Time-Dependent Hygro-Thermal Creep Analysis of Pressurized FGM Rotating Thick Cylindrical Shells Subjected to Uniform Magnetic Field

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1 Jounal of Solid Mhanis Vol. 9, No. 3 (17) pp Tim-Dpndnt Hygo-Thmal Cp Analysis of Pssuizd FGM Rotating Thi Cylindial Shlls Subjtd to Unifom Magnti Fild A. Bahshizadh 1, M. Zamani Njad 1,*, M. Davoudi Kasholi 1 Dpatmnt of Mhanial Engining, Yasouj Univsity, Yasouj, Ian Dpatmnt of Mhanial Engining, Shahid Chaman Univsity of Ahvaz, Ahvaz, Ian Rivd 9 Jun 17; aptd 3 August 17 ABSTRACT Tim-dpndnt p analysis is psntd fo th alulation of stsss and displamnts of axisymmti thi-walld ylindial pssu vssls mad of funtionally gadd matial (FGM). Fo th pupos of tim-dpndnt stss analysis in an FGM pssu vssl, matial p bhavio and th solutions of th stsss at a tim qual to zo (i.. th initial stss stat) a ndd. This osponds to th solution of th poblm onsiding lina lasti bhavio of th matial. Thfo, using quations of quilibium, stss stain and stain displamnt, a diffntial quation fo displamnt is obtaind and subsquntly th initial lasti stsss at a tim qual to zo a alulatd. Assuming that th Magnto-hygo-thmolasti p spons of th matial is govnd by Noton s law, using th at fom of onstitutiv diffntial quation, th displamnt at is obtaind and thn th stss ats a alulatd. On th stss ats a nown, th stsss at any tim a alulatd itativly. Th analytial solution is obtaind fo th plan stain ondition. Th pssu, inn adius and out adius a onsidd to b onstant and th magnti fild is unifom. Matial poptis a onsidd as pow law funtion of th adius of th ylind and th poisson s atio as onstant. Following this, pofils a plottd fo diffnt valus of matial xponnt fo th adial, iumfntial and fftiv stsss as a funtion of adial dition and tim. Th in-homognity xponnt hav signifiant influn on th distibutions of th p stsss. 17 IAU, Aa Banh. All ights svd. Kywods: Thi ylindial pssu vssl; Magnto-hygo-thmolasti-p; Tim-dpndnt; Funtionally gadd matial (FGM). 1 INTRODUCTION F UNCTIOALLY gadd matials (FGMs) a miosopially inhomognous omposit matials, in whih th matial poptis vay smoothly and ontinuously fom on sufa to th oth (Xi, Dai & Rao [1]). Th main advantag is that th mhanial poptis vay ontinuously aoss th shll thinss, whih liminats stss disontinuitis typial of omposit laminats. Today, FGMs find ngining appliations in aospa stutus, hmial and mhanial industis, t (Lvyaov & Kuzntsov []). Sval studis hav * Cosponding autho. Tl.: ; Fax: addss: m_zamani@yu.a.i (M. Zamani Njad). 17 IAU, Aa Banh. All ights svd.

2 664 A. Bahshizadh t al. bn pfomd by sahs on th stati bhavio of FGM stutus (Njad & Fathi [3], Njad, Jabbai & Ghannad [4-5-6], Njad, Rastgoo & Hadi [7]). Thi shlls involving ylindial pssu vssls, in nt yas, a widly usd in spa vhils, aiafts, nula pow plants and many oth ngining appliation (Njad, Jabbai, & Ghannad [8-9], Jabbai, Njad & Ghannad [1-11]). Elasti, plasti and p stss analysis of ths omponnts has attatd wid attntion du to th ombination of diffnt matial poptis and loading onditions. Cp stsss in FGM thi ylindial shlls und thmal loading hav bn analyzd xtnsivly with gad to th lasti matial bhavio in th past yas. Attia, Fitzgog and Pop [1] invstigatd th sidual stsss podud in ast ion ylind by th p-laxation of thmal stsss. In this study a sis of thi-hollow ylinds a subjtd to a adial flow of hat by hating th bo and wat-ooling th out diamt fo a hosn piod of tim, duing whih th thmal stsss a laxd by p. Wi [13] invstigatd p stsss in pssuizd thi walld tubs. Assuming th simply suppotd and fixd-nd bounday onditions fo th ylinds, Wah [14] dvlopd a thoy fo th ollaps of ylindial shlls und stady-stat p and und xtnal adial pssu and high tmpatu (3 to 5 F.). Bhatnaga and Gupta [15] obtaind solution fo an othotopi thi-walld intnally pssuizd ylind by using onstitutiv quations of anisotopy p and Noton s p law. Bssling [16] invstigatd th fasibility of a numial analysis of nonstationay p poblms fo thi-walld tubs und axially symmti loading. Pai [17] studid th stady-stat p of a thi-walld othotopi ylind subjtd to intnal pssu. Thy obsvd that th p anisotopy has a signifiant fft on th ylind bhavio patiulaly in tms of p ats whih may diff by an od of magnitud ompad to an isotopi analysis. Sanaanaayanan [18] studid th stady p bhavio of thin iula ylindial shlls subjtd to ombind latal and axial pssus. Th analysis is basd on th Tsa ition and th assoiatd flow ul. Assuming that th total stain is onsist of lasti and p omponnts, Muaami and Iwatsui [19] invstigatd th tansint p analysis of iula ylindial shlls on th basis of th stain-hadning and tim-hadning thois. Muaami and Suzui [] dvlopd a numial analysis of th stady stat p of a pssuizd iula ylindial shll on th basis of Miss ition and th pow law of p. Sim and Pnny [1] studid th dfomation bhavio of thi-walld tubs subjtd to a vaity of loadings duing stss distibution ausd by p. Muaami and Iwatsui [] invstigatd th stady stat p of simply suppotd iula ylindial shlls with opn nds und intnal pssu by using Notons s law. Kasholi and Njad [3] invstigatd th fft of hat flux on p stsss of thi-walld ylindial pssu vssls by using Notons s law. Using finit-stain thoy Bhatnaga and Aya [4] studid th p bhavio of a thi-walld ylind und lag stains. Muaami and Tanaa [5] invstigatd th p buling of lampd iula ylindial shlls subjtd to axial ompssion ombind with intnal pssu with spial mphasis on th onpt of p stability and th auay of th analysis. Jons and Sullivan [6] studid th advantags and limitations of a ptubation mthod of analysis fo th p buling of shlls by xamining th patiula as of a long ylindial shll subjtd to a unifom xtnal pssu. Aya, Dbnath and Bhatnaga [7] invstigatd th poblm of p in a thin iula ylindial shll mad of a homognous, inompssibl and othotopi matial using a non-stady p law. Cp damag simulation of thi-walld tubs using th thta pojtion onpt was invstigatd by Loghman and Wahab [8]. Thy obtaind a losd-fom solution fo stady stat p stsss in FGM ylinds. Yang [9] psntd an analytial solution fo th alulation of stsss in FGM fo th lasti and p bhavio of th matials. This solution an b usd to study th dpndn of stss on tmpatu and tim fo FG stutus. Finally, th analytial sults w ompad with FEM. Gupta and Patha [3], studid thmo p analysis in a pssuizd thi hollow ylind. Jahd and Bidabadi [31] psntd a gnal axisymmti mthod fo an inhomognous body fo a dis with vaying thinss. An appoximation has bn mployd duing thi solution algoithm. It mans that thy avoid onsiding th diffntiation onstitutiv tms of govning quations fo p analysis. Chn, Tu, Xuan and Wang [3] studid th p bhavio of a funtionally gadd ylind und both intnal and xtnal pssus. Thy obsvd that an asymptoti solution an b divd on th basis of a Taylo sis xpansion if th poptis of th gadd matial a axisymmti and dpndnt on adial oodinat. In od to invstigat p pfoman of thi-walld ylindial vssls o ylinds mad of funtionally gadd matials, You, Ou and Zhng [33] poposd a simpl and auat mthod to dtmin stsss and p stain ats in thi-walld ylindial vssls subjtd to intnal pssu. Basd on th pow law onstitutiv quation, Altnbah, Goash and Naumno [34] psntd th lassial solution of th stady-stat p poblm fo a pssuizd thi-walld ylind. In this pap thy applid an xtndd onstitutiv quation whih inluds both th lina and th pow law stss dpndnis. Singh and Gupta [ ] dvlopd a mathmatial modl to dsib th stadyp bhavio of funtionally gadd omposit ylinds ontaining linaly vaying silion abid patils in a matix of pu aluminum involving thshold stss-basd p law. Th modl dvlopd is usd to invstigat th fft of gadint in distibution of SiCp on th stady-stat p spons of th omposit ylind. Njad and 17 IAU, Aa Banh

3 Tim-Dpndnt Hygo-Thmal Cp Analysis of Pssuizd 665 Kasholi [39] invstigatd tim-dpndnt thmo-lasti p spons fo isotopi otating thi-walld ylindial pssu vssls mad of FGM, taing into aount th p bhavio of th pssu vssls, as dsibd in Noton s modl. Assuming total stains to b th sum of lasti, thmal and p stains, Loghman, Aani, Ami and Vajdi [4] studid th tim-dpndnt p stss distibution analysis of a thi-walld FGM ylind plad in unifom magnti and tmpatu filds and subjtd to an intnal pssu. Following Noton s law fo matial p bhavio and using quations of quilibium, stain displamnt and stss-stain lations in th at fom and onsiding Pandtl-Russ lations fo p stain at-stss quation, thy obtaind a diffntial quation fo th displamnt at and thn alulatd th adial and iumfntial p stss ats. Singh and Gupta [41] invstigatd th stady stat p in tansvsly isotopi funtionally gadd ylind, opating und intnal and xtnal pssus. In this pap th fft of anisotopy on p stsss and p ats in th FGM ylind has bn analyzd and ompad with an isotopi FGM ylind. Assuming that th thmo-lasti p spons of th matial is govnd by Noton s law, Kasholi and Njad [4] psntd an analytial solution fo th alulation of stsss and displamnts of FGM thi-walld sphial pssu vssls. Dai and Zhng [43] psntd th p buling and post-buling analyss of th visolasti FGM ylindial shll with initial dfltion subjtd to a unifom in plan load by adopting th Boltzmann lina supposition pinipl. Shama, Sahay and Kuma [44] invstigatd th p stsss in thi-walld iula ylinds und intnal and xtnal pssu, using tansition thoy, whih is basd on th onpt of gnalizd pinipal stain masu. Jamian, Sato, Tsuamoto and Watanab [45] invstigatd th p analysis fo a thi-walld ylind mad of funtionally gadd matials (FGMs) subjtd to thmal and intnal pssu. Singh and Gupta [46] studid th stady stat p bhavio in a funtionally gadd thi omposit ylind subjtd to intnal pssu in th psn of sidual stss. Hoffman s yild ition is usd, to dsib th yilding of th ylind matial in od to aount fo sidual stss. Assuming plan stain ondition, Njad, Hosini, Ninjad and Ghannad [47] psntd an xat solution fo th analysis of FGM otating thi ylindial pssu vssls subjtd to a stady stat p ondition. Noton's pow law of p is mployd to div gnal xpssions fo stsss and stain ats. Kasholi, Tahan and Njad [48-49] psntd a thotial solution fo tim-dpndnt thmo-lasti p analysis of funtionally gadd (FG) and homognous thi-walld ylinds basd on th fist-od sha dfomation thoy. Shama, Yadav and Shama [5] invstigatd th p bhavio of a funtionally gadd ylind in tosion und intnal and xtnal pssu whih is subjtd to thmal loading. Loghman, Shaystmoghadam and Loghman [51] studid th p bhavio of stains, stsss, and displamnt ats in a thi-walld ylind mad of polypopyln infod by funtionally gadd (FG) multi-walld abon nanotubs (MWCNTs) using Bugs visolasti p modl. Loghman and Atabahshian [5] studid th tim-dpndnt p bhavio of otating ylinds mad fom xponntially gadd matial using Baily-Noton p onstitutiv modl. Th sults show that using xponntially gadd matial signifiantly dass p stains, stsss and dfomations of th EGM otating ylind. Ghobanpou Aani, Kolahhi, Mosallai Bazoi and Loghman [53] studid th histoy of stsss, dfomation and lti potntial of a thi hollow FGM otating ylind mad of adially polaizd anisotopi pizolti matial, (.g., PZT-7A), using a smi-analytial mthod bas on Mndlson s mthod of sussiv lasti solution. Alayoub and Loghman [54] invstigatd th tim-dpndnt p stss distibution analysis of thi-walld FGM sphs subjtd to an intnal pssu and a unifom tmpatu fild. Th sults show that adial stss distibutions a not signifiant fo diffnt matial poptis. Howv, majo distibutions ou fo tangntial and fftiv stsss. In this atil, assuming that th Magnto-hygo-thmolasti-p spons of th matial is govnd by Noton s law, an analytial solution is psntd fo th alulation of stsss and displamnts of FGM thiwalld ylindial pssu vssls. Th govning oupld diffntial quations a xatly solvd. Th ylind is subjtd to intnal and xtnal pssu and unifom magnti fild. Th tmpatu and moistu onntation distibutions a obtaind spaatly in an unoupld hygo-thmal analysis by solving hat ondution and moistu diffusion quations. GEOMETRY, LOADING CONDITION, MATERIAL PROPERTIES AND CREEP CONSTITUTIVE MODEL A thi-walld, ylindial vssl mad of funtionally gadd matial with an inn adius a and out adius b with pft ondutivity is onsidd. Lt th ylindial oodinats of any psntativ point b,, z and 17 IAU, Aa Banh

4 666 A. Bahshizadh t al. assum that th ylind subjtd to a adially hanging of tmpatu Th ylind is plad in a unifom magnti fild H H T and moistu onntation M.,, z and subjtd to intnal pssu and xtnal pssu, and also an intia body fo du to th otation of th ylindial vssl with a onstant angula vloity of. Th matial poptis a assumd to b adially dpndnt, In this study, Poisson s atio, v, is onsidd to b a onstant and modulus of lastiity, E, dnsity,, thmal xpansion offiint,, moistu P onntation offiint,, thmal ondutivity,, moistu diffusivity offiint, pmability,, a assumd to oby th pow-law vaiation as: T M P i and magnti E E b 1 (1) b () 3 o b (3) T T b 4 (4) M M b 5 (5) 6 o b (6) 7 b (7) i H is th in-homognity onstants dtmind mpiially. Th uniaxial p onstitutiv modl is th Noton s law, n B (8) wh is th adial oodinat, and n a th adial-dpndnt matial p paamts as: a fftiv stss and fftiv stain, sptivly and B and B ( ) b1 b (9) n( ) n (1) wh b, b1and n a matial onstants fo p. 17 IAU, Aa Banh

5 Tim-Dpndnt Hygo-Thmal Cp Analysis of Pssuizd HEAT CONDUCTION AND MOISTURE DIFFUSION FORMULATION In th stady stat as, th hat ondution quation fo th on-dimnsional poblm in pola oodinats simplifis to, 1 T T (11) wh is th thmal ondutivity. It is to b notd that th thmal ondutivity is also hangd though th adial dition of th ylind aoding to Eq. (6). W may dtmin th tmpatu distibution in th ylindial vssl by solving Eq. (11) and applying appopiat bounday onditions. Eq. (11) may b intgatd twi to obtain th gnal solution, T 4 T W W (1) 1 Th bounday onditions fo tmpatu a, T T Ti, a T To, b (13) Applying ths onditions to th gnal solution, w thn obtain, W W i o 1 a 4 b 4 T T a a T T b 4 4 o i 4 4 b (14) In modling, th tansint moistu diffusion quation is analogous to th tansint hat ondution quation. It an b dsibd by th following quation, 1 M M (15) wh th moistu diffusivity offiint is also hangd aoss th thinss of th ylind aoding to Eq. (7). Th gnal solution of th moistu diffusion quation is, 5 1 M M S S (16) Th bounday onditions fo th moistu onntation a, M M i, a M M o, b (17) Applying ths onditions to th gnal solution, w thn obtain, S S 1 M i M o 5 5 a b M a M b a b 5 5 o i 5 5 (18) 17 IAU, Aa Banh

6 668 A. Bahshizadh t al. 4 FORMULATION OF THE MAGNETO-HYGRO-THERMOELASTIC CREEP ANALYSIS 4.1 Solution fo lina lasti bhavio of FGM thi ylindial pssu vssl Fo th stss analysis in an FGM thi ylindial pssu vssl, having matial p bhavio, th solutions of th stsss at a tim qual to zo (i.. th initial stss stat) a ndd. This osponds to th solution of matials with lina lasti bhavio. Assuming total stains to b th sum of lasti, thmal, moistu and p stains thn th stss-stain lation fo axisymmti plan stain ondition in ylindial oodinat may b wittn in tms of adial displamnt as: u u T 1M (19) u u T 1 M () wh and a adial and iumfntial stsss, sptivly, u is adial displamnt and adial and iumfntial p stains, sptivly. Also, and a z (1) z () E 1 (3) z (4) z (5) Considing z and z thn, E 1 11 E E 1 E 1 E 1 (6) (7) (8) (9) (3) Th quation of th stss quilibium insid th FGM ylindial pssu vssl is, 17 IAU, Aa Banh

7 Tim-Dpndnt Hygo-Thmal Cp Analysis of Pssuizd 669 Fz (31) wh F z is th Lontz s fo whih an b wittn in tms of adial displamnt as: u Fz H z (3) Substituting Eqs.(19),() and (3) into quilibium Eq.(31), and assuming that ( ; i 1,,..., 7 ) th following i diffntial quation fo displamnt is obtaind, d u du dt dm I L u I T M d d 1 b d d d I 1 d E 1 H z 1v 1 v (33) wh, I E E H v v z (34) v L 1 v I (35) Ignoing th p stains in Eq. (33) and onsiding a unifom tmpatu fild T and a unifom moistu fild M th following diffntial quation fo magntothmolasti analysis is obtaind, 1 ( 1) ( 1) ( ) d u du dt dm I L u I T M d d 1 b d d 3 (1 )(1 ) E H v v (1 ) z (1 )(1 ) (36) Th gnal solution of th displamnt u is, m1 m u u u C C G G G (37) wh, p m G I 4L b 3m 3m 1,, G1 I S W 1 1 I W 1 S , G 3 1 b 3 m1 3 m E 1 H z 1 v 1 v 5 m1 5 m (38) Th osponding stsss a, 17 IAU, Aa Banh

8 67 A. Bahshizadh t al. ( m ) C ( m ) C (3 ) G m11 m (( 3) ) G (5 ) G ( W S ) ( W S ) (39) ( m ) C ( m ) C (3 ) G m11 m (( 3) ) G (5 ) G ( W S ) ( W S ) (4) To dtmin th unnown onstants a, C 1 and C in ah matial, bounday onditions hav to b usd, whih Pi, a Po, b (41) Ths onstants a wittn as follows, m 1 m 1 m 1 m 1 C1 Poa Pib G 1 a b a b m1 m G a b a b m m 1 m 1 m G 3 a b a b 1W 1 1S 1 a b a b m1 m1 m1 1 1W 1S a b a b 1 m 1 m 1 m 1 11m1 1 a b a b m1 1 m1 1 m1 1 m1 1 C Poa Pib G 1 a b a b m11 m G a b a b m m1 1 m1 1 m G 3 a b a b 1W 1 1S 1 a b a b m11 m11 m 1 1 1W 1S a b a b 1 m 1 m 1 m 1 11m 1 a b a b (4) (43) 4. Solution fo p bhavio of FGM thi ylindial pssu vssl Considing th tmpatu and moistu filds to b stady th diffntial Eq. (33) ontaining p stains may b wittn in tms of p stain ats as follows: d u du 1 d I 1 L 1u I d d 1 d (44) Cp stain ats a latd to th stsss and th matial uniaxial p onstitutiv modl by th wll nown Pandtl-Russ quations as:.5 z (45).5 z (46) z z.5 (47) 17 IAU, Aa Banh

9 Tim-Dpndnt Hygo-Thmal Cp Analysis of Pssuizd 671 wh and plan stain ( a th fftiv p stain at and th fftiv stss sptivly. Considing th as of z ) and Noton s law (Eq. (8)), th adial and iumfntial stain ats an b wittn as follows, 3 n 1 B 4 (48) 3 n 1 B 4 (49) Considing th Von Miss quivalnt stss in th as of plan stain, 3 (5) Thn Eqs. (48) and (49) may b wittn as follows: 3 n B (51) 3 n B (5) Substituting Eqs. (51) and (5) into diffntial Eq. (44) th following diffntial quation is obtaind, n d u du 3 1 b11 n b1 d I 1 L 1u Ib b1 d d 1 d (53) Th gnal solution of th displamnt u is, wh m1 m m1 m u D D u u (54) u m R d m1 m (55) w, u m1 R m1 m (56) w,, 1 m 1 m m 1 m w m m 1 (57) 1 d 1 d n 3 b11 n b1 R Ib b1 (58) Th osponding stss ats a, 17 IAU, Aa Banh

10 67 A. Bahshizadh t al. m D m D m u m1 1 m 1 m u u 3 ( m ) u b m 1 1 m1 m b m D m D m u m1 1 m 1 m u u 3 m u b m 1 1 m1 m b n n (59) (6) To dtmin th unnown onstants and in Eq. (54), bounday onditions hav to b usd. Sin insid and outsid pssus do not hang with tim, th bounday onditions fo stss ats at th inn and out sufas may b wittn as: D 1 D, a, b (61) Th unnown onstants D 1 and D a as follows: m 1 m1 1 m1 1 m 1 D m u b a b u a a b m u u u b a b u a a b a b a b b a m 1 m 1 m 1 m 1 1 m 1 m1 1 m1 m 1 11 u 11 a b b m1 m m1 1 m 1 m 1 m1 1 11m1 1 a b a b u m m 1 3 n m 1 b1 n b1 m 1 a b b b a b b a b a m1 1 m 1 m 1 m1 1 D m u b a b u a a b m u u u b a b u a a b a b a b b a m1 1 m1 1 m1 1 m1 1 m1 1 m m m u 11 a b b 1 m11 m1 1 m 1 m1 1 m1 1 m 1 11m 1 a b a b u m1 m1 1 3 n m1 1 b1 n b1 m1 1 a b b ( b) a b b a b a (6) (63) whn th stss at is nown, th alulation of stsss at any tim t i should b pfomd itativly. wh i i 1 i i ij, ti ij, ti 1 ij, ti dt (64) t i i dt (65) 17 IAU, Aa Banh

11 Tim-Dpndnt Hygo-Thmal Cp Analysis of Pssuizd NUMERICAL RESULTS AND DISCUSSION In th pvious stions, th analytial solution of p stsss fo FGM thi-walld ylindial vssls subjtd to unifom pssus on th inn and out sufas and unifom magnti fild w obtaind. In this stion, som pofils a plottd fo th adial stss, iumfntial stss, fftiv stss, adial stss at and iumfntial stss at as a funtion of adial dition fo diffnt tims aft p. An FGM thi-walld ylindial vssl with p bhavio und intnal and xtnal pssu and unifom magnti fild is onsidd. Mhanial poptis of th ylind, suh as modulus of lastiity, dnsity, thmal xpansion offiint, moistu onntation offiint, thmal ondutivity, moistu diffusivity offiint and magnti pmability a assumd to oby th pow-law vaiation. In-homognity onstants ang fom to. Th following data fo loading and matial poptis a usd in this invstigation b, E GPa,. 33, 1. 1 K,. 81 m, 786 g a g m H, H 31 A z, T i C, T 5 C, M i, M 3 m m 36 Pi 8MPa, P MPa, 1ad, b. 111, b 1 5, n 3 s Th distibutions of p stss omponnts 1, and and fftiv stss at zo tim fo valus of, a plottd in Figs. 1,,3, sptivly. It an b sn fom Fig. 1 that th bounday onditions fo adial stss at th inn and out sufas of th ylind a satisfid fo all matial poptis. As this figu shows maximum valus of adial stss blong to and th minimum valus blong to. Aoding to Fig. and Fig. 3, iumfntial and fftiv stsss fo,1, 1 is tnsil thoughout thinss whil fo is ompssiv. Th absolut maximums of adial and iumfntial stsss fo, 1,1 ou at th inn dg, whih mans th maximum sha stss fo, 1,1 sufa of th vssl and an aus yilding to ou. whih is max will b vy high on th inn 3 Fig.1 Nomalizd adial stss vsus dimnsionlss adius at zo tim. Fig. Nomalizd iumfntial stss vsus dimnsionlss adius at zo tim. 17 IAU, Aa Banh

12 674 A. Bahshizadh t al. Fig.3 Nomalizd fftiv stss vsus dimnsionlss adius at zo tim. Radial and iumfntial stss ats distibutions fo diffnt valus of, a shown in Figs. 4 and 5. Fig. 4 shows that th adial stss ats fo all valus of, a zo at th inn and out sufas of th ylind whih satisfy th bounday onditions. Th maximum at of hang of adial stsss blongs to and th minimum at blongs to. Thfo minimum hangs in adial stsss with tim will ta pla fo th matial idntifid by as will b xplaind nxt in adial stss distibutions. It an b sn fom Fig. 5 that, at th inn adius of th ylind th iumfntial stss ats inass as dass. Fig.4 Radial stss at vsus dimnsionlss adius at zo tim. Fig.5 Ciumfntial stss at vsus dimnsionlss adius at zo tim. Tim-dpndnt p stss distibutions fo a shown in Figs Fig. 6 shows th tim-dpndnt adial stss fom initial lasti at zo tim to fouth sltd tim intval. Th is an inas in th valu of th adial stss as tim inass. It also satisfis th bounday onditions at th inn and out sufas of th ylind. Ciumfntial stss distibution fo this as fom initial lasti at zo tim to fouth sltd tim intval is shown in Fig. 7. It is la that th valu of th iumfntial stss inass as tim inass. Th fftiv stss along th adius is plottd in Fig. 8. It an b sn that th valu of th iumfntial stss inass as tim inass and also th maximum hang in th valu of fftiv stss ou at th out sufa of th ylind. It is notd that th maximum sha stss at th inn and out sufas of th vssl is inasing with tim. 17 IAU, Aa Banh

13 Tim-Dpndnt Hygo-Thmal Cp Analysis of Pssuizd 675 Fig.6 Nomalizd adial p stss distibution fom initial lasti at zo tim to fouth sltd tim intval fo th as. Fig.7 Nomalizd iumfntial p stss distibution fom initial lasti at zo tim to fouth sltd tim intval fo th as. Fig.8 Nomalizd fftiv p stss distibution fom initial lasti at zo tim to fouth sltd tim intval fo th as. Figs. 9 and 1 show th adial and iumfntial stss ats along th thinss of FGM ylind, fom initial lasti at zo tim to fouth sltd tim intval fo th as. It an b sn fom Figs. 9 and 1 that th ats of th adial and iumfntial stss dass as tim inass. Fig.9 Radial stss at distibution fom initial lasti at zo tim to fouth sltd tim intval fo th as. 17 IAU, Aa Banh

14 676 A. Bahshizadh t al. Fig.1 Ciumfntial stss at distibution fom initial lasti at zo tim to fouth sltd tim intval fo th as. Figs show th ffts of adding intnal pssu, angula vloity and inn sufa tmpatu on fftiv stss fo th as aft yas fom p. It an b sn fom ths figus that fftiv stss inass as intnal pssu and angula vloity inas and inn sufa tmpatu dass. It must b notd that in ths figus th xtnal pssu is and out sufa tmpatu is P MPa T 5 C. Fig.11 Th fft of adding intnal pssu on fftiv stss fo th as aft yas fom p. Fig.1 Th fft of adding angula vloity on fftiv stss fo th as aft yas fom p. Fig.13 Th fft of adding inn sufa tmpatu on fftiv stss fo th as aft yas fom p. 17 IAU, Aa Banh

15 Tim-Dpndnt Hygo-Thmal Cp Analysis of Pssuizd 677 Tmpatu and moistu distibutions of fou diffnt valus of is shown in Figs. 14 and 15. Fig.14 Nomalizd tmpatu distibution of FGM thi walld ylindial vssl fo valus of. 1, Fig.15 Nomalizd moistu distibution of FGM thi walld ylindial vssl fo valus of 1,. 6 CONCLUSIONS In this atil, assuming that th thmo-p spons of th matial is govnd by Noton s law, an analytial solution is psntd fo th alulation of stsss of FGM thi-walld ylindial pssu vssls. Fo th stss analysis in a ylind, having matial p bhavio, th solutions of th stsss at a tim qual to zo (i.. th initial stss stat) a ndd, whih osponds to th solution of matials with lina magntohygothmolasti bhavio. Th analytial solution is obtaind fo th ondition of plan stain. It is assumd that th matial poptis hang as gadd in adial dition to a pow law funtion. To show th fft of inhomognity on th stss distibutions, diffnt valus a onsidd fo in-homognity onstants. Th govning oupld diffntial quations a xatly solvd. Th ylind is subjtd to intnal and xtnal pssu and unifom magnti fild. Th tmpatu and moistu onntation distibutions a obtaind spaatly in an unoupld hygothmal analysis by solving hat ondution and moistu diffusion quations. Fo th stss analysis aft ping fo a long tim, th itativ podu is nssay. It ould b sn that th in-homognity onstants hav signifiant influn on th distibutions of th p stsss. Th bst matials is idntifid by 1, in whih th minimum sha stss distibution will ou thoughout th thinss of th FGM ylind. REFERENCES [1] Xi H., Dai H. L., Rao Y. N., 13, Thmolasti dynami bhavios of a FGM hollow ylind und nonaxisymmti thmo-mhanial loads, Jounal of Mhanis 9: [] Lvyaov S. V., Kuzntsov V. V., 14, Nonlina stability analysis of funtionally gadd shlls using th invaiantbasd tiangula finit lmnt, Jounal of Applid Mathmatis and Mhanis 94: [3] Njad, M. Z. Fathi, P. 15, Exat lasto-plasti analysis of otating thi-walld ylindial pssu vssls mad of funtionally gadd matials, Intnational Jounal of Engining Sin 86, IAU, Aa Banh

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