Where is a slope tangential to the section. Herewith, the semi-axes of the. contour, a 3b.

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1 ISSN: 77-7 ISO 9: Cerified Inernionl Jornl of Engineering nd Innovive Tehnolog (IJEIT) Volme Ie 6 Fige Filre of Ovl Cro Seion Primi Br he Pling Torion Lif Kh Tll Nigr Ngiev Aerijn Nionl Adem of Siene Inie of hemi nd ehni B Vgde S 9 Bk Aerijn AZ Ar The nmer of pling orion whih he fir dmge nd fige filre of ovl ro eion primi r hppen re deermined The r' meril h no renghening nd he pli re enirel over he eion' onor A he iniil eli-pli orion of he r from he nrl e VV Sokolovk' olion [] w ed Ineni of reidl deformion i eped for deermining fige filre prmeer [] Ke word: ovl eion r pling orion dmge fige filre I INTRODUCTION Here nder fige or li drili i i ommonl nderood dioniniie of he meril of onrion nder oniderion nder li hnge of pli deformion [] A ome eperimenl inveigion [] rried o infrred peroop nd oi emiion i i noed he f h fir fige dmge of rrl elemen pper fer erin nmer of le fer he r of li deformion proe The nmer of le of loding preeding o formion of fir dmge nd o filre re ommenrle vle Here ing he r of li dmge ondiion nd li rengh oined in [] prolem of fige filre of n ovl ro eion primi r pling orion i olved Herewih he renghening of he r meril i elded (idell eli-pli deformle meril i onidered) nd i i med h he pli re enirel over he r onor he known olion of VV Sokolovk [] i ed In he nloding proe pperne of he re of eondr pli deformion i llowed [] The reidl re nd rin re fond ing he VV okviin heorem on eondr pli deformion [] II STATEENT AND SOLUTION OF VV SOKOLOVSKY PROBLE ON INITIAL TORSION OF A BAR Le onider n ovl ro eion (lmo ellipi) primi r whoe onor i deermined he following eqion in he prmeri form [] in in o o o o Where i lope ngenil o he eion onor Herewih he emi-e of he ovl (ellipe) will e: A qrer of he onidered ovl ro eion i given in Fig Le he onidered r in he nrl e e jeed o he orqe Under he ion of he orqe in he r here pper he ree rin nd diplemen em The oordine whoe origin oinide wih he ener of he ovl he i wih he i of he r he i wih gre mll emi- i he i wih mll emi-i i ed The prolem i in deerminion of re rin nd diplemen field in eli nd pli domin nd lo in elihmen of ondrie eween hee domin The given prolem he prolem of iniil elipli orion of r nder ppoiion h he r meril h no renghening nd he pli re enirel over he ro eion w olved VVSokolovk he peifi o lled he invere mehod I i med h he hpohei of plne ro eion hold Therewih he diplemen m e repreened in he form f () Where i he wi ngle of ni lengh r re f i orion fnion hreriing he wrping Relive orion ngle i onidered o e poiive for definiene Sine he ree nd rin hen he eqilirim eqliie nd ondiion of rin ompiili ep he form : 76

2 ISSN: 77-7 ISO 9: Cerified Inernionl Jornl of Engineering nd Innovive Tehnolog (IJEIT) Volme Ie 6 () () The Ch kinemi relion hold e flfilled: () Sine he lerl rfe of he r i re free on he ro eion onor he ngenil re veor hold e direed long he ngen o he onor: d g d The prinipl momenm of ngenil ree ing on he r ro eion eql he orqe : F df () Here F i he re of r ro eion In he eli domin he Hook' lw hold: (6) where i her modl In he pli domin he von ie ield rierion i flfilled: / (7) where i he ield poin in her epreed he ield poin in enion he relion: / Frhermore he onini ondiion of ngenil re nd il diplemen omponen when ping hrogh he ondr of eli nd pli domin i flfilled Aording o [] he olion of he eli prolem h he form: () or (9) f () () () The diplemen re deermined from forml () king ino on () nd () In onformi o he onidered prolem he ineni of he rin forml: / () i epreed he B ing () in forml () he ineni of eli deformion will e: () / Le for on he ondr of he ro eion onor fir he pli deformion pper I i ler h he wi fir will ep he vle he poin A hi poin he ield ondiion (7) h he form Allowing for he fir forml of () we ge: () The orqe whih pli deformion pper ed on relion (9) nd() re epreed he forml: (6) For in he ro eion of he r here rie pli deformion re In he e when he pli re enirel over he ro eion i hold he VV Sokolovk olion [] Aording 77

3 ISSN: 77-7 ISO 9: Cerified Inernionl Jornl of Engineering nd Innovive Tehnolog (IJEIT) Volme Ie 6 7 o VVSokolovk' invere mehod he elipli ondr i given in he form of n ellipe d Herewih for he emi-i nd d he following forml re oined: d ( 7) Where he wi nd he orqe re onneed wih he forml: 9 / () In he eli re (inide he ellipe) he diplemen i repreened he forml: / (9) in he pli re he forml: in o / () In he eli re he deformion nd re epreed he following forml: / () / () in he pli re we hve o () The ree nd hve he following epreion in he eli re / () / () in he pli re in o (6) Herewih long ll he ro eion i hold he eqion: in o in Noe h he olion of (9) - (6) w oined VVSokolovk [] nd i re provided or / III RESIDUAL STRESSES AND STRAINS OF THE OVAL BAR Define now reidl ree nd rin h re preerved in he ovl r fer removing he orqe And we will onider h in he nloding proe in he r here rie he re of eondr pli deformion To hi end we will e he VVokviin heorem on eondr pli deformion [] Aording o hi heorem he reidl ree rin nd diplemen of he wi m e deermined he forml: (7)

4 Here ISSN: 77-7 ISO 9: Cerified Inernionl Jornl of Engineering nd Innovive Tehnolog (IJEIT) Volme Ie 6 re he ree rin diplemenwi repeivel efore he eginning of nloding h re deermined he forml () () () () ()(9) () () The qniie re he ree rin diplemen wi h hold in fiiio ovl r eli-pli orion he me orqe Unlike he onidered r he ield poin he her of meril of he fiiio r i Beween he qniie nd i : hold he relion () hnging 9 6 / () Herewih he eli-pli ondr in he ro eion of he fiiio r will e n ellipe wih he emi-e nd d : d (9) In he eli domin he ro eion of he fiiio r will e: () / () / ) / ( / () In he pli domin of he ro eion of he fiiio r we hve: o in () o ( ) In eli nd pli domin he epreion will e wrien imilrl for nd We define he reidl wi forml he Herewih he qni i fond from he eqion () he qni from () The ellipe epring he re of eli nloding nd eondr pli deformion re deermined he emi-e nd d oiniding wih he emi- e nd d repeivel d (6) Herewih he following ondiion hold e flfilled: d (7) B dereing he orqe from o here will hppen eli nloding B dereing he orqe from o ero he re of eondr pli deformion eend from he poin o he ener of he ro eion of he onidered r Herewih he eernl onor of he re of eondr pli deformion will he pr of he onor of he ovl ro eion wih he emi-e he inernl onor will e he pr of he ellipe wih he emi-e nd d defined he relion (6) (fig ) Aording o he heorem on eondr pli deformion [] nd forml (6)he eond 79

5 ISSN: 77-7 ISO 9: Cerified Inernionl Jornl of Engineering nd Innovive Tehnolog (IJEIT) Volme Ie 6 pli deformion in he nloding proe will pper in he e if Herewih he ondiion of pperne of eondr pli deformion ol nloding will e: () I i ler h je o ondiion () he ondiion (7) will e flfilled A i vle of he orqe he pli re iniil orion fill ll he ro eion nd eli kernel degenere ino ome eion whoe lengh ording o forml (7) will e For he qni ording o forml () we hve: 9 (9) Herewih he vle of he orqe hold e le hn of Sje o (9) nd ineqli () he ineqli flfilled in he e if he ovl ro eion prmeernd if he ondiion: () Condiion () hold je o he ineqli: 6 / Тh in he r mde of idell pli meril wih ro eion in he form of n ellipe pperne of eondr pli deformion i poile onl he e if he rio of i gre emi-i o he mll one doen' eeed 6 Sje o hi ondiion he eond pli deformion nloding will neeril rie if he vle of orqe will if ondiion () Clle now he reidl ree nd rin Herewih we hold er in mind he eiene of heir differen re in he ro eion (fig) Are of eondr pli deformion Aording o forml (6) nd () nd forml () nd ()in hi re we hve: in o () o () в The re eween he ellipe wih emi-e d nd d In hi re ording o forml (6) nd () () nd lo forml () nd () () we hve: o / () in / () o / () / (6) сthe re inide he ellipe wih emi-e nd d In hi re ording o forml () () nd () () nd lo forml () () nd () () we hve: / / (7) / / ()

6 ISSN: 77-7 ISO 9: Cerified Inernionl Jornl of Engineering nd Innovive Tehnolog (IJEIT) Volme Ie 6 / / (9) / / () In he onidered re he forml of reidl replemen m e wrien in he imilr w Forml () () re vlid in he e when ondiion () nd () re flfilledie if ol nloding here pper eondr pli deformion B if one of he ondiion () nd ()i no flfilled hen removing he orqe here pper eli nloding Herewih he reidl ree nd rin m e lled AA Ilhin eli nloding heorem [] Aording o hi heorem he reidl ogh-for vle re deermined forml (7) Herewih he qniie re deermined from () - (6) () () () (9) () () B he qniie re he ree rin diplemen nd wi eiing in ome fiiior of ovl ro eion i eli orion he orqe h w pplied o he onidered r efore nloding Aording o whp h een id nd on he e of forml (9) nd () we wrie epreion for nd : () () () The reidl wi will e: where nd re deermined hrogh he orqe forml () nd () The reidl ree nd rin in he oide he ellipe wih he emi-e nd d ording o formle (6) () nd lo forml () () will e: in o o In he re inide he ellipe wih he emi-e nd d ording o forml () () () nd lo forml () () () we hve: / / / / IV FATIUE FAILURE OF OVAL BAR AT PULSATIN TORSION Now le onider fige filre of he onidered r nder pling orion We will e li dmge nd he li drili ondiion Aording o [] he li dmge ondiion h he following form:

7 ISSN: 77-7 ISO 9: Cerified Inernionl Jornl of Engineering nd Innovive Tehnolog (IJEIT) N N N N N N N k N k N () The li drili ondiion i wrien follow: N () N N N dk N N N N k N k k In relion () nd () i he ineni of reidl deformion in he к-he loding irle Ni he nmer of loding le whih in he meril he dmge mlion proe egin for k N i he nmer of loding le o filre for k N N nd N N re of he fnion eperimenll defined for eh meril herewih N nd N re he loding le efore pperne of dmge in he eperimenl mple o filre In relion () nd () Nnd N re he ogh for qniie The fnion N nd N m e pproimed in he form: N A N A (6) When proeing eperimenl d of he pper [] for he eel of mrk he following vle were oined: Volme Ie 6 dk A 9 A N / N B In [] i w Herewih menioned h he rio N / N B on hold for everl meril Proeeding from hi f ondiion () nd () re rnformed o he form: N N dk N k N k dk B (7) Ue ondiion (7) for deermining he mon of pling orion le o he fir dmge nd filre of he onidered ovl ro eion r To hi end we define he ineni of reidl deformion of orion k where к i he rren mon Sine he r meril i idell pli hen he reidl deformion for n к le of ol nloding will e he me in he fir ol nloding Thi onlion i vlid lo in he e if in he fir ol nloding here rie eondr pli deformion We lle he ineni of reidl deformion he forml: / () In he ovl r he eion i dngero for filre From phil reoning' i follow h filre of he ovl r egin he Frhermore poin eiene of eondr pli deformion ol nloding elere he mel filre proe Proeeding from hi f lling he ineni of reidl deformion we e forml () For we hve: Tking ino on he l relion in () we ge: (9) Forml (9) hold in n k-h le of he pling orion ie he qni i independen of he nmer of pling orion Herewih from(7) i follow h he qniie N nd Noinide wih he qniie N nd B N repeivel Coneqenl we hve: N BA N A A he ove menioned vle B A for he eel of mrk nd lo he vle nd 6 for he qni N nd N we ge he following

8 vle: N ISSN: 77-7 ISO 9: Cerified Inernionl Jornl of Engineering nd Innovive Tehnolog (IJEIT) N 67 le le V CONCLUSION Condiion of pperne of eondr pli deformion ol nloding fer preinr eli-pli orion of ovl ro eion r were deermined The reidl ree nd deformion eli nloding pli deformion h re preerved in he ovl r eli-pli orion re deermined The nmer of he le of pling elipli orion whih he fir dmge pper nd filre egin were deermined Volme Ie 6 Fig Diriion of pli deformion re in he qrer of n ovl ro eion r orion nd ol nloding REFERENCES [] Sokolovkii VV Theor of plii oow: VhhShkol p [] Ilhin АА Plii Pr I Eli-pli deformion - Моow: oehid 9-76 p [] Моkviin VV Plii vrile loding oow: МU Pl 96-6 p (in Rin) [] Ivnov VS RgoinYI Voroev NА Reglriie of mel filre i nd li lod In he ook: Termo plihno meril: konrkivnikh elemen Ie IV Kiev NkovDmk 967 pp77- (in Rin) [] ТаllLKh To mll le nd herml fige Pro of he I Replin Conferene on mehni nd mhemi devoed o er of NAS of Aerijn Bk Jne 96 Pr I ehni Bk 99 - pp 9-96

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