Analysis of Dynamics of Boundary Shape Perturbation of a Rotating Elastoplastic Radially Inhomogeneous Plane Circular Disk: Analytical Approach

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1 Alid Mtmtics tt://dxdoiorg/436/m3568 Pulisd Onlin My (tt://wwwscirporg/journl/m) Anlysis of Dynmics of Boundry S Prturtion of Rotting Elstolstic Rdilly Inomognous Pln Circulr Disk: Anlyticl Aroc Dmytro М Lil А А Mrtynyuk Crksy Ntionl Bodn Kmlnytsky Univrsity Crksy Ukrin Stility of Procsss Drtmnt S P Timosnko Institut of Mcnics Ntionl Acdmy of Scincs of Ukrin Kyiv Ukrin Emil: dim_l@ukrnt Rcivd Mrc 5 ; rvisd Mrc 6 ; cctd Aril 3 ABSTRACT For rotting inomognous circulr disk wy of clculting dynmics of oundry s rturtion nd filur of ring ccity is roosd in trms of smll rmtr mtod Crctristic qution of lstic zon criticl rdius is otind s first roximtion A formul of criticl ngulr vlocity is drivd wic dtrmins t stility loss of t disc ccording to t slf-lncd form Efficincy of t roosd mtod is sown y n illustrtiv xml considrd in Sction 7 Vlus of criticl ngulr vlocity of rottion r found numriclly for diffrnt rmtrs of t disc Kywords: Axisymmtric Elstolstic Prolm; Mtod of Boundry S Prturtion; Rotting Inomognous Circulr Disc; Stility Loss; Filur of Bring Ccity; Criticl Angulr Vlocity Introduction T filur of ring ccity of quickly rotting lstic disc [-3] ovrlodd y cntrifugl dilting forcs is ssocitd wit t dynmics of its oundry s rturtion [4] Aftr t disc tks u nw lncd s du to considrl growt of lstic zons [5] t instility [6] dvlos tontilly noug wit t incrs of rottion vlocity [7] Tis is stiultd y t rsons of t intrnl oints to t disc contour rsing nd outrunning growt of vril rdius of t rturd lstolstic oundry s comrd wit t vrition of its cunt rdius for stl disc [89] To study t stility loss nd vlocity dynmics of rotting disc t rturtion mtod cn lid [-] In t nlysis of ln strss strin stt tis mtod ws mloyd to otin roximt criticl vlus of t lstic zon dimnsions nd ngulr vlocity of continuous omognous circulr discs [34] ring-sd discs [5] including tos lodd long t contour y dditionl rdil forcs [6] std discs nd som rity rofil discs [7] s wll s simlst inomognous discs Tis rovs fficincy of t nlyticl mtod of oundry s rturtion (wit t us of simlst numricl rocdurs t crtin stg) wic rducs ssntilly t mount of clcultions nd t t sm tim fcilitts fruitful liction of vrious numricl tcniqus [8-] for stility nd strngt clcultion of turin nd otr mssiv discs Mnwil tr is still n on rolm of stlising y t smll rmtr mtod t rltionsi twn t vlu of oundry s rturtion lstic zon rdius nd rottion vlocity of unstl con- tinuous inomognous circulr disc cosonding to t indictd stt of rturd lstolstic oundry Tis is t sujct of our rsnt invstigtion Sttmnt of t Prolm W considr disc D consisting of two omognous nd isotroic ln discs D nd D Continuous circulr disc D osssss rdius wic coincids wit t intrnl rdius of t ring-sd circulr disc D T xtrnl rdius of disc D quls to Discs D nd D md from diffrnt mtrils r rigidly connctd into on disc D long t circumfrnc r W dsignt y s t mtril yild limit of disc D E is t lsticity modulus is t dnsity nd is t Poisson cofficint T cosonding mtril rmtrs of disc D r dsigntd y s E nd rsctivly It is ssumd tt constnt ngulr rottion vlocity of disc D is igr tn its criticl vlocity Tis mns t rsnc of rturtion of t initil contour circumfrnc r rturtion of t cunt Coyrigt SciRs

2 45 D М LILA А А MARTYNYUK rdius of lstolstic oundry r r nd in gnrl rturtion of strss strin stt of t wol (unstl) disc W focus our ttntion on t slf-lncd form of t stility loss of disc D wic is littl diffrnt from t circulr form T disc oundry qution u to t first ordr infinitsiml is rsntd in t form r = dcosn d const n or = cosn () wr = r is dimnsionlss cunt rdius is smll rmtr n is olr ngl On tis sis lt us dtrmin criticl vlus r = r nd = wic ccomny rcing of t ov mntiond circumfrnc r = y t rturd lstolstic oundry in t lstic zon of disc D T criticl vlus cosonding to rcing of t disc dg y t lstolstic oundry i its contct wit curv () sould scilly clcultd W rcll tt for solution of ts rolms it is ncssry first of ll to nlyticlly stlis t condition of contct of t lstolstic oundry nd circumfrnc of givn rdius i to construct crctristic qution wit t rmtr wit rsct to r ving solvd first t systm of linr qutions d u =for r = dr d u r =for r = d =for r = r r =for r = r d u =fo = r dr wit rsct to ur nd rity constnts found in t xrssions for strss nd dislcmnt comonnts r nd u dtrmining rturd strss strin stt of t rotting disc D T ov mntiond linrizd rturtions of t first ordr of smllnss stisfy diffrntil lnc qutions of ln rolm nd rtil diffrntil qutions of rltionsi twn strsss nd dislcmnts [] Unrturd strss stt (dsigntd y t ur indx ) is dtrmind y ordinry diffrntil qutions of qusisttic quilirium nd constrint qutions in t lstic zon or y t yild Sint-Vnnt condition [5] in t lstic zon In viw of t instility dvlomnt mcnism of t inomognous disc undr considrtion t sttd rolm will solvd for c of t four css: () DD (Figur ()); (s) DD (Figur ()); () DD (Figur (c)); (c) DD (Figur (d)) 3 Solution in t Cs DD In ordr to us oundry nd conjugtion conditions Au =for = () du A =for = (3) d =for = (4) =for = (5) Au 3 =for = (6) for rturtions of t first ordr of smllnss nd of rdil contct nd tngntil strsss rltd to t yild limit s w rcll tt in D (Figur ()) nd in D II = I cos n III IV = c ( ) c ( ) I II c ( ) c ( ) sin n III IV = I II III IV = I II III IV = ci c II ciii civ nd sids = q q q 3 q 4 = q q q q cos n cos n sin n Hr = r = nd r indfinit cofficints nd q q 8 r t cofficints xrssd vi n nd = E E ; I IV I IV ci civ r known functions [6] Morovr Coyrigt SciRs

3 D М LILA А А MARTYNYUK 453 r r () () r r r (c) (d) Figur Prturd lstolstic oundry rcing givn circumfrnc r = A = C 6 3 x A = A 4 x (7) A3 = 8 3 x C = s m 3 x x = 4q = s k m 3 l 3 k k s = s s = q = Coyrigt SciRs s k = k 3 3 m = wr l= For = rturtion of t first ordr of smllnss of t rdil dislcmnt rltd to is known from () u = cos n Trfor du d = n sin n

4 454 D М LILA А А MARTYNYUK Conditions ()-(6) in t xtndd form [6] r wr A = na = q q q q = 3 4 q q q q = w w w w cos n 3 4 Au 3 = IV IV w = II q q5 w = q q 6 I II IV w3 = II q3 q7 w4 = III II q4 IV q 8 = Hnc u = Ucosn wr U= w ww 3 w4 A 3 nd sids = A = na = q q q q q q qq qq = q q q q q q q q q q As consqunc t crctristic qution wit rsct to t lstic zon rdius cosonding to t momnt of contct of t rturd lstolstic oundry nd t mntiond circumfrnc = = coms U = (8) T criticl vlu of ngulr vlocity cosonding to t criticl vlu of rdius of t lstic domin = ( is criticl rdius of t lstic zon D for wic t disc loss its stility) is otind in trms of (7) 4 Solution in t Cs DD Now in contrst to Sction 3 t lstic domin is omognous nd snts zon D (Figur ()) trfor [45] = cos n n n n na nb n C n D n = n n na nb n n C n n D c n os n n n n = na nb nc nd sin n wr A B C nd D r indfinit cofficints Coltions (7) r writtn s wr of A =C6 3 x A = A 4 x A = C 6 3 x 3 3 x C = s x= = r (9) Trfor t systm of qutions for dtrmintion u ( ) s t form na nb n C n D A = A BC D A = n An Bn Cn D= n n n n A B C D n n n n = n An B n C n D cosn Au n n n n 3 Hnc wr U = u = U cosn = na nb n C n D A nd morovr n n n n 3 n A= A n n A n n n n n N n B= A n n n A n n n n N n C = A n ( n) n An ( n) ( n) N n D= A n n n An n n N Coyrigt SciRs

5 D М LILA А А MARTYNYUK 455 N = n n n n Tus crctristic Eqution (8) is constructd T crctristic qution wit rmtr wit rsct to criticl rdius of t lstic domin wic rcd t xtrnl dg of t disc D rds [9] wr U = () x= k5 k k k 3 = r U for 6 Solution in t Cs DD 5 Solution in t Cs DD Hving comrd strss stts (unrturd nd rof t disc D in css (s) nd () (Figur (c)) turd) of instility dvlomnt w conclud out t ncs- 4 ving sity to rsrv r ll clcultions of Sction x = 4q cngd t xrssion for x = In ordr to follow t rturtion dynmics of lstolstic oundry twn D nd D (Figur (d)) w will tk into ccount (s Sction 4) t fct tt x C = = r s k 3 k k 3 3 k 3 wr c c cc c = 4 3 c = 3 s k k c = 3 k k k c3 =3 3 3 s s k Similrly to Sction 5 t rst of clcultions including gnrl form of crctristic Equtions (8) nd () coincid wit t rsults rsntd in Sction 4 7 Exmls nd Concluding Rmrks For inomognous disc wit t rmtrs n = =9 =93 =3 =3 = =99 s =99 nd = s E wic loss its stility ccording to cs () for =748 nd q =685 t vlus of criticl rdius of lstic zon D nd rltiv criticl rottion vlocity q r rsntd in Tl Tl rsnts crctristic criticl vlus otind in trms of solution of crctristic Eqution () for t disc wit rmtrs n = = =4 =3 = = s =8 nd E = s wic loss its stility ccording to cs (s) for = = = 767 nd q =6674 T sm rolm is solvd for t disc wit n = =5 =3 = = s = nd = s E wos instility dvlos ccording to () (Tl 3) for =8 =76 nd q = 73 nd ccording to (c) (T l 4) for = =889 = 7 nd q = 76 Tl Vlus of criticl rdius nd rltiv criticl vloc- ity dnding on δ δ q Tl Vlus of criticl rdius nd rltiv criticl vlocity dnding on δ δ q Tl 3 Vlus of criticl rdius nd rltiv criticl vlocity dnding on δ δ q Coyrigt SciRs

6 456 D М LILA А А MARTYNYUK Tl 4 Vlus of criticl rdius nd rltiv criticl vlocity dnding on δ δ q T rltionsis stlisd twn t vlu of oundry s rturtion loction nd ty of r- of turd lstolstic oundry nd rottion vlocity unstl continuous circulr disc llow qulittiv nd quntittiv conclusions to md out culiritis of t disc sur-ig-s d dynmics T liction of t otind rsults nls us to forcst t dvlomnt of unstl stt nd to clcult ossil loss of stility nd filur of ring ccity of rotting discs It sould notd tt sic qutions of stility tory of stil dforml odis drivd y linriztion of nonlinr qutions contin trms scifid vi t comonnts of unrturd ground stt Tis cuss som difficultis in t rolm on loss of stility sinc loding rmtr ssocitd wit t criticl fforts ntrs t sic qutions Aliction of t roximtd roc rsntd in t r for stility invstigtion of stil lstic odis simlifis t rolm cus ot t rturtions ij stisfy t initil lnc qutions nd t loding rmtr is introducd into oundry conditions on t rturd initil surfc of t ody T loding rmtr is dtrmind y ssntilly mor siml crctristic qutions REFERENCES [] K B Bitsno nd R Grmml Tcnicl Dynmics Vol Gosudrstvnno Izdtlstvo Tkniko-Torticskoy Litrtury Moscow nd Lningrd 95 [] A E H Lov A Trtis on t Mtmticl Tory of Elsticity Dovr Pulictions Nw York 97 [3] S P Timosnko nd J N Goodir Tory of Elsticity McGrw-Hill Nw York 934 [4] А N Guz nd Yu N Nmis Mtod of Boundry Form Prturtion in t Mcnics of Continu Vysc Skol Kiv 989 [5] V V Sokolovsky Plsticity Tory Vyssy Skol Moscow 969 [6] A N Guz nd I Yu Bic Tr-Dimnsionl Stility Tory of Dforml Bodis Nukov Dumk Kiv 985 [7] A Ndi Plsticity nd Frctur of Solid Bodis Vol Izdtlstvo Inostrnnoy Litrtury Moscow 954 [8] D D Ivlv Continuum Mcnics Vol Pismtlit Moscow [9] D D Ivlv On t Loss of Bring Ccity of Rotting Discs Clos to Circulr Ons Izvstiy Akdmii Nuk SSSR Otdlni Tknicskik Nuk No [] D V Gorgivskii Smll Prturtions of n Undformd Stt in Mdi wit Yild Strss Dokldy Pysics Vol 48 No doi:34/63545 [] D D Ivlv nd L V Yrsov Prturtion Mtod in t Tory of Elstolstic Body Nuk Moscow 978 [] L V Yrsov nd D D Ivlv On t Stility Loss of Rotting Discs Izvstiy Akdmii Nuk SSSR Otdlni Tknicskik Nuk No [3] D M Lil Eccntric Form of Stility Loss of Rotting Elstolstic Disc Rorts of t Ntionl Acdmy of Scincs of Ukrin No [4] D M Lil nd А А Mrtynyuk Aout t Stility Loss of Rotting Elstolstic Circulr Disc Rorts of t Ntionl Acdmy of Scincs of Ukrin No 44-5 [5] D M Lil nd А A Mrtynyuk Dvlomnt of Instility in Rotting Elstolstic Annulr Disk Intrntionl Alid Mcnics Vol 48 No 4-33 [6] D M Lil On t Instility of Rotting Elstolstic Std Annulr Disc Lodd ovr t Boundry in t Middl Pln Intrntionl Alid Mcnics (to ulisd) [7] D M Lil nd А A Mrtynyuk Stility Loss of Rotting Elstolstic Discs of t Scific Form Alid Mtmtics Vol No doi:436/m577 [8] I V Dminusko nd I A Birgr Strss Clcultion of Rotting Discs Msinostroyniy Moscow 978 [9] M Mzièr J Bsson S Forst B Tnguy H Clons nd F Vogl Ovrsd Burst of Elstoviscolstic Rotting Disks Prt I: Anlyticl nd Numricl Stility Anlyss Euron Journl of Mcnics A/Solids Vol 8 No doi:6/juromcsol878 [] M Mzièr J Bsson S Forst B Tnguy H Clons nd F Vogl Ovrsd Burst of Elstoviscolstic Rotting Disks: Prt II Burst of Surlloy Turin Disk Euron Journl of Mcnics A/Solids Vol 8 No doi:6/juromcsol8 Coyrigt SciRs