Application of spectral elements method to calculation of stress-strain state of anisotropic laminated shells

Transcription

1 IOP Cofrc Sris: Matrials Scic ad Egirig PAPER OPE ACCESS Alicatio of sctral lmts mthod to calculatio of strss-strai stat of aisotroic lamiatd shlls To cit this articl: K A Ptrovsiy t al 16 IOP Cof. Sr.: Matr. Sci. Eg Viw th articl oli for udats ad hacmts. Rlatd cott - Som rsults of msh covrgc stimatio for th sctral lmt mthod of diffrt ordrs i FIDESYS idustrial acag V S Karo, A V Vrshii, V A Lvi t al. - A mass ad rgy cosrvig sctral lmt atmoshric dyamical cor o th cubd-shr grid M A Taylor, J Edwards, S Thomas t al. - Larg-scal lctromagtic modligs basd o high-ordr mthods: aoscic alicatios Misu Mi, Paul Fischr, Jaso Motgomry t al. This cott was dowloadd from IP addrss o 6/11/17 at 1:8

2 11th Itratioal Cofrc o "Msh mthods for boudary-valu roblms ad alicatios" IOP Publishig IOP Cof. Sris: Matrials Scic ad Egirig 158 (16) 177 doi:1.188/ x/158/1/177 Alicatio of sctral lmts mthod to calculatio of strss-strai stat of aisotroic lamiatd shlls K A Ptrovsiy 1, A V Vrshii 1 ad V A Lvi 1 LLC "Fidsys" Moscow Stat Uivrsity Scic Par amd aftr M. V. Uivrsity, offic 4, Lisi gory d. 1, str. 77, Moscow 11934, Russia Moscow Stat Uivrsity, GSP-1, Lisi gory d. 1, Moscow , Russia costati.89@yadx.ru Abstract. A modl of a high-ordr triagular sctral shll lmt is rstd to calculat a strss-strai stat of a shll costructio udr small dformatios cosidrig th matrial ihomogity. Cosidratio of shar dformatios, chags i th thicss of a shll is imlmtd through th us of 7-aramtr modl of shll lmts. Th rsults of roblms solvig ar to illustrat th ossibilitis of th lmt. 1. Itroductio Most of th matrials studid i modr tchology xric dformatio durig oratio. Th dvlomt of w structural matrials rquir a availability of adquat mchaical modls ad aalytic comlxs o thir basis for th assssmt of th strgth of structural lmts mad of such matrials. For a umrical simulatio of a wid rag of hysical homa th fiit lmt mthod is o of th most owrful tools for accurat, fficit ad stabl aroximat solutios. At rst, th fiit lmt mthod is widly ow as th basic calculatio mthod for th umrical simulatio of roblms of solid mchaics. Sic th bgiig of th dvlomt of ractical rocdurs usig th fiit lmt aalysis, o of th mai objctiv was to dvlo mthods for th calculatio of th comlx shll structurs. I modr ractic, th shll structurs ar usd i may brachs of idustry, icludig automotiv, aviatio ad civil costructio domais. I fiit lmt aalysis, thr ar svral basic aroachs to shll lmts modlig [1]. Th bhavior of shlls of th first ty is cosidrd as a surositio of mmbra ad bdig bhaviors. Fiit lmts ar built by siml combiig of rigidity matrics for lats bdig ad la strss stat. Such lmts hav low accuracy, sic th bdig ad mmbra bhavior ar combid at th odal oits oly. Th scod catgory icluds lmts basd o th us of a articular thory of shlls ad discrtizatio of a corrsodig variatioal formulatio. O of disadvatags of this aroach is that if th shll thory is alicabl to a scific gomtry or aalysis coditios oly, th th comositio of fiit lmts is subjct to th sam rstrictios. Th third aroach to shlls modlig is basd o th us of 3D lastic lmts. Thrdimsioal lastic lmts ar th most commo lmts, but thir us bcam o logr racticabl for subtl ad multi-layrd mmbras. Cott from this wor may b usd udr th trms of th Crativ Commos Attributio 3. licc. Ay furthr distributio of this wor must maitai attributio to th author(s) ad th titl of th wor, joural citatio ad DOI. Publishd udr licc by IOP Publishig Ltd 1

3 11th Itratioal Cofrc o "Msh mthods for boudary-valu roblms ad alicatios" IOP Publishig IOP Cof. Sris: Matrials Scic ad Egirig 158 (16) 177 doi:1.188/ x/158/1/177 At rst, th most commo ad rsarchd lmts ar th shlls cotiuum lmts ad lmts of th D thory of shlls. Shll cotiuum lmts ar built by dislayig two-dimsioal mastr-lmt to a surfac i 3 costitutig th mid-surfac of a shll. A aroximatd thrdimsioal gomtry of th shll lmt ca usually b rcovrd throught th cotiuum aroach, scifyig th uit ormal at ach od, which is itrolatd usig th stadard basic fuctios of a lmt. Th formulatio is comltd through alyig th rlvat imatic assumtios about th dislacmt fild. Although o o of thoris of shlls is usd, th rsultig formulatio ca b dfid as a shll lmt with rortis corrsodig to a shll modl with shar dformatios of th first-ordr. Uli th cotiuous shll lmts, th shll lmts basd o D thory of shlls, ar formulatd usig a xact aalytical dscritio of udformd mid-surfac. I th thory of shlls th basic imatic assumtios, masurs th strss ad strai, costitutiv quatios ar xrssd i gral curviliar coordiats,,, which ar usd to dscrib th gomtry of a thrdimsioal shll. Formulatios of lmts of th thory of shlls ad lmts basd o cotiuous mdium ar basd o th sam mchaical assumtios ad diffr i th ty of dcomositio to th lmts oly. I th thory of shlls of Kirchhoff-Lov ty ad i a 5-aramtr shlls of Rissr-Midli th thr-dimsioal costitutiv quatios ar simlifid through a assumtio of strss quality to zro i th thicss dirctio. Rtur to th thr-dimsioal cotiuous mdium is rquird if th strss ad strai i th thicss dirctio ar of grat valu. For xaml, i cas of dlamiatio of comosit. To do this, a assumtio of costat ormal strss i th thicss dirctio is addd to th thory of shlls, ad th formulatio is xtdd to 6 aramtrs. But this lads to comlx quatios of dformatio ad iability to draw th xlicit xrssio of dfiig rlatios. Th diffrc btw th distributio of strsss ad strais ovr th thicss forcd to add a liarly chagig strai mmbr, i.. xad th formulatio to 7-aramtr thory. A liar dformatio i th thicss dirctio is itroducd as a additioal variabl ad discrtizatd i th usual way via a fiit lmt rocdur. Th mai advatag of th shll 7-aramtr modl is th us of dfiig rlatios for th thr-dimsioal body with o chags []. For may roblms of mchaics of a dformabl solid th fiit-lmt rocdurs of high-ordr offr a varity of thortical ad ractical advatags as comard with fiit-lmt mthods of lowr ordr, which for th ast fw dcads bga to domiat i rsarchs ad commrcial softwar [3]. I articular, it is ossibl to avoid various forms of locig, which without adquat stabilizatio oft soil th fiit lmt modl of lowr ordr i th wa formulatio of th Galri mthod for lastic ad ilastic solids. Lots of disadvatags that may b coutrd i th fiit lmt modl basd o th ricil of ucoditioal miimizatio ca b avoidd maily or comltly through th us of adquat olyomial of -ordr wh cratig a fiit lmt aroximatio u isid ach lmt. I articular, o ca obtai ffctiv fiit lmt rocdurs which do ot rquir th us of comlicatd scial tchiqus, which ar oft rquird i th fiit lmt formulatio of th lowr ordr to imrov th accuracy of umrical solutios. As a rsult, o ca us th formula of full itgratio ad fuctioal sac of fiit-lmt of a high ordr avoids ay icosistcis culiar to a low-ordr aroximatio, which oft lad to locig.. Sctral triagular lmt Th basis of sctral lmts mthod is a itgratio schm of Gauss-Lobatto [4]. It is ot oly a mthod of umrical itgratio. Itgratio oits also dtrmi a od basis, which givs th aroximatly diagoal matrix of th masss. This mas it ossibl to us th ustady simulatios which do ot rquir th comutatio of ivrs matrix of masss, but to obtai th otimal covrgc rat. Ths rortis allow th mthod of sctral lmts to achiv a high ordr of accuracy. Howvr, this rul of itgratio dos ot aly to triagls. h

4 11th Itratioal Cofrc o "Msh mthods for boudary-valu roblms ad alicatios" IOP Publishig IOP Cof. Sris: Matrials Scic ad Egirig 158 (16) 177 doi:1.188/ x/158/1/177 O of th altrativs to th Gauss-Lobatto-Lgdr(GLL) oits for triagls ar th Ft oits [5]. Ths oits ar th solutio of xtrm roblms ad should b calculatd umrically, but i most cass th miimizatio will b a simlr tas tha th calculatio of otimal quadratur oits. Ft oits o th bordr match with th GLL oits ad hav atural xtsio to thr dimsios. Thy wr calculatd for th ttrahdro, ad for a wdg o ca us th tsor roduct of Ft oits o a triagl with GLL oits i th third dirctio. Figur 1 shows th locatio of th ods i th triagular lmt of th 5th ad 8th ordrs. Figur 1. Locatio of odal oits i sctral triagl of th 5th ad 8th ordrs Ft oits ar dtrmid i accordac with th fiit-dimsioal sac i which th itrolatig fuctio is built. For triagl a olyomial sac of multil two-dimsioal olyomials is commoly usd with a dimsio t 1, ad i ordr to build itrolatio olyomials xactly t oits ar rquird. For th slctd sac th Ft oits ar dtrmid by th Vadrmod matrix V. Lt D - is th basis for a st of olyomials of dgr, whr Th q, 1,..., t, ad, D - comot of, - is a st of oits i a right triagl with bas ad hight 1, 1. qvadrmod matrix. Th Ft oits ar th oits maximizig th Vadrmod dtrmiat of th matrix, whr th maximum is ta ovr all ossibl sts of oits i th triagl. For th slctio of th orthogoal basis i th sac of olyomials o th triagl th Dubir olyomials ar usd. To us th Ft oits for th sctral lmt mthod it is cssary to calculat th Lagrag itrolatio fuctios ad thir drivativs [4]. Lt's xrss ths itrolatio fuctios through Dubir olyomials: t, ad,. (1) i i i1 Cofficits a i rrst th i, lmts of th ivrs Vadrmod matrix. Drivativs of Lagrag itrolatio fuctios ar as follows:, t, t Di, Di, ai, ai. () i1 i1 3

5 11th Itratioal Cofrc o "Msh mthods for boudary-valu roblms ad alicatios" IOP Publishig IOP Cof. Sris: Matrials Scic ad Egirig 158 (16) 177 doi:1.188/ x/158/1/177 I ordr to calculat itgrals of som fuctio f,,, that is rquird i th mthod of sctral lmts, ovr ach whr lmt, th followig formula is usd: 1 t h t,, d,,,, d r,, r,, r f w J f w w J f 1 1 r 1 w - quadratur wight qual to th first Dubir cofficit i itrolatio olyomial of ordr; J - Jacobia associatd with th lmt maig wight of o-dimsioal Lgdr olyomial of r ordr with 1, 1. w r - quadratur GLL oit o th itrval to rfrc triagl; 1 3. Shll lmt modl By dfiitio a shll is a solid with o gomtric dimsio cosidrably smallr i rlatio to th othr two os. Lt's cosidr a covtioal shll lmt Shll lmt gomtry. Th gomtry of shll sctral lmt i th iitial momt of tim is dscribd by th formula: X, X M, h v (3) 1 1 whr X M - coordiats of a matrial oit o th mid-surfac of th shll, h - thicss, v - ormal to th mid-surfac at that oit (figur ). Dformd gomtry at tim t is dscribd as follows: t t t t, t t x xm, h v, h Q v (4) Hr t v - ormal vctor at th tim t, t h - shll thicss at th oit at th tim t. t Q - dgr of frdom corrsodig to th quadratic distributio of dislacmts i th dirctio of t v (figur 3). h, Figur. Shll lmt gomtry at th iitial momt of tim. Figur3. Quadratic fuctio of dislacmt i th od. Th dislacmt icrmt from th ositio of th lmt at th tim t to ositio at th tim t t is qual to 4

6 11th Itratioal Cofrc o "Msh mthods for boudary-valu roblms ad alicatios" IOP Publishig IOP Cof. Sris: Matrials Scic ad Egirig 158 (16) 177 doi:1.188/ x/158/1/177 u x x x x v v tt t tt t tt tt t t 1 1, M M, h h 1 tt tt t t, h Q v Q v I th dgrs of frdom at th tim t th dislacmt icrmts ca b xrssd as follows: tt t x x u u v w (6) M M M 1 3 tt t h h h a tt t Q Q q (5) (7) (8) Formula (6) dscribs th dislacmts i th global Cartsia coordiat systm, (7) - chag i thicss of th lmt, (8) - quadratic icrmt of dislacmts ovr th thicss. Icrmt of dirctig vctors is xrssd i th trms of rotary dgr of frdom,, accordig to th formula tt t vi Q v i, i1,,, (9) whr Q - rotatio tsor. si si 1 Q I S S,, S, t t Hr, - Icrmt of turs aroud th vctors v1, v, rsctivly. By substitutig (6) - (9) to th formula (5) w obtai th icrmt of dislacmts as follows u,,, u L (1) whr: t t t t t t t t t t ul um h a h h 1 h q Q Q 1 v v v v v v Th modl uss th Lagragia formulatio for small dformatios. Th covariat comots of Gr tsor of strais ar dfid as follows: t 1 t t ij gi g j gi g j. (11) I this statmt of th covariat comots formulatio, all mmbrs highr tha liar by ar discardd. 3.. Aisotroic matrials Th shlls usd i modr tchology as structural lmts ar mostly atural or structurally aisotroic. Morovr, most of aisotroic shlls ar layrd. Th widsrad distributio of such shlls is of grat itrst with rsct to th thory of aisotroic shlls. Thr ar may forms of rlatioshi btw strai ad strss for aisotroic matrials. I gral, th gralizd Hoo's law for homogous aisotroic body is as follows: 1 5

7 11th Itratioal Cofrc o "Msh mthods for boudary-valu roblms ad alicatios" IOP Publishig IOP Cof. Sris: Matrials Scic ad Egirig 158 (16) 177 doi:1.188/ x/158/1/ xx C C C C C C xx yy C C C C C C yy zz C C C C C C zz (1) xy C C C C C C xy C C C C C C xz C C C C C C xz Thr ar 1 iddt lastic costats. If th body faturs a aisotroic lastic symmtry, th quatios of gralizd Hoo's law ar simlifid. A imortat class of aisotroic bodis ar orthotroic bodis with thr las of lastic symmtry. I this cas, rlatios btw strai ad strss may b xrssd as follows: 1 xy xz Ex Ey Ez xy 1 xx Ey Ey Ez xx yy xz 1 yy zz Ez Ez E z zz. xy 1 xy G xy xz 1 xz G 1 Gxz Hr w ar gttig 9 iddt aramtrs: E x. E y. Ez - lastic moduli; xy.. xz - Poisso moduli; G xy. G. G xz - shar moduli. Layrd comosit matrials cosist of two or mor matrials faturig i sum th dsirabl rortis, which do ot occur wh usig ths matrials idividually. Lt's cosidr th layrd shll, whri ach layr is a orthotroic matrial with th ricial dirctios of lasticity ˆ, ˆ, ˆ. Wh itgratig by th thicss it is cssary to calculat th lasticity tsor C for 1 3 ach layr i th global coordiat systm. Th, i ordr to writ dow th lasticity tsor C comots for o layr i th global coordiats systm 1,, 3 it is cssary to rform th covrsio ijl C T T T T C, whr th comots T ij ar calculatd as follows: im j lq mq i T g ˆ. ij j 4. umrical xamls I all th xamls it is blivd that ach layr of th shll is mad of orthotroic matrial, ad th rlatioshi btw th comots of th lastic tsor is dfid by Tabl 1. Tabl 1. Orthotroic rortis adotd i th xamls C C C C

8 11th Itratioal Cofrc o "Msh mthods for boudary-valu roblms ad alicatios" IOP Publishig IOP Cof. Sris: Matrials Scic ad Egirig 158 (16) 177 doi:1.188/ x/158/1/ C C C C C C C C C C C C Shll udr th uiform load, ormal to th ur surfac. A simly suortd shll udr th uiformly distributd ormal static load q is cosidrd. umrical rsults rstd i Tabl ar obtaid for th homogous shll with th asct ratio ab 1. ad a thicss ha.1. Tabl. Dflctios ad strss of homogous shll / q 1111 C w hq / x q y Exact solutio Triagular sctral shll Shll udr th uiform load, ormal to th ur surfac. A simly suortd shll udr th uiformly distributd ormal static load q is cosidrd. Th shll cosists of thr layrs with idtical to ad bottom layrs. Th rsults ar rstd for ratio of layrs CT CM, whr C T - tsor of lasticity cofficit for th to layr, ad C M - th cofficit for th middl layr. umrical rsults rstd i Tabl 3 ar obtaid for th homogous shll with th asct ratio ab 1. ad a thicss ha T M Tabl 3. Dflctios ad strss of thr-layr shll C C 5 1 Exact solutio Triagular sctral shll Exact solutio Triagular sctral shll 1111 C w hq / q x Th ur surfac of th to layr Th ur surfac of th bottom layr / q y Th ur surfac of th to layr Th ur surfac of th bottom layr Coclusio Th 7-aramtr modl of triagular sctral shll lmt is dvlod. Th modl is itdd for strss aalysis of shlls udr small strais usig th mthod of sctral lmts. Th thicss 7

9 11th Itratioal Cofrc o "Msh mthods for boudary-valu roblms ad alicatios" IOP Publishig IOP Cof. Sris: Matrials Scic ad Egirig 158 (16) 177 doi:1.188/ x/158/1/177 chag of shll thicss is allowd, ad it is assumd that strais varis liarly with th thicss of a shll. Ths faturs of th modl rmit o to us this modl for th aalysis of comlx comosit structurs. Th rsults of comutatios for som roblms that ca b solvd aalytically show a good accuracy of th roosd mthod. Acowldgmts This wor was fiacially suortd by th Miistry of Educatio ad Scic of th Russia Fdratio i th framwor of th agrmt o (uiqu idtifir of th rojct RFMEFI57915X11). Ivstigatios wr carrid out withi th Fidsys comay a grat of th Miistry of Educatio ad Scic of th Russia Fdratio. Rfrcs [1] Ziiwicz OC, Taylor RL ad Zhu JZ 13Th fiit lmt mthod:its basis ad fudamtals, Svth ditio (Oxford: Elsvir) [] Kim D ad Bath K J 8 A 4-od 3D-shll lmt to modl shll surfac tractios ad icomrssibl bhaviorcom. ad Struct [3] LviV A advrshii A V 15 oliar comutatioal mchaics of strgth. Volum. umrical mthods. Paralll comutig o a comutr (Moscow: Fizmatlit) [4] Komatitsch D, Marti R, Trom J, Taylor M A ad Wigat B A 1 Wav roagatio i - D lastic mdia usig a sctral lmt mthod with triagls ad quadragls J. Comut. Acoustics [5] Taylor M A, WigatB A ad Vict R E A algorithm for comutig Ft oits i a triaglsiam J. umr. Aal [6] VallalaV P ad RddyJ 13 High-ordr sctral/h fiit lmt tchology for shlls ad flows of viscous icomrssibl fluids Math. ad Com. A [7] Sriivas S ad Rao A K 197 Bdig, vibratio ad buclig of simly suortd thic orthotroic rctagular lats ad lamiats It. J. Solids Structurs