Keywords: Functionally graded material; Piezoelectric material; Sphere; Elasticity solution; Nonlinear differential equation.

Transcription

1 M. Saviz * ssistat Pofsso. Ghobapou ai * Pofsso Elastiity Solutio ppoah fo Futioally Gadd Sphial Shll with Pizolti Poptis I this pap, a aalytial mthod is adoptd, basd o lastiity appoah to aalyz th hollow FGM sph with pizolti poptis. Th lto-mhaial poptis xpt th Poisso s atio a assumd to b pow futios of adius. Loadig is a ombiatio of pssus ad a distibutd lti fild. Fo axisymmti poblm, 3-D govig quatios a dud to a -D sod od olia auhy-typ difftial quatio, i tms of adial displamt. Th solutio of olia difftial quatio is optd as a pow law futio. By satisfyig fiv sts of bouday oditios ad iopoatig thm ito govig quatio, a systm of algbai quatios is obtaid that dlivs th ukow ostats. Stati sposs of FG shll to ltomhaial loads with difft ad th fft of siz a ivstigatd. Th idud adial ad iumftial stsss of a imposd lti pottial a ompad to th sidual stsss lokd i th homogous sph. Kywods: Futioally gadd matial; Pizolti matial; Sph; Elastiity solutio; Nolia difftial quatio. Itodutio Pizolti matials hav b widly usd as distibutd ssos ad atuatos i th fild of smat stutus ad ativ stutual otol. smat stutu typially ompiss of o o mo ativ o futioal matials. Th lightwight high-stgth shlls with pizolti poptis a famous fo thi apability of povidig th xptd bhaviou at th smat stutus lvl. awly [] potd a ovviw of appliatios of pizolti matials fo itlligt ad aospa stutus. Futioally gadd matials FGMs a miosopially ihomogous omposits usually mad fom a mixtu of mtals ad amis. By gadually vayig th volum fatio of osistt matials, thi matial poptis xhibit a smooth ad otiuous hag alog o o mo ditios to obtai optimum spos to xtally applid loads. * ospodig utho, ssistat Pofsso, Mhaial Egiig Dpatmt, zabaija Shahid Madai Uivsity, Tabiz, Ia, saviz@azauiv.a.i Pofsso, Dpatmt of Mhaial Egiig, Uivsity of Kasha, Kasha, Ia, ghobapo@yahoo.om

2 04 Iaia Joual of Mhaial Egiig Vol. 8, No., Mah 07 Niio [] itodud th opt of futioally gadd matial to satisfy th dmad of ulta-high-tmpatu viomt ad to limiat th stss sigulaitis. Yamada t al. [3] pstd a futioally gadd pizolti plat atd by fomig a tmpatu vaiatio aoss th plat with lativly low ui tmpatu. h t al. [4] pstd th stati aalysis of a stadily otatig pizolti sphial shll with a futioally gadd popty. Eltomhaial sposs of ompositioally gadd pizolti lays w aalysd by Lim ad H [5]. Siha [6] obtaid th solutio of th poblm of stati adial dfomatio of a pizolti sphial shll ad ud a giv voltag diff btw ths sufas, oupld with a adial distibutio of tmpatu fom th i to th out sufa. Thdimsioal lastiity stati aalysis of a multilayd lasti sphial hollow shll with sphial isotopy was pstd by h ad Dig [7]. Two idpdt stat quatios w divd aft itoduig th displamt futios ad two stss futios. Ghobapou t al. [8] ivstigatd th stss ad lti pottial filds i pizolti hollow sphs. Th stss fild i pizolti hollow sph ud thmal viomt was dvlopd by Saadatfa ad astgoo [9]. Shao t al. [0] divd aalytial solutios fo mhaial stsss of a futioally gadd iula hollow ylid with fiit lgth. I-homogity was osidd i a umb of studis. Elasti aalysis of itally pssuizd thik-walld sphial pssu vssls of futioally gadd matials FGMs ivstigatd by You t al. []. Sladk t al. [] divd Loal itgal quatios fo umial solutio of 3-D poblms i lia lastiity of FGMs viwd as -D axisymmti poblms. Th mshlss loal Ptov Galki mthod was applid to tasit dyami poblms i 3D axisymmti pizolti solids with otiuously o-homogous matial poptis subjtd to mhaial ad thmal loads by Sladk t al. [3]. Wag ad Xu [4] studid th fft of matial ihomogity o ltomhaial bhavios of futioally gadd pizolti sphial stutus. Magtothmolasti poblms of FGM sphs a studid by Ghobapou t al. [5]. Tasit aalysis of odiay futioally gadd ylidial shlls subjtd to ital dyami pssu was pstd by Stoodh t al. [6], implmtig a pow law distibutio futio i th thikss. Ghobapou ai t al. [7] solvd hollow sph mad fom futioally gadd pizolti matial, usig ussay dimsiolss quatitis. Thy osidd a patiula solutio, ot satisfyig th govig quatio ad a xpotial futio fo gal solutio of auhy-eul quatio, istad of a pow law futio, whih is giv i mathmatial Hadbooks.g. [8]. Buklig of shallow futioally gadd sphial shlls with sufa-bodd pizolti atuatos ud thmal load was studid by Sabzika Booujdy ad Eslami [9]. It was assumd that poptis of th futioally gadd matial vay though th thikss aodig to a pow law distibutio of th volum fatios of th ostitut matials. Th stati ad dyami sposs of a simply-suppotd, thik lamiatd othotopi ylidial shll with pizolti atuato ad sso lays, basd o a 3-D lastiity solutio appoah was publishd by Shaki t al. [0]. Hafzalkotob ad Eslami [] pstd th thmomhaial buklig of simply suppotd thi shallow sphial shlls mad of futioally gadd matial. I this pap, th fist-od shll thoy of Lov ad Kihhoff, th Doll- Mushtai-Vlasov kimatis quatios, ad th alulus of vaiatios was usd. Thfo, i ojutio with th pvious woks, w appliatios of pizolti ssos ad atuatos a big itodud fo a w gomti ofiguatio. This sah attmpts to aalys a hollow sph omposd of a adially polaizd tasvsly isotopi futioally gadd pizolti matial, subjtd to uifom stati load togth with a pottial diff idud by ltods attahd to th i ad out sufas of th aula sph.

3 Elastiity Solutio ppoah fo Futioally Gadd 05 ll mhaial ad pizolti poptis of th FGM hollow sph, xpt fo th Poisso s atio, a assumd to dpd o th adius ad xpssd i tms of its pow futio with matial i-homogity lvl. Th 3-D govig quilibium quatios of adially polaizd sph a dud to a systm of sod-od odiay difftial quatios, yildig a -D sod od olia auhy-typ difftial quatio i tms of adial displamt, whih th is solvd aalytially. By satisfyig fou difft sts of bouday oditios ad iopoatig thm ito govig quatio, a systm of algbai quatios is obtaid that dlivs th ukow ostats. Th auay ad omputatioal ffiiy of th poposd appoah a vifid by ompaig th sults with thos obtaid fo homogous matial i th litatu. Fomulatio ad thoy osid a hollow FGPM sph with isid adius i, outsid adius o, ad total shll thikss H. Th shll gomty xposd to uifom ital ad xtal pssus Pi ad Po, ad distibutd lti pottial V is show i Figu. Baus of th losd gomty, th ltods hav to b attahd to th out ad i sufas duig th maufatuig poss. This lads to havig a lti fild i adial ditio. Futhmo, th ditio of polaizatio is stablishd duig th idutio poss by mas of th lti fild applid btw two ltods ad its quatity is dtmid by lti displamt. So, i a thikss-wis polaizd pizolti sph, th will b oly adial ompots of lti fild ad displamt. Dtails with spt to dfiitio ad dtmiatio of th ostats dsibig ths matials hav b stadadizd by th Istitut of Eltial ad Eltois Egis []. Fo t-symmti stss ad displamt oditios, th lia ostitutiv latios fo a othotopi matial, with adially polaizd pizolti popty a b witt as follows, [3], [4]: EF ED 3 Δ E 3 -a -b Wh ij, Δij, ad ij a th lasti, dilti, ad pizolti ostats, sptivly, whih lat th ompots of stss, stai, lti fild EF ad lti displamt vtos ED. It is assumd that th futioally gadd matial has tasvsly isotopi poptis with spt to axis of otatio oitd i th adial ditio, th lastiity ad pizolti offiit tsos a xpssd as,,, -a H, it has oly th idpdt matial paamts, i additio to a pizolti offiit. I as of havig th mhaial poptis ditioal lasti moduli ad poisso's atios, by usig mio-mhais, ul of mixtus o xpimt, th lasti offiits fo thikss-wis FGM sph a obtaid as follows, [3]

4 06 Iaia Joual of Mhaial Egiig Vol. 8, No., Mah 07 Figu FGPM shll gomty subjt to uifom pssu ad applid voltag. E 3 3 E E 3 3 3, E 3 3, , -b wh, Poisso s atio is assumd to b ostat though th shll thikss Fo isotopi poptis, th FGM lasti offiits a summaizd to th followig latios, [7] E 33, E Th t-symmti quatios of motio 0 i th abs of body fo a 0 3-a Th hag quatio of ltostatis is giv by Tist [4]. ED ED 0 3-b Th t-symmti stai-displamt ad th lti fild-lti pottial latios of th pizolti lasti mdium a witt as u u, u,, 4-a EF 4-b By ombiig Eqs. 4 with Eqs., th stss ad ltial displamt ompot will b obtaid as follows u u 3 z 3 u u u u 3 5-a

5 Elastiity Solutio ppoah fo Futioally Gadd 07 u u ED 3 Δ 5-b ft substitutig ths ompots ito th govig Eqs. 3 ad fatoizig th simila divativs of u ad ψ, th quatios of quilibium i tms of displamt ad lti pottial fo sphial shll bom: u Δ d u d u d d d 0 d dδ Δ 0 d d u d 3 d d u u 6-a 6-b Fo isotopi FG matial, th mhaial quatio of motio is simplifid as u d u d d d d 0 d u By substitutig ompots of th lasti ostats fom th abov quatios ito Eq. 6- a, th quilibium quatio is dvlopd i tms of th displamt ad lti pottial fild of th futioally gadd sphial shll, whil th ltial quilibium Eq. 6-b mais th sam, as follows: E u u u u d 0 d. u de d 7 O th oth had, th FGM poptis hag though th -ditio, whih a b a ombiatio of amis ad mtals. Th mixig atio is vaid otiuously ad smoothly aoss th thikss. Th followig modl is tak fo matial popty distibutio [3], [9] q qi 8 i q is matial popty that is otolld by volum fatio as a futio of, is th o-gativ pow-law xpot ad subsipts i ad o stad fo i ad out sufas. q a b substitutd fo Youg's modulus E, sha modulus G, lti offiits, ad mass dsity ρ. Poissio's atio is osidd ostat though th thikss. Vaiatio of matial poptis i tms of volum fatio of xtal / itio matial with omalizd adial dista i thikss ditio fo difft valus of a show i Figu.

6 08 Iaia Joual of Mhaial Egiig Vol. 8, No., Mah 07. Bouday oditios Baus of symmti gomty, th is o sstial bouday oditio i this poblm. Th loadig oditio of th out/ i sufa of th shll is osidd to b f of sha/ i pla tatio ad th lto-stati pottial is assumd to b zo o th out sufa. Ud ths iumstas, fou sts of mhaial ad ltial loadig bouday oditios of th FGP sph a witt as follows: I i Pi, o 0, i 0, o 0 9-a II i 0, o 0, i Vi, o 0 9-b III i 0, o Po, i 0, o 0 9- IV i Pi, o 0, i Vi, o 0 9-d V i Pi, o 0, i 0, o Vo 9- I as I, th FGPM hollow sph is subjtd to a ital uifom pssu without ay imposd lti pottial ad xtal pssu. Howv, i this as th idud lti pottial is xistd aoss th thikss. I this as, th sph ats as a sso. I th sod as, a ltial pottial diff is applid btw th i ad out sufas of th sph without ay ital ad xtal pssus. I this as, th sph ats as a atuato. I as III, th FGPM hollow sph is xposd to a xtal uifom pssu without ay imposd lti pottial ad ital pssu. ass IV ad V a th suppositio of ass I ad II, alog with a ltial pottial o th xtal ltod o th shll sufa. 3 alytial solutio mthod Th tivial solutio of lti quilibium Eq. 3-b is as follows, [7] D ED 0 wh, D is a ukow ostat that should b dtmid usig bouday oditios. Usig this solutio, ombiig quatios 6-a ad 6-b ad olltig h simila divativs of adial displamt, yilds as follows Δ D Δ u dδ d Δ d d d Δ d Δ Δ Δ d 3 Δ d d dδ Δ d Δ d d d Δ 0 dδ d u Δ d d u -a

7 Elastiity Solutio ppoah fo Futioally Gadd 09 Figu Matial popty vaiatio i tms of xpot Fo isotopi FG matial, th mhaial stati quilibium is simplifid as E Δ u de E Δ d Δ u d d dδ Δ d Δ d Δ d D Δ dδ d Δ de E d Δ dδ d Δ d Δ d d d Δ 0 u -b t this stag, th lastiity ad ltiity offiits a obtaid fom Eq.8 ij = ij /i, E= E/i, = /i, = /i, = /i -a Th Poisso s atios υ ij a assumd ostat i difft ditios. E E wh, 3 3,, -b fo isotopi as: E E By substitutig Eqs. ito Eq. -a, th followig o-lia patial difftial quatio mgs

8 0 Iaia Joual of Mhaial Egiig Vol. 8, No., Mah 07 u u - 3 D u 0 3 Fo isotopi FG matial, th o-lia patial difftial quatio is witt as E E u E D 3 u 0 u 3-a 3-b Multiplyig Eq.3-a with -, yilds a o-homogous auhy-eul typ difftial quatio as follows wh, u u u 3D Fo isotopi FG matial, ths multiplis a 3 5-a E 3 E E 5-b Th gal solutio fo Eq. 4 has th followig wll kow fom, [8]: wh u g g g u u K K 6, 4 Subsqutly, th patiula solutio of Eq. 4 a b dvlopd by mployig th mthod, so alld vaiatio of paamt [8], as u p 7 ug ug ad a b dtmid by substitutig Eq. 7 ito quilibium Eq. 4 as

9 Elastiity Solutio ppoah fo Futioally Gadd d du u d du u W F d u F d u F g g g g g g,,.,. 8 Wh W is th xpssio o th ight had sid of Eq. 4. Substitutig Eq. 8 ito Eq. 7, th ovall solutio is foud as follows 3 D K K u 9 Wh K, K a ukow ostats that a dtmid usig bouday oditios. Now, by substitutig th displamt fom Eq. 9 ito Eqs. 5 th adial stss is obtaid. lso, substitutig u ito Eq. 5-b, th ombiig with Eq. 4-b ad pfomig itgatio, lti pottial alog with adial stss a witt as D K K a b is th itgatio ostat. Fo all th fou load ass mtiod i bouday oditios, th systm of lia algbai quatios fo th ostats K, K, D ad of Eqs. 0 a b witt i th followig fom o i o i o o o i i i o o o i i i D K K

10 Iaia Joual of Mhaial Egiig Vol. 8, No., Mah 07 Th ompots of vto o th ight sid a dfid i bouday oditio Eqs. 9. O th vto of ostats is alulatd, by usig Eqs. 9 ad 0 th displamt, lti pottial ad stss would b obtaid. osqutly, th iumftial stss is alulatd by usig th followig latio K K D Evaluatio ad Numial sults Th fftivss of th dvlopd fomulatio has b dmostatd though th aalysis of xampls ad makig ompaisos with th publishd sults i bhmak poblms. I th fist stp, a sphial shll with o/i=, ad mad of homogous isotopi matial with th followig stl-lik poptis is osidd, whil puttig th pizolti latd ostats qual to a gligibl valu: E= 07 GPa, υ = 0.9. Th shll is subjtd to a xtal uifom pssu load as III. I Figus 3a ad b th distibutios of th odimsioal adial stss σ /Po ad omalizd adial displamt u E / + υ o Po, sptivly a ompad to th sults obtaid by h ad Dig [7]. Sodly, th followig tasvsly isotopi shll is ivstigatd: Shll Sphial shll mad of pizolti ami PZT-5 with th matial poptis giv i Tabl, [8]. I th fist stp, th sults obtaid fo fou difft stati loadigs of shll a ompad to th potd sults [8]. Th gomtial poptis of th symmti sph a osidd: o/i=.3,.4 ad 4.0. I Tabl, th valus of adial stss, iumftial stss ad lti pottial o th mid-adius of th shll o / i / a ompad to thos obtaid by Ghobapou t al. [8]. Without supis, it is s that th is a good agmt btw th pst sults ad th potd os. Th asos fo ths diffs a dissimila fomulatio ad omputatioal appoah. I th pst wok, th Mapl softwa has b implmtd to solv th govig quatios i paamtial fom, osumig lss tim ad omputatioal ffot. It is s that th pst appoah, to som xtt ovstimats th sults. Havig do th afomtiod study, th apability of th poposd fomulatio ad dvlopd omput od to aalys th FGPM sphial shll has b assssd.

11 Elastiity Solutio ppoah fo Futioally Gadd 3 a adial displamt b adial stss Figu 3 ompaiso of thikss-wis vaiatios, as III Tabl Matial poptis of pizolti PZT-5 Elasti ostats, Gpa Pizolti ostats, /m Pmittivity,0-9 / Nm 3 35 Δ Tabl ompaiso btw sults i th middl of shll thikss fo Shll Eltomhaial Loadig as I as II as III as IV o / i pst f [7] pst f [7] pst f [7] Sodly, adial displamts of shll ospodig to th abov last th load ass a pstd i Figus 4a to 4d. s it is xptd, th adial displamts fo as I a positiv, whil adial displamts of th oth ass a gativ, havig imtal vaiatio tds. Thi shll o/i=.3 has a difft dfomatio patt fom th thik os. It is ifd fom Figus 4 ad 4d that load ass III ad IV dliv vy simila sults, du to th shikig fft of lti load. Nxt, th followig sphial shll is ivstigatd:

12 4 Iaia Joual of Mhaial Egiig Vol. 8, No., Mah 07 Shll B FGPM mad of pizolti ami PZT-4, whih has b sltd baus of its thial appliatios. Mhaial ad ltial poptis of pizolti matial, PZT_4 a tabulatd i Tabl 3. Pstd sults a latd to th fiv ass of difft bouday oditios with aspt atio o / i =.3. Th umial sults a daw i Figus 5 to 9, showig th vaiatio of stsss, lti pottial ad displamt aoss th thikss of th FGPM sph fo difft matial ihomogity paamt. Tabl 3 Matial poptis of pizolti PZT-4 Elasti ostats, Gpa Pizolti ostats, /m Pmittivity,0-9 / Nm 3 35 Δ Δ as I sults of th fist as a illustatd i Figus 5. adial stsss fo difft matial ihomogity paamts a show i Figu 5a. adial stsss satisfy th mhaial bouday oditios at th i ad out sufas of th FGPM sph. Th maximum absolut valus of adial stsss blogs to a matial idtifid by i-homogity paamt =.5 th miimum absolut valus of whih blog to = -.5. I this as th is o imposd lti pottial. Howv, th idud lti pottials fo difft matial i-homogity paamts a show i Figu 5b. Elti pottials satisfy th goudd ltial bouday oditios at th i ad out sufas of th shll B. It is also obvious that high idud lti pottials blog to high absolut valus of ompssiv adial stsss. Hoop stsss Figu 5 a highly tsil though thikss. adial displamts a illustatd i Figu 5d fo all matial poptis. Displamts a positiv thoughout th thikss ad thy smoothly das fom thi maximum valu at th i sufa to thi miimum valu at th out sufa of th shll B. Maximum valus of displamts blog to = -.5 ad miimum valus blog to =.5. a as I b as II

13 Elastiity Solutio ppoah fo Futioally Gadd 5 as III d as IV V o =V i =0 Figu 4 Though thikss vaiatio of adial displamts, o / i =.3,, 4, a adial stss b Elti pottial Hoop stss d adial displamt Figu 5 Though thikss vaiatio of sults fo fiv xpots, o / i =.3, as I, ΔV = 0volt as II sults of th fully atuato as a illustatd i Figus 6a to 6. I whih, th is o applid pssu at th i ad out sufas of th sph howv th idud ompssiv adial stsss satisfy th tatio f mhaial bouday oditios. Itstigly, th maximum absolut valus of th idud ompssiv adial stsss blog to th sam maximum valu of lti pottial. I this as, th imposd lti pottial satisfis th ltial bouday oditios at th i ad out sufas of th sph. This mas that th diff btw lti pottials o i ad out sufas ΔV= V o - V i a impotat ath tha thi idividual valus. That is, V o =0, V i = podus th sam sults as V o = -, V i =0. It is obsvd that th gat lti pottials blog to = -.5, whil th small valus of whih blog to =.5.

14 6 Iaia Joual of Mhaial Egiig Vol. 8, No., Mah 07 iumftial idud stsss a both tsil ad ompssiv thoughout thikss fo difft matial i-homogity paamts. Howv, fo gativ paamts th miimum valus of iumftial stsss a loatd at th i sufa, whil fo positiv paamts of thi miimum valus a loatd at th out sufa of th FGPM sph. This stss tds to vaish o th mid-adius fo all paamts. Th idud adial displamt is gativ aoss th thikss fo all matial paamts ad about t tims lag tha thos of as I Figu 5d, i spit of whih, th miimum valus a loatd at th i ad thi maximum valus a at th out sufas of th shll B. It is obsvd fom Fig. 6 that th ovall bhavio of idud adial ad iumftial stsss lokd i th sph duig th autofttag poss of sphs mad of uifom matial [5] a typially simila to thikss-wis stss distibutios fo load as II Fig. 6, i whih adial stss ausd by lti fild vaishs at f sufas of FGPM sph. iumftial idud stss is ompssiv o th i adius ad has a small tsil valu o th out adius. Fo gativ paamts, th miimum valus of iumftial stss a loatd at th i sufa, whil fo positiv paamts of its miimum valus a loatd at th out sufa of th FGPM sph. This stss tds to vaish o th midadius fo all paamts. as III sults of th thid load as a illustatd i Figus 7a to 7d. adial stsss fo difft matial i-homogity paamts a show i Figu 7a. adial stsss satisfy th mhaial bouday oditios at th i ad out sufas of th shll B. Th lagst absolut valus of adial stsss blog to a matial idtifid by = -.5 ad th smallst valus of whih blog to =.5. I this as, th is o xtal lti pottial; howv, th idud lti pottials fo difft matial i-homogity paamts a show i Figu 7b. Elti pottials satisfy th fully goudd ltial bouday oditios at th i ad out sufas of th shll B. a adial stss b Elti pottial Hoop stss d adial displamt

15 Elastiity Solutio ppoah fo Futioally Gadd 7 Hoop ad adial sidual stss, [4] Figu 6 Though thikss vaiatio of sults fo fiv xpots, o / i =.3, as II, ΔV = -volt It is also la that high idud lti pottials ospod to thos with lag absolut valus of ompssiv adial stsss. iumftial stsss a ompltly ompssiv aoss th thikss fo difft matial i-homogity paamts Figu 7. Howv, fo gativ paamts th miimum valus of iumftial stsss a loatd at th i sufa, whil fo positiv os thi miimum valus a at th out sufa of th FGPM sph. It is itstig to s that th ompssiv iumftial stsss i this as is vy simila to th idud iumftial stsss sultd fom imposig a lti pottial as II. adial displamts a illustatd i Figu 7d fo all matial poptis. Displamts a agai gativ thoughout th thikss ad thy smoothly hag fom thi lss valu at th i sufa to thi lativly ostat valus at th out sufa of th shll B. Maximum valus of displamts blog to = -.5 ad miimum valus blog to =.5. as IV as fou is suppositio of th ass I ad II. Th sults of this as a giv i Figus 8a to 8d. adial stsss ad th lti pottials satisfy th mhaial ad ltial bouday oditios. positiv lti pottial is applid o th ital sufa, whih is qual to imposig a gativ lti pottial o th xtal sufa. Th maximum ompssiv valus of adial stsss blog to a matial idtifid by = -.5 th miimum valus of whih blog to =.5. It is also la that high lti pottials ospod to thos with lss valu of ompssiv adial stsss. iumftial stss distibutio is simila to thos of as II fo difft matial i-homogity paamts. Howv, fo gativ paamts th miimum valus of iumftial stsss loatd at th i sufa whil fo positiv paamts thi miimum valus loatd at th out sufa of th shll B.

16 8 Iaia Joual of Mhaial Egiig Vol. 8, No., Mah 07 a adial stss b Elti pottial Hoop stss d adial displamt Figu 7 Though thikss vaiatio of sults fo fiv xpots, o / i =.3, as III, ΔV = 0 volt a adial stss b Elti pottial Hoop stss d adial displamt Figu 8 Though thikss vaiatio of sults fo fiv xpots, o / i =.3, as IV, ΔV = -volt

17 Elastiity Solutio ppoah fo Futioally Gadd 9 adial displamt is gativ aoss th thikss fo all matial paamts, smblig Figu 6d, xpt that th iwad displamt is dud slightly. Thi miimum valus loatd at th i ad thi maximum valus loatd at th out sufas of th shll B. as V Th afomtiod ltial loadigs a ot vy patial, baus it is mo alisti to atuat th ltod istalld o th xtal sufa of th losd shll. Figu 9a shows that although mhaial pssu is lik th pvious as, adial stss distibutio has th opposit td fo difft matial paamts. Similaly, lti pottial dmostats bhavious opposit to thos giv i Figu 8b. iumftial stss distibutio is simila to thos of as I fo difft matial i-homogity paamts. Howv, a t tims ias is idud i this stss Figu 9, du to applid voltag. Dspit all th pvious ass, adial displamts a positiv thoughout th thikss ad thy vay losly fom thi maximum valu o th i sufa to thi miimum valus o th out sufa. ompaig Figu 9d with Figu 5d ifs that th positiv lti load o th out sufa buks th vaiatio td i adial displamt with gad to matial xpot, as wll as itsifyig th xpasio of sphial shll B du to mhaial pssu, mo tha t tims ad this might ot b ally dsid i tms of shap otol. a adial stss b Elti pottial Hoop stss d adial displamt Figu 9 Though thikss vaiatio of sults fo fiv xpots, o / i =.3, as V, ΔV = volt

18 0 Iaia Joual of Mhaial Egiig Vol. 8, No., Mah 07 5 olusio aalytial appoah is dvlopd fo th aalysis of futioally gadd pizolti sphial shlls with pizolti poptis subjtd to difft load. This study hlps to dp udstadig of th bhaviou of adially polaizd FGPM smat stutus. Th ffts of th gomtial paamts, bouday oditios ad volum fatio xpot o stati bhaviou of th FGPM sph a dmostatd. olatios btw yildd sults ad xistig solutios suppot th auay ad vsatility of dvlopd fomulatios. Vaiatio of stsss, lti pottial ad displamt of fiv sts of bouday oditios fo difft matial i-homogity paamts a plottd agaist adius. I gal, adial stsss ad lti pottials satisfy th mhaial ad ltial bouday oditios at th i ad out sufas of th FGPM sph. High absolut valus of ompssiv adial stsss a assoiatd with th high idud lti pottials thoughout th thikss i all ass. Th oth obsvatios of this sah a b outlid as follows:. Th xat solutio is oly sposibl to aptu th ot though-thikss distibutios of displamt. Whil, th ostitutiv quatios a aoutabl fo th ovall auay of th stss ompots.. Th pst fomulatio is apabl fo stati aalysis of both thi ad thik FGPM sps, as wll as pvtig sha lokig i thi shlls. 3. Maigful lss amout of omputatioal ffot of th dvlopd appoah, with spt to th fiit lmt mthod is witssd. 4. It is s that th ompssiv adial stss du to ital pssu tds to bom tsil by imposig a gativ lti pottial, fo shll with high FGM xpots. 5. Elti xitatio has a sigifiat fft o th distibutio of stss ad displamt filds i a FGPM shll. So that, hoop stss boms ompssiv o th ital adius ad th amout of this stss sultd fom ital pssu i FGPM sph a b dud istad of big positiv by applyig a pop amout of lti fild as wll as applyig a xtal pssu, whih a b substatial i tms of duability ad sufa fatigu ak gowth i th shll. This saio bigs about vy simila adial displamts fo load ass III ad IV. fs [] awly, E. F., "Itlligt Stutus fo ospa: a Thology Ovviw ad ssssmt", J. I, Vol. 3, pp , 994. [] Niio,., ad Mada, S., "t Dvlopmt Status of Futioally Gadit Matials", It. J. ISI Vol. 30, pp , 990. [3] Yamada, K., Yamazaki, D., ad Nakamua, K., " Futioally Gadd Pizolti Matial atd by a Ital Tmpatu Gadit", Jp J. ppl. Phys. Vol., No. 40, pp. 49 5, 00. [4] h, W. Q., Dig, H. J., ad Liag, J., "Th Exat Elastolti Fild of a otatig Pizoami Sphial Shll with a Futioally Gadd Popty", It. J. Solids Stut., Vol. 38, pp , 00. [5] Lim,. W., ad H, L. H., "Exat Solutio of a ompositioally Gadd Pizolti Lay ud Uifom Stth Bdig ad Twistig", It. J. Mh. Si., Vol. 43, pp , 00.

19 Elastiity Solutio ppoah fo Futioally Gadd [6] Siha, D.K., "Not o th adial Dfomatio of a Pizolti, Polaizd Sphial Shll with a Symmtial Distibutio", J. oust. So. Vol. 34, pp , 96. [7] h, W.Q., ad Dig, H.J., " Stat-spa-basd Stss alysis of a Multilayd Sphial Shll with Sphial Isotopy", J. pplid Mh. Vol. 68, pp. 09 4, 00. [8] Ghobapou,., Golabi, S., ad Saadatfa, M., "Stss ad Elti Pottial Filds i Pizolti Smat Sphs", J. Mh. Si. Thol. Vol. 0, pp , 006. [9] Saadatfa, M., ad astgoo,., "Stss i Pizolti Hollow Sph ud Thmal Eviomt", J. Mh. Si. Thol. Vol., pp , 008. [0] Shao, Z. S., Fa, L. F., ad Wag, T. J., "alytial Solutios of Stsss i Futioally Gadd iula Hollow ylid with Fiit Lgth", J. Ky Egg Mat., Vol. 6 63, pp , 004. [] You, L.H., Zhag, J.J., ad You, X.Y., "Elasti alysis of Itally Pssuizd Thikwalld Sphial Pssu Vssls of Futioally Gadd Matials", It. J. Ps. Vs. Pip. Vol. 8, pp , 005. [] Dig, H.J., Wag, H.M., ad h, W.Q., "alytial Solutio fo a No-homogous Isotopi Pizolti Hollow Sph", h. ppl. Mh. Vol. 73, pp. 49 6, 003. [] Sladk, V., Sladk, J., ad Zhag, h., "Tasit Hat odutio alysis i Futioally Gadd Matials by th Mshlss Loal Bouday Itgal Equatio Mthod", omput. Mat. Si. Vol. 8, pp , 003. [3] Sladk, J., Sladk, V., Solk, P., ad Saz,., "Dyami 3D xisymmti Poblms i otiuously No-homogous Pizolti Solids", It. J. Solids Stut. Vol. 45, pp , 008. [4] Wag, H.M., ad Xu, Z.X., "Efft of Matial Ihomogity o Eltomhaial Bhavios of Futioally Gadd Pizolti Sphial Stutus", omput. Mat. Si. Vol. 48, pp , 00. [5] Ghobapou,., Salai, M., Khadmizadh, H., ad fmash,., "Magto Thmolasti Poblms of FGM Sphs", h. ppl. Mh. Vol. 43, pp , 00. [6] Stoodh,.., Tahai, M., ad Slahi, E., "Hybid Laywis-difftial Quadatu Tasit Dyami alysis of Futioally Gadd xisymmti ylidial Shlls Subjtd to Dyami Pssu", omposit Stutus, Vol. 93, pp , 0. [7] Ghobapou ai,., Kolahhi,., Mosallai Bazoki,.., ad Loghma,., "Elto-Thmo-mhaial Bhavios of FGPM Sphs usig alytial Mthod ad NSYS Softwa", ppl. Math. Modlig, Vol. 36, pp , 0. [8] Zill, D.G., " Fist ous i Difftial Equatios with Modlig ppliatios", Books/ol Publishig ompay, alifoia, 00.

20 Iaia Joual of Mhaial Egiig Vol. 8, No., Mah 07 [9] Sabzika Booujdy, M., ad Eslami, M.., "Usymmtial Buklig of Pizo-FGM Shallow lampd Sphial Shlls ud Thmal Loadig", Joual of Thmal Stsss, Vol. 38, pp , 05. [0] Shaki, M., Saviz, M.., ad Yas, M.H., "Th-Dimsioal Elastiity Solutio fo Thik Lamiatd ylid with Pizolti Lay", Iaia Joual of Mhaial Egiig, Tasatio of ISME, Vol.6, No., pp. 4-0, 005. [] Hafzalkotob,., ad Eslami, M.., "Thmomhaial Buklig of Simply Suppotd Shallow FGM Sphial Shlls with Tmpatu-dpdt Matial", Iaia Joual of Mhaial Egiig, Tasatio of ISME, Vol., pp , 00. [] Istitut of Eltial ad Eltois Egis, Stadad o Pizoltiity, Std IEEE, Nw Yok, 978. [3] ddy, J.N., "Mhais of Lamiatd omposit Plats ad Shlls: Thoy ad alysis", Boa ato: Pss, 004. [4] Tist, H.F., "Lia Pizolti Plat Vibatios", Plum Pss, Nw Yok, 969. [5] Malki, M., Faahi, G.H., Haghpaah Jahomi, B., ad Hossiia, E., "sidual Stss alysis of uto fttagd Thik-walld Sphial Pssu Vssl", It. J. Pssu Vssls ad Pipig, Vol. 87, pp , 00. Nomlatu ij = lasti ostats at adius ij = matial ostat fo lasti ostats {ED} = lti displamt vto ij = pizolti ostats at adius ij= matial ostat fo pizolti offiits {EF}= lti fild vto Eii = ditioal modulus of lastiity at adius E = matial ostat fo modulus of lastiity Gij = sha modulus at adius H= total shll thikss K, K, D ad = ostats i solutios of quatios = pow-law xpot Pi ad Po= ital ad xtal pssus i = isid adius, o = outsid adius u =adial displamt V = distibutd lti pottial Gk symbols {} = stss vto {} = stai vto = lti pottial Δij = dilti ostats at adius Δij = matial ostat fo dilti ostat

21 Elastiity Solutio ppoah fo Futioally Gadd 3 چکيده براساس رویکرد حل الاستیسیته مخزن كروي توخالی ساخته شده از ماده هدفمند به صورت تحلیلی در راستاي شعاعی بررسی شده است. تغییرات خواص مکانیکی برحسب نسبت حجمی به صورت تابع مدل توانی از مختصه راستاي شعاعی كنترل می شوند. مخزن مزبور تحت فشار داخلی یا خارجی و بار الکتریکی گسترده با گرادیان پتانسیل الکتریکی مثبت یا منفی و به صورت مجزا و همزمان قرار گرفته است. مدل توانی براي در نظر گرفتن تغییرات خواص الکترومکانیکی به جز نسبت پواسون در امتداد ضخامت می باشد. براي تقارن محوري معادلات سه بعدي حاكم به یک معادله دیفرانسیل غیر خطی كوشی اویلر برحسب جابجایی شعاعی تبدیل می شوند. این معادله بصورت تحلیلی حل شده است. حل عمومی معادله دیفرانسیل بصورت تابع توانی و حل خصوصی متفاوت از كارهاي قبلی چاپ شده می باشد. با نوشتن پنج شرط مرزي مختلف و اعمال آنها به حل بدست آمده براي معادله مذكور یک دستگاه معادلات جبري برحسب چهار ثابت مجهول بدست می آید. بدین ترتیب میدان جابجایی و تنش در كره حل می شود. و ابعاد هندسی مختلف به بارهاي مکانیکی و الکتریکی جهت پاسخ استاتیکی كره با مواد مقایسه با مراجع موجود استفاده شده است. سپس پاسخ كره با توان هاي تابع هدفمندي و ابعاد هندسی مختلف تحت بارهاي مکانیکی و الکتریکی متنوع بررسی شده است. تنشهاي بوجود آمده در اثر بار الکتریکی قابل مقایسه با تنشهاي پسماند محبوس در یک كره ساخته شده از مواد غیر