THREE-DIMENSIONAL ELASTIC AND ELASTO-PLASTIC FRICTIONAL CONTACT ANALYSIS OF TURBOMACHINERY BLADE ATTACHMENTS

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1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 3,39,2001 THREE-DIMENSIONAL ELASTIC AND ELASTO-PLASTIC FRICTIONAL CONTACT ANALYSIS OF TURBOMACHINERY BLADE ATTACHMENTS Grzegorz Zbońsk Wesłw Ostchowcz Insttute of Flud Flow Mchnery, Polsh Acdemy of Scences e-ml: The pper presents summry of the reserch on three-dmensonl contct nlyss of turbomchnery blde ttchments n the elstc nd elsto-plstc rnge. In tht context the pper dels wth theoretcl, methodologcl nd phenomenologcl descrpton of the problem. The pper focuses on the ppled vrtonl formulton of contct problems of elstcty nd elsto-plstcty nd the correspondng fnte element methods, utlzes contct mechncs lgorthms s well s descrbes dsplcement, stress, contct nd slp sttes wthn rel turbomchnery blde ttchments. Non-lner chrcter of contct mechncs procedures nd nfluence of these nonlnertes on stress nd deformton wthn the ttchments re treted wth specl cre. Key words: fnte elements, contct mechncs, elstcty, elsto-plstcty 1. Introducton We wll strt ths pper wth presentton of three dfferent spects delt wth n the topc. Frstly we wll express some generl remrks concernng complexty of of the nlyss under consderton. Then development nd stte ofthertwllbepresented.wewllendupwthreserchssuesnduthors contrbuton to the subject Generlremrks Contct nlyss of turbomchnery blde ttchments consttutes very good exmple of how recent dvnces of contemporry mechncl nd com-

2 770 G. Zbońsk, W. Ostchowcz puter scence cn nfluence our pproch to the soluton of complex technologcl problems. The nlyss of turbomchnery blde ttchments s not new tsk for turbomchnery engneeres. The numercl tools, however, tht re t therdsposlrestllnprogress.thtswhythequltyoffnteelement (FE) nlyss of turbomchnery blde ttchments s stll beng mproved. Note tht the mn dffcultes n modelng nd nlyss of such objects rse from ther complexty,.e. hghly complcted geometrcl form s well s strong non-lnerty of the relted phenomen cused by plstcty nd frcton(physcl non-lnerty) s well s the chngng contct zone(geometrcl non-lnerty) of two prts of the ttchment,.e. blde root nd grooved dsc sector.itsclerthtnlyssofsuchproblemscnbechllngefromboth the engneerng nd scentfc ponts of vew. Defnng precsely our requrements for proper modelng nd suffcently ccurte FE nlyss of blde ttchments, we should menton the followng fctors ffectng the numercl soluton: three-dmensonl chrcter of the ttchment geometry nd lodng s well s of the resultng strn nd stress sttes, elstc or elsto-plstc propertes of the bodes n contct(blde root nd dsc sector), geometrclly non-lner chrcter of contct(generlly unknown contct zone) due to unlterl contct constrnts, frcton occurng between contctng bodes, three-dmensonl chrcter of cyclo-symetrc nd support constrnts on the outer boundres of the dsc sector of the ttchment Stte of the rt Therewere(orshouldwesy,therere)fourmnstepsnthedevelopment of pplcton of the fnte element methods to the nlyss of blde ttchments. The frst step strted n erly 1970s wth plne elstc models loded by concentrted forces nd subject to smplfed boundry nd stble frctonless contct condtons(gontrovskǐ et l., 1978; Ngn, 1976). At the sme tme the plne elstc models were enrched by ncluson of frcton (ChnndTub,1971).Thenextstepwhchbegnnerly1980sccounted for elsto-plstc plne models, frstly wthout(gontrowskǐ nd Krkch, 1982) nd lter wth frcton(bloch nd Orobnskǐ, 1983). The thrd stge strted n lte 80s nd ws cused by dvnces n computer technology(ncrese of nternl memores of the vlble mchnes) nd resulted n trnston

3 Three-dmensonl elstc nd elsto-plstc frctonl to three-dmensonl ttchment models. Agn, elstc models wthout frcton for very smple geometry(alderson et l., 1967) nd for complcted ones (Iked et l., 1989; Teper nd Moore, 1989; Robertson nd Wlton, 1990) were nlysed frst, nd then frcton ws ncluded(zbońsk, 1993d). At the present stge, three-dmensonl elsto-plstc models wthout frcton re often employed(ro et l., 2000; Megud et l., 2000). Incluson of frcton to such models hs become fct(zbońsk nd Ostchowcz, 1997,b). Two mporttnt comments concernng the pst nd contemporry FE clcultons re necessry. Frstly, there hs been gret del of work done n ths re by vrous mnufcturers, usully by mens of commercl computer codes. Much of ths work s of consderbly hgh qulty. However, mnufcturershvegenerllynotdecdedtopresenttheresultsoftherworknthe ccessble lterture. And secondly, the nowdys vlble very effcent nd generl commercl tools for technologcl FE clcultons my not fulfll the specfc needs of the user when non-stndrd, non-lner models of mterls, frcton nd contct re of nterest. For exmple, comprng frcton modelng cpbltes of NASTRAN(1995), ABAQUS(1995) nd ADINA(1999) wth the cpbltes of the code utlzed n ths pper(zbońsk nd Ostchowcz, 1997b), one cn notce tht only the ltter one ccounts for chngng frcton coeffcent Contrbuton of the uthors Contrbuton of the uthors to the subjects concernng the FE contct nlyss of turbomchnery elements cn be descrbed s follows. The generl ncrementl locl nd vrtonl formultons for elstc contct problems whch cn be ppled to dervton of the correspondng fnte element methods ws presented by Zbońsk(1993c, 1995). The specfc vrtonl prncple nd the bss of the correspondng fnte element method ppled to contct problems of turbomchnery blde ttchments cn be found n the pper by the sme uthor(1993e). The lgorthms nd computer progrms correspondng to ths method re descrbed by Zbońsk n(1993) nd(1993b), respectvely. Vrous spects of stress, strn nd contct nlyss of rel turbomchnery blde ttchments n the elstc rnge re ncluded n the followng ppers: Zbońsk(1992, 1993d, 1995b). The methods, lgorthms nd computer progrms were then extended by Zbońsk nd Ostchowcz(1997,b) to elsto-plstc contct problems. Ths pper completes the presentton of the reserch on pplcton of elsto-plstc contct nlyss to turbomchnery elements. The mn objectve here s to show some spectculr results of elsto-plstc stress nd contct sttes wthn rel turbomchnery blde

4 772 G. Zbońsk, W. Ostchowcz ttchments. Some qulttve comprsons of these results wth the cse of ttchments opertng n the elstc rnge wll lso be gven. 2. The method nd lgorthm of the nlyss 2.1. Locl nd vrtonl formulton We consder the ncrementl contct problem of two elsto-plstc or elstc bodes n common non-nertl reference frme wth globl Crtesn coordntesx.themotonnthereferencesystemsssumedtobecusedbysmll deformtons. The locl mtrx formulton of the problem conssts of the equton of moton, consttutve reltons of elsto-plstcty(or elstcty), nd knemtcreltons (X V) σ j,j + f ρ[ +(Ω Ω+E) q+( Ω Ω+ E)X]=0 σ= D( ε α Tg) (2.1) ε=γ q whch hold n volumes Vofechbody =1,2.Themtrxofelsto-plstc propertes(see Zbońsk nd Ostchowcz, 1997) nd the sx-component stress ndstrnncrementvectorsredenoted D, σ, ε,respectvely.theterm σ j sgloblvectorcomponent (j=1,2,3)ofthestressncrementvector, whle Γ stnds for the mtrx dervtve opertor trnsformng the dsplcement ncrement vector q nto the strn vector. Increments f nd re due to the body force nd trnslton ccelerton, respectvely. The known skew-symmetrc mtrces of ngulr velocty nd ccelerton ncrements re denoted Ω nd E whle ther totl vlues from the prevous ncrementl stepredenotedwthωnde.thentlncrementlstrnsduetothesclr sttonry ncrementl temperture feld T re defned by the coeffcent ofthermlexpnson αndsx-componentcrtesnmetrcvector g. The stress boundry condtons holdng on the prt Pofthebodysurfce cn be wrtten n the followng mtrx form σν= p X P (2.2) where the qunttes σ, p nd ν re the stress ncrement mtrx, surfce trcton ncrement vector nd the unt exteror norml vector, respectvely.

5 Three-dmensonl elstc nd elsto-plstc frctonl Addtonlly, on the prt Q of the body surfce the ncrementl knemtc boundry condtons nd stress reltons hold q= d σν= r X Q (2.3) Terms dnd rnthebovereltonsstndfortheknownvluesof dsplcement ncrements nd the unknown vlues of surfce stress recton vector ncrements, respectvely. The ncrementl contct mechncs equtons vld on the common contct surfce Kofthetwobodescontnvectors nnd t:untexterorvector normlto Kndthemtrxofvectorsoftwounttngentsofthelower numberedbody (=1).Theone-componentvectorofgp h n sexpressed ntermsofthentlgpvector h 0n ndnormldsplcementncrements q n ndtotlvluesq n.notethtboththelttervectorsredefneds dfferences of the proper qunttes of the frst nd second body, for exmple: q n = 1 q n 2 q n.fourterms: r n,r n, r t,r t denotencrementl nd totl(known from the prevous ncrementl step) vlues of norml nd tngentl components of the surfce recton vectors. The two-component vectorofslpncrement s t sexpressedntermsoftngentldsplcement ncrements q t defnedsdfferenceofthedsplcementsoftwobodes ncontct: q t = 1 q t 2 q t.thefuncton grepresentsncrement of the tngentl trcton bound, whle C s generl surfce consttutve mtrx of ether Coulomb or elsto-coulombn slp rule. Usng such notons thementonedequtonsofcontctmechncscnbewrttens (X K) n σν= r n t σν= r t q n +q n +h 0n =h n h n r n =0 h n r n =0 r n 0 st s t ( r t r t + g)=0 r n 0 s t = q t st s t ( r t r t +µr n )=0 rt r t + g+b 0 b= r t r t +µr n 0 st s t 0 s t =C r t (2.4) Oneshoudnotcethtthebovesetofreltonscorrespondtothecseof Coulombn slp rule for whch one cn dstngush between dheson, slp nd gp sttes. For the cse of the model bsed on elsto-coulombn slps, for whchdhesonsttedoesnotexstfourlstbutoneequtonshvetobe

6 774 G. Zbońsk, W. Ostchowcz rejected. The specfc form of the surfce consttutve lws for both mentoned cses cn be found n(zbońsk nd Ostchowcz, 1997). A vrtonl formulton correspondng wth the bove locl reltons (2.1)-(2.4) tkes the ncrementl verson of the Hmlton prncple s bss. The formulton presented here ssumes tht two lst equtons of(2.1), frst equlty(2.3) nd ll except the frst, second, sxth, thrteenth nd ffteenth reltons(2.4) hold. Thus the proper vrtonl nequlty(see Zbońsk nd Ostchowcz, 1997) cn be wrtten s 2 { {δ( ε ) D( ε α Tg) δ( q ) f =1 V δ( q ) ρ[ +(Ω Ω+E) q+( Ω Ω+ E)X]}dV δ( q ) pdp δ( q ) rdq } (2.5) P Q [δ( s t s t ) g+δ( q n+q +h 0n h n) r n + K +δ( q t s t) r t ]dk 0 The nequlty form of the bove prncple s cused by unlterl norml contct condtons nd exstence of tngentl trcton bound n the cse of Coulombn slps. For the cse of elsto-coulombn slps, one should replce n(2.5)theterm δ( s t s t ) gccountngforetherdhesonortheslp stteswth δ( s t) r t whchcorrespondstotheslpstteonly.aftersuch chnge, the nequlty form of the prncple s stll preserved due to unlterl norml contct constrnts. InordertoderveonegenerlmethodofsolutonoftheCoulombnnd elsto-coulombn problems under consderton we cn propose the ntroducton of the proper reduced contct problem t ech ncrementl step, s one of the possbltes. The suggested method s bsed on the ssumpton tht the potentl contct re whch s dvded ether nto dheson, slp or gp (Coulombn model) or slp nd gp(elsto-coulombn one) prts, does not chnge wthn the ncrementl or tertve step of the soluton procedure. Note thtthethreeprtsofthepotentlcontctreredenotedbyletters A, S nd G, respectvely. The mentoned dvsons need employng the proper crter bsed on nequltes from the set(2.4). In prtculr, we use the sxth nd eghth reltons to choose between the gp nd contct sttes, whle the twelvth nd fourteenth reltons re used to dstngush between the dheson

7 Three-dmensonl elstc nd elsto-plstc frctonl nd slp sttes. Addtonlly, n our formulton we hve to tke nto ccount tht:theworkoftngentltrctonssdoneonlyontheprt S,forprts And Stherelton h n =0holds,whereson Awehve s t =0. Moreover, we ssume tht the tngentl trcton bound ncrement nd slp ncrements re known n ech tertve step of the soluton process. We denote theknownvluesofthesequnttesby g µ p n nd s t z t.use ofthebovenotons,expressonoftheslpncrements δ( s t )throughthe tngentldsplcementncrements δ( q t ),ndchngeofsurfcedegreesof freedom(dofs) q n, q t ntoglobldofs qn X-drectonsbymens ofreltons: q n =n q, q t =t qllowstowrtethepropersttonrty prncple correspondng to Coulomb slps n the followng form 2 { {δ( ε ) D( ε α Tg) δ( q ) f =1 V δ( q ) ρ[ +(Ω Ω+E) q+( Ω Ω+ E)X]}dV δ( q ) pdp δ( q ) rdq (2.6) P ( 1) +1[ S + A Q δ( q ) t z tµ p n zt z t ds+ ]} δ( q ) rda =0 whle for the elsto-coulombn cse we cn wrte S δ( q )n r n ds+ 2 { {δ( ε ) D( ε α Tg) δ( q ) f =1 V δ( q ) ρ[ +(Ω Ω+E) q+( Ω Ω+ E)X]}dV δ( q ) pdp δ( q ) rdq (2.7) P ( 1) +1[ S S Q δ( q )t t t ds+ δ( q )tk t t qds=0 S ]} δ( q )n r n ds

8 776 G. Zbońsk, W. Ostchowcz It s worth notcng tht we replced dfferences q wth ncrements q ofechoftwobodesncontctn(2.5)nd(2.6),whentwsposssble.note lsothtnthelstrelton, k t nd t t representthesurfceconsttutve mtrx correspondng to slp ncrements nd tngentl trcton ncrement vector due to norml trcton ncrement(compre Zbońsk nd Ostchowcz, 1997). We should underlne tht two bove reltons consttute strtng pont for ntroducton of one generl lgorthm for two fnte element methods of contct problems of elstoplstcty. Elucdton of these methods cn be found n the pper cted bove Some detls of the lgorthm A full lgorthm for the FE soluton of the problems under consderton conssts of generton of the element stffness mtrces nd force ncrements vectors, structure stffness nd force ncrement formton, soluton of the set of nonlner lgebrc equtons nd frctonl contct lgorthms. The former two prts of the procedure hve the property tht they correspond wth the stndrd FE lgorthms for non-lner elstoplstc problems, whle the ltter two nclude the non-stndrd, non-lner spects of the contct nlyss.thesmplfedflowdgrmofthelgorthmspresentednfg.1.note tht two dfferent procedures correspondng to Coulomb(C-model) nd elsto- Coulombn(EC-model) models re combned n one generl lgorthm wth the proposed pproch. Ths lgorthm s utlzed n the computer progrm PLAST presented n(zbońsk nd Ostchowcz, 1997b) whch s ppled to the reserch conducted n ths pper. Detls of the bove lgorthm were presented n the prevous work by the uthors(zbońsk nd Ostchowcz, 1997). Here we restrct ourselves to some orgnl spects concernng the clculton of frcton stffnesses nd forces whch both reflect the physcl non-lnerty of the contct. The frst mportnt feture of the proposed pproch s ntroducton of ll frcton terms on the globl level,.e. through nodl surfce stffness nd nodl frcton forces of the structure nsted of defnng them on the element level. We should stress tht the globl nodl descrpton of frcton forces hs been succesfully ppled to problems wth Coulombn models ccountng only for non-reversble (Coulombn slps) nd neglectng elstc(reversble) slps. On the contrry, ntroducton of the frcton terms to elsto-coulombn models hs been performed on the element level through element vectors of frctonl stresses nd element surfce stffnesses, so fr. The reson for ths s dependence of the elstc prt of the surfce stffness on dscretzton of the contct re. Hence, nordertontroducethefrctontermsonthegloblnodllevel,wehveto

9 Three-dmensonl elstc nd elsto-plstc frctonl Fg. 1. Generl lgorthm for frctonl elstc nd elsto-plstc contct nlyss

10 778 G. Zbońsk, W. Ostchowcz overcome the problem of dscretzton, or should we sy, of ssgnment of the the contct re to globl(structure) contct node. The proper method s gven below. The second feture of the proposed pproch dels wth the soluton method of the contct problems, whch ssumes reduced problem t ech ncrementl nd tertve step of the soluton process. Such n ssumpton leds to lnerzton of the FE element equlbrum equtons. In prctce t mens tht thegloblsurfcestffness K t (ncludedonlyforthecseofelsto-coulombn frcton)defnedfornyprofnodes Ind Jofthefrstndsecondbody, respectvely,swellsthegloblncrementlfrctonforces T t tnode I, whch both depend on the unknown dsplcement ncrements, re prescrbed tnystep ofthesolutonprocedure.thessumedtobeknownvlues ofthenodlvectorsofthenormltrctonncrements Pn,totlsurfce trctons P=col[ P t, Pn]ndslpncrements Z t reusedforthepurpose of frcton terms determnton nd re ssumed to be equl to the proper resultntvluesfromtheprevousstep 1,.e. Pn= 1 Rn Pn= 1 Rn P t = 1 T t Z t = 1 S t (2.8) For the elsto-coulombn model of frcton, contrbuton of the surfce consttutvestffnessoftheprofnodes Ind Jtothegloblequlbrum equton s δ( q t) K t q t = [δ( q 1 t),δ( q 2 ] t) whereechmtrxblock K t sgvenby K t = Kt 11 Kt 12 Kt 21 Kt 22 K t K t K 1 q t (2.9) t Kt q 2 t (2.10) nd terms Kt kl (k,l,m=1,2)redefnedsfollows ( Kt kl = ka I δ kl β ka I P Pt k Pt l Pt m Pt m ) (2.11)

11 Three-dmensonl elstc nd elsto-plstc frctonl wth δ kl bengthekroneckersymbol,whlecontrbutonofthencrements offrctonforcestnode Itothesmegloblequlbrumequtonscnbe defned by δ( q 1 t ) T t = [δ( q 1 t1 ),δ( q 1 t2 ) wthterms Tt k gvenbythefollowngformul Tt k = β ka I P Pt k ] Pt m Pt m µ Tt 1 Tt 2 (2.12) Pn (2.13) Inthebovereltonsβ=0defnesthecseofelstcslpswheresforβ=1 the slps re elsto-coulombn. Constnt k chrcterzes the surfce elstc properteswhletheuxlryterm P cn be clculted ccordng to P=kA I + µ St c m St c m Zt m Pt m Ztm Ztm Pt m Pt m Pn (2.14) Notethttheterm µ/ St c m St c m representsknownreltonbetween the frcton coeffcent nd the Coulombn prt of slp. Ths relton correspondstoelsto-coulombnslprulewthhrdenng.theterm A I stnds forthecontctressgnedfornodeior J(A I A J )ndcnbeobtned throughsummtonofcontctres ei ssgnedtonodesoftheelements e I meetngtnode I A I = E I e I =1 ei (2.15) where E I stndsfortotlnumberofelementsformngthecontctreunder consderton. The elementl prt of the contct re cn be defned s follows ei = N I N I da e (2.16) A e wth A e stndngfortheelementfcewthncontctsurfcend N I beng the shpe functon ssgned to element node I. Subsequently, for the Coulombn model of frcton we hve to ntroduce only the vector of ncrements of frcton forces updted wthn ech step of the

12 780 G. Zbońsk, W. Ostchowcz soluton procedure. The components of ths vector re Tt k = k µ Pt Pt m Pt m Pn (2.17) Note tht for del Coulombn slp rule lso nother equvlent representton s possble Tt k = Zt k Ztm Ztmµ Pn (2.18) Note tht when necessry(chnge from dheson to slp), n order to stblzendthustospeedupthetertonprocessforthelttercseofcoulombn model, one cn use the relxton method whch ssumes wth α stndng for the relxton fctor. Tt k =(1 α) 1 Tt k +α Tt k (2.19) 3. Elstc nd elsto-plstc contct nlyss of rel blde ttchments 3.1. Appled FE models of rel ttchments The ppled exmples of rel turbomchnery blde ttchments(see e.g. Fg.2)sutlzednthsreserchrethesmesnourprevousttempts (Zbońsk, 1993d, 1995b; Zbońsk nd Ostchowcz, 1997b). The correspondngfemodelsofsngleblderfolndrootswellsdscsectorwth sngle groove consst of ether bout 40 thousnd DOFs nd fve thousnd soprmetrc sold elements or bout therts thousnd DOFs nd four thousnd elements. These two cses correspond to the ttchments opertng wthn elstc or elsto-plstc rnges, respectvely. In both models the frst-order nd second-order sold elements s well s mxed-order trnston elements(from frst- to second-order sold elements) re employed. The second-order elements re ppled n the most-stressed regons of the ttchment(mnmum neck sectonsoftherootnddscswellscontctsurfcesofhooks).forother less-stressed prts of the ttchment, frst-order elements re ppled. One lyer of the trnston elements s utlzed n order to jon the frst-order nd

13 Three-dmensonl elstc nd elsto-plstc frctonl Fg.2.Exemplrymeshoftheblde,rootnddscsectorofthettchment

14 782 G. Zbońsk, W. Ostchowcz second-order regons together. The elements re put nto 11 lyers prllel to thettchmentdscmd-plne.thefemeshdetlsrethoseshownnfg.3 for the ttchment blde root nd dsc sector. Fg. 3. Mesh detls wthn cross-secton of sngle contct surfce 3.2. Elstc nlyss of frctonl contct problem The frst exmple dels wth the ncrementl elstc frctonl contct nlyss of n xl entry, curved, fr-tree ttchment of the blde, wth four prs ofrootndgroovehooksonechsdeofthettchment.thettchmentcurvturerduss0.125m,shftofthecurvturecenterfromthedscmddle plnes0.025m.thettchmentslodedbytheblde0.39mlongofthe frststgeofthelowpressureprtofthe200mwstemturbne.theblderottngtthengulrveloctyof /s(3000rpm)produces thecentrfuglforceequlto knctngontherootpltform0.0078m thck.theouterrdusofthedscs0.519m,thennerones0.250m,thedsc (nd ttchment) thckness s m, whle the number of the blde ttchmentsonthedsccrcumferences82.young smodulusoftherootndblde mterlequls N/m 2,wheresthemodulusofthedscmterl s N/m 2.ThemterldenstesndPosson srtosressumed tobeequltothevluesof kg/m 3 nd0.3,respectvely.fourcses

15 Three-dmensonl elstc nd elsto-plstc frctonl correspondng to geometrclly nonlner contct problem(unknown geometry of contct) re ncluded. The frst cse dels wth the physclly lner (frctonless) contct problem, whle the rest three cses refer to physclly non-lner(frctonl) contct problems. The constnt nd unform Coulomb frcton coeffcent µ for the ltter three cses equls 0.05, 0.10 nd 0.15, respectvely. An exmple of effectve stress dstrbuton wthn the ttchment for the cse1sshownnfg.4.itcorrespondstothemddlecross-sectonofthettchment. Effectve stress solnes from 1 to 8 correspond to the vlues chngng proportonllyfrom to N/m 2.Notethtfortherest three cses stress dstrbuton s qulttvely the sme. For ll four cses the mxml norml contct stresses wthn the mxmlly loded contct surfces sfunctonoffrctoncoeffcentrepresentedbelowntble1.forcses1 nd 4, ddtonlly, the ntl nd resultnt contct sttuses of contct nodes wthn mxmlly stressed contct surfces re shown n Tble 2. Note tht the equvlent ntl nd resultnt sttuses correspondng to dheson, slp nd gp re denoted wth symbols: A, S nd G, respectvely, whle trnston fromdhesontoslp,fromdhesontogpndfromslptogpredenoted s: A/S, A/G, S/G, respectvely. Tble 1. Mxml norml contct stresses for the elstc cses 1-4 Normlcontctstresses[10 8 N/m 2 ] Cse Locton(lowest left hooks) Locton(hghest rght hooks) Leftedge Mddle Rghtedge Leftedge Mddle Rghtedge Tble2.Mpofntlndresultntcontctsttusforelstccses1 nd4 Node Intl nd resultnt sttus of contct nodes Cse loc- Node lyer(lowest left hooks) Node lyer(hghest rght hooks) ton Left S S S S S S S/G S/G S/G S S S 1 Mddle S S S S S S S S S S S S Rght S/G S S S S S S S S S S S Left A/S A/S A/S A/S A/S A/S A/G A/G A/G A/S A/S A/S 4 Mddle A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S Rght A/G A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S

16 784 G. Zbońsk, W. Ostchowcz Fg. 4. Elstc effectve stress n the cross-secton of the ttchment

17 Three-dmensonl elstc nd elsto-plstc frctonl It s worth notng tht the stress dstrbuton wthn the ttchment s typcl. It mens tht hgh vlues of effectve stresses pper t the mnmum necksectonsorthedjcentcontctsurfcesoftherootnddsc,ndthe mxml vlues correspond to the frst left root nd fourth rght dsc hooks. Itsnterestngtonotcethtnclusonoffrcton (µ=0ncomprson to µ=0.05)chngesthevluesofcontctstressesmuchmorethnfurther ncreseoffrctoncoeffcent (µ=0.10nd µ=0.15).notelsotht ncluson of frcton dmnshes the contct stresses nd does not ffect the tendency to gp ppernce Elsto-plstc nlyss of frctonl contct problem The second exmple dels wth the elsto-plstc frctonless contct nlyss of n xl entry, curved, fr-tree ttchment of the blde, wth three prsofrootndgroovehooksonechsdeofthettchment.thettchmentcurvturerduss0.185m,theshftofthecurvturecenterfromthe dscmddleplnes0.024m.thettchmentslodedbytheblde0.71m longofthepenultmtestgeofthelowpressureprtofstemturbne.the blderottngtthengulrveloctyof /s(3000rpm)produces thecentrfuglforceequlto knctngontherootpltform0.0075m thck.theouterrdusofthedscs0.705m,thennerones0.250m,thedsc (nd ttchment) thckness s m, whle the number of the blde ttchmentsonthedsccrcumferences53.young smodulusoftherootndblde mterlequls N/m 2,wheresthemodulusofthedscmterl s N/m 2.ThemterldenstesndPosson srtosressumed tobeequlto kg/m 3 nd0.3,respectvely.theyeldpontsof theblderootnddscmterlsre N/m 2 nd N/m 2, respectvely.thecommonvlueoftheplstcmoduluss N/m 2. Agn four cses re nlysed, the frst one of whch corresponds to geometrclly non-lner contct problem wth µ = 0, whle the next three cses to geometrclly nd physclly nonlner contct wth µ equl to: 0.05, 0.10 nd An exmple of effectve stress dstrbuton wthn the ttchment for the csesshownnfg.5.itcorrespondstothemddlecross-sectonofthettchment. Effectve stress solnes from 1 to 5 correspond wth vlues chngngproportonllyfrom to N/m 2.Thelstsolne (denoted wth number 6) corresponds to the vlue of effectve stress equl to N/m 2.Hencetheregonsofelsto-plstcdeformtoncnbeesly recognzed s those neghborng the lst solne. As before, for ll four cses the mxml norml contct stresses wthn the mxmlly loded contct sur-

18 786 G. Zbońsk, W. Ostchowcz Fg. 5. Elsto-plstc effectve stress n the cross-secton of the ttchment

19 Three-dmensonl elstc nd elsto-plstc frctonl fces s functon of frcton coeffcent re presented(see Tble 3 below). For cses1nd4lsothentlndresultntcontctsttusesofcontctnodes wthn mxmlly stressed contct surfces re shown n Tble 4. Tble 3. Mxml norml contct stresses for the elsto-plstc cses 1-4 Normlcontctstresses[10 8 N/m 2 ] Cse Locton(lowest rght hooks) Locton(hghest left hooks) Rghtedge Mddle Leftedge Rghtedge Mddle Leftedge Tble 4. Mp of ntl nd resultnt contct sttus for elsto-plstc cses1nd4 Node Intl nd resultnt sttus of contct nodes Cse loc- Node lyer(lowest rght hooks) Node lyer(hghest left hooks) ton Rght S S S S S S S/G S S S S S/G 1 Mddle S S S S S S S S S S S S Left S S S S S S S S S S S S Rght A/S A/S A/S A/S A/S A/S A/G A/S A/S A/S A/S A/G 4 Mddle A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S Left A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S The stress dstrbuton wthn the ttchment s typcl, tht s hgh vlues of effectve stresses pper t the mnmum neck sectons or the djcent contct surfces of the root nd dsc, nd the mxml vlues correspond to the frst root nd thrd dsc hooks. Note, however, tht the elsto-plstc deformtons occur only for the ltter hooks. It s nterestng to notce gn tht nclusonoffrcton (µ=0ncomprsonto µ=0.05)chngesthevlues of contct stresses much more thn further ncrese of the frcton coeffcent (µ=0.10nd µ=0.15).agn,nclusonoffrctondmnshesthecontct stresses nd does not ffect the tendency to gp ppernce. Comprng the results of the vlues of the norml contct stresses from Tbles1nd3onecnnotcethtnclusonoffrctonchngesthesevlues more sgnfcntly n the elstc cse. Also, tendency to gp ppernce s stronger for tht cse(compre Tbles 2 nd 4). These two observtons cn be trbutted to smller effectve stresses wthn the ttchment opertng n the elstc rnge.

20 788 G. Zbońsk, W. Ostchowcz 4. Conclusons It s possble to generte one generl FE lgorthm conformng both to the Coulombn nd elsto-coulombn slp rules for ncrementl contct problems of elstcty nd elsto-plstcty s ppled to three-dmensonl turbomchnery blde ttchment nlyss. Physcl non-lnerty of contct problems due to frcton plys n mportnt role n the stress nlyss both n the elstc nd elsto-plstc rnge. In the cse of the ttchments under consderton, ncluson of frcton chnges theeffectvestressestthehooknotchesndtendencytoreducethemprevls. Also tendency to chnge the dstrbuton nd to reduce the norml contct stresses ppers. Onthecontrry,nclusonoffrctondoesnotseemtoffectthegpppernce wthn contct surfces for both the cses of elstc nd elsto-plstc nlyss. In the cse of hghly stressed ttchments(whch my nclude lso elstoplstc deformton), chnges of the norml contct stresses due to ncluson of frcton re weker thn n the cse of less-stressed ttchments. A tendency to gp ppernce s weker for hghly stressed ttchments. References 1. Alderson R.G., Tn M.A., Tree D.J., 1967, Three-Dmensonl Optmzton of Gs Turbne Dsk nd Blde Attchment, J. Arcrft, 13, Bloch M.V., Orobnskǐ A.V., 1983, O Modfkc Metod Konechnykh Elementov dlj Resheny Dvumernykh Uprugkh Plstcheskkh Kontktnykh Zdch, Problemy Prochnost, 5, Chn S.K., Tub I.S., 1971, A Fnte Element Method for Contct Problems of Sold Bodes. Prt II Applcton to Turbne Blde Fstenngs, Int. J. Mech. Scences, 13, Gontrovskǐ P.P., Krkč B.N., 1982, Issledovne Npryzhenno- Deformrovnnogo Sostoyny Zmkovykh Soednenǐ Loptok Turbomshn Metodom Konechnykh Elementov, Problemy Prochnost, 8, Gontrovskǐ P.P., Krkč B.N., Mrcenko G.A., 1978, Prblzhennyǐ Metod Rschet Zmkovykh Soednenǐ Loptok Turbomshn, Problemy Mshnostroeny, 6, 52-55

21 Three-dmensonl elstc nd elsto-plstc frctonl IkedT.,HsS.,MtsuurT.,1989,TheDevelopmentofTtnum Lst-Stge Blde for 3600 rpm Stem Turbnes, Ltest Advnces n Stem Turbne Desgn, Bldng, Reprs, Condton Assessment, nd Condenser Intercton, PWR-7, 69-76, ASME, New York 7. Megud S.A., Knth P.S., Czeknsk A., 2000, Fnte Element Anlyss of Fr-Tree Regon n Turbne Dscs, Fnte Elements n Anlyss nd Desgn, 35, Ngn A.A., 1976, Rshchët Elochnogo Zmk, Izv. Vysshykh Uceb. Zvedenǐ. Mshnostroene, 2, Ro J.S., Rmkrshnn C.V., Gupt K., Sngh A.K., 2000, Elsto- Plstc Stress Anlyss of LP Stem Turbne Bldes under Centrfugl Lodng, J. Turbomchnery(to pper) 10. Robertson M.D., Wlton D., 1990, Desgn Anlyss of Stem Turbne Blde Roots under Centrfugl Lodng, Journl of Strn Anlyss for Engneerng Desgn, 25, TeperB.,MooreC.T.,1989,TheRedesgnofFrTreeBldeRootforthe Penultmte Stge of 500 MW Turbne, Ltest Advnces n Stem Turbne Desgn, Bldng, Reprs, Condton Assesment, nd Condenser Intercton, PWR-7, 17-22, ASME, New York 12. Zbońsk G., 1992, Numercl Contct Anlyss of Turbomchnery Blde Attchments, Archves of Mechncl Engneerng, 32, Zbońsk G., 1993, FE Algorthm for Incrementl Anlyss of Lrge 3D Frctonl Contct Problems of Lner Elstcty, Computers nd Structures, 46, Zbońsk G., 1993b, FE Computer Progrm for Incrementl Anlyss of Lrge 3D Frctonl Contct Problems of Lner Elstcty, Computers nd Structures, 46, Zbońsk G., 1993c, Incrementl Vrtonl Prncples for Frctonl Contct Problems of Lner Elstcty, ASME J. Appl. Mech., 60, Zbońsk G., 1993d, Numercl Reserch on 3D Contct Problems of Turbomchnery Blde Attchments n the Elstc Rnge, Int. Journl of Mechncl Scences, 35(1993), Zbońsk G., 1993e, The Incrementl Vrtonl Prncple nd Fnte Element Dsplcement Approxmton for Frctonl Contct Problem of Lner Elstcty, Journl of Non-Lner Mechncs, 28, Zbońsk G., 1995, Dervton of the Vrtonl Inequltes of the Incrementl Frctonl Elstc Contct Problems, Archves of Mechncs, 47,

22 790 G. Zbońsk, W. Ostchowcz 19. Zbońsk G., 1995b, Physcl nd Geometrcl Non-Lnertes n Contct Problems of Elstc Turbne Blde Attchments, Proc. of the Instn. of Mechncl Engneers. Prt C: Journl of Mechncl Engneerng Scences, 209, Zbońsk G., Ostchowcz W., 1997, A Generl FE Algorthm for 3D Incrementl Anlyss of Frctonl Contct Problems of Elsto-Plstcty, Fnte Elements n Anlyss nd Desgn, 27, Zbońsk G., Ostchowcz W., 1997b, A Generl FE Computer Progrm for 3D Incrementl Anlyss of Frctonl Contct Problems of Elsto-Plstcty, Fnte Elements n Anlyss nd Desgn, 27, , ABAQUS. Theory Mnul, Verson 5.5, Hbbt, Krlsson nd Sorensen, Inc., Pwtucket, RI(USA) , MSC/NASTRAN. Relese Notes for Verson 68, The McNel- Schwendler Corporton, Los Angeles, CA(USA) , Automtc Dynmc Incrementl Nonlner Anlyss. Theory nd ModelngGude.VolumeI:ADINA,ReportARD99-7,ADINAR&D,Inc., Wtertown, MA(USA) Trójwymrow nlz kontktow z trcem zmocowń łoptek mszyn wrnkowych w zkrese sprężystym sprężysto-plstycznym Streszczene Nnejsz prc przedstw podsumowne wynków bdń nd trójwymrową nlzą kontktu w zmocownch łoptek mszyn wrnkowych w zkrese sprężystym sprężystoplstycznym. W tym kontekśce zprezentowne zostły stotne spekty nlzy zwązne z teoretycznym, metodologcznym fenomenologcznym opsem problemu. Prc koncentruje sę n zstosownych sformułownch wrcyjnych problemów kontktowych sprężystośc sprężysto-plstycznośc, odpowdjących m metodch elementów skończonych, zproponownych lgorytmch kontktowych, tkże opsch stnów przemeszczeń, nprężeń, kontktu poślzgów w rzeczywstych zmocownch łoptek turbnowych. Ze szczególną uwgą potrktowno problemy nelnowego chrkteru lgorytmów kontktowych orz wpływu tych nelnowośc n nprężen odksztłcen w zmocownu. Mnuscrpt receved December 11, 2000; ccepted for prnt Februry 20, 2001