MATHEMATICAL AND COMPUTER HOMOGENIZATION MODELS FOR BULK MIXTURE COMPOSITE MATERIALS WITH IMPERFECT INTERFACES

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1 Materal Phyc and Mechanc 37 (218) Receved: December 22, 217 MATHEMATICAL AND COMPUTER HOMOGENIZATION MODELS FOR BULK MIXTURE COMPOSITE MATERIALS WITH IMPERFECT INTERFACES A.A. Naedkna 1 *, A. Rajagopal 2 1 I.I. Vorovch Inttute of Mathematc, Mechanc and Computer Scence, Southern Federal Unverty, Mltchakova Str., 8a, Rotov-on-Don, 3449, Rua 2 Department of Cvl Engneerng, Indan Inttute of Technology, Hyderabad, 5225, Inda *e-mal: naedkna@math.fedu.ru Abtract. The paper decrbe the homogenzaton procedure for two-phae mxture elatc compote that cont of two otropc phae. It aumed that on the boundary between the phae, pecal nterface boundary condton are held, where the tre jump over the nterphae boundary are equal to the urface tree at the nterface. Such boundary condton are ued for decrpton of nanocale ect n elatc nanobode and nanocompote. The homogenzaton problem are olved ung the approach of the ectve modul method, the fnte element method and the algorthm for generatng the repreentatve volume that cont of cubc fnte element wth random dtrbuton of element materal properte. To provde a numercal example, a wolfram-copper compote condered, where the nterface condton are modeled by urface membrane element. Keyword: compote materal; homogenzaton problem; ectve modul method; fnte element method. 1. Introducton The proce of producng mxture compote often reult n compote tructure wth looely adherng phae conttuent. For example, th happen whle nterng the mxture of pezoceramc powder wth more dene granule. In th cae, we obtan mxture compote made of pezoceramc and elatc ncluon, where the nterphae boundare can contan ar layer. In addton, multphae mxture compote can experence detachment of materal of dfferent phae whle ther explotaton. In th and other cae, t of nteret to conder compote tructure wth mperfect contact condton at the nterphae boundare [1]. Popular recent model of nanomechanc wth urface or nterface ect for the compote wth nanocale nhomogenete conder the boundary condton wth urface tree on the boundare between the phae (ee for example revew [2 4]). Thu, n the Gurtn-Murdoch model, the boundare of a nanocale body are covered by elatc membrane, n whch nternal force are decrbed by the urface tree. Elatc membrane can be alo located nde a compound body on the nterphae boundary, whch allow modelng the mperfect nterface boundare wth tre jump [5 9]. The Gurtn-Murdoch model alo frequently ued for mulatng elatc nanocale compote. For example, n [1 2] and other work n the framework of the elatcty theory wth urface tree, the mechancal properte of the compote wth phercal ncluon, fber and other compote were nvetgated. The technque of fnte element approxmaton for elatc 218, Peter the Great St. Peterburg Polytechnc Unverty 218, Inttute of Problem of Mechancal Engneerng RAS

2 Mathematcal and computer homogenzaton model for bulk mxture compote materal wth mperfect nterface 35 materal wth urface ect and numercal example wa demontrated n [8, 16, 21 26] and other. For the ectve modul determnaton,.e. for the homogenzaton of the compote tructure, th work ue a general approach, baed on the ue of the ectve modul method (wth modfcaton that take nto account addtonal factor on the nterphae boundare), the mulaton of the repreentatve volume and the fnte element method for numercal oluton of the homogenzaton problem [21 24]. 2. Homogenzaton procedure for a mxture compote wth mperfect nterface Let u conder a two-phae compote wth nanocale ncluon. Let V a repreentatve ( j) (1) (2) volume; V are the volume occuped by eparate phae ( j = 1, 2); V = V V ; S = V the external boundary of the volume; S the et of the boundary urface of materal wth dfferent phae; n the vector of the unt normal to the external boundary (1) wth repect to the volume of the man materal V ; x the radu-vector of a pont n a (1) (2) Cartean coordnate ytem. Let u aume that the volume V and V are flled wth dfferent otropc elatc materal. Then nto framework of clac tatc lnear elatcty theory, we formulate the followng ytem of equaton n the volume V wth repect to the component of the dplacement vector u k= u k (x) : lσ =, σ = lε mmδ + 2mε, ε = ( luk + kul ) / 2, (1) where σ are the component of the tre tenor; ε are the component of the tran tenor; λ, µ are the Lame cocent; µ = G alo called the hear modulu. A t known, an elatc materal decrbed by two ndependent materal modul. The mot frequently ued are the Young modulu E and the Poon rato ν. Then the Lame cocent λ, µ = G, the bulk modulu K and the tffne modul c 44 can be expreed n term of the Young modulu and the Poon rato by well-known formulae: νe E E λ =, µ = G =, K = = λ + 2µ = λ 44 = G. (2) ( 1+ ν )(1 2ν ) 2(1 + ν ) 3(1 2ν ) ( j) ( j) For a two-phae medum, thee modul dffer for each phae: E = E, ν = ν, and o ( j) on for x V, j = 1, 2. In accordance wth the Gurtn-Murdoch model, we adopt that on the nanocale nterphae boundare S the followng condton held: n [ ] =, x S, (3) l where l (1) (2) [ σ ] σ σ = the tre jump over the boundary between the phae; l = l nl ( nm m ) the urface nabla-operator. Here σ are the component of the tenor of nterface tree, whch are related to the tran by the nterface Hook law: = l ε mmδ + m ε, ( l k ) / 2 σ 2 ε = u + u, u k = ( δ k m n kn m ) um, (4) where λ, µ are the Lame cocent for the nterface. We note that ntead of λ and µ other par of modul can be ued to formulate the Hook law. For example, we can ue E and ν, through whch we can expre the Lame cocent or elatc tffne ung the expreon mlar to the plane tre formula. Thu, a two-phae compoteontng of two otropc elatc phae wth nterface (1) (1) (2) (2) boundare, characterzed by four materal modul, for example, E, ν, E, ν, E

3 36 A.A. Naedkna, A. Rajagopal and ν. If the repreentatve volume ha not a pronounced anotropy n the component dtrbuton, then a uch called equvalent homogeneou materal wll alo be otropc and wll be decrbed by only two ndependent modul, for example, E and ν. In order to determne thee ectve modul, t enough to olve only one boundary-value problem (1) (4) n the repreentatve volume of the compote wth the boundary condton ( ε = cont ) = x ε, x S. (5) uk 1 δ1k Then, from the oluton of problem (1) (5) mlar to [21 24], we can fnd c = σ /e = σ 22 /e, (6) where the angular bracket denote the averaged by the volume V and by the nterface S value: 1 < (...) >= ( (...) (...) ) dv + ds. (7) V V S To check that the homogenzed materal otropc, we can olve problem (1) (5), whch oluton hould gve: c σ 33 /e ; σ k m, k m. For addtonal control of the qualty of the repreentatve volume, t make ene to olve problem (1) (4) wth a hear boundary condton: uk = ε ( x3δ 2k + x2δ 3k ) / 2, x S. If we obtan c 44 = σ 23 /e from the oluton of th problem, then th value hould be approxmately equal to c44 ( c c ) / 2, where the modul c are determned from the oluton of the prevou problem. In concluon of th ecton, we would lke to note that for nanotructured compote takng nto account the nterface urface ect lead to the appearance of pecal boundary condton (3), (4) on the urface S and to the necety of calculatng the averaged tree n (7) not only by the volume V, but alo by the urface S. In a mlar way we can conder the problem of homogenzaton for the compote wth mperfect contact on S, for example, wth the nterface condton of prng type [1]. 3. Numercal example Problem (1) (5) wa numercally olved by the fnte element method n the fnte element package ANSYS ung the technque mlar to the one decrbed n [21 23]. The repreentatve volume V wa choen n the hape of a cube wth the de L, whch wa evenly dvded nto maller geometrcally dentcal cube. Thee cube were eght node 3 hexahedral fnte element SOLID45. A a reult, n the volume V there were n V = n fnte element, where n the number of element along one of the ax. For the mulaton of a two-phae compote, the fnte element had properte of one of the phae. At the begnnng, all element had the properte of the frt phae. Then, baed on the requred percentage of the materal of the econd phae p a, for the randomly choen n p = NINT( nv pa /1) fnte element, ther materal properte were changed to the materal properte of the econd phae (here NINT the roundng functon to nearet nteger n ANSYS APDL programmng language). We alo note that the reultng percentage of ncluon p = 1n p / nv can dffer from the expected value p a, although th dfference uually neglgbly mall. Then the boundare of the element wth dfferent materal properte were automatcally [21 23] covered by four-node hell element SHELL63 wth the opton of

4 Mathematcal and computer homogenzaton model for bulk mxture compote materal wth mperfect nterface 37 only membrane tree. At the end all nterface edge were covered by membrane elatc fnte element, whch mulated the preence of nterface tree (3), (4) on the boundare S. At the next tage, for the generated repreentatve volume, we olved tatc problem (1) (5), and after that n ANSYS potproceor we calculated the averaged tree by both volume and urface element. Latly, ung formulae (6), (7) and the obtaned averaged tree, we calculated the ectve modul of the compote, takng nto account the nterface ect. We note that for the ame ze L of the volume V, dependng on the number of element n along the axe, the ze l = L / n of fnte element can be dfferent, hence, the ze of ncluon can be dfferent. Therefore, for a fxed percentage of ncluon p and ncreang n, the ze of ncluon wll be maller, but the total number of ncluon n p wll ncreae, and thu the total area of the boundary S wll alo ncreae [21 23]. To provde an example, we conder the problem of fndng the ectve modul of a two-phae compote, where the frt (man) phae wolfram and the econd (ncluon) phae copper. A t known, both phae can be adopted a otropc materal. For numercal reult, we took the followng value of bulk materal modul of the frt and (1) econd phae: E = 4 1 N/m 2 (1) (2) ; ν =. 28; E = N/m 2 (2) ; ν =. 34. It eay to note that the materal of the frt phae almot 4 tme tffer than the materal of the econd phae. However, the Poon rato of the materal of the frt phae maller than that of the econd phae. A the nterface tree were mulated by the fnte element of elatc membrane, t wa neceary for them to et the thckne h. The Young modulu wa taken ~ (1) (2) proportonal to the dfference between the Young modul of two phae E = k ( E E ), where k a dmenonle factor and the Poon rato determned by the formula (1) (2) ν = ( ν + ν ) / 2. Such approach lead to takng nto account the tree (3), (4) wth the ~ (1) (2) Young modulu E = h E = h k ( E E ),.e. n th cae the value h and k are not mportant ndependently, but ther product h k mportant. In connecton to th, n further numercal calculaton the thckne h wa fxed, h = 1 m, and for the analy of the 4 nterface tree nfluence only the cocent k vared from 1 to 1. Fgure 1, 2 how the dependence of the ectve modul wth repect to the factor n the logarthmc cale ( lg log1 ): the Young' modulu k E (Fg. 1a), the hear modulu G (Fg. 1b), the bulk modulu K (Fg. 2a), and the Poon' rato ν (Fg. 2b). Here the percentage of ncluon (materal of the econd phae) reman unchanged p = 1 %, and the curve 1, 2, 3, 4 correpond to the value n = 1, n = 2, n = 3, and n = 4, repectvely. A we can ee, the behavor of the ectve modul E, G and K, dependng on the nterface modul value, qualtatvely concde, but the modul E and G ncreae lghtly more rapdly when ncreang k, and the Young' modulu ncreae the mot rapdly. 3 For k 1, the urface ect have only a lght nfluence on the value of the tffne 2 modul. Approxmately tartng wth the value of k 1, the tffne modul begn to grow qute rapdly. Thu for the fxed percentage of ncluon p = 1 % wth ncreang n = 1, 2, 3, 4, the ze of ncluon decreae, but the number of ncluon and ther total area ncreae. A a reult, the nterface ect are manfeted n a greater extent, and the tffne modul (Fg. 1 and 2a) ncreae (curve 1 are located lower than curve 2urve 2 are located lower than curve 3,and curve 3 are located lower than curve 4). It hould be noted that for large value of the nterface tffne modul, the overall ectve tffne

5 38 A.A. Naedkna, A. Rajagopal modul of compote nanomateral wth nterface may exceed the modul of the tffet phae n the compote. In Fg. 1 and 2a, th ect oberved, when the value of the modul are larger than the modul of the tffet materal n the compote (dahed lne). (a) Fg. 1. Dependence of the ectve modul E (a) and G (b) veru the factor p = 1 %: curve 1 for n = 1; curve 2 for n = 2; curve 3 for n = 3; curve 4 for n = 4. (b) k for (a) Fg. 2. Dependence of the ectve modul K (a) and ν (b) veru the factor p = 1 %: curve 1 for n = 1; curve 2 for n = 2; curve 3 for n = 3; curve 4 for n = 4. (b) k for Fgure 2b how that the behavor of the Poon rato wth ncreang nterface modulu oppote to the behavor of the tffne modul E, G and K. Th can be explaned by an ncreae n the overall tffne of the materal. In order to analyze the nfluence of the nterface ect on the ectve modul wth dfferent percentage of ncluon, we have calculated the ectve tffne modul wth the fxed number of element n = 2, but wth dfferent percentage of ncluon and wth

6 Mathematcal and computer homogenzaton model for bulk mxture compote materal wth mperfect nterface 39 dfferent, but not too large, value of the factor hown n Fg. 3 and 4. k. The reult of thee calculaton are (a) Fg. 3. Dependence of the ectve modul ncluon for n = 2 and for the factor curve 3 for E (a) and k : curve 1 for k =.1; curve 4 for (b) G (b) veru the percentage of k =.1; curve 2 for k =.1. k =.5; (a) Fg. 4. Dependence of the ectve modul ncluon for n = 2 and for the factor curve 3 for K (a) and k : curve 1 for k =.1; curve 4 for (b) ν (b) veru the percentage of k =.1; curve 2 for k =.1. k =.5; A thee fgure demontrate, for mall value of the factor k (curve 3 and 4) the nterface ect do not affect the tffne modul. However, for any percentage of ncluon the nterface tree are larger than the ectve tffne of the compote materal. Moreover, a t wa mentoned earler, there are cae when the compote materal wth nterface can have greater tffne than the tffet materal n the compote. Th tuaton

7 4 A.A. Naedkna, A. Rajagopal take place when k =. 1 for the Young' modulu G, f p 63 %, and for the bulk modulu E, f p 75 % K, f p 63 %, for the hear modulu, (ee curve 1 and 2, whch are located hgher that the dahed lne n Fg. 3 and 4a). When k =. 5, the reult are more content wth the expermental data for other materal, and mlar tuaton take place for the Young' modulu E, f p 65 %, for the hear modulu G, f p 33 %, and for the bulk modulu K, f p 5 %. On the contrary, the Poon rato of the compote tructure can be maller than the Poon rato of the man materal for large value of k n a wde range of percentage of ncluon p (Fg. 4b). We note that the percentage of ofter ncluon and the nterface ect have the oppote nfluence on the ectve tffne: a mple ncreae of the percentage of ofter ncluon lead to a decreae n the tffne modul, however, the nterface ect ncreae the tffne. The growth of the percentage of the ofter ncluon for nanocompote materal wth nterphae ect ental an ncreae of the boundare area wth nterface tree. Therefore, wth the ncreae of the percentage of ofter ncluon, the tffne modul of nanocompote materal wth nterface may ncreae. For example, for large value of nterface modul an ncreae of the percentage of ncluon for not very large p ( p 45 % for k =. 1 and p 35 % for k =. 5) lead to a growth of the ectve Young' modulu, and wth further ncreae of p the ectve Young' modulu tart to decreae. Thu, a t wa mentoned n [2, 21 24, 27, 28] for porou nanocompote, from the reult of computatonal experment the followng trend have been oberved. If we compare two mlar bode, one wth ordnary dmenon and the other wth nanocale dmenon, then for the nanozed body at the expene of urface tree, the ectve tffne wll be greater than for the body of ordnary ze. Furthermore, for the compote body wth ofter ncluon of the macrocopc ze and wthout nterface, the ectve elatc tffne decreae wth an ncreae of the percentage of ncluon. Meanwhle, for the ame percent of ncluon the ectve tffne of the nanocompote body wth ofter ncluon and wth nterface may ether decreae or ncreae dependng on the value of the nterface modul, dmenon and number of ncluon. Th ect can be explaned by the fact that the ze of the urface nterface wth nterface tree depend not only on the overall percentage of ncluon, but alo on ther confguraton, ze and number. 6. Concluon The homogenzaton model wa decrbed for a two-phae elatc compote wth urface tree at the nterphae boundare, whch reflected the nanocale ect for nanotructured compote. Th model wa appled for the calculaton of the ectve modul of a two-phae wolfram-copper compote wth nterface boundary condton on the boundare between the phae. The oluton of the homogenzaton problem were obtaned by the fnte element method n ANSYS fnte element package for a cubc repreentatve volume wth mapped mehng n hexahedral fnte element and random dtrbuton of ncluon. For the condered compote wth tffer keleton and ofter ncluon, we have noted that the ectve modul were dependent on the percentage of ncluon, ther confguraton and ze. Thee dependence are mlar to known dependence for nanoporou compote. Acknowledgement. Th reearch wa performed n the framework of the Ruan-Indan RFBR-DST Collaboratve project wth RFBR grant number IND_om and DST grant number DST/INT/RFBR/IDIR/P-/216.

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