Creep damage evaluation of thick-walled spheres using a long-term creep constitutive model

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1 Jounal of Mhanal Sn and Thnology (009) 577~58 Jounal of Mhanal Sn and Thnology DOI 0.007/s x Cp damag valuaton of thk-walld sphs usng a long-tm p onsttutv modl Abbas Loghman and Nad Shokouh Dpatmnt of Mhanal Engnng, Faulty of Engnng, Unvsty of Kashan, Kashan, I. R. Ian Dpatmnt of Mhanal Engnng, Shaf Unvsty of Thnology, Than, I. R. Ian (Manuspt Rvd August 8, 007; Rvsd May 0, 009; Aptd May 0, 009) Abstat Ths pap dsbs a numal modl dvlopd fo th omputaton of p damags n a thk-walld sph subjtd to an ntnal pssu and a thmal gadnt. Th modl pdts th p damag hstos dung th lf of th sph, owng to vaatons n stsss wth tm and though-thknss vaatons. Th p damag faton s basd on th Robnson s lna lf faton damag ul, whh has bn nopoatd n a nonlna tm-dpndnt stss analyss. Followng th stss hstos, th fftv stss hstos a obtand and th p damags a alulatd and summd dung th lf of th sph. Th matal long-tm p popts up to th uptu and p uptu data a dfnd by th Θ pojton onpt []. Th damag hstos up to 8 yas a alulatd and th sults show that th maxmum damags a always loatd at th nn sufa of th sph, whl th out sufa of th vssl sustans mnmum damags. Kywods: Cp damags; Lf assssmnt; Thta pojton; Thk-walld sphs Intoduton Ths pap was ommndd fo publaton n vsd fom by Assoat Edto Youngsog L * Cospondng autho. Tl.: , Fax.: E-mal addss: aloghman@kashanu.a. KSME & Spng 009 Thk-walld sphs ontanng hgh pssu n hgh- tmpatu nvonmnts a usd xtnsvly n th aas of pow gnaton, ol, and hmal ndusts. Ths vssls oftn fal by p uptu akng [] and n gnal, falus a always atastoph []. Whl n- sv xamnatons povd usful nfomaton about matal ondton, btt undstandng of th sph damag bhavo s ual bfo ths nfomaton an b usd to povd auat pdtons of th sph futu pfoman [4]. If an auat tm-dpndnt damag an b modld fo th sph, thn th sph xamnatons an b shduld n a sltv mann. A majo dffulty n th sph dsgn lf o assssmnt of ts manng lf s that stss dstbuton ous dung th lf of th sph owng to tm-dpndny of stans. Howv, p stan ats a latd to th nstantanous stss ondton and th matal un-axal p onsttutv modl by th wll-known Pandtl-Russ quatons. Impovd knowldg of long-tm matal p popts hav shown that th p uv shap vas sgnfantly wth hangng stss and tmpatu [5] and th tadtonal assumpton of onstant p at gnos a onsdabl amount of nfomaton. Th man objtv of ths pap s to valuat th damag hstos of a thk-walld sph usng a long-tm matal p onsttutv modl dfnd by th Θ pojton onpt.. Matal p onsttutv modl On of th ky lmnts of an nlast falu analyss s th matal onsttutv quatons that dsb th matal dfomaton and uptu bhavo und load. Th matal sltd fo th psnt

2 578 A. Loghman and N. Shokouh / Jounal of Mhanal Sn and Thnology (009) 577~58 Tabl. Coffnts a, b, and onstants fo p onsttutv modl. d of th matal 4 ε f a b d study s C, Mo, 4V ft stl. Th stan-tm bhavo of ths matal has bn dsbd usng th Θ pojton onpt as follows: ε =Θ( ) +Θ ( ) () Θt Θ4t wh ε s th p stan and t s th tm. Thn th p stan at, ε s Fg.. Cp uvs pdtd by th Θ pojton onpt fo C, Mo, V stl. 4 =ΘΘ +ΘΘ () Θt Θ4t 4 wh Θ = a + bt + + dσt =,,, 4 () log0 σ wh T and σ a tmpatu and stss lvls, sptvly. Coffnts a, b, and d a matal onstants. Fo ths matal, ths onstants a shown n tabl Th unts of tm tmpatu and stss a sonds, dg K and MPa sptvly. It has also bn shown that th p fatu stans an b modld as follows: ε = a + bt + σ + dσt = 5 (4) f On ths bass Eqs. () and () an b usd to onstut a pdtd p uv at any stss lvl and tmpatu. Th lvant uptu lf s thn dfnd as th tm takn to ah th appopat falu stan as gvn by Eq. (4). Th full p uvs pdtd fo vaous stss lvls a shown n Fg. and th p uptu ontous a shown n Fg... Thotal analyss Consd a sph wth nn adus a and out adus b, subjtd to an ntnal pssu P and a tmpatu dstbuton sultng fom an outwad Fg.. Cp uptu ontous pdtd by Eq. (4) fo C, Mo, V stl. 4 flow of hat owng to an nn sufa tmpatu of T and out sufa tmpatu of T o. Equatons of qulbum and ompatblty n sphal oodnat and n dmnsonlss fom [6] a wttn as: ( ) ds S S + = 0 (5) d d + = 0 (6) d Th total stans a mad up of last, p, and thmal stans wttn n dmnsonlss fom as: S µ S ( µ ) τ ( ) S S ( ) = + + = + + µ µ µ τ (7)

3 A. Loghman and N. Shokouh / Jounal of Mhanal Sn and Thnology (009) 577~ wh th nondmnsonal paamts n Eqs. (5) (7) a dfnd as follows: b P EαT P τ a a σ o ( µ ) σo σ σ Eε Eε S S σ o σo σo σo Th bounday ondton of pssu and tmpatu at th nn and out sufas of th sph a = a T = T, σ = P = b T = T, σ = 0 o Tmpatu dstbuton fo stady-stat hat onduton s as follows: τo τ ( τ τo) τ = + ( ) (8) wh τ and τ o a dmnsonlss nn and out tmpatus, sptvly. Wth th abov bounday ondtons and tmpatu dstbuton smultanous soluton of qulbum, ompatblty and stssstan quatons sult n th followng latons fo th stsss [6]. = τ + + S d d µ C P S = τ P+ τd C d ( µ ) µ (9) (0) 4. Cp flow ul Cp stsss a funtons of total p stans as shown n Eqs. (9) and (0). Cp stans a tm, tmpatu, and stss-dpndnt. Thfo, nmnts of p stans must b aumulatd dung th lf of th sph along appopat loadng path to obtan total p stans. Cp stan ats a latd to th nstantanous stss tnso and th matal unaxal p popts by th wll-known Pandtl-Russ quatons as: = ( S S ) S = φ = () wh S s th fftv stss and ε s th fftv p stan at dfnd by th followng quatons: S = S S = () Th abov quatons and th matal p onsttutv modl a usd n a numal podu, whh gvs th stss and stan hstos. Ths numal podu s xpland n dtal lat n ths pap. Th adal and tangntal stss hstos and th fftv stss hstos a obtand and shown n Fgs. and 4. Ths stss hstos a usd to obtan damag hstos and th manng lf of th sph. Th damag hstos up to 8 yas a obtand and shown n Fg. 5. wh = + µ C τd d P In th abov quatons th nompssblty ondton ( ε + ε = 0 ) s usd and th tangntal p stans a plad by th adal p stan ε ( ε = ). Fg.. Radal and tangntal stss dstbuton fo 5, 0, and 8 yas.

4 580 A. Loghman and N. Shokouh / Jounal of Mhanal Sn and Thnology (009) 577~58 ε f Θ ( ) Θ ( ) = 0 (4) Θt Θ4t wh ε f s th uptu stan. It an b alulatd fo any tmpatu and stss lvl usng Eq. (4). Thn th uptu tms an b valuatd numally fo any stss lvl and tmpatu usng Eq. (4). Th manng lf at any pont s thn gvn by RL = ( D) t (5) Fg. 4. Efftv stss hstos fo th nn, mddl, and out sufas of th sph. 6. Numal podu Th stp-by-stp podu of th numal mthod s wttn as follows: () A thk-walld sph of = s loadd wth an ntnal pssu of 50MPa and a thmal 0 gadnt of t = 0 C s onsdd. Th tmpatu at th nn sufa of th sph 0 s 550 C. () Fo th fst tmng stp, an appopat tm nmnt s sltd. In ths soluton th fst tm 6 nmnt s t = 0 S. Th total tm s th sum of tm nmnts as th p poss pogsss n tm. Fo th th tmng stp th total tm s t = t + t k Fg. 5. Hstoy of damags vsus thknss of th sph fo 5, 0, and 8 yas. 5. Cp damag and mnant lf Fo p uptu pdtons, th Robnson's lna lf-faton damag ul s usd. Th damag aumulatd at any pont s gvn by th followng quaton: () Th thknss of th sph s dvdd nto N qual dvsons. Intal valus of ε j, = fo adal p stan nmnts at all dvson ponts a assumd. Ths a addd to th aumulatd p stans obtand fom th pvous tmng stp at all dvson ponts thoughout th wall thknss of th sph. j, = kj, + j, ε ε ε D t = () t wh t s th duaton and t s th p uptu tm at th quvalnt stss and tmpatu of that pont. Th p uptu tm s th tm takn to ah th uptu stan dfnd by Eq. (4). Thfo, Eq. () may b wttn n tms of uptu-stan and uptu tm as follows (4) Cp stan nmnts n and φ dtons a obtand fom th symmty and nompssblty ondtons as ε, j = εφ, j = ε, j. Th total p stans n and φ dtons a thn, j =, kj +, j ε ε ε φ, j = φ, kj + φ, j ε ε ε

5 A. Loghman and N. Shokouh / Jounal of Mhanal Sn and Thnology (009) 577~58 58 (5) Wth th assumd p stan dstbuton, th ntgals of Eqs. (9) and (0) a valuatd. Thfo th ntal stmats of p stsss a alulatd. (6) Efftv stsss a thn alulatd at all dvson ponts. S = S S j, j,, j (7) Tmpatu dstbutons a alulatd usng τo τ ( τ τo) τ j = + ( ) (8) Wth th abov tmpatu dstbuton and fftv stsss, p stan ats a alulatd at all dvson ponts thoughout th thknss usng th matal's p onsttutv modl Eq. (). j fom stp untl onvgn s obtand at all dvson ponts thoughout th thknss of th sph. () Aft onvgn s obtand, th uptu stans a alulatd at all dvson ponts usng th fftv stss hstos and tmpatu dstbuton as follows: ε = a + bt + σ + d σ T f, j 5 5 j 5, j 5, j j () Ruptu tms a thn alulatd usng th matal p onsttutv quaton as follows: ε ( ) ( ) = 0, jt, j 4, jt, j f, j, j, j (4) Damags a thn alulatd at all dvson ponts and addd to th pvous aumulatd damags as follows =Θ Θ +Θ Θ Θ, jtj Θ, j 4tj j,, j, j, j 4, j wh Θ, Θ, Θ and Θ 4 a tmpatu, and stss-dpndnt as follows: LogΘ = a + b T + σ + d σ T k =,,, 4 k, j k k j k, j k, j j (9) Radal p stan ats a thn alulatd usng th Pandtll-Russ quaton = ( ) j, j, Sj, S, j Sj, (0) Nw valus fo adal p stan nmnts at all dvson ponts a thn alulatd usng th abov p stan ats and th tm nmnt as follows: ε = t nw, j, j, () Ths nw obtand valus of th p stan nmnts a ompad wth th ntal assumd valus of th p stan nmnts fo th onvgn of th poss. If onvgn has not oud, thn ths nw obtand valus a assumd as th ntal valus of th p stan nmnts and th podu s patd t Dj = t j, (5) Th mnant lf an b valuatd fo all dvson ponts as follows: RL = ( D } t, j j, j (6) Tm s advand on nmnt and th podu s patd fo th nw tm nmnt. 7. Conlusons and dsusson An mpovd p onsttutv quaton ontanng full p uv up to th uptu and th p uptu data has bn mployd to obtan th damag vaaton wth tm as wll as though thknss vaatons of damag n a thk-walld sph. Th poblm ontans a vaabl stss and a dstbutd tmpatu fld. Robnson's lna lf faton damag ul has bn nopoatd nto a nonlna tm-dpndnt p stss analyss to alulat damags. Fo ths loadng ondton, sults show that th nn sufa of th sph s th most damagd aa and th out sufa of th sph sustans mnmum damags (s Fg. 5). Whl mnmum fftv stss s loatd at th nn sufa of th sph lat n th lf of th vssl, Fg. 4, maxmum damagd aa s always loatd at th nn sufa of th sph, Fg. 5. Owng to th hangs n th slop of th p uvs, a vaabl tm nmnt should

6 58 A. Loghman and N. Shokouh / Jounal of Mhanal Sn and Thnology (009) 577~58 [4] A. Loghman and M. A Wahab, Cp damag smulaton of thk-walld tubs usng th thta pojton onpt, Int. J. Pss. Vs. & Ppng 67(996) -05. [5] R. W. Evans and B. Wlsh, Cp of Mtals and Alloys. Inst. of mtals, London, (985). [6] A. Mndlson, Plastty: Thoy and Applaton, Th Mamllan Company, Nw Yok, USA, (968). Fg. 6. Tm nmnt vaatons wth tm fo onvgn of th podu. b mployd fo apd onvgn of th numal podu. Fo aly stags of th p lf of th sph wh th slop of th p uvs a vy small, th podu onvgd wth a tm nmnt 6 of t = 0 sonds. Howv, lat n th lf of th sph wh th slops of th p uvs a vy hgh, th podu wll onvg wth a tm nmnt of t = 0 sonds. Vaatons n tm nmnt wth tm fo onvgn of th podu a shown n Fg. 6. Rfns [] R. W. Evans, J. D. Pak and B. Wlsh, Th thta pojton onpt a modl basd appoah to dsgn and lf xtnton of ngnng plant. Int. J. Ps. Vs. and Ppng, 50 (99) [] C. A. Jask, Lf assssmnt of hot hat stam pp. ASME J. Ps. Vs. Thnology, (990) 7-0. [] F. A. Smson and C. A Jask, a Computatonal modl fo pdtng th lf of tubs usd n ptohmal hat sv. ASME J. Ps. Vs. Thnology, 07(985) Abbas Loghman vd hs BS dg fom Shaf Unvsty of Thnology, Than, Ian, n 980, H thn vd hs MS dg fom th Amkab Unvsty of Thnology, Than, Ian, n 986 and hs PhD dg fom th Unvsty of Adlad, South Austala, n 995. D. Loghman s an Assstant Pofsso n th Mhanal Engnng Dpatmnt of Kashan Unvsty, Kashan, Ian. Hs unt sah ntsts a p and p-fatgu lf assssmnt of pssu vssls. Nad Shokouh vd hs BS dg fom Kashan Unvsty, Ian, n 00 and hs MS dg fom th Amkab Unvsty of Thnology, Ian, n 004. H s untly wokng towads hs PhD dg n th Dpatmnt of Mhanal Engnng, Shaf Unvsty of Thnology, Ian. Hs unt sah ntsts a n al vhl dynams modlng. H s Dto of th Rsah and Dvlopmnt Dpatmnt n Iankhodo Ral Tanspot Industs Company (IRICO).