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82 TAP CHI KHOA HQC TRUONG DAI HQC MO TP.HCM - SO (35) 2014 PHAN TICH ON DINH CiJA TAM FGM Sir DUNG LA"( THUYET BIEN DANG CAT BAC CAO Ngay nh4n bai:01/12/2013 Ngay nh'On lai:18/02/2014 Ngay du)* &dug:10/03/2014 Ngo Phat Dot' Ngo Thanh Phong Trim Trung Dung TOM TAT Trong bai bao nay, phrearzg phap ph an tri ha.0 hgn duoc tich hop vai 1.); thuyn bien dang cat bac cao logi C° de phcin tich on dinh ca va nhiet cria tam vat lieu bien doi chirc nang (tam FGM,) Trong tam FGM, thuac tinh vat lieu duoc gia dinh phcin too khac dgc theo chieu day bai mat luat phcin phoi dan gicin cac phan the tich On dinh cfra tam FGM chiu tac clang crier tai ca va nhiet se duoc phcin tich so chi tiet flq chinh xac va tin eery cua phrrang phap hien tgi dug.c kiem thong bang cach so scinh vai ket gua cfra nhang 161 gicii khac da tong b6 tru.g.c day Tir khoa: Tam FGM, ly thuyet Wen dang cat bac cao (HSDT), phan tich on dinh ABSTRACT In this paper, the finite element method is intergated with the C°-type higher-order shear deformation theory (HSDT) for mechanical and thermal buckling analyses of functionally graded material plates (FGM) In the FGM, the material properties are assumed to vary through the thickness by a simple power rule of the volume fractions of the constituents The buckling behavior of FGM plates under mechanical and thermal loads is numerically analyzed in detail The accuracy and reliability of the present method is verified by comparing with those of other published solutions in the literature Keywords: functionally graded material plade, higher-oder shear deformation theory (HSDT), buckling analysis chiu throe tac d6ng hoc kha tot Vi vay TONG QUAN moi tnaing Nam 1984, met nhom nha khoa hoc FGM co the lam rat phil hop cho cac cau Nhat Ban [1] da tim mot m6 hinh vat nhiet d6 cao va rat trdc hang khong vu tru, nha may dien lieu mm vbi nhiing thu6c tinh vuot tr6i so vai cac vat lieu truck day va duqc g9i la hat nhan hay c6ng nghiep ban an, v.v Val nhang thu6c tinh uu viet cua vat lieu bien doi chire nang (functionally graded material - FGM) Mat ten FGM FGM nhieu *rig dung thuc tien, thuOng duoc lam tir g6m va mat duoi la FGM da duqc cac nha khoa hoc tren the kim loai Gom each nhiet rat tot va chiu giai quan tam va nghien cuu bang nhieu duoc nhiet de cao, kim loai phucrng phap khac Huang va Shen I Truthig DO hoc Khoo hoc to nhien, Doi hoc Quac gia Tp.HCM Diving Dai hoc Khoa hoc ty nhien, Dpi hoc Queic gia Tp.HCM Throng Dai hoc Mir Tp.HCM 83 KHOA HOC KY THU 'AT [2] da nghien ciru dap irng dao dOng phi tuyen cita tam FGM moi truErng nhiet Yang va Shen [3] da phan tich dab ding to tam FGM moi tn.rang nhiet Najafizadeh [4] da phan tich on dinh tam ton FGM dua tren IST thuyEt bien dang cat bac cao Vel va Batra [5] da sir dung lei giai 3D,de phan tich dap,Ung luc dm tam FGM bang nhieu lyr thuyet tam khac Matsunaga [6,7] da sir dung ly thuyet bac cao va m6 hinh 2D de phan tich dao deng to va on dinh dm tam FGM Them vao do, mot so phucmg phap phan tir hitu han [8-26] hoac phuang phap khong huh (meshfree methods) [27-29] ding da dugc nghien ciru d giai cho tam va tam FGM Tuy nhien phan ion ,cac nghien cint teen deu sir dung thuyet bien dang cat bac cao lien tuc C' (xap xi phan tir hint hp bac cao) Ben canh do, hau het cac ly thuyet tam dEu bat bu6c phan tir lien tuc C' vi phucmg trinh vi phan cita tam la phtrcrng trinh vi phan bac b6n Vi vay, viec chia luai va xap xi tnrEmg chuyen vi se kha phirc tap, doi hoi,chi phi tinh toan khd cao De khac phuc van de teen, 13'7 thuyet bien dang cat bac cao loci C° (C°-HSDT) [30, 28] da duac ,da xudt.,Tuy nhien, phan tir tam giac tuyen tinh ket hop C°-HSDT cho phan tich on dinh ca nhiet cita tam FGM van han 6E Trong bai bao nay, chiing toi da ap dung phtrang pita!) ten (phan tir tarn gide ba nut ket hop IS, thuyet bien dang cat bac cao loci C°) de phan tich lap bai toan ve on dinh ca nhiet cua tarn FGM Anh htrErng cita tai ca va tai nhiet den on dinh cua tam FGM se duccc khao sat so chi tiet va so sanh vai ket qua,cita cac phircrng phap khac da ducrc cong bo trade day de danh gia chinh xac va tin cay cita phucrngphap Cac thu6c tinh vat lieu doc theo chieu day tam se phu thu6c vao 1114 phan phoi ti le the tich cua cac phan cau tao nen tam FGM PHOING TRINH DANG YEU VA CONG THIYC PHAN TO IltrU HAN 2.1 V4t lieu FGM Vdt lieu FGM, throng dirge hinh tir hal hay nhieu loci vat lieu khac va dubc phan bo doc theo chieu day dm tam theo mot ti le nhat dinh nhu the hien Hinh la Thu6c tinh vat lieu FGM se phu thu6c vao ti le phan phoi gift de vat lieu va &roc cho bai ding thirc sau [3 1] +P ; V = (- + -) (n>_0) (1) dO m va c la 147 hieu cho kim loci (metal) va Om (ceramic); P la thu6c tinh mo dun,dan hoi Young E, vat lieu bao ting lu rieng p, he so Poisson v, he so clan nhiet k va he so gidn na nhiet a; P va va kim e P la 147 hieu thu6c tinh loci; V la ti le the tich cua g'6m; z la toa d6 doc theo chieu day dm tam va nam khoang tir -t/2 den, t/2; n la he ti le the tich He so ti re the tich phan 116 doc theo chieu day dirge the hien Hinh lb P(z).(P, - Hinh (a) V4t lieu FGM; (b) Ti le the tich V doc theo chieu day tam OS Os 0S or: -03 -0.1 -0.3 -03 -0., 0.1 0.2 Si 0.+ Nor—tOmenvonal mid, sit (a) (b) TAP CHI KHOA HOC TRUONG DAI HOC MO TP.HCM - So (35) 2014 84 Khi n = 0, tam se hoan toan lam bang gem va n co , tam se hoan toan la vat lieu kim loai 2.2 Phtrang trinh 4ng yeu va cling fink phan tir him h4n cho tam FGM Trong 1ST thuyet tam, phtrang trinh vi phan cua tam la phuang trinh vi phan bac bon Do vay de xap xi tridyng chuyen vi u =1,10 dm tam doi hai phai xap xi, phan tir bac cao tuc C') Nhu the rat khO ap dung phan tir him han thong thuOng de xap xi tru6ng chuyen vi ciia tam vi phan to huu han thong thir6ng chi lien tuc De khac phuc khO khan giao sir Reddy [30,31], da xay dung m6 hinh lien tuc cho phan tir tam va tnr6ng chuyen vi cua tam se ducic dinh nghia theo cong thirc sau _ 4z3 4z3 3i20-; v=v + z 3t2 4z3 (Z -312 4z3 ,v _ ;w=w 3/ (2) t chieu day tam, uo Iuo vo r va wa la chuyen vi mang va vong tai mat phang trung hoa; va 13 ={A A,}1 la goc xoay xung quanh truc y va x Trong phucmg trinh (2) gido su Reddy da cong them hai bien goc xoay = {O OAT de chuyen vector chuyen vi có bac tir tai nut cho phan dr lien tuc C' vector chuyen vi có bac to m8i nut cho phan tir lien tuc phii hop vai ly thuyet phan tir huu han u =1uo vo wo fl fl, 0,, yiT • Bien dang mat phang cho tam Mindlin cho bai cong thirc sau [Exx En, yr,,]T = Co ZKI (3) Z3K2 bien dang mang dirgc tinh bai du av a., avo Co =I T; a, ay (4) = V sU va bien dang uon &roc tinh theo cong thirc sau 1{vo + 0713Y 1; I (5) {(v+ + (vor ) + (v13+ (vpy )} vai = -7- 2 va bien clang cat &roc tinh b6i + ?Ks ; [rsz Yyz] Es (6) =Vw+13; Ks =C(3+41) veri v =[a i ax a i ay]' la toan tir dao ham Quan he img sat va Wen dang tir dinh luat Hook 6= E(z) - v(z)2 v(z) V(Z) 0 (I—P(r)) [ 0( V( Z)) ] k E( z ) (E + Z1(1 + Z31(2); = z 2K ) (7) 85 KHOA HQC KY THUAT led E(z) la mo dun dan hell Young; v(z) la he so Poisson plm thuOc luat phan cho bed phtrcrng trinh (1) Phtrong trinh dang yeu cho phan tich on dinh tam FGM dirge cho bai L 17, D*8 pd.) + LgyrDyd0+ +1LS [v ro/3 r7/ Tip kV/3.v a o Vfiy v °PY 12 n veri En , a VTowiko Vwc152+tin[Vrguo VT5vo ri; idn 12 ^ [vra0„ v780[6° o n vuo 0 VO LLL voy dci=° (8) co dang = {K )7' = {Ks K2 K1 (9) K,.} ' va ma tran hang so vat lieu if va DS &rot cho bai D =[A B E; B D F; E (10) D*s, =[As B'; F 1-1] , h/2 = f (Ay ,130 ,Du ,Ey (l,z,z ,z i,j =1,2,6 (11) rh/2 , , ,z ,z6 )adz i,j = 4,5 (A'4 ,B;J ,D;)= j_hi2 (1,z2 ,z4 )adz va o Q, ao =[ o (12) o Dual tac &Ong cua tai ca, cac ung suat có dang N,,`) N°y N° (13) Duai tac Ong cua tai nhiet thi N0 J-'12 E(z) k(z)Anz ; V(Z) (14) do, k(z) la he so dart nhiet dm tam Trong phtrong phap phan tir hau han, mien bai toan cl se &rot raj rac j Trtrirng chuyen vi cho tam FGM dirge phan tir cho S2= U52 va 52 n QI #0, /3, 0, O,, ]' va dtrcrc xap xi theo cOng thirc sau W0 fix cho bai uh = [ u0 vo uh = Ediag(N„N„N„N„N„N„N; )d, =Nd (15) vai N„ la tong so nut mien bai town duvc rOri rac; N, la ham dang tuyen tinh cua phan tir tam giac ba nut tai nut thir i; d = [u, v, 14'1 fixi /30 0,,lT la vector chuyen vi tai 86 TAP CHI KHOA HOC TRUONG DAI HQC MO TP.HCM - SO (35) 2014 The phtrong trinh (15) vao (3), (6), bin clang cong thirc (9) c6 the &rot vi6t 141 nhu sau: r yi,T=EB,d, E Eh =LE (16) i=1 vai ma tran bin clang chuye'n vi dugc cho bari 13: = [(B: )7 (B,' )7 (13:'' )' (Bs,' )T (Bs/ )T (17) dO Ar, y 0 0 0 0 N,, N,o, 0 0 , B = 0 0 N,„ 0 0 0 0 N,o, 0 N,„ N, Ni y N,„ =[ B:" = [0 0 0 jr, = c 0 N, N, 0 ' ' 0 0 0 N,, Bb, = 0 0 0 N, ), vai Ni, No No, N,, N,o, 0 0 N,„ 0 Ni N; 0 N (18) N,„_ va No, la cac ciao ham dm ham clang theo htrang x va y Phucmg trinh red rac cho phan tich on dinh cua tam FGM co clang (K - /1„Kg)d = (19) kr la tai tad han co hoc nh* cua tam va K, Id ma tran •:I() cimg hinh hpc dugc tinh bai K g = LBgrinBgc1S2 (20) vai m dugc cho bed Mo 0 0 0- too 0 0 t60 0 0 23 (10 0 L6 12 0 m= 11.6 12 0 t' 0• sym (21) 12 0_ va K la ma tran dO cfmg town cuc dugc 1ap ghep tir cac ma tran dO cirng phAn to K Q c6 dang sau N, N, K = EK,=E(f B,TD*Bidn +L s,TD:s,m) K, (22) 87 KHOA HOC KY THU;?■T B vaductc xac dinh bed B, = [(Br, )1 ( )1 (B (23) s, =[(138,°)T (Bs; )1 VI DV SO Trong phan nay, chung toi khao sat di) chinh xac va tinh hiau qua cua phuong phap hien tai cho phan tich on dinh tam FGM Tam có dieu kin bien goi tua (simply supported — S) va ngam (clamped - C) Ky hiau CCCC va SSSS la diau kian ngam va gOi Ara doc theo canh cua tam chit nhat 3.1 On dinh co Chung t8i xet tam vuong Al/Zr02-2 co chiau dai moi canh L = 0.2, chiau day t = 0.01 chiu mot tai nen mat phang (24) Oxy Hinh the hien tai tai han cua tam FGM vbi dieu.kian bien CCCC va SSSS Ka qua cho thay rang, nghiam cua phiremg phap de xuat rat triing khop vbi nghiam kpRitz [28] (sir dung phucmg phap Meshfree, xap xi truang chuyan vi bac cao) cho cac gia tri phan phoi the tich n = 0, 0.2, 0.5, 1, 2, Ngoai ra, chi:mg to co the thay rang, tai tai han se giam ti le the tich n tang cling cua tam len bed vi n tang.thi se giam Ngoai ra, bon dang dao damg on dinh dau tien cua tam cling dugc the hien Hinh Hinh Tai tea han on dinh cua tam vuOng 90 80 `I, 200 E E 70 CU to re cr, 150 60 CO co 50 100 40 Volume fraction exponent (a) Dang dao dOng thin Volume fraction exponent (b) Dang dao Ong thir 88 TAP CHI KHOA HOC TRUONG DAI HOC MO TP.HCM - SO (35) 2014 Hinh Biin dang dao thing on dinh clan tiers cua tam FGM 4`,440+.1,■"%=- *14,Vo 46.■ 44,1, k*OpV4:./ 11**46‘ 41714:97::, , -s,-2,4sougyilcir Vitogr.4 -ats.1.-4,,,,o 404 Ave 4.1p v -v •P.A.;,,e4APorap474 AN, 4.1PN v,P4 "a4V■vAPP,00, ihrkAO, 0.2 ••••• -•••••• 0.2 gi.* ,tt'■ 4I■1P /51 4746,c N°40 AAkkk,-siffrva,4 Ao•- •=4.44, 44,1,:• vosi P 4121b fr''' 04144 )0 • 10 ^ 40" VOW I: -"OP 4* -4* 4,■0 k•k+0‘ -1-1-'•••■•••■•04; =1:7 ■41N7w, ' • 'A 40,1,17'&`44 4.trta4/1-61■40.4ze" VAlf41PP -4-7,:■■0, 1/ -St*vA■4, 4.4t0AvA 4041 4.11r - *5 NNNI 40 10 4.„ ;1/4.■ 02 02 ww.v01,), ,10 • (d) Dang 3.2 On Binh nhqt n = 0, 1, Tir ket qua to thay rang nghiem cua phucmg phap hien tai rat trung khap voi nghiem dua ten ly thuyet hien dang cat bac nhat (FSDT) va 1y thuyet co dien diroc,nghien cfm ,beri Lanhe [32] (tnramg chuyen vi &roc xap xi lien tuc C') Trong vi du nay, chUng Col xet met tam hinh vuong Al/A1203 vai ti Lit = 10 chiu tac (Rog boi nhiet phan be deu doc theo chieu day dm tam Hinh the hien nhiet dO tai han ciia tam FGM Ung veri Hinh Nhiet tiri han ciia tam FGM FSDT Ti le the tich n KHOA HOC KC/ THUAT KET LUAN Bai bao the hien met tiep can so dun gian va hieu qua, Oa ten su ket hop gifra phan tir tam gide ba nut va ly thuyet bien Bang cat bac cao loai C° de phan tich on dinh ea nhiet dm tam FGM Thuec tinh vat lieu FGM phu thuee vao ti le cac phan the tich cua gom Ira kim loai va tuan 89 theo luat phan be ham se mil Anh huemg tai so mil ctia ham phan bo,,tac deng co va tai nhiet den on dinh cila tam FGM da dugc khao sat va so sanh vo i ket qua cac phucmg phap khac da dugc nghien cfru tn.rac day Cac ket qua dat dirge cho thay de tin cay cao cua phuong phap hien tai doi veri lap bai toan TAI LIEU THAM KHAO Koizumi M FGM activities in Japan Composites, 28 (1997) 1-4 Huang XL, Shen SH 2004, 'Nonlinear vibration and dynamic response of functionally graded plates in thermal environments', International Journal of Solids and Structures, 41, pp 2403-2427 Yang J, Shen HS 2002, 'Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments', Journal of Sound and Vibration, 255, pp 579-602 Najafizadeh MM and Heydari HR 2004 'Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory', European Journal of Mechanics - A/Solids, 23, pp.1085-1100 Vel SS, Batra RC 2004, 'Three-dimensional exact solutions for the vibration of functionally graded rectangular plate', Journal of Sound and Vibration, 272, pp.703-730 Matsunaga H Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory Composite Structures, 82 (2008) 499-512 Matsunaga H Thermal buckling of functionally graded plates according to a 2D higher-order deformation theory Composite Structures, 90 (2009) 76-86 Croce LD, Venini P Finite elements for functionally graded Reissner—Mindlin plates Computer Methods in Applied Mechanics and Engineering, 193 (2007) 705-725 Nguyen-Thoi T, Phung-Van P, Nguyen-Xuan H, Thai-Hoang C 2012, A cellbased smoothed discrete shear gap method using triangular elements for static and free vibration analyses of Reissner—Mindlin plates', International Journal for Numerical Methods in Engineering, 91(7), pp 705-741 10.Nguyen-Thoi T, Bui-Xuan T, Phung-Van P, Nguyen-Xuan H, Ngo-Thanh P Static, free vibration and buckling analyses of stiffened plates by CS-FEMDSG3 using triangular elements Computers & Structures, 125 (2013) 100-113 11.Nguyen-Thoi T, Phung-Van P, Thai-Hoang C, Nguyen-Xuan H 2013, 'A cell-based smoothed discrete shear gap method (CS-DSG3) using triangular elements for static and free vibration analyses of shell structures', International 90 TAP CHI KHOA HQC TRUONG DAI HOC MO TP,HCM - SO (35) 2014 Journal of Mechanical Sciences, 74, pp 32-45 12.Phung-Van P, Nguyen-Thoi T, Le-Dinh T, Nguyen-Xuan H Static and free vibration analyses and dynamic control of composite plates integrated with piezoelectric sensors and actuators by the cell-based smoothed discrete shear gap method (CS-FEM-DSG3) Smart Materials and Structures, 22 (2013) 095026 13.Nguyen-Thoi T, Bui-Xuan T, Phung-Van P, Nguyen-Hoang S, Nguyen-Xuan H 2013, 'An edge-based smoothed three-node Mindlin plate element (ESMIN3) for static and free vibration analyses of plates', KSCE Journal of Civil Engineering, (2013) at press 14.Phung-Van P, Nguyen-Thoi T, Tran V Loc, Nguyen-Xuan H 2013, 'A cellbased smoothed discrete shear gap method (CS-FEM-DSG3) based on the CO-type higher-order shear deformation theory for static and free vibration analyses of functionally graded plates', Computational Materials Science, 79, pp 857-872 15.Nguyen-Thoi T, Phung-Van P, Rabczuk T, Nguyen-Xuan H, Le-Van C 2013, `Free and forced vibration analysis using the n-sided polygonal cell-based smoothed finite element method (nCS-FEM)', International Journal of Computational Methods, 10(1) (2013) 1340008 16.Nguyen-Thoi T, Phung-Van P, Rabczuk T, Nguyen-Xuan H Le-Van C An application of the ES-FEM in solid domain for dynamic analysis of 2D fluidsolid interaction problems International Journal of Computational Methods, 10(1) (2013) 1340003 17.Nguyen-Thoi T, Phung-Van P, Ho-Huu V, Le-Anh L 2014, 'An edge-based smoothed finite element method (ES-FEM) for dynamic analysis of 2D fluidsolid interaction problems', KSCE Journal of Civil Engineering, (2014) at press 18.Nguyen-Thoi T, Phung-Van P, Nguyen-Hoang S, Lieu-Xuan Q 2014, 'A smoothed coupled NS/nES-FEM for dynamic analysis of 2D fluid-solid interaction problems', Applied Mathematics and Computation, 232, pp 324346 19.Phung-Van P, Nguyen-Thoi T, Luong-Van H, Lieu-Xuan Q 2014, 'Geometrically nonlinear analysis of functionally graded plates using a cell-based smoothed three-node plate element (CS-MIN3) based on the C°-HSDT', Computer Methods in Applied Mechanics and Engineering, 270, pp.15-36 20 Phung-Van P, Nguyen-Thoi T, Dang-Trung H, Nguyen-Minh N A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using layerwise theory based on the CO-type higher-order shear deformation for static and free vibration analyses of sandwich and composite plates Composite Structures, (2014), DOI: 10.1016/j.compstruct.2014.01.038 21 Phung-Van P, Nguyen-Thoi T, Luong-Van H, Thai-Hoang C, Nguyen-Xuan H 2014, 'A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using layerwise deformation theory for dynamic response of composite plates resting on viscoelastic foundation', Computer Methods in Applied Mechanics KHOA HOC K? 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