Pankaj Thakur 1, Monika Sethi 1, Shivdev Shahi 2, Satya Bir Singh 2, Fadugba Sunday Emmanuel 3

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1 Pakaj Thaku Moika Sethi Shivdev Shahi Satya Bi Sigh Fadugba Suday Emmauel EXACT SOLUTION OF ROTATING DISC WITH SHAFT PROBLEM IN THE ELASTOPLASTIC STATE OF STRESS HAVING VARIABLE DENSITY AND THICKNESS TAČNO REŠENJE PROBLEMA ROTIRAJUĆEG DISKA SA OSOVINOM PROMENLJIVE GUSTINE I DEBLJINE PRI ELASTOPLASTIČNIM NAPONIMA Oigiali auči ad / Oigial scietific pape UDK /UDC: 59.9 Rad pimlje / Pape eceived: Keywods disc iclusio stesses displacemet desity thickess Abstact The pupose of this pape is to peset study of the elastoplastic state of stess i a otatig disc with a shaft havig a thickess ad desity pofile by usig Seth s tasitio theoy. It has bee obseved that the otatig disc made of the compessible mateial with a iclusio equies highe agula speed to yield at the iteal suface as compaed to the disc made of icompessible mateial ad a much highe agula speed is equied to yield with the icease i adii atio. The thickess ad desity paametes decease the value of agula speed at the iteal suface of the otatig disc of compessible as well as icompessible mateials. The model poposed i this pape is used i mechaical ad electoic devices. They have extesive pactical egieeig applicatios such as i steam ad gas tubies tubo geeatos flywheel of iteal combustio egies tubojet egies ecipocatig egies cetifugal compessos ad bake discs. INTRODUCTION Theoetical ivestigatio of defomatios i discs iduced by cetifugal foces is a impotat topic due to its vaious applicatios i stuctual egieeig compoets such as i steam ad gas tubie otos tubo geeatos ad iteal combustio egies castigs of ship popelles tubojet egies ecipocatig ad cetifugal compessos. The stess aalysis of thi otatig discs made of isotopic mateial has bee discussed extesively by Timosheko et al. /5/ i the elastic age ad by Chakabaty /7/ ad Heyma // fo the plastic age. Thei solutios fo the poblem of fully plastic state do ot ivolve the plae stess coditio that is to say oe ca obtai the same stesses ad agula speed ecessay fo the fully plastic stess of the disc without usig the plae stess Adesa autoa / Autho's addess: Depat. of Mathematics Faculty of Sciece ad Tech. ICFAI Uivesity Baddi Sola Idia pakaj_thaku5@yahoo.co.i Depatmet of Mathematics Pujabi Uivesity Patiala Pujab Idia Depatmet of Mathematics Faculty of Sciece Ekiti State Uivesity Ado Ekiti Nigeia Ključe eči disk uključak apoi pomeaje gustia debljia Izvod Cilj ada je pedstavljaje elastoplastičog aposkog staja u otiajućem disku sa osoviom datog pofila debljie i gustie pimeom teoije pelazih apoa Seta. Uočeo je da otiajući disk poizvede od stišljivog mateijala sa uključkom zahteva veću ugaou bziu za pojavu tečeja a uutašjim povšiama u poeđeju sa diskom od estišljivog mateijala. Poteba je i veća ugaoa bzia za pojavu tečeja pi poastu odosa polupečika. Paameti debljie i gustie smajuju veličiu ugaoe bzie a uutašjoj povšii otiajućeg diska i kod stišljivih i estišljivih mateijala. Model pedstavlje u adu se koisti kod mehaičkih i elektičih ueđaja. Ovi ueđaji imaju izazit začaj u paktičim ižejeskim pimeama a p. kod paih i gasih tubia tubogeeatoa zamajca u SUS motoima mlazim i klipim motoima cetifugalim kompesoima i disk kočicama. coditio (i.e. T zz. Elastic-plastic defomatios i vaiable thickess otatig discs have bee ivestigated extesively by Game /6/ Güve et al. /9/ You et al. // Easla et al. //. O the othe had themally iduced elastic-plastic defomatios of statioay discs have bee studied aalytically by Güve et al. /9 / employig diffeet bouday coditios. Güve /8/ aalysed elasticplastic stesses i a otatig hypebolic disc with a igid iclusio ude the assumptio of Tesca s yield coditio its associated flow ule ad liea stai hadeig. Pefect elasticity ad ideal plasticity ae two exteme popeties of the mateial ad the use of a ad-hoc ule like the yield coditio amouts to divide the two exteme popeties by a shap lie which is ot physically possible. Whe a mateial passes fom oe state to aothe qualitatively diffeet state a tasitio takes place. Sice this tasitio 8

2 is oliea i chaacte ad difficult to ivestigate wokes have take cetai ad hoc assumptios like yield coditio icompessibility coditio ad a stai law which may o may ot be valid fo the poblem. Gupta et al. /4 5/ ivestigated the poblem of elastic-plastic ad ceep tasitio i a thi otatig disc with the iclusio by usig Seth s tasitio theoy. Seth s tasitio theoy // does ot equie these ad-hoc assumptios ad thus solves a moe geeal poblem fom which cases petaiig to the above assumptios ca be woked out. This theoy utilizes the cocept of geealized stai measue ad asymptotic solutio at the citical poits o tuig poits of the diffeetial equatios defiig the defomed field ad has bee applied to a lage umbe of poblems successively (Gupta et al. /4-6/ Seth / / Thaku et al. / 7-4. This pape ivestigates the ifluece of desity ad thickess pofile o the elastic-plastic distibutio i a otatig thi disc with a igid shaft usig Seth's tasitio theoy. The thickess h ad desity of the disc ρ ae assumed to vay alog the adius i the fom: ( / k h h b ad ρ ρ ( / m b ( whee h ad ρ ae thickess ad desity at b espectively m ad k ae desity ad thickess paametes espectively. GOVERNING EQUATIONS Coside a thi otatig disc of vaiable desity ad thickess with the cetal boe of adius a ad exteal adius b. The aula disc is mouted o a igid shaft. The disc is otatig with agula speed ω about a axis pepedicula to its plae ad passig though the cete. The disc is thi so that it is effective i a state of plae stess T zz ad the vaiatio of the thickess is adial ad symmetic with espect to the mid-plae. Displacemet coodiates The compoets of displacemet i cylidical pola coodiates ae give usig: u ( β ; v ; w dz ( whee: β is positio fuctio depedig o x + y oly; ad d is a costat. The geealized compoets of stai ae give by /4/ as: ( β e β + θθ e β ( e ( zz d eθ eθz ez. whee: is the measue ad β dβ/d. Stess-stai elatio The stess-stai elatios fo isotopic mateial ae give by //: ij λδ ij I + µ e ij (i j (4 whee: λ ad µ ae lame s costats; ad e kk is the fist stai ivaiat δ ij is the Kocecke s delta. Equatio (4 fo this poblem becomes: λµ [ e + eθθ ] + µ e λ + µ λµ θθ [ e + eθθ ] + e θθ (5 λ + µ θ θz z zz. Stai compoets i tems of stesses ae obtaied as: u u ( β β C e + θθ E C u u C eθθ [ β ] θθ E C (6 w w ( C ezz [ ( ] [ θθ ] d z z E( C eθ eθz ez µ (λ + µ µ whee: E ; ad ( λ + µ C λ + µ. Substitutig Eq.( ito Eq.(5 we get µ β { ( ( C C+ C P+ } µ θθ Cβ { C+ ( C( P+ } (7 θ θz z zz. Equatio of equilibium The equatios of equilibium ae all satisfied except: d ( h h θθ + ρω h (8 d whee: ρ is the desity; ad h be the thickess of the mateial i the otatig disc. Citical poits o tuig poits By substitutig Eq.(7 ito Eq.(8 we get a oliea diffeetial equatio with espect to β: + dp ρω h ( C β PP ( + + dβ µ h Cβ {( C + ( C( P+ } + + β ( P + p{ C + ( C( P + } whee: β βp (P is fuctio of β ad β is fuctio of oly. Tasitio poits Tasitio poits of β i Eq.(9 ae P ad P ±. Bouday coditio The bouday coditios of the poblem ae give by: a u b. ( (9 9

3 SOLUTION THROUGH THE PRINCIPAL STRESSES Fo fidig the plastic stess the tasitio fuctio is take though the picipal stess (see / 4-4/ at the tasitio poit P ± we defie the tasitio fuctio ζ as Eq.(: { } ζ θθ ( C β C+ ( C( P + ( µ whee: ζ be the tasitio fuctio of oly. Takig the logaithmic diffeetiatio of Eq.( with espect to ad usig Eq.(9 we get Eq.(:. ( Cξθ µ ρω { } { } C h β ( ( ( β ( ( ( P + C P ( β (log ζ C C C P C P d C µβ h d Cβ { C+ ( C( P+ } Takig the asymptotic value of Eq.( as P ± ad m ρω itegatig we get a h A ( R ( A / h C m b ( m. ( Substitutig the values of costats A ad A fom Eqs.(4 (5 ad (6 espectively we get the tasitioal stesses ad displacemet as: whee: A is a costat of itegatio which ca be detemied fom the bouday coditio. Fom Eqs. ( ad ( ad usig Eq.( we have Ab ( C θθ h. (4 Substitutig Eq.(4 ito Eq.(8 ad usig Eq.( the itegatig we get m Ab C ( C ρω A + b k m h C b ( m h (5 whee: A is a costat of itegatio which ca be detemied fom the bouday coditio. Substitutig Eq.(4 ad Eq.(5 ito the secod equatio of Eq.(6 we get ( C β E( C b ( m h Substitutig the value β fom Eq( we have m ρω Ab m ( C u E( C b ( m h m ρω Ab m. (6 µ ( C whee: E is the Youg s modulus. Usig ( C bouday coditio Eq.( i Eq.(5 ad Eq.(6 we get m m ρω ( Ch ( b a A C C m ( C( mb b ρω ( m + k b C a C θθ ( m( C b b ρω m + k m m b a C a + ( m b m k b b b m m C ρω ( a u (7 E m C b ( m ad ρω m b a θθ ( m b C Iitial yieldig + k m m C a + b m k b b (8 Fom Eq.(8 it is see that θθ is maximum at the ie suface (at a theefoe yieldig of the disc takes place at the iteal suface ad Eq.(8 may be witte as k ρω m m b a C a a θθ + (say ( Y a m k C b b b Fully-plastic state whee: Y is the iitial yieldig stesses. The agula speed ecessay fo iitial yieldig is give by: i ρω b m Ω i Y k m k m a C a a + C b b b (9 whee: ωi ( Ωi / b Y / ρ. The disc becomes fully plastic (C at the exteal suface ad Eq.(8 becomes: ρω m m b a a θθ + b m b b Y (say whee: Y * is yieldig stess. The agula speed equied fo fully plastic state is give by:

4 ρω f b ( m Ω f m Y ( a/ b ( ad ρω f b ( m Ω f m Y ( a/ b k m. (4 σ * whee: ωf ( Ω f / b Y / ρ. We itoduce the followig o-dimesioal compoets: R /b R a/b σ /Y σθ θθ/y u u/b Ω i ρ ω i b /Y ad Ω f ρ ω f b /Y *. Elastic-plastic tasitioal stesses agula speed ad displacemet fom Eqs.(7 ad (9 i o-dimesioal fom become: Ω ( k i C σ m θ R R C ( m( C Ω k m k i m { m R C } + R R R ( m R m m Y C Ωi ( R R u RR E C ( mr ad Ω i ( m k / ( m m C R C ( R R + [ ] ( k m k m. Stesses displacemet ad agula speed fo fully plastic state (C ae obtaied fom Eqs.( ad ( as: σ Ω f k σ m θ R R ( m Ω m f k m m R { } + R R R ( m R ( Ω m k m k Y f R R u RR E ( mr ( ( k m NUMERICAL RESULTS AND DISCUSSION Fo calculatig stesses displacemet ad agula speed based o the above aalysis the followig values have bee take: C (icompessible mateial i.e. ubbe; C.5 (compessible mateial i.e. satuated clay; ad C.5 (compessible mateial i.e. coppe; k (flat disc. (betwee values ad ; ad m ad i espect. I Fig. cuves ae daw betwee agula speed equied fo iitial yieldig ad vaious adii atios R a/b fo C.;.5;.5; k.; ad m i espect. It has bee obseved that the otatig disc made of compessible mateial with a iclusio has equie highe agula speed to yield at the iteal suface as compae to the disc of icompessible mateial ad a much highe agula speed is equied to yield with icease i adii atio. The thickess ad desity paametes decease the value of agula speed at the iteal suface of the otatig disc of compessible as well as icompessible mateial. It ca also be see fom Table that fo highe pecetage of icease i agula speed is equied to become a fully plastic state fo k. as compaed to the disc of thickess k ad m. Cuves ae poduced betwee stesses alog the adii atio R /b at iitial yieldig (see Fig. fo the otatig disc made of compessible as well as icompessible mateial. k k k. Figue. Agula speed equied fo iitial yieldig alog the adio atio R a/b.

5 Table. Agula speed equied fo iitial yieldig ad fully plastic state. Desity Agula k k (flat disc k. m speed C C.5 C.5 C C.5 C.5 C C.5 C.5 Ωi Ωf Ωi Ωf Ωi Ωf P % P % P % whee / Ωi Ωf P is the pecetage icease i agula speed fom iitial yieldig to fully plastic state havig thickess k. ad desity m. k m m m k m m m m m m k. Figue. Stesses fo iitial yieldig state.

6 k k k. Figue. Stesses fo the fully-plastic state. It is also obseved fom Fig. that the adial stess has a maximum value at the iteal suface of the otatig disc made of the compessible mateials (i.e. satuated clay o coppe as compaed to the disc made of the icompessible mateial (i.e. ubbe fo k ad m i espect. Whe the value of thickess paamete k vaies betwee ad the cicumfeetial stess is maximum at the exteal suface of the disc made of icompessible mateial as compaed to the disc of compessible mateial fo m ad ad adial stesses ae maximum at the iteal suface fo compessible mateial fo m i espect. Cuves ae poduced betwee stesses alog the adii atio R /b at fully-plastic state (see Fig. fo k. ad m. It has bee see fom Fig. that adial stesses ae maximum at the iteal suface at m fo k but whe the value of thickess paamete k lies betwee ad the esults ae evesed. The thickess paamete deceases ad the desity paamete iceases the values of stesses at the ie ad oute sufaces of the otatig disc of icompessible as well as compessible mateial. ACKNOWLEDGEMENT The authos gatefully ackowledge UGC New Delhi fo povidig fiacial suppot to cay out this eseach wok ude UGC-Majo Reseach Poject Scheme (MRP- MAJOR-MATH--46. REFERENCES. Sokolikoff I.S. Mathematical Theoy of Elasticity d Ed. McGaw-Hill Ic. New Yok 95.. Heyma J. (958 Plastic desig of otatig discs Poc. Ist. Mech. Egs 7(: doi.og/.4/pime_proc_ 958_7_45_. Seth B.R. (96 Tasitio theoy of elastic-plastic defomatio ceep ad elaxatio Natue 95: doi:.8/ 95896a. 4. Seth B.R. (966 Measue cocept i mechaics It. J Noliea Mech. (: Timosheko S.P. Goodie J.N. Theoy of Elasticity d Ed. New Yok McGaw-Hill Book Co. Lodo Game U. (984 Elastic-plastic defomatio of the otatig solid disk Igeieu-Achiv 54(5: Chakavoty J. Theoy of Plasticity McGaw-Hill Book Ic. New Yok Güve U. (998 Elastic-plastic stess distibutio i a hypebolic disk with igid iclusio It. J Mech. Sci. 4(: doi.og/.6/s-74( Güve U. Altay O. (998 Liea hadeig solid disk with igid casig subjected to a uifom heat souce Mech. Res. Comm. 5(6: doi.og/.6/s9-64( You L.H. Tag Y.Y. Zhag J.J. Zheg C.Y. ( Numeical aalysis of elastic-plastic otatig disks with abitay vaiable thickess ad desity It. J Solids ad Stuc. 7(5: doi:.6/s-768(998-x. Güve U. Altay O. ( Elastic-plastic solid disk with ouifom heat souce subjected to exteal pessue It. J Mech. Sci. 4(5: doi:.6/s-74(99-6. Easla A.N. Ageso H. ( Limit agula velocities of vaiable thickess otatig disks It. J Solids ad Stuc. 9 (: 9-. doi:.6/s-768(49-4. Thaku Pakaj (6 Some poblems i elastic-plastic ad ceep tasitio Ph.D. thesis Depatmet of Mathematics H.P.U. Shimla Gupta S.K. Thaku P. (7 Ceep tasitio i a thi otatig disc with igid iclusio Defece Sci. J 57(: Gupta S.K. Thaku P. (7 Themo elastic-plastic tasitio i a thi otatig disc with iclusio Themal Sci. (: -8. doi:.98/tsci7g 6. Gupta S.K. Thaku P. (8 Ceep tasitio i a isotopic disc havig vaiable thickess subjected to iteal pessue Poc. Nat. Acad. of Sciece Idia Sectio-A Vol.78 Pat- pp Thaku P. (9 Elastic-plastic tasitio stesses i a tasvesely isotopic thick-walled cylide subjected to iteal pessue ad steady-state tempeatue Themal Sci. (4: 7-8. doi:.98/tsci947p

7 8. Thaku P. ( Elastic-plastic tasitio stesses i otatig cylide by fiite defomatio ude steady-state tempeatue Them. Sci. 5(: Thaku P. ( Defomatio i a thi otatig disc havig vaiable thickess ad edge load with iclusio at the elasticplastic tasitioal stess Stuc. Iteg. & Life (: Thaku P. ( Themo ceep tasitio stesses i a thick walled cylide subjected to iteal pessue by fiite defomatio Stuc. Iteg. & Life (: Thaku P. Sigh S.B. Thaku J.K. ( Elastic-plastic tasitioal stesses i a thi otatig disc with shaft havig vaiable thickess ude steady state tempeatue Stuc. It. & Life (:9-6.. Thaku P. ( Fiitesimal defomatio i a tasvesely isotopic thi otatig disc with igid shaft J Tech. of Plast. 8(: Thaku P. (4 Steady themal stess ad stai ates i a otatig cicula cylide ude steady state tempeatue Them. Sci. 8 (Suppl. : S9-S6. doi:.98/tsci 879P 4. Thaku P. (4 Steady themal stess ad stai ates i a cicula cylide with o-homogeeous compessibility subjected to themal load Themal Sci. 8 (Supp. : S8- S9. doi:. 98/TSCI58P 5. Thaku P. (5 Aalysis of themal ceep stesses i tasvesely thick-walled cylide subjected to pessue Stuc. Iteg. ad Life 5(: Thaku P. Sigh S.B. Lozaović Šajić J. (5 Themo elastic-plastic defomatio i a solid disk with heat geeatio subjected to pessue Stuc. Iteg. ad Life 5(: Thaku P. Kuma S. Sigh J. Sigh S.B. (6 Effect of desity vaiatio paamete i a solid disk Stuc. Iteg. ad Life 6(: Thaku P. Kuma S. (6 Stess evaluatio i a tasvesely isotopic cicula disk with a iclusio Stuc. Iteg. ad Life 6(: Gupta N. Thaku P. Sigh S.B. (6 Mathematical method to detemie themal stai ates ad displacemet i a thick-walled spheical shell Stuc. Iteg. ad Life 6(: Kau J. Thaku P. Sigh S.B. (6 Steady themal stesses i a thi otatig disc of fiitesimal defomatio with mechaical load J Solid Mech. 8(: 4-.. Thaku P. Kau J. Sigh S.B. (6 Themal ceep tasitio stesses ad stai ates i a cicula disc with shaft havig vaiable desity Eg. Comput. (: doi. og/.8/ec Thaku P. Gupta N. Sigh S.B. (7 Ceep stai ates aalysis i cylide ude tempeatue gadiet mateials by usig Seth s theoy Eg. Comput. 4(: -. doi.og/.8/ec Thaku P. Pathaia D. Vema G. Sigh S.B. (7 Elastic-plastic stess aalysis i a spheical shell ude iteal pessue ad steady state tempeatue Stuc. Iteg. ad Life 7(: Thaku P. Shahi S. Gupta N. Sigh S.B. (7 Effect of mechaical load ad thickess pofile o ceep i a otatig disc by usig Seth s tasitio theoy AIP Cof. Poc. Ame. Ist. of Physics USA 859(: 4. doi.og/.6/ Thaku P. Pathaia D. Vema G. Sigh S.B. (7 Themal ceep aalysis i o-homogeeous spheical shell Stuc. Iteg. ad Life 7(: Thaku P. Pathaia D. Vema G. Sigh S.B. (7 Ceep stesses i a otatig disc havig vaiable desity ad mechaical load ude steady-state tempeatue Stuc. Iteg. ad Life 7(: Thaku P. Pathaia D. Vema G. Sigh Lozaović Šajić J. (7 No-homogeeity effect i the spheical shell by usig Seth s theoy Stuc. Iteg. ad Life 7(: Thaku P. (7 Ceep stesses i a cicula cylide subjected to tosio Stuc. Iteg. ad Life 7(: Shama S. Thaku P. Shama R. et al. (7 Themoelastic-plastic tasitio i tosio of composite thick-walled cicula cylide subjected to pessue Stuc. Iteg. ad Life 7(: Thaku P. Sigh S.B. Pathaia D. Vema G. (7 Themal ceep stess ad stai aalysis i a o-homogeeous spheical shell J Theo. ad Appl. Mech. Wasaw 55(4: doi:.56/jtam-pl Thaku P. Mahaja P. Kuma S. (8 Ceep stesses ad stai ates fo a tasvesely isotopic disc havig the vaiable thickess ude iteal pessue Stuc. Iteg. ad Life 8(: Thaku P. (8 Stai ates aalysis i a otatig disc havig vaiable thickess paamete ad subjected to mechaical load accepted fo publicatio: J Theo. ad Appl. Mech. Wasaw 8. 8 The Autho. Stuctual Itegity ad Life Published by DIVK (The Society fo Stuctual Itegity ad Life Pof. D Stoja Sedmak ( This is a ope access aticle distibuted ude the tems ad coditios of the Ceative Commos Attibutio-NoCommecial-NoDeivatives 4. Iteatioal Licese 4