatio clusters (radom fractals) above the elasticity threshold exhibit power-law (scalig) behavior with the size L of the system: C L τ. Also, it is widely accepted that percolatio clusters are isotropic fractals, ad differet compoets of effective stiffess tesor C scale with the same expoets. P. Sheg ad R. Tao [4] first ivestigated elastic scalig behavior of a cotiuous aisotropic regular fractal - the 2D Sierpiski carpet (Fig. 1). I the case of porous i a elastic matrix (elastic problem), calculatio of effective moduli was based o a iterative solutio of the Dyso equatio for elastic wave scatterig i a heterogeeous media. Due to the square symmetry of the carpet, three moduli determie tesor C. It was foud that axial [8], C 11, C 11 C 12, ad shear, C 44, moduli exhibit scalig behavior with correspodig expoets: τ 0. 27, τ 0. 25, τ (1) Usig differet approach, S. A. Patlazha [5] has obtaied aother estimates of the same expoets:

2 τ 025., τ 026., τ (2) I the case of rigid iclusios (superelastic problem), scalig law C L s was foud, but with equal expoets: s1 s2 s (3) It follows from qualitatively cosisted estimates (1) ad (2) that, cotrary to the isotropic case, axial ad shear moduli of the Sierpiski carpet exhibit distict elastic scalig behavior. The ature of such discrepacy has remaied uclear. However, it was proposed [4] two possible origis. The first oe is i the square symmetry of the carpet. Aother cotributig factor may be the breakdow of aalogy betwee the discrete ad cotiuum elasticity. Several importat questios have arise from previous study. I both papers, liear log(c) log(l) depedecies were obtaied for the three iitial geeratios of the carpet uder the uique value of the matrix Poisso ratio ν = 0.2. However, the scalig elastic respose should be expected at large L. Whether the same scalig would be asserted for the developed fractal structure? Would it be varied with the host parameters? heterogeeity: void or rigid iclusio a elastic matrix l (a) = 1, L = 3 (b) = 2, L = 9 (c) = 3, L = 27 Fig. 1. The three iitial geeratios of a Sierpiski carpet. The size L of the system is defied as the ratio of the side l of the outer square to the miimal thickess a of the matrix ad is determied by the umber of the geeratio: L = l/a = 3. Recetly [6], we have computed tesor C for the three iitial geeratios of the carpet umerically, usig the fiite elemet method (FEM). It was foud that: (i) the expoets deped o ν; (ii) the liearity of log(c) log(l) depedecies is broke as matrix approaches to the limit of icompressible media with ν close to uity [9]. So, we must coclude that estimates (1) ad (2) obtaied for the iitial fractal geeratios are determied by specific relatios betwee matrix moduli ad may be icosistet at large L. Ufortuately, direct FEM-based calculatio is ot applicable at large L because of computer memory ad time cosumig. Costructio of the positio-space reormalizatio group (PSRG) trasformatio ad its applicatio for aalysis of scalig behavior of 2D Sierpiski carpet elastic moduli at large L is the subject of the paper. A idea of the PSRG trasformatio is i the substitutio of effective moduli of a fiite fractal geeratio back ito the same geeratio as the matrix moduli. This procedure was suggested, but ot performed i

3 the paper [4]. To do this, effective matrix should be regarded as a elastic cotiuum complied with the same square symmetry as a fiite geeratio of the carpet. If this coditio is satisfied, the trasformatio ca be iterated as far as ecessary. Therefore, arbitrary size of the fractal ca be aalyzed. We have foud that both for elastic ad superelastic problems the PSRG trasformatio has correspodig stable fixed poits, P e = ( Σ e, Α e ) ad P s = ( Σ s, Α s ), o a plae of the effective Poisso ratio ad coefficiet of aisotropy Α = Σ = C12 C11 (4) ( C C ) ( C ) (5) A existece of the fiite fixed poits provides a commo scalig for differet compoets of tesor C with expoets idepedet of matrix mechaical properties: τ 0.29 ad s The rest of the paper is orgaized as follows. The descriptio of the PSRG trasformatio forms a cotet of the ext sectio. The results of simulatios are discussed i the fial part of the paper. PSRG TRANSFORMATION. Let us defie dimesioless size L of the system as a ratio of the side l of outer square to the miimum thickess a of the matrix (Fig.1). It is determied by the umber of the fractal geeratio: L= la=3. (6) Followig the defiitio (6), the size of the homogeeous square without iclusios ( = 0) is equal to 1. So, we deote iitial stiffess tesor of matrix as C(1). The FEM code elaborated i the paper [6] uder assumptio of isotropic matrix (Α = 1, idepedet C 11 ad C 44 ) was used for computatio of compoets C 11, C 12 ad C 44 of effective stiffess tesor C(L,C(1)) at L = 3, 9, 27 ad 81. (Below, where it is ot lead to ambiguity, the secod argumet will be dropped, i.e. C(L,C(1)) = C(L)). I that case, we ca defie trasformatio f mappig matrix moduli o effective moduli of the th fractal geeratio: ( 3 ) ( () 1) C = f C. (7) Ufortuately, the trasformatio (7) does ot coserve the isotropy of C ad thereby caot be iterated. Geeralizatio of the FEM code o a matrix complyig with the square symmetry (idepedet C 11, C 12 ad C 44 ) allows us to defie PSRG trasformatio f iduced by the th geeratio i the form suitable for iteratios. : ( 3 L) ( ( L) ) C = f C. (8) Accordig to the defiitio (8), umerical PSRG techique cosists i the followig. Give arbitrary values of iitial matrix moduli C 11 (1), C 12 (1) ad C 44 (1), we apply the FEM code for computatio C 11 (3 ), C 12 (3 ) ad C 44 (3 ). The we repeat computatio of effective moduli of

4 C(L) C(3L) Fig. 2. Oe step of the PSRG trasformatio (8) iduced by the 1st fractal geeratio. the th fractal geeratio usig the compoets of previously obtaied tesor C(3 ) as the ew matrix moduli. As a result, we obtai tesor C(9 ). This homogeizatio procedure may be cotiued util arbitrary preassiged value of L will be reached. After k iteratios, tesor C(3 k ) effectively accouts a ifluece of iclusios of the size less tha L = 3 k. A scheme of the procedure described is illustrated o Fig. 2 at = 1. Obviously, the result of the PSRG techique applicatio should be depedet of the two parameters. Accuracy of umerical method is the first oe. This factor ca be excluded by the choice of sufficietly large FE mesh, which is supposed to be doe. The secod parameter is the level of the trasformatio (8), which caot be excluded ad seems to be importat. For example, C(9) ca be approximated usig maps f 1 : C ( ( )) () 9 f f C() 1 =, (9) 1 1 or f 2 : ( ) () 9 () 1 C = f C. (10) 2 It is ot a priori kow whether the left sides of the Eqs. (9) ad (10) are close to each other. This problem will be discussed i the ext sectio o the basis of the umerical data obtaied. Due to the ifiite ratio betwee stiffess of matrix ad iclusios, right side of Eq. (8) is a homogeeous fuctio of order oe for both elastic ad superelastic fractals: ( ) ( ) f αc = αf C, forarbytraryreal α. (11) Coditio of homogeeity (11) ad defiitios (4) ad (5) allow us to defie reduced map r i a plae of the Poisso ratio Σ ad coefficiet of aisotropy Α: ( 3 L) = ( r ) ( ( ) ( ) L, L ) Σ ( 3 L) = ( r ) ( ( L), ( L) ) Σ Σ Α Α Σ Α where (r ) Σ ad (r ) Α are give throw compoets of the map f : ( r ) Σ ( f) ( f ) Α ( r ) Α ( f) ( f) 2( f ), (12) =, =. (13) 11 44

5 Existece of the fiite fixed poit ( Σ Α) ( ) ( ) ad equality ( Σ(), Α() ) ( Σ, Α ) 1 1 = scalig law for effective moduli:, of map (12): Σ, Α = r Σ, Α, Σ 0, ad Α 0,, (14) where the uiversal expoets τ ad s are give by the formula: provide size idepedet stiffess ratios ad exact τ C L or C L s, (15) lg (( f) ( C )) 11 τ, s = lg 3, (16) ad idepedet compoets of C fixed poit: ( C ) 11 = 1, ( C ) 12 = are determied by stiffess ratios correspodig to the Σ, ( C ) = ( 1 ) ( 2 ) 44 Σ Α. It will be show below that r is a cotractio map. This implies covergece o (Σ, Α) plae of the iterative procedure described above: r k ( Σ() 1, Α() 1) = ( Σ( L), Α( L) ) ( Σ, Α ), k, L, (17) ad asymptotic validity of scalig law (15) for arbitrary iitial matrix moduli. Uique fixed poit values ( Σ Α) RESULTS AND DISCUSSION., of the r have bee foud for every tested level. Obtaied results are summarized i Table 1. Table 1. Fixed poits of the map (12) ad correspodig expoets (16). Elastic fractal Σ Α τ Superelastic fractal Σ Α s Covergece (17) of the flow diagrams of iterated map r has bee checked by umerical aalysis i wide rage of iitial matrix values both for elastic (Fig. 3a) ad superelastic (Fig. 3b) fractals.

6 6 Aisotropy coefficiet (a) 1-0,4-0,2 0,0 0,2 0,4 6 Aisotropy coefficiet (b) 1-0,4-0,2 0,0 0,2 0,4 Poisso ratio Fig. 3. The flow diagrams of iterated maps r 2 (12) (solid lies) for elasic (a) ad superelastic (b) fractals ad correspodig eigedirectios of the liearized maps R 2 (18) (dotted lies). Near fixed poit, cotractio property of the r was directly proved by its liearizatio: r ( Σ, Α) ( Σ, Α) R Σ Σ, (18) Α Α ad computig of eigevalues, λ 1, λ 2, ad correspodig eigevectors, e 1, e 2, of the liearized map R. The results of the aalysis for = 2 (Fig. 3) are give i Table 2. It is see from Fig. 3 ad Table 2 that cotractio coditio 1> λ 1 λ 2 is satisfied ad flow diagrams ted to the eigedirectios correspoded to the maximum eigevalues.

7 Table 2. Eigevalues, λ 1, λ 2, ad eigevectors, e 1, e 2, of liearized maps R 2 (18) for elastic ad superelastic fractals. λ 1 e 1 λ 2 e 2 Elastic fractal 0.76 (-0.02,1) 0.52 (0.37,-0.93) Suprelastic fractal 0.9 (-0.03,1) 0.8 (0.46,-0.89) lg(l) 0 lg(l) -2 C C 44-4 C 11-6 C 12-6 C lg(c ) (a) lg(c ) C 12 (b) Fig. 4. The log(c) log(l) depedecies of elastic carpet. Matrix Poisso ratio Σ(1) = 0.2 (a) ad Σ(1) = 0.8 (b). I both cases, matrix coefficiet of aisotropy Α(1) = 1. As it was already metioed i the previous sectio, existece of the fixed poit ad cotractio property of the r provide ot oly covergece (17) i the ratio plae, but also uiversal asymptotic scalig law (15). Validity of the last was directly checked by drawig of log(c) log(l) plots (Fig. 4). It is iterestig to ote that liearity of the plots is almost valid i the hole rage of the sizes icludig small oes i the case of Σ(1) = 0.2, A(1) = 1(Fig. 4a). This feature was foud ad discussed i the first publicatios [4,5], cocerig elastic scalig properties of the Sierpiski carpet of iitial geeratios. However, liearity of the iitial portios of the plots does ot hold for Σ(1) = 0.8, A(1) = 1 (Fig. 4b). The reaso of such differece is caused by the differet distace from matrix values of the ratios to that of the fixed poit, which is much less i the first case (see Fig. 3a). I particular, iequality C 44 > C 12, which is a case of fixed poit, holds for Σ(1)= 0.2, but becomes opposite if the matrix becomes close to icompressible media.

8 Naturally, the limit, which correspods to the ifiitely developed fractal sets, is iterestig. These limit have bee estimated by 2-power polyomial approximatio of the umerical data of Table 1 with respect to ε = 1/ ad ε = 1/lg() (Table 3). There is see certai discrepacy betwee the estimates obtaied by the differet approximatios. The data o larger iteratio level is ecessary to defie expoets more exactly. However, the divergece is small, so data of Table 2 ca be used as rough estimates. It is attractive to assume that scalig properties of the fractal are primary determied by the dimesio of the set ad thereby to estimate the expoets for elastic ad superelastic percolatio problem o the basis of the results obtaied for regular fractals. Let us use Sierpiski carpet as a model of percolatio cluster for the speculatio suggested. Size L of the system is related to the matrix volume fractio p (m) as L d /L D =L (d D), where d = l8 / l3 ad D = 2 be fractal ad cofiguratioal dimesios, correspodigly. It is widely accepted that i the eighborhood of percolatio threshold p e elastic or superelastic behavior of highly disordered system is maily determied by the properties of rigid or soft ifiite cluster. If oe supposes that fractio p (m) of the matrix i fractal is proportioal to p, the elastic, T, ad superelastic, S, Table 3. Estimates of scalig expoets ad fixed poit values for 2D Sierpiski carpet. Elastic fractal Limit τ Σ Α ε = 1/ ε = 1/lg() Superelastic fractal Limit s Σ Α ε = 1/ ε = 1/lg() T C ( p p e ) (19) p e ( ) C p p e S (20) expoets ca be estimated o the basis of the data of Table 3: τ s T S D 255., d D d 15. (21) Sure, much more covicible argumets should be doe to groud the estimates (21). However, the relatio T > S, which holds [1-3,7] for 2D disordered elastic systems, is also supported by simulatios o regular elastic ad superelastic fractals - 2D Sierpiski carpet.

9 REFERENCES. 1. S. Feg, P. N. Se, Percolatio o Elastic Networks: New Expoet ad Threshold, Physical Review Letters, Vol. 52, No. 3, 1984, pp ; Y. Kator, I. Webma, Elastic Properties of Radom Percolatig Systems, ibid., No. 21, pp ; D. J. Bergma, Y. Kator, Critical Properties of a Elastic Fractal, ibid.,vol. 53, No. 6, 1984, pp E. Duerig, D. J. Bergma, Scalig properties of the elastic stiffess moduli of a radom rigid-origid etwork ear the rigidity threshold: Theory ad simulatios, Physical Review B, Vol. 37, No. 16, 1988, pp ; E.Duerig, D.J.Bergma, Physica A, Vol. 157, 1989, pp S. Arbaby, M. Sahimi, Mechaics of disordered solids. I. Percolatio o elastic etworks with cetral forces, Physical Review B, Vol. 47, No. 1, 1993, pp ; S. Arbaby, M. Sahimi, Mechaics of disordered solids. II. Percolatio o elastic etworks with bodbedig forces, ibid., pp P. Sheg, R. Tao, First-priciples approach for effective elastic-moduli calculatio: Applicatio to cotiuous fractal structure, Physical Review B, Vol. 31, No. 9, 1985, pp S. A. Patlazha, Elastic properties of filler aggregates, Proceedigs of the Iteratioal Coferece o Mechaics of Polymer Composites - MPC 91, Prague, 1991, pp V. G. Oshmya, S. A. Patlazha, S. A. Tima, Scalig of 2D Sierpiski carpet elastic properties, Abstracts of the Iteratioal Coferece o Problems of Codeced Matter Theory, Moscow, 1997, p S. A. Tima, V. G. Oshmya, FEM based simulatio of percolatio features of disordered heterogeeous media elastic properties, Proceedigs of the MRS Symposia, Pittsburgh, 1995, Vol. 367: Fractal Aspects of Materials, F. Family, B. Sapoval, P. Meaki, R. Wool, Eds., pp Below, we use the accepted otatios for compoets C ijkl of tesor C i a 2D cotiuum of square symmetry: C 1111 =C 2222 =C 11, C 1122 =C 12, C 1212 =C Recall that Poisso ratio for 2D media takes values o the iterval [-1,1]