Journal of Simulation & Analysis of Novel Technologies in Mechanical Engineering 10 (4) (2017) 0049~0064

Transcription

1 Jounl of Siultion & Anlysis of Novel Technologies in Mechnicl Engineeing (4 (7 49~64 ISSN: Theo-elstic Anlysis of Functionlly Ge Thick- Wlle Cyline with Novel Tepetue Depenent Mteil Popeties using Petution Technique Hi Mohi Hooyeh, Aliez Nfoskouie Deptent of Soli Mechnics, Fculty of Mechnicl Engineeing, Univesity of Eyvnekey, Gs, In Astct (Mnuscipt Receive M. 8; Revise --- My 8; Accepte July 8 In this wok, theo elstic nlysis fo functionlly ge thick wlle cyline with tepetue - epenent teil popeties t stey conition is cie out. The length of cyline is infinite n loing is consist of intenl hyosttic pessue n tepetue gient. All of physicl n echnicl popeties expect the Poisson's tio e consiee s ultiplie n exponentil function of tepetue n powe function of ius. With these ssuptions, the nonline iffeentil equtions fo tepetue istiution t cylinicl coointe is otine. Tepetue istiution is chieve y solving this eqution using clssicl petution etho. With consieing stin isplceent, stess stin n equiliiu eltions n tepetue istiution tht poucte pevious, the constitutive iffeentil eqution fo cyline is otine. By eploying echnicl ouny conition the il isplceent is yiel. With hving il isplceent, stesses istiution long the thickness e chieve. The esults of this wok show tht y incesing the oe of tepetue petution seies the convegence t cuves is occue n lso iensionless il stess ecese n othe stesses with iensionless il isplceent incese. Keywos: Infinite thick wlle cyline, nonline het tnsfe, clssicl petution etho, tepetue-epenent popeties, functionlly ge teil. - Intouction Functionlly ge teils (FGMs e icoscopiclly inhoogeneous in which the echnicl popeties vy soothly n continuously fo one sufce to the othe [-]. FGMs e ege s one of the ost poising cnites fo futue vnce coposites in ny engineeing sectos such s the eospce, icft, utooile, n efense inusties, n ost ecently the electonics n ioeicl sectos [4-6]. All of these inusties use piping systes n thickwlle cylines. Mny stuies fo thel stesses of functionlly ge cylines e ville in the litetue. Ji et l. [7] epesente genel nlysis fo one-iensionl stey-stte thel stesses in hollow thick wlle FGM cyline. They consiee tht the teil popeties, except Poisson s tio, e ssue to epen on vile the ius n they e expesse s powe function. Thei esults evele tht the gnitue of the il stess is incese s the teil pete ( is incese. Sho n M [8] stuie theoechnicl nlysis of functionlly ge hollow cicul cylines sujecte to echnicl los n linely incesing ouny tepetue. They ssue tht the theo-echnicl popeties of FGM to e tepetue inepenent n vy

2 5 H.Mohi et l./ Jounl of Siultion & Anlysis of Novel Technologies in Mechnicl Engineeing (7 49~64 continuously in the il iection of cyline. They otine tie-epenent tepetue n unstey theoechnicl stesses y Lplce tnsfo techniques n seies solving etho. Di n Fu [9] pesente exct solutions fo stesses n petutions of the gnetic fiel vecto in FGM hollow cylines using the infinitesil theoy of gnetotheoelsticity. Thei esults illustte tht the inhoogeneity constnt hs jo effect on the gnetotheoelstic stesses, negtive vlues yiels copessive il n cicufeentil stesses in the whole FGM hollow cyline, while positive vlues gives conty esult. Axisyetic isplceents n stesses in functionllyge hollow cylines sujecte to unifo intenl pessue using plne elsticity theoy n Copleenty Functions etho escie y Tutuncu n Teel []. Sho n Wng [] investigte nlyticl solutions of the thee-iensionl tepetue n theoelstic stess fiels in the FGM cylinicl pnel with finite length. They consiee tht the thel n echnicl popeties of the FGM to e tepetue inepenent. Thei esults showe tht the xil stess is lge thn the il stess n is slle thn the cicufeentil stess. Tutuncu [] lso otine powe seies solutions fo stesses n isplceents in FG cylinicl vessels sujecte to intenl pessue lone using infinitesil theoy of elsticity. It is conclue fo his esults tht negtive inhoogeneity constnt woul cete stess plifiction effect. His stuy y e useful fo specific pplictions to contol the stess istiution. Moh n Mhooeh Azi [] cie out nonline tnsient het tnsfe n theoelstic stess nlyses of thick-wlle FGM cyline with tepetue - epenent teils using the Heitin tnsfinite eleent etho. The tepetue istiution n the il n cicufeentil stesses e investigte vesus tie, geoeticl petes n inex of powe lw in thei esech. Thei esech epesente tht y incesing of thickness, stesses ecese ue to ecesing of tepetue gient. A novel etho fo theoelstic nlysis of cylinicl vessel of FGMs pesente y Peng n Li [4].The thel n theoelstic petes ssue to itily vy long the il iection of the hollow cyline in thei wok. They coine the ouny vlue pole with theoelstic pole n then convete to Fehol integl eqution. Accoingly, y nueiclly solving the esulting eqution, the istiution of the thel stesses n il isplceent ws otine. They tht ppopite gient cn ke the istiution of thel stesses oe gentle in foun the whole stuctue. Exct n ppoxite solutions of theoelstic stesses in FG cylines with powe lw n exponentilly vitions of teil popeties eive y Seifi [5]. He conclue tht effects of high tepetue on the stesses e oe ipotnt thn the high intenl pessue. Exct theoelstic nlysis of FG nisotopic hollow cylines with ity teil gtion povie y Vel [6]. He solve the iffeentil equtions of het conuction n theoelsticity using powe seies solution technique. He foun tht the tepetue, isplceents n stesses e sensitive to the teil gtion. Loghn n Ps [7] nlyze the gneto-theo-elstic esponse fo thick oule-wlle cyline e fo FGM intelye n hoogeneous oute lye. They showe tht tht une theogneto-echnicl loing iniu effective stess istiution n the iniu il isplceent cn e chieve y selecting n ppopite teil pete in the FGM lye. Ghonpou Ani n his esech goup [8] escie electo-theo-echnicl ehvio of illy polize otting functionlly ge piezoelectic cyline. They consiee tht the teil

3 H.Mohi et l./ Jounl of Siultion & Anlysis of Novel Technologies in Mechnicl Engineeing (7 49~64 5 popeties except Poisson s tio n thel conuction coefficient to e exponentilly istiute long ius. It is conclue fo thei esults tht the inhoogeneity exponent plys sustntil ole in il n cicufeentil stess istiutions. Theo-electoechnicl ehvio of FG piezoelectic hollow cyline une nonxisyetic los stuie y Atin et l. [9] Aefi n Rhii [] investigte the effect of nonhoogeneity n en suppots on the theo elstic ehvio of clpe-clpe FG cyline une echnicl n thel los. They eploye Hilton pinciple n fist oe she efotion theoy (FSDT fo eivtion of the pinciple iffeentil equtions. Thei esults showe tht the solute vlue of xil isplceent of the cyline eceses with incesing the non hoogenous inex. Thel stesses nlysis of FG cylinicl shell une thel shock se on iffeentil qutue etho chieve y Zhng et l. [] They illustte tht the thel stesses coul e llevite y ens of chnging the volue fctions of the constituents. Theo-elstic nlysis of clpe-clpe thick FGM cylines y using thi-oe she efotion theoy inicte y Ghooni et l. [] They solve the set of nonhoogenous line iffeentil equtions fo the cyline with clpe-clpe ens. It is oseve fo thei esults tht using negtive inhoogeneities in the cyline cuses sll eceses une thel lo n consiele incese une echnicl lo in il isplceents. Two iensionl theoelstic nlysis of FG cyline fo iffeent functionlities y using the highe oe she efotion theoy pesente y Aefi []. His esults showe tht the il isplceent eceses with incesing powe low inex ue to n incese in the cyline stiffness with incesing powe low inex. Aefi n his esech goup [4] epesente two-iensionl theoelstic nlysis of FG cylinicl shell esting on the Pstenk fountion sujecte to echnicl n thel los se on FSDT foultion. They use enegy etho n Eule equtions fo govening iffeentil equtions of syste. They showe tht y incesing the nonhoogeneity inex oth, il n xil isplceent eceses. In ll of pevious stuies tht conucte on the FGM, siply oeling fo echnicl n physicl chnges popeties ws selecte. These oels wee exponentilly istiute o expesse y powe function long the ius. In cuent wok, we ie to consie chnges ll physicl n echnicl popeties (except fo Poisson's tio s powe function vesus ius n function of exponentilly vesus tepetue. This ction le to nonline iffeentil eqution fo het tnsfe tht y solving of it using petution technique we cn use the tepetue istiution t theoelstic nlysis. - Geoety, loing conition n ssuptions An infinitely long, functionlly ge thick wlle hollow cyline with inne ius n oute ius is consiee. This cyline is sujecte to n intenl hyosttic pessue P n unifo tepetue fiel with inne sufce tepetue T n oute tepetue of T. One iensionl n stey stte het conuction e selecte fo tepetue istiution. Cente cylinicl coointes is on the cente of the cyline n xil syety in geoety n loing will e consiee. Figue shows the schetic view of the cyline n its loing.

4 5 H.Mohi et l./ Jounl of Siultion & Anlysis of Novel Technologies in Mechnicl Engineeing (7 49~64 Figue. Schetic of FG cyline une echnicl n thel los - Het conuction pole The het conuction eqution with eg to the ove-entione ssuptions n without ny het souce is witten in cylinicl coointe s follow: T k ( In which, k n T e ius, noinl het conuctivity coefficient n tepetue istiution espectively. It is ssue tht the theophysicl n echnicl popeties expect the Poisson's tio e consiee s pouct of n exponentil function of tepetue n powe function of ius. Futheoe, the noinl het conuctivity coefficient is ssue to epen on tepetue s follow [5]: k, T k ( k ( T ( The exponentil function of tepetue is ssue s follow fo: 6 k( T kexp( ( T T ( The powe function istiution in the il iection consiee s [5], [6]: k ( k 4( Accoingly, fo noinl het conuctivity coefficient cn e witten s follow: 6 k, T k exp( ( T T k k k 5( Whee k,, e the physicl constnts chcteizing the teil ehvio. By sustituting Eq. ( in to Eq. ( we hve: T T ( k (, T k (,T k (, T T T k (, T 6( k (, T In which is clculte using the pouct ule s follow: k (,T 6 ( k exp( ( T T 6 (exp( ( T T k 7( 6 k exp( ( T T 6 6 T k exp( ( T T Sustituting Eq. (7 in to Eq. (6 n then siplify, the following nonline iffeentil eqution of the secon oe is otine: T 6 T T 8( - - Petution technique Petution technique is one of the ost efficient ppoch to solving vious ouny issues in elstic stuctues. This etho s ens of ppoxite - nlyticlly useful is eploye to solve lge pt of the nonline poles. Accoing to the petution etho, coplex nonline iffeentil eqution is ivie to unliite nue of eltively siple equtions (Petution eqution. Accoingly, the solution of oiginl eqution is the sution of solving ech Petution equtions with the ising powe of sll petution pete s coefficient is expesse. Theefoe, the fist few tes, epesent the view of solving the pole. It is ipotnt t petution technique tht the sll petution pete, select the ppopite. In using this etho it is necessy tht the sll petution

5 H.Mohi et l./ Jounl of Siultion & Anlysis of Novel Technologies in Mechnicl Engineeing (7 49~64 5 pete to e consiee slle thn one. This etho is use in continue. By iviing Eq. (8 on pete we hve: 6 T T T 9( 6 The sll tio of is to ppe in the nonline iffeentil eqution (9 tht we cn now efine this tio equl to sll pete. Theefoe the Eq. (9 cn e ewitten s follow: T T T ( Petution theoy les to n expession fo the esie solution in tes of fol powe seies in soe sll pete, known s petution seies, tht quntifies the evition fo the exctly solvle pole. The leing te in this powe seies is the solution of the exctly solvle pole, while futhe tes escie the evition in the solution, ue to the evition fo the initil pole. Folly, we hve fo the ppoxition to the full solutiont, seies in the sll pete, like the following [7-]: T T T T... In Eq. (, T woul e the known solution to the exctly solvle initil pole nt, T,... epesent the higheoe tes which y e foun itetively y soe systetic poceue. Fo sll these highe-oe tes in the seies ecoe successively slle. An ppoxite petution solution is otine y tuncting the seies, usully y keeping only the fist thee tes, the secon-oe petution coection cn e witten s []: T T T T By sustituting Eq. ( into Eq. ( n gouping ll tes with the se powe of gives: T T ( T T ( T T T ( T T ( With setting zeo coefficient of vious powes of (the coefficient of will e ignoe set of iffeentil eqution cn e otine s follow: T T O ( : ( T T T O ( : ( 4 T T T O ( : ( Fist, the O ( pole is to e solve n then it esults in highe-oe ppoxition is use. The nswe of O ( pole cn e chieve s: c T ( c 5 c n c e the integtion constnts which cn e eteine y the following thel ouny conitions: T( T T( T 6 Iposition of these ouny conitions onto the Eq. (5 gives the following eltions fo the integtion constnts c n c :

6 54 H.Mohi et l./ Jounl of Siultion & Anlysis of Novel Technologies in Mechnicl Engineeing (7 49~64 c T T T c T, 7 By sustituting Eq. ( into the O ( t Eq. (, the iffeentil eqution fo O ( pole will e otine s follow: T T c 8 This is n inhoogeneous Cuchy Eule oiny iffeentil eqution whose genel n pticul solutions cn e foun s: c c T( c4 9 The integtion constnts c n c 4 cn e chieve y pplying thel ouny conitions (6 to the solution (9 s: c ( ( T T c ( c 4 ( c T ( ( c T ( As efoe, we sustituting Eq. (9 into the O ( t Eq. ( n the O ( iffeentil eqution will e epesente s follow: T T c c ( Eq. ( is gin n inhoogeneous Cuchy-Eule iffeentil eqution whose genel solution is pesente s the su of the hoogeneous solution n pticul solution s follow: c5 4 4 T ( c6 c cc c Appliction of ouny conitions (6 to the genel solution ( yiels the integtion constnts c5 nc 6 s follow: c 5 6 c ( ( 4 c c ( ( 6 c ( ( T T c ( c ( 4 c c 4 4 ( (6c T c 4 ( 4 4 c cc ( (6c T ( ( 4 Theefoe, the tepetue istiutions t iffeent oes of ppoxition cn e otine s follow: c T ( ( c O ( T ( ( c c c ( c c O ( 4 c T ( ( c c c ( c4 c ( c c c c c O ( Theoelstic nlysis 4- - Stess stin eltions The stess stin eltions t cylinicl coointes syste t genel fo cn e witten s follow []: 4

7 H.Mohi et l./ Jounl of Siultion & Anlysis of Novel Technologies in Mechnicl Engineeing (7 49~64 55 ( ( z T ( E ( ( z T ( E ( z ( z z T ( E,, G G G z z z z 5 In which, n z enotes il, cicufeentil n xil iections, espectively. i n i ( i,, z e the stess n stin, n E e the Poisson's tio n oulus elsticity, espectively. G is the young oulus n i ( i,, z is the coefficient of thel expnsion tht consiee to e se t ll coointes. Accoing to the ssuptions, ll she stins n cicufeentil eivtives e zeo n fo plne stin conitions the xil stess is consiee s follow: E T ( z z 6 With sustituting Eq. (6 into Eq. (5 n siplify, the stess stin eltions fo functionlly ge thick wlle cyline t one iensionl het conuction e foun s follow: T ( 7 T ( The coefficients in Eq. (6 e in the powe n exponentil functions fo n e efine s elow: T T E ( ( ( 6 exp( ( 6 exp( ( T T E ( ( ( T T 6 exp( ( ( z Stin isplceent eltions The stess isplceent eltions t cylinicl coointes syste t genel fo cn e otine s follow []: u u u, uz u u u, z u u z u u z z, z z z 9 Whee ui ( i,, z is the isplceent vecto. These eltions with consieing ssuptions ewitten s elow: u u, 4- - Stess isplceent eltions Sustituting Eq. ( into Eq. (7, the stess isplceent eltions yiels s follow: u u T ( u u T ( 4-4- Equiliiu eltions The equiliiu equtions of the hollow cyline t genel fo e s follow [], []: z z F t z z u F t z z z u z z Fz t u

8 56 H.Mohi et l./ Jounl of Siultion & Anlysis of Novel Technologies in Mechnicl Engineeing (7 49~64 Whee Fi ( i,, z e the oy foces n is the teil ensity. The equiliiu eqution of the FGM hollow cyline, with consieing pevious ssuptions n stey conitions in the sence of oy foces, is expesse s: 4-5- Theoeltic constitutive eqution By sustituting the esulting tepetue istiution fo Eq. (4 into Eq. ( n then into Eq. ( the theoelstic constitutive eqution fo FG thick wlle hollow cyline with tepetue epenent teil popety is eive s follow: u u ( u ( c c c c c4 c c c5 c6 4 4 cc ( c c c 4 4 c 4 cc c c5 4 The oots of the chcteistic eqution e: So, the genel solution of the inhoogeneous eqution (4 cn e expesse s su of pticul solution n the genel solution of the hoogeneous one s elow: u ( u ( u ( C C g g 4 4 g g g In which: p c c4 c6 g ( c c g ( ( ( 4 4 c g 4 ( 4 ( ( 4 cc g 4 ( ( ( 7 8 Cn C e the integtion constnts which will e foun s follow Integtion constnts By sustituting il isplceent Eq. (7 into Eq. ( the il stess cn e ewitten s: This eqution is the fili Cuchy-Eule iffeentil eqution fo il isplceent tht chcteistic eqution fo it is s follow: 5

9 H.Mohi et l./ Jounl of Siultion & Anlysis of Novel Technologies in Mechnicl Engineeing (7 49~64 57 exp( ( T T ( C C g g g g exp( ( T T ( C C g g g g 4 exp( ( T T ( c c 6 c c c c c c c c 5 c6 The theoelstic pole is sujecte to the following echnicl ouny conitions. The cyline is loe with intenl hyosttic pessue: 9 P( P P( 4 Appliction of these ouny conitions to the il stess (9 yiels the following eltions fo the integtion constnts: C (( ( ( c c 4(( g ( ( c c 4( ( g 4 ( c 48 g (( 4 4 ( cc 9g 4( ( ( exp( ( T T P 5 ( c c c ( c 48 g (( 4 4 ( cc 9 g 4(( ( ( c c c ( ( c c c g ( ( 4 4

10 58 H.Mohi et l./ Jounl of Siultion & Anlysis of Novel Technologies in Mechnicl Engineeing (7 49~64 C (6( ( c c 6 4 c 4(( 4(( g ( g 48 g (( ( cc 9 g 4(( ( ( ( c c 5 ( c c c 4 exp( ( T T P 4 4 ( c 48 g (( ( ( c5 c c 6 4 ( ( ( cc 9 g 4(( ( ( c c c g 4 With fining unknown constnts the ll stesses n il isplceent cn e clculte. Also the Von Mises stess cn e otine s follow: eff ( z ( z 5- Nueicl esults n iscussion 4 In the following nueicl clcultions, the echnicl n geoetic popeties fo FGM cyline e consiee s [- 5]: E (G P..e 6( C P (MP 4 It shoul e note tht tepetue vlues t inne n oute sufces n othe pete is escie fte thn ny figue. Fo ette nlysis of esults the iensionless pete e pesente s follows: i i i,, z, eff P 44 u u A istiute tepetue fiel ue to stey-stte het conuction fo thickwlle FGM cyline vesus iensionless ius with iffeent oes of ppoxition illustte in figue. It is shown fo this figue tht y incesing the oes of ppoxition fo zeothoe (i.e., line pole until the secon- oe solution the convegence of petution seies is occus. Also, it is oseve esily tht the thel ouny conitions hve een stisfie. Figues n 4 inicte the influence of physicl constnts chcteizing the teil ehvio (, on the tepetue istiution (secon oe. It is cle fo figues n 4 tht y incesing the vlue of n the tepetue istiution inceses long the thickness of FGM cyline. Figue 5 epesents the iensionless il stess istiution vesus iensionless ius with iffeent oes of ppoxition. It is conclue tht iensionless il stesses eceses with incesing oes of ppoxition. Fo this figue lso convegence of petution seies is otine. Futheoe, it is copessive t the inne pt of the cyline which hs to stisfy the intenl pessue ouny conition. Diensionless cicufeentil n xil stesses long the thickness of cyline with incesing the oes of ppoxition e eonstte y Figues 6 n 7, espectively. It cn e conclue fo these figues tht y

11 T( o C T( o C T( o C H.Mohi et l./ Jounl of Siultion & Anlysis of Novel Technologies in Mechnicl Engineeing (7 49~64 59 incesing iensionless ius the iensionless cicufeentil n xil stesses inceses. It cn e seen tht xiu n iniu stesses hppens t inne n oute sufces, espectively. Figues 8 n 9 epicts the effects of incesing the oes of ppoxition on the iensionless effective stesses n istiution of iensionless il isplceent long the thickness of cyline, espectively. It is shown tht y incesing oes of ppoxition the iensionless effective stess n iensionless il isplceent inceses. Also it is oseve tht xiu effective stess n il isplceent e t inne sufce n it ens tht xiu ge occus t in this sufce. The influence the influence of physicl constnts chcteizing the teil ehvio (, on the iensionless effective stess e epesente in figues n. It is conclue fo figue tht fo constnt vlue of, y incesing the vlue of the vlues of iensionless effective stess t the inne lye to the ile lye of the cyline hs een euce n the iensionless effective stess vlues of the oute lye is e. Figue show tht y incesing the vlue of, t constnt vlue of, the iensionless effective stess eceses T +T +T Figue. Tepetue istiution stess long the thickness of FGM cyline with iffeent oes of ppoxition T ( C, T ( C, 4, = - = - = - = = = Figue. Tepetue istiution stess long the thickness of FGM cyline with iffeent vlue of T ( C, T ( C, =.6 =.7 =.8 = Figue4. Tepetue istiution stess long the thickness of FGM cyline with iffeent vlue of T ( C, T ( C, +T +T +T Figue5. Diensionless il stess long the thickness of FGM cyline with iffeent oes of ppoxition T 5( C, T 5( C,,.

12 6 H.Mohi et l./ Jounl of Siultion & Anlysis of Novel Technologies in Mechnicl Engineeing (7 49~ T +T +T u Figue6. Diensionless cicufeentil stess long the thickness of FGM cyline with iffeent oes of ppoxition T 5( C, T 5( C,, z T +T +T Figue7. Diensionless xil stess long the thickness of FGM cyline with iffeent oes of ppoxition T 5( C, T 5( C,, eff 9 x T +T +T Figue9. Diensionless il stess long the thickness of FGM cyline with iffeent oes of ppoxition eff T 5( C, T 5( C,,. = = = = Figue. Diensionless effective stess long the thickness of FGM cyline with iffeent vlue of.5 T 5( C, T 5( C,. =. =.5 =. =.5 +T +T +T eff Figue. Diensionless effective stess long the thickness of FGM cyline with iffeent vlue of T 5( C, T 5( C, Figue8. Diensionless effective stess long the thickness of FGM cyline with iffeent oes of ppoxition 5- Conclusion T 5( C, T 5( C,,. In this ppe theoelstic nlysis of functionlly ge thick wlle cyline

13 H.Mohi et l./ Jounl of Siultion & Anlysis of Novel Technologies in Mechnicl Engineeing (7 49~64 6 with tepetue epenent teil popety is investigte. This ticle epesente suitle ppoch fo nlysis teil tht thei theophysicl n echnicl popeties expect the Poisson's tio e consiee s pouct of n exponentil function of tepetue n powe function of ius. Fo this pupose, petution technique is use to solving nonline iffeentil eqution fo tepetue istiution. This tepetue istiution is eploye t giving ehvio of thel stesses. The esults of this ticle cn e liste s follows: - By incesing the oes of ppoxition fo zeoth-oe (i.e., line pole until the secon- oe solution the convegence of petution seies t ll figues is occus. - By incesing the vlue of n the tepetue istiution inceses long the thickness of FGM cyline. - The iensionless il stesses eceses n iensionless cicufeentil, xil n effective stesses n il isplceent inceses with incesing oes of tepetue ppoxition. 4- Mxiu effective stess n il isplceent e locte t inne sufce. It ens tht xiu ge occus t in this sufce. The otine esults fo this esult cn e use to esign of cylinicl stuctues. 5- By incesing the vlue of the iensionless effective stess t the inne lye to the ile lye of the cyline hs een euce n the iensionless effective stess t the oute lye inceses. Also y incesing the vlue of the iensionless effective stess eceses. Refeences: [] S. Suesh, n A. Motensen, Funentls of functionlly ge teils, Bnes n Nole Pulictions, 998. [] M.Ynouchi, n M. Koizui, Functionlly gient teils. Poceeing of the fist intentionl syposiu on functionlly ge teils, Seni, Jpn, 99. [] M.M. Njfizeh, n H.R. Heyi, An exct solution fo uckling of functionlly ge cicul pltes se on highe oe she efotion plte theoy une unifo il copession, Int. J. Mech. Sci., vol. 5, pp. 6 6, 8. [4] H.J. Xing, n J. Yng, Fee n foce vition of linte FGM Tioshenko e of vile thickness une het conuction, Copos. Pt B (Eng, vol. 9, pp. 9, 8. [5] R. Ansi, n M. Dvizeh, Peiction of ynic ehviou of FGM shells une ity ouny conitions, Copos. Stuct., vol. 85, pp. 84 9, 8. [6] A. Allhveizeh, M.H. Nei, n M. Nikkhh Bhi, Vition plitue n thel effects on the nonline ehvio of thin cicul functionlly ge pltes, Int. J. Mech. Sci., vol. 5, pp , 8. [8] M. Ji, S. Sohpou, n M.R. Eslic, Mechnicl n thel stesses in functionlly ge hollow cyline ue to illy syetic los, Int. J. Pessue Vessels Piping, vol. 79, pp ,. [9] Z.S. Sho, n G.W. M, Theoechnicl stesses in functionlly ge cicul hollow cyline with linely incesing ouny tepetue, Copos. Stuct., vol.8 pp , 8. [] H.L. Di, n Y.M. Fu, Mgnetotheoelstic intections in hollow stuctues of functionlly ge teil sujecte to echnicl los, Int. J. Pessue Vessels Piping vol. 84 pp. 8, 7.

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