NUMERICAL EVALUATION OF PERIODIC BOUNDARY CONDITION ON THERMO-MECHANICAL PROBLEM USING HOMOGENIZATION METHOD

Transcription

1 THE 9 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS NUMERICAL EVALUATION OF PERIODIC BOUNDAR CONDITION ON THERMO-MECHANICAL PROBLEM USING HOMOGENIZATION METHOD M.R.E. Nasuton *, N. Watanabe, A. Kondo,2 Department of Aerospace Engneerng, Toyo Metropoltan Unversty, Hno, Toyo, Japan 2 e-xstream Engneerng, MSC Software Company, Shnjuu, Toyo, Japan * nasuton-rdlo@sd.tmu.ac.jp Keywords: perodc boundary condton, thermo-mechancal problem, homogenzaton Introducton Composte structure s composed of very complex and heterogeneous materal consttuents. The heterogenety of ts mcrostructure leads the dffculty on the analyss of composte materal. In the analyss, heterogeneous mcrostructure can be dealzed such that possesses a perodc pattern. Such nd of dealzaton enables composte structure to be effectvely and effcently analyzed by usng asymptotc expanson homogenzaton (AEH) method []. Ths method nvolves representatve volume element (.e., unt-cell) n two scales, those are mcroscopc and macroscopc. The unt-cell model s usually consdered to be repeated nfntely n three-dmenson (n-plane and out-ofplane drectons). Guedes and Kuch [2] used ths method for evaluatng the averaged elastc constants and equvalent stress of composte materals. Chung et al [3] employed AEH method for analyzng heterogeneous meda. Unt-cell of 2-D plan weave composte was used n the numercal example. Mechancal response of the unt-cell due to the applcaton of perodc boundary condton was fgured out by localzaton method whch can be vewed as the nverse of homogenzaton method. Localzaton analyss was also conducted by Pnhoda-Cruz et al [4]. The study ncluded the formulaton of AEH method n strong form wth the mplementaton on the mcroscopc structure whch can be found n [5]. The mentoned studes exclude the calculaton of coeffcents of thermal expanson (CTE) and thermal resdual stresses. However, the thermal effect should be consdered due to ts potentalty to affect composte damage behavor [6, 7]. CTE of fber renforced compostes are numercally studed usng fnte element method by Karadenz and Kumlutas [8]. Ths study deals wth mcromechancal modelng and excludes macromechancal modelng. Some other AEH studes nvolvng thermo-mechancal propertes were conducted by Shabana and Noda [9] and Dasgupta et al []. However, thermal resdual stresses dstrbuton was not ncluded n the studes. In ths paper, thermo-mechancal propertes and thermal resdual stresses wll be evaluated n the analyss usng AEH method. AEH method uses perodc boundary condton whch s usually assumed to be perodc n threedmenson. However, composte lamnates, especally n aerospace applcaton, are very thn. It should not be consdered nfntely n thcness drecton. The effect of fnte thcness n composte structure s suggested by Woo and Whtcomb [] as a future study. In ths paper, fnte thcness effect wll be evaluated by relevng perodc boundary condton n thcness drecton. Asymptotc analyss usng AEH method s carred out by mplementng two nds of perodc boundary condtons. The frst nd of the boundary condton apples the perodcty n three-dmenson whereas the second one has only n-plane perodcty (.e., perodcty n thcness drecton s omtted). Homogenzed thermo-mechancal propertes as well as characterstc dsplacement and thermal resdual stress dstrbuton wll also be analyzed to understand the effect of fnte thcness. 2 Asymptotc Expanson Homogenzaton (AEH) Method 2. General Concept In ths paper, AEH method wll be classfed nto two nds based on the perodc boundary condton that s used. The frst s ordnary method whch ncludes perodc boundary condton n threedmenson. The second s new method whch releves perodc boundary condton n thcness drecton (.e., out-of-plane drecton). More detal explanaton about perodc boundary condton wll be presented n the subchapter 2.3.

2 Homogenzaton method regards perodc and heterogeneous mcrostructure as a homogeneous macrostructure. In Fg., the elastc body Ω s subjected to tracton t and body forces f, where the dsplacement s prescrbed on Γ d. Ths fgure shows how a heterogeneous unt-cell s taen as a representatve volume element of a homogeneous elastc body. Heterogeneous unt-cell conssts of two parts, those are sold part ( ) and hole part (θ) as seen n Fg. 2. The unt-cell tself can be vewed from two doman scales (.e., macroscopc and mcroscopc scales). Problems n equlbrum of structures can be solved by prncple of vrtual wor. Eq. () s the mathematcal expresson of the prncple of vrtual wor by ncludng the thermal effect represented n stran term. In homogenzaton method, the equaton nvolves the functon of total doman by ncludng mcrostructure regon ndcated by superscrpt ε. The term p s tracton on the boundary between sold part and hole part (.e., surface S). u v E T d f v d jl l xl x j t v d p v ds t S () AEH method expresses the dsplacement feld by usng asymptotc expanson seres as follows x, y x, y x, y 2 2 u xy, u u u (2) Mcroscopc dsplacement u s obtaned by nvolvng soluton of varatonal problem (.e., characterstc dsplacement) [2]. u x u x, y x, y x, y (3) p xq where χ and ψ are the characterstc dsplacement vectors for elastc and thermal problem respectvely. Homogenzaton method consders the varaton of mcrostructure varables wthn the unt-cell n both mcroscopc and macroscopc scales. The varaton s delvered by perodc vector functon g expressed n Eq. (4).,, g x g x y g x y (4) where ε = x/y and s the unt-cell dmenson. The rgorous formulaton of ordnary homogenzaton method whch consders the unt-cell to be repeated nfntely n three drectons (x -, x 2 -, and x 3 - drectons) can be found n [2]. 2.2 Formulaton of New AEH Method In ths subchapter, subsequent steps summarze the formulaton of proposed new method whch omtted the perodc boundary condton n thcness drecton wheren the detal can be found n [2] and [3]. The new method consders that the n-plane dmenson of composte structure s much larger than the out-of-plane dmenson. Ths consderaton leads the unt-cell should not be repeated nfntely n thcness drecton (see Fg. 3). In ths study, the consequence of omttng the perodcty n thcness drecton s proposed by nvolvng the use of through thcness unt-cell. It means that the mcrostructure varables n thcness drecton are vared wthn the mcroscopc scale, not n the macroscopc scale. It yelds the dfferent perodc vector functon g whch s represented n Eq. (5).,,,, g x g x x y y y (5) Perodc vector functon n Eq. (5) yelds the dervatves wth respect to macroscopc coordnate x as follows g g g x x y g g g x x y g x g y 3 3 (6) (7) (8) Perodc and heterogeneous mcrostructure wll be regarded as macroscopcally homogeneous structure by tang the lmt of perodc vector functon as ε approaches zero as expressed by the equaton below lm g xd g, dd x y (9) lm g xd g, dsd x y () S S where d = dy dy 2 dy 3 and dω = dx dx 2. Substtutng the asymptotc expanson seres nto the prncple of vrtual wor equaton and nvolvng the

3 dfferentaton wll results n three herarchcal equatons based on the order of ε as follows 2 u t 2 u v Ejl d () y y u v u Ejl yl x j x l l yl y j S l v T d p v ds u u v Ejl lt xl yl x j 2 u u v d f vd xl yl y j t v d j (2) (3) The followng procedures are carred out to solve Eq. ()-(3): Order of ε -2 : Multplyng Eq. () by ε 2 and tang the lmt usng expresson (9) obtans u v Ejl dd (4) y y l j Eq. (4) s solved by determnng the vrtual dsplacement v. If v = v(x), Eq. (4) wll automatcally be satsfed. Choosng v = v(y) wll need further treatments. Applyng ntegraton by parts and Gauss dvergence theorem as well as consderng n-plane perodc boundary condtons and free tracton boundary at the top and bottom of unt-cell surfaces yelds u Ejl v yd d y j y l (5) Eq. (5) mples that u u x, x. 2 Ths statement asserts that the macroscopc problem s reduced nto two-dmensonal problem n the new method. Order of ε - : Multplyng Eq. (2) by ε and tang the lmt usng expresson (9) and () yelds u v u u Ejl l T y l x j xl y l v dd yj S p v dsd (6) Choosng v = v(x) and consderng u u x, x mples S 2 pvds (7) Choosng v = v(y) and representng the mcroscopc dsplacement by Eq. (3) wll obtans two mcroscopc equlbrum equatons as follows For elastc problem: y l p v Ejl Ej d y q yj For thermal problem: y p v Ej T d y q yj (8) (9) where, j,, p, q =, 2, 3 and l =, 2. Due to the symmetrc property of elastc characterstc dsplacement vector χ, ndexes n Eq. (8) wll also be only and 2. It mples that there are only three modes of χ, those are χ, χ 2, and χ 22. Order of ε: Tang the lmt of Eq. (3) usng expresson (9) wll results the followng equaton u u v Ejl l T x l yl x j 2 u u v dd fvdd xl yl y j t v d t (2) Choosng v = v(x) and representng the mcroscopc dsplacement by Eq. (3) wll obtans macroscopc equlbrum equaton as follows u v v t v Ejl d j d j d xl x j x j x j (2) b v d t v d t

4 where:,, p, q =, 2, 3 j, l =, 2 Macroscopc homogenzed elastc constants: l p Ejl Ejl Ej d y (22) q Averaged stresses due to nternal forces: p j Ej d y (23) q Averaged thermal stresses: t j j Averaged body forces: E T d (24) b fd (25) Macroscopc homogenzed coeffcents of thermal expanson: where t l Sjl j j T Sjl (26) s the macroscopc homogenzed complance tensor. The above formulaton mples the new AEH method by omttng the perodc boundary condton at the top and bottom of unt-cell surfaces wll results n homogenzed n-plane propertes. Macroscopc equlbrum equaton wll then be used for solvng the macroscopc dsplacement u. Afterward, stresses on each pont n the doman are approxmated by the frst approxmaton of stresses as represented by the followng equaton: l p u j j Ejl Ej y q xl E E T p j j yq 2.3 Perodc Boundary Condton (27) Perodc boundary condton s mplemented n the calculaton procedures of elastc characterstc dsplacement vector (χ) and thermal characterstc dsplacement (ψ). Both characterstc dsplacements are calculated by solvng Eq. (8) and (9), respectvely. Perodc boundary condton s appled on the untcell surfaces. Ordnary method consders the untcell to be perodc or repeated nfntely n three drectons. Such nd of boundary condton can be represented by the followng equaton. and, y2, y, y2, y,, y, y2, y, y2, y, y2,, y2, y, y2, y,, y, y2, y, y2, y, y2, (28) (29) New method proposes that the perodc boundary condton n thcness drecton should be omtted. It s reached by relevng perodc boundary condton so that the top and bottom of unt-cell surfaces do not apply perodc boundary condton. Ths consderaton can be represented as follows and, y2, y, y2, y,, y, y2, y, y2, y, y2,, y2, y, y2, y,, y, y2, y, y2, y, y2, 3 Numercal Results (3) (3) In ths study, two nds of analyss wll be done nvolvng ordnary AEH method for a unt-cell wth three drectons perodcty and new AEH method for the unt cell wth only n-plane perodcty. Numercal evaluaton of perodc boundary condton wll be performed by modelng the unt-

5 cell of 3-D orthogonal nterloced composte. Fg. 4 represents the schematc of the unt-cell of 3-D orthogonal nterloced composte represented n sngle stac model. The unt-cell conssts of x-tow, y-tow, z-tow, and resn rch regon. Each tow s composed of T3 fber and Epoxy 828 wth vf = 54.6%. Tow and resn propertes can be shown n Table [4, 5]. Characterstc dsplacement vectors are compared between the results obtaned from ordnary and new methods. To understand the effects of releasng perodc boundary condton n thcness drecton, numercal evaluaton s performed by buldng the unt-cell wth ncreasng number of stacs (, 2, and 4 stacs). Fg. 5 shows that the perodc boundary condton at the top and bottom of unt-cell model are successfully omtted by the new method. It ndcates wth the dfferent dsplacement vector at the top and bottom of unt-cell surfaces as shown n Fg. 5(b) whle the dfferent result s obtaned by ordnary method n Fg. 5(a). For a better understandng, the results of characterstc dsplacement obtaned by model wth 4 stacs are shown by Fg. 6(a) and 6(b). It shows that the repeatng pattern n thcness drecton s obtaned from the result of ordnary method (Fg. 6(a)) whle the dfferent dstrbuton s produced by new method (Fg. 6(b)). Homogenzed propertes are analyzed by nvolvng the model wth ncreasng number of stacs n the new method. Table 2 shows the results of ordnary method wth three drectons perodcty and new method wth only n-plane perodcty. The obtaned homogenzed n-plane propertes of new method wll be normalzed wth the result of ordnary method by usng a sngle stac. Fg. 7 shows the normalzed homogenzed thermo-mechancal propertes as nfluenced by the ncreasng number of stacs. It shows that n the case of 3-D orthogonal nterloced composte, the releasng of perodc boundary condtons affects the Posson s rato and coeffcents of thermal expanson. The ncrease of number of stacs tends to mnmze the dscrepancy between the results of ordnary and new method. However, the elastc and shear modulus are nsenstve to the fnte thcness effect. The thermal resdual stresses are also analyzed n ths study. The results are shown by usng the 4 stacs model. Thermal resdual stress dstrbuton (σ ) obtaned by applcaton of perodc boundary condton n three drectons and only n-plane perodc boundary condton are dsplayed by Fg. 8(a) and 8(b) respectvely. From the obtaned results, the sgnfcant dfferent pattern of stress dstrbuton s occurred n regon near z-tows. Fg. 9(a) and 9(b) represent the stress dstrbuton along thcness drecton of lne A-A (see Fg. 8(a) and 8(b)). These fgures show that the repeatng pattern along thcness drecton (.e., from Stac to 4) s occurred n the result obtaned from ordnary method whle the pattern s vanshed when the perodc boundary condton n thcness drecton s omtted. 4 Conclusons Formulaton of new AEH method has been proposed by ncludng the thermal and fnte thcness effects. The study consders fnte thcness effect by relevng the perodc boundary condton at the top and bottom of unt cell surfaces. The formulaton of new method asserts that the macroscopc problem s reduced nto two-dmensonal problem. In ths paper, numercal evaluaton was performed n the case of 3-D orthogonal nterloced composte. Results of new method (wthout perodcty n thcness drecton) were compared to the ordnary method (three-dmenson perodcty). Calculaton of homogenzed thermo-mechancal propertes shows that the elastc and shear modulus obtaned from new method are nsenstve to the ncrease of number of unt-cell stacs whle the Posson s rato and coeffcents of thermal expanson are affected. The numercal example shows that the ncrease of number of stacs tends to mnmze the dscrepancy between the results of ordnary and new method. Dstrbuton of characterstc dsplacement and thermal resdual stresses are also shown to understand the effect of relevng perodc boundary condton at the top and bottom of unt-cell surfaces. Dfferent dstrbuton pattern of characterstc dsplacement was obtaned whch ndcates that the perodcty n thcness drecton has been successfully omtted. References [] A. Bensoussan, J. Lon and G. Papancolaou Asymptotc analyss for perodc structures. North- Holland Publ Comp, 978. [2] J.M. Guedes and N. Kuch Preprocessng and post processng for materal based on the homogenzaton method wth adaptve fnte element methods. Computer Methods n Appled Mechancs and Engneerng, Vol. 83, pp 43-98, 99.

6 [3] P.W. Chung, K.K Tamma and R.N. Namburu Asymptotc expanson homogenzaton for heterogeneous meda: computatonal ssues and applcatons. Compostes: Part A, Vol. 32, pp 29-3, 2. [4] J. Pnho-da-Cruz, J.A. Olvera and F. Texera-Das Asymptotc homogenzaton n lnear elastcty. Part I: Mathematcal formulaton and fnte element modellng. Computatonal Materals Scence, Vol. 45, pp 73-8, 29. [5] J.A. Olvera, J. Pnho-da-Cruz and F. Texera-Das Asymptotc homogenzaton n lnear elastcty. Part II: Fnte element procedures and multscale applcatons. Computatonal Materals Scence, Vol. 45, pp 8-96, 29. [6] T. Hobbebrunen, B. Fedler, M. Hojo, S. Ocha and K. Schulte Mcroscopc yeldng of CF/epoxy compostes and the effect on the formaton of thermal resdual stresses. Compostes Scence and Technology, Vol. 65, pp , 25. [7] L. ang,. an, J. Ma and B. Lu Effects of nterfber spacng and thermal resdual stresses on trasverse falure of fber-renforced polymer-matrx compostes. Computatonal Materals Scence, Vol. 68, pp , 23. [8] Z.H. Karadenz and D. Kumlutas A numercal study on the coeffcents of thermal expanson of fber renforced composte materal. Composte Structures, Vol. 78, pp -, 27. [9]. Shabana and N. Noda Numercal evaluaton of the thermomechancal effectve propertes of a functonally graded materal usng the homogenzaton method. Internatonal Journal of Solds and Structures, Vol. 45, pp , 28. [] A. Dasgupta, R.K. Agarwal and S.M. Bhandarar Three-dmensonal modelng of woven-fabrc compostes for effectve thermo-mechancal and thermal propertes. Compostes Scence and Technology, Vol. 56, pp , 996. [] K. Woo and J.D. Whtcomb Effects of fber msalgnment on the engneerng propertes of plan weave textle compostes. Composte Structures, Vol. 37, No. 3/4, pp , 997. [2] N. Watanabe, S. Taahash, M.R.E. Nasuton and A. udhanto Formulaton of 3-D homogenzaton method for fabrc composte lamnates by consderng the effect of fnte thcness. Proceedngs of the 5 th European Conference on Composte Materals, Vence, Italy, paper ID 426, 22. [3] M.R.E. Nasuton and N. Watanabe Fnte thcness effect on thermo-mechancal propertes of 3-D orthogonal nterloced fabrc composte. Proceedngs of the 5 th US-Japan Conference on Composte Materals, Texas, Unted States of Amerca, pp , 22. [4] E. Myaosh Analyss of damage propagaton of textle compostes usng homogenzaton method. Master Thess, Department of Aerospace Engneerng, Toyo Metropoltan Unversty, Toyo, Japan, 29. [5] H. Mbayash Damage development analyses of 3D- CFRP by usng node separaton method. Master Thess, Department of Aerospace Engneerng, Toyo Metropoltan Unversty, Toyo, Japan, 23. Fg.. Elastc body wth heterogeneous and perodc mcrostructure Fg. 2. Unt-cell vewed from macroscopc and mcroscopc scales Fg. 3. Composte structure wth only n-plane perodcty

7 (a) Fg. 4. Schematc of the unt-cell of 3-D orthogonal nterloced composte (b) (a) Fg. 6. Thermal characterstc dsplacement : (a). ordnary method (three drectons perodcty), (b). new method (only n-plane perodcty) (b) Fg. 5. Elastc characterstc dsplacement : (a). ordnary method (three drectons perodcty), (b). new method (only n-plane perodcty) 2 Fg. 7. Normalzed homogenzed thermo-mechancal propertes as nfluenced by the ncreasng number of stacs

8 (b) (a) Fg. 9. Thermal resdual stress along lne A-A : (a). ordnary method (three drectons perodcty), (b). new method (only n-plane perodcty) (b) Fg. 8. Thermal resdual stress : (a). ordnary method (three drectons perodcty), (b). new method (only nplane perodcty) Propertes T3 (vf=.546) Epoxy828 Unt E L GPa E T GPa G LT GPa G TT GPa ν LT ν TT α L 6.4E E-5 (/⁰C) α T 4.8E E-5 (/⁰C) Table. Tow and resn propertes Prop. Ordnary Method New Method Stac Stac 2 Stacs 4 Stacs E, E 2 (GPa) ν G 2 (GPa) α, α 2 (⁰/C) E E E-6 4.7E-6 Table 2. Homogenzed thermo-mechancal propertes (a)