Transient Thermal Stress Problem of a Functionally Graded Magneto-Electro-Thermoelastic Hollow Sphere

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1 Mtls 6-5; o:.9/m6 Atcl OPE AESS mtls ISS Tnsnt Thml Stss Polm of Functonlly G Mnto-Elcto-Thmolstc Hollow Sph Yoshho Ooto * n Msyuk Ishh Dptmnt of Mchncl Ennn Gut School of Ennn Osk Pfctu Unvsty - Gkun-cho k-ku Sk Jpn; E-Ml: shh@m.oskfu-u.c.p * Autho to whom cosponnc shoul ss; E-Ml: ooto@m.oskfu-u.c.p; Tl.: ; Fx: Rcv: ovm ; n vs fom: Dcm / Accpt: 7 Dcm / Pulsh: Dcm Astct: Ths tcl s concn wth th thotcl nlyss of th functonlly mnto-lcto-thmolstc hollow sph u to unfom sufc htn. W nlyz th tnsnt thmolstc polm fo functonlly hollow sph constuct of th sphcl sotopc n ln mnto-lcto-thmolstc mtls usn lmnt compost mo s on of thotcl ppoxmton n th sphclly symmtc stt. As n llustton w cy out numcl clcultons fo functonlly hollow sph constuct of pzolctc n mntostctv mtls n xmn th hvos n th tnsnt stt. Th ffcts of th nonhomonty of mtl on th stsss lctc potntl n mntc potntl nvstt. Kywos: thml stss; mnto-lcto-thmolstc mtl; functonlly mtl; hollow sph; tnsnt stt. Intoucton Functonlly mtls FGMs nw nonhomonous mtl systms tht two o mo ffnt mtl nnts chns contnuously n ully. Th concpt of FGMs s pplcl to mny nustl fls such s ospc nucl ny chmcl plnt lctoncs n so on. On th oth hn t hs cntly n foun tht composts m of pzolctc n mntostctv mtls xht th mntolctc ffct whch s not sn n pzolctc o mntostctv mtls [].

2 Mtls 7 Ths mtls known s multfoc composts []. Ths composts xht coupln mon mntc lctc n lstc fls. In th pst vous polms n mnto-lcto-lstc m tht xht nsotopc n ln coupln mon th mntc lctc n lstc fls w nlyz. Exmpls fo th plts n ms w nlyz n th pps [-5]. Exmpls fo th shll typ stuctus w nlyz n th pps [6-8]. Exmpls of functonlly mnto-lcto-lstc m s follows. Wn n Dn [9] tt sphclly symmtc tnsnt sponss of functonlly mnto-lcto-lstc hollow sph. M n L [] nlyz n n-pln polm n functonlly mnto-lcto-lstc mtls. Yu n Wu [] nlyz th popton of ccumfntl wv n mnto-lcto-lstc functonlly cylncl cuv plts. Wu n Lu [] nlyz th D ynmcs sponss of functonlly mnto-lcto-lstc plts. Hun t l. [] nlyz th sttc polm of n nsotopc functonlly mnto-lcto-lstc ms suct to ty lon. L n M [] nlyz th two-mnsonl polm of two on ssml hlf-plns fo functonlly mntolctolstc mtls suct to nlz ln focs n scw sloctons. Exmpls of th thml stss polms of lcto-mnto-lstc m s follows Gnsn t l. [5] nlyz th spons of ly multphs mntolctolstc cyln suct to n xsymmtc tmptu stuton usn fnt lmnt pocus. Kumvl t l. [6] nlyz th spons of th-ly mntolctolstc stp suct to unfom tmptu s n non-unfom tmptu stuton usn fnt lmnt pocus. Hou t l. [7] otn D funmntl solutons of sty pont ht souc n nfnt n sm-nfnt othotopc lcto-mnto-thmo-lstc plns. Wth to tnsnt thml stss polms of lcto-mnto-lstc m Wn n ul [8] nlyz tnsnt thml fctu n tnsvsly sotopc lcto-mnto-lstc cylns. Th xct soluton of tnsnt nlyss of multly mnto-lcto-thmolstc stp suct to nonunfom ht supply ws otn n th pp [9]. Th xct soluton of tnsnt nlyss of multly mnto-lcto-thmolstc hollow cyln suct to unfom ht supply ws otn n th pp []. Thouh svl tnsnt thml stss polms of th functonlly hollow sphs [] usn lmnt compost mol w nlyz ly thss stus on t cons coupln mon mntc lctc n thmolstc fls. Howv to th utho s knowl th tnsnt thml stss polm fo functonlly mnto-lcto-thmolstc hollow sphs un unsty ht supply consn coupln mon mntc lctc n thmolstc fls hs not n pot. In th psnt tcl w hv nlyz th tnsnt hvo of functonlly mnto-lcto-thmolstc hollow sph u to unfom sufc htn. W ssum tht th mnto-lcto-thmolstc mtls polz n mntz n th l cton. W nlyz th tnsnt thml stss polm fo functonlly hollow sph constuct of th sphcl sotopc n ln mnto-lcto-thmolstc mtls usn lmnt compost mol s on of thotcl ppoxmton. W c out numcl clcultons fo functonlly hollow sph compos of pzolctc n mntostctv mtls n xmn th ffcts of th nonhomonty of mtl on th stsss lctc potntl n mntc potntl.

3 Mtls 8. Anlyss W cons functonlly hollow sph constuct of th sphcl sotopc n ln mnto-lcto-thmolstc mtls. W nlyz th tnsnt thml stss polm usn multly compost hollow sph mol wth num of homonous lys. Th hollow sph s nn n out snt n spctvly. s th out us of th th ly. Thouhout ths tcl th ncs ssoct wth th th ly of compost hollow sph fom th nn s... Ht onucton Polm W ssum tht th multly hollow sph s ntlly t zo tmptu n ts nn n out sufcs sunly ht y suounn m hvn constnt tmptus T n T wth ltv ht tnsf coffcnts h n h spctvly. Thn th tmptu stuton s on-mnsonl n th tnsnt ht conucton quton fo th th ly s wttn n th follown fom: T κ τ T T ; Th ntl n thml ouny contons n mnsonlss fom τ ; T ; T ; H T H T R ; T T ; T T R ; ; 5 T H T H T 6 ; In Equtons 6 w ntouc th follown mnsonlss vlus: T T T T T T / T R / τ κ t / κ κ /κ 7 / H H h h wh T s th tmptu chn; t s tm; s th thml conuctvty n th l cton; κ s th thml ffusvty n th l cton; n T n κ typcl vlus of tmptu thml conuctvty n thml ffusvty spctvly. To solv th funmntl quton w ntouc th Lplc tnsfomton wth spct to th vl τ s follows; T * p p T τ τ τ 8 Pfomn th Lplc tnsfomton on Equton un th conton of Equton vs

4 Mtls 9 wh * T T * * μ p μ T 9 κ Th nl soluton of Equton 9 s T * A μ B y μ wh n y zoth-o sphcl Bssl functons of th fst n scon kn spctvly. Futhmo A n B unknown constnts tmn fom th ouny contons. Susttutn Equton nto th ouny contons n th tnsfom omn fom Equtons 6 ths qutons psnt n mtx fom s follows: A B [ kl ] { c k } p A B Mkn us of m s fomul th constnts A n B cn tmn fom Equton. Thn th tmptu soluton n th tnsfom omn s * T [ A μ B y μ ] pδ wh Δ s th tmnnt of mtx [ kl ] n th coffcnts A n B fn s tmnnts of mtx sml to th coffcnt mtx [ kl ] n whch th th column o th column s plc wth th constnt vcto { c k } spctvly. Usn th su thom w cn ccomplsh th nvs Lplc tnsfomton on Equton. Bcus th snl-vlu pols of Equton cospon to p n th oots of Δ n whch th su fo p vs soluton fo th sty stt. Accomplshn th nvs Lplc tnsfomton of Equton th soluton of Equton s wttn s follows: B xp μ τ T A [ A μ B y μ ] ; F μ Δ μ wh F th tmnnts of mtx [ kl ] n th coffcnts A n B fn s tmnnts of mtx sml to th coffcnt mtx [ kl ] n whch th th column o th column s plc wth th constnt vcto {c k } spctvly. Th nonzo lmnts of th coffcnt mtcs [ kl ] n [ kl ] n th constnt vcto {c k } vn fom th Equtons 6. In Equton Δ μ s Δ Δ μ μ μ μ 5

5 Mtls n μ s th th postv oot of th follown tnscnntl quton.. Thmolstc Polm Δ μ 6 W vlop th nlyss of multly mnto-lcto-thmolstc hollow sph s sphclly symmtc stt. Th splcmnt-stn ltons xpss n mnsonlss fom s follows: u ε u ε θθ ε φφ θ φ θφ 7 wh th comm nots ptl ffntton wth spct to th vl tht follows. Fo th sphcl sotopc n ln mnto-lcto-thmolstc mtl th consttutv ltons xpss n mnsonlss fom s follows: σ θθ σ ε ε θθ H φφ T E q σ ε εθθ θ T E q H 8 wh α α θ α α θ 9 Th consttutv qutons fo th lctc n th mntc fls n mnsonlss fom vn s D ε ε θθ η E H p T B q ε qε θθ E μ H m T Th lton twn th lctc fl ntnsty n th lctc potntl φ n mnsonlss fom s fn s E φ Th lton twn th mntc fl ntnsty n th mntc potntl ψ n mnsonlss fom s fn s H Th qulum quton s xpss n mnsonlss fom s follows: ψ σ σ σ θθ If th lctc ch nsty s snt th qutons of lctosttcs n mntosttcs xpss n mnsonlss fom s follows: D D 5

6 Mtls B B 6 In Equtons 7 6 th follown mnsonlss vlus ntouc: σ σ kl α kl YT ε kl kl ε kl kl u α T u α k kl α k kl T α α Y D D B α YT k η k η q Y Y B κ φ ψ φ ψ α T α T κ α Y k T qkκ μ κ Y μ 7 κ p α Y p m κ m α E E α T H H κ α YT wh σ kl th stss componnts; ε kl kl th stn componnts; u s th splcmnt n th cton; α k th coffcnts of ln thml xpnson; kl th lstc stffnss constnts; D s th lctc splcmnt n th cton; B s th mntc flux nsty n th cton; k th pzolctc coffcnts; η s th lctc constnt; p s th pyolctc constnt; q k th pzomntc coffcnts; μ s th mntc pmlty coffcnt; s th mntolctc coffcnt; m s th pyomntc constnt; n α Y n typcl vlus of th coffcnt of ln thml xpnson Youn s moulus n pzolctc moulus spctvly. Susttutn Equtons 7 n nto Equtons 8 n n lt nto Equtons 6 th ovnn qutons of th splcmnt u lctc potntl φ n mntc potntl ψ n th mnsonlss fom wttn s u u u q ψ q q ψ u u u φ φ ˆ 8 θ T T η φ η φ ψ ˆ ψ p T T 9 q u q q u q u φ m T T φ μ ψ μ ψ If th nn n out sufcs of th multly mnto-lcto-thmolstc hollow sph tcton f n th ntfcs of ch onn ly pfctly on thn th ouny contons of nn n out sufcs n th contons of contnuty t th ntfcs cn psnt s follows:

7 Mtls R ; σ ; σ σ u u ; ; σ Th ouny contons n th l cton fo th lctc n mntc fls xpss s R ; D ; D B o φ ψ D B B φ φ ψ ψ ; ; D B o φ ψ Th solutons of Equtons 8 ssum n th follown fom: u uc up φ φc φ p ψ ψ c ψ p In Equton th fst tm on th ht-hn s vs th homonous soluton n th scon tm vs th ptcul soluton. W now cons th homonous soluton n ntouc th follown quton: xps hnn vl wth th us of Equton th homonous xpsson of Equtons 8 [ D c D α ] uc [ D D D] Φc [ D D q D] Ψ 5 [ D [ D c D D ] uc D D Φc D D Ψ 6 c D q D ] uc D D Φc δ D D Ψ 7 wh By lmntn out u c : Φ Φ n c c φ c Ψ c ψ c q q q D 8 s q α q η μ δ q 9 Ψ twn Eqs. 5-7 w cn otn n ony ffntl quton c D D uc D uc Th soluton of Equton cn xpss s follows whn / >. u c Fom Equtons 6 7 n w cn otn n ony ffntl quton out Ψ : c

8 Mtls s s c c Ψ Ψ D Usn Equton 9 th soluton of Equton s 5 q c ψ Fom Equtons 6 n w cn otn n ony ffntl quton out c Φ : s s c c Φ Φ D 6 Usn Equton 9 th soluton of Equton s 7 6 c φ 5 In Equtons 5 / ± α [ ] q q δ δ ] [ ] [ ] [ q δ ] [ q δ 6 In Equtons n 5 k 7 k unknown constnts. W hv th follown lton. 5 7 q α 7 Th homonous solutons whn / omtt h fo vty. It s ffcult to otn th ptcul solutons usn th tmptu soluton of Equton. In o to otn th ptcul solutons ss xpnsons of Bssl functons vn n Equton us. Equton cn wttn n th follown wy:

9 Mtls T n n τ [ n τ n τ ] 8 n wh A δ F n τ n B τ δ F n n n xp μ τ μ A μ Δ μ n! xp μ τ B μ Δ μ n μ n! n n 9 H δ n s th Konck lt. Th ptcul solutons u p φ p n ψ p otn s th functon systm lk Equton 8. Thn th stss componnts lctc splcmnt n mntc flux nsty cn vlut fom Equtons n 5. Dtls of th solutons omtt fom h fo vty. Th unknown constnts n th homonous solutons tmn so s to stsfy th ouny contons n n.. umcl Rsults To llustt th foon nlyss w cons th functonlly hollow sph compos of pzolctc n mntostctv mtls. Th pzolctc mtl s m up of BTO n th mntostctv mtl s m up of of O. umcl pmts of ht conucton n shp psnt s follows: H H. T T.7 R R. m / Th fst ly s pu pzolctc mtl n th th ly s pu mntostctv mtl. It s ssum tht th volum fctons of th pzolctc phs V p n th mntostctv phs V m fo oth lys vn y th ltons V p V p / M f M M f M Vm V p Th vlu of V p n th ly s otn y clcultn th vlu of V p n Equton t th cnt pont of ch ly fn y R R /. To stmt th mtl popts of FGM w pply th smplst ln lw of mxtu. Th mtl constnts cons fo BTO n of O shown n th pp []. Th typcl vlus of mtl pmts such s κ α Y n us to nomlz th numcl t s on thos of BTO s follows: κ κ θ 5 5 α α Y 6GP 78 / 5 In th numcl clcultons th ouny contons t th sufcs fo th lctc n mntc fls xpss s

10 Mtls 5 ; D B ; φ ψ 5 Fus 5 show th numcl sults fo M n. Th vtons of tmptu chn n splcmnt u lon th l cton shown n Fus n spctvly. Fom Fus n t s cl tht th tmptu n splcmnt ncs wth tm n hv th lst vlus n th sty stt. Th vtons of noml stsss σ n σ θθ lon th l cton shown n Fus n spctvly. Fu vls tht th mxmum tnsl stss of σ occus n th tnsnt stt n th mxmum compssv stss of σ occus n th sty stt. Fom Fu t s cl tht th mxmum tnsl stss occus n th out sufc. Th vtons of lctc potntl φ n mntc potntl ψ lon th l cton shown n Fus n 5 spctvly. Fu vls tht th solut vlu of th lctc potntl ncss wth tm n ttns ts mxmum vlu n th sty stt. Th lctc potntl s lmost zo n th tnth ly.. th pu mntostctv ly. Fom Fu 5 t s cl tht th solut vlu of th mntc potntl ncss wth tm n ttns ts mxmum vlu n th sty stt. Th mntc potntl s lmost constnt n th fst ly.. th pu pzolctc ly. Fu. Vton of th tmptu chn M. Fu. Vton of th splcmnt u M

11 Mtls 6 Fu. Vton of th thml stsss M noml stss σ. θθ : noml stss σ ; Fu. Vton of th lctc potntl M. Fu 5. Vton of th mntc potntl M.

12 Mtls 7 In o to ssss th ffct of th nonhomonous pmt M on th stsss lctc potntl n mntc potntl th numcl sults fo shown n Fus 6 8. M shows pzolctc mtl ch n M. 5 shows mntostctv mtl ch. Th vtons of stsss σ n σ θθ shown n Fus 6 n 6 spctvly. Fom Fu 6 t s cl tht th mxmum compssv stss of σ css whn th pmt M ncss n th sty stt. Fom Fu 6 t s cl tht th mxmum tnsl stss of σ θθ css whn th pmt M css n th sty stt. Th vtons of lctc potntl n mntc potntl shown n Fus 7 n 8 spctvly. Fom Fus 7 n 8 th solut vlu of th lctc potntl n th nn sufc s mxmum whn th pmt M n th sty stt whl tht of th mntc potntl s mxmum whn th pmt M. 5 n th sty stt. Fu 6. Vton of th thml stsss : noml stss σ ; noml stss σ θθ. Fu 7. Vton of th lctc potntl.

13 Mtls 8 Fu 8. Vton of th mntc potntl. In o to ssss th ffct of lxton of stss vlus n functonlly mnto-lcto-thmolstc hollow sph th numcl sults fo th two-ly hollow sph shown n Fu 9. Fus 9 9 9c n 9 show th vtons of stsss σ σ θθ lctc potntl n mntc potntl spctvly. Fom Fus n 9 th ffct of lxton of stss stutons fo th functonlly hollow sph cn clly sn comp wth th two-ly hollow sph. Fom Fus 5 n 9 t s cl tht th mxmum solut vlus of th lctc potntl n mntc potntl fo functonlly hollow sph t thn thos fo th two-ly hollow sph. Fu 9. umcl sults fo th two-ly hollow sph : noml stss σ ; noml stss σ θθ ; c lctc potntl; mntc potntl.

14 Mtls 9 Fu 9. ont. c. onclusons In ths stuy w nlyz th tnsnt thml stss polm fo th functonlly mnto-lcto-thmolstc hollow sph u to unfom sufc htn usn lmnt compost mo y solvn th ovnn qutons of th splcmnt lctc potntl n mntc potntl. As n llustton w c out numcl clcultons fo functonlly hollow sph compos of pzolctc BTO n mntostctv of O n xmn th hvos n th tnsnt stt fo tmptu chn splcmnt stss lctc potntl n mntc potntl stutons. W nvstt th ffcts of th nonhomonty of mtl on th stsss lctc potntl n mntc potntl. Futhmo th ffct of lxton of stss vlus n functonlly mnto-lcto-thmolstc hollow sph ws nvstt. W conclu tht w cn vlut not only th thmolstc spons of th functonlly mnto-lctothmolstc hollow sph ut lso th lctc n mntc fls of functonlly mnto-lcto-thmolstc hollow sph quntttvly n tnsnt stt. Rfncs. Hsh G.; Douhty J.P.; wnhn R.E. Thotcl moln of multly mntolctc compost. Int. J. Appl. Elctomn Mt n.w. Mntolctc ffct n composts of pzolctc n pzomntc phss. Phys. Rv. B Pn E.; Hyl P.R. Exct solutons fo mnto-lcto-lstc lmnts n cylncl nn. Int. J. Sols Stuct Annkum R.; Ann R.; Gnsn.; Swnmn S. F vton hvo of multphs n ly mnto-lcto-lstc m. J. Soun V Wn R.; Hn Q.; Pn E. An nlytcl soluton fo multly mnto-lcto-lstc ccul plt un smply suppot ltl ouny contons. Smt Mt. Stuc

15 Mtls 5 6. B M.H.; hn Z.T. Exct solutons fo lly polz n mntz mntolctolstc ottn cylns. Smt Mt. Stuc Yn J.; Wn H.M. Mntolctolstc fls n ottn multfoc compost cylncl stuctus. J. Zhn Unv. Sc. A Wn H.M.; Dn H.J. Tnsnt sponss of mnto-lcto-lstc hollow sph fo fully coupl sphclly symmtc polm. Eu. J. Mch. A Sols Wn H.M.; Dn H.J. Sphclly symmtc tnsnt sponcs of functonlly mnto-lcto-lstc hollow sph. Stuct. En. Mch M..; L J.M. Thotcl nlyss of n-pln polm n functonlly nonhomonous mntolctolstc mtls. Int. J. Sols Stuct Yu J.; Wu B. cumfntl wv n mnto-lcto-lstc functonlly cylncl cuv plts. Eu. J. Mch. A Sols Wu.P.; Lu Y.. A mof pno mtho fo ynmc sponss of functonlly mnto-lcto-lstc plts. ompount. Stuct Hun D.J.; Dn H.J.; hn W.Q. Sttc nlyss of nsotopc functonlly mnto-lcto-lstc ms suct to ty lon. Eu. J. Mch. A Sols L J.M.; M.. Anlytcl solutons fo n ntpln polm of two ssml functonlly mntolctolstc hlf-plns. Act Mch Gnsn.; Kumvl A.; Sthumn R. Fnt lmnt moln of ly multphs mntolctolstc cyln suct to n xsymmtc tmptu stuton. J. Mch. Mt. Stuc Kumvl A.; Gnsn.; Sthumn R. Sty-stt nlyss of th-ly lcto-mnto-lstc stp n thml nvonmnt. Smt Mt. Stuct Hou P.F.; Y T.; Wn L. D nl soluton n funmntl soluton fo othotopc lcto-mnto-lstc mtls. J. Thml Stsss Wn B.L.; ul O.P. Tnsnt thml fctu nlyss of tnsvsly sotopc mnto-lcto-lstc mtls. J. Thml Stsss Ooto Y.; Tnw Y. Tnsnt nlyss of multly mnto-lcto-thmolstc stp u to nonunfom ht supply. ompount. Stuct Ooto Y.; Ishh M. Exct soluton of tnsnt thml stss polm of multly mnto-lcto-thmolstc hollow cyln. J. Sol Mch. Mt. En Ooto Y.; Tnw Y. Tnsnt thml stss nlyss of nonhomonous hollow sph u to xsymmtc ht supply. Tns. Jpn. Soc. Mch. En A Ooto Y.; Tnw Y. Th-mnsonl tnsnt thml stss nlyss of nonhomonous hollow sph wth spct to ottn ht souc. Jpn. Soc. Mch. En. 99 6A y th uthos; lcns MDPI Bsl Swtzln. Ths tcl s n opn ccss tcl stut un th tms n contons of th tv ommons Attuton lcns

Exam 2 Solutions. Jonathan Turner 4/2/2012. CS 542 Advanced Data Structures and Algorithms

Exam 2 Solutions. Jonathan Turner 4/2/2012. CS 542 Advanced Data Structures and Algorithms CS 542 Avn Dt Stutu n Alotm Exm 2 Soluton Jontn Tun 4/2/202. (5 ont) Con n oton on t tton t tutu n w t n t 2 no. Wt t mllt num o no tt t tton t tutu oul ontn. Exln you nw. Sn n mut n you o u t n t, t n