ELASTIC-PLASTIC STRESSES IN A THIN ROTATING DISK WITH SHAFTHAVING DENSITY VARIATION PARAMETER UNDER STEADY-STATE TEMPERATURE

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1 Kagujevac J. Sc. 36 (4) 5-7. UDC 53.5 ELSTIC-PLSTIC STRESSES IN THIN ROTTING DISK WITH SHFTHVING DENSITY VRITION PRMETER UNDER STEDY-STTE TEMPERTURE Pakaj Thaku, Satya B Sgh ad Jatde Kau 3 Depatmet o Mathematcs, IEC Uvesty Badd, Sola, Hmachal Padesh 743, Ida E-mals: pakaj_thaku5@yahoo.co., d_pakajthaku@yahoo.com Depatmet o Mathematcs, Pujab Uvesty Patala, Pujab 47, Ida 3 Depatmet o ppled Scece, Rayat Isttute o Egeeg & Iomato Techology, Ropa, Pujab, Ida (Receved Jue 8, 3) BSTRCT. Steady themal stesses a otatg dsc wth shat havg desty vaato paamete subjected to themal load have bee deved by usg Seth s tasto theoy. Nethe the yelds cteo o the assocated low ule s assumed hee. Results ae depcted gaphcally. It has bee see that compessble mateal equed hghe pecetage ceased agula speed to become ully-plastc as compae to otatg dsc made o compessble mateal. Ccumeetal stesses ae maxmal at the oute suace o the otatg dsc. Wth the toducto o themal eect t deceases the value o adal ad ccumeetal stesses at e ad oute suace o ully-plastc state. Key wods: Stesses, dsplacemet, otatg dsc, agula speed, shat, tempeatue, desty. INTRODUCTION Dsc plays a mpotat ole mache desg. Stess aalyss o otatg dscs has a mpotat ole egeeg desg. Rotatg dscs ae the most ctcal pat o otos, tubes moto, compessos, hgh speed geas, lywheel, sk ts, tubo jet eges ad compute s dsc dve etc.the poblem o th otatg lat dscs made o sotopc mateal has bee studed extesvely [-3]. CHKRBRTY [] ad HEYMN [] solved the poblem o the plastc state by utlzg the soluto the elastc state ad cosde the plastc age wth the help o Tesca s yeld codto. Futhe, to obta the elastc-plastc stesses, these authos matched the elastc ad plastc stesses at the same adus = c o the dsc. Peectly elastcty ad deal plastcty ae two exteme popetes o the mateal ad the use o ad-hoc ule lke yeld codto amouts to dvde the two exteme popetes by a shap le, whch s ot physcally possble. Seth s tasto theoy[4] does ot equed ay assumptos lke a yeld cteo, compessblty codto, assocated low ule ad thus poses ad solves a moe geeal poblem om whch cases petag to the above assumptos ca be woked out. Ths theoy [4] utlzes the cocept o geealzed sta measue ad asymptotc soluto at ctcal pots o tug pots o the

2 6 deetal equatos deg the deomed eld ad has bee successully appled to a lage umbe o poblems [4-9].SETH [5]has deed the geealzed pcpal sta measue as: e e = e d e = e, (=,,3) () whee s the measue ad e s the almas te sta compoets. Fo =-, -,,, t gves Cauchy, Gee Hecky, Swage ad lmas measues, espectvely. I ths eseach pape we dscusselastc-plastc tastoal stesses a th otatg dsk wth shat havg desty vaato paamete ude steady state tempeatue by usg Seth s tasto theoy. The desty o dsc s assumed to vay alog the adus the om: ( b) / ρ = ρ () whee ρ s the costat desty at = b ad ms the desty vaato paamete. Result obtaed have bee umecally ad depcted gaphcally. MTHEMTICL MODEL Cosde a th dsc o sotopc ad homogeeous mateal havg vaable desty wth cetal boe o e adus a ad exteal adus b. The aula dsc s mouted o a gd shat. The dsc s otatg wth agula speed ω o gadually ceasg magtude about a axs pepedcula to ts plae ad passed though the cete as show Fg.. The thckess o dsc s assumed to be costat ad sucetly small so that t s eectvely a state o plae stess, that s, the axal stess Tzz s zeo. We assume that steady state tempeatue Θ s appled o the teal suace o the dsc. Fgue. Geomety o Rotatg Dsc. Bouday codtos The dsk cosdeed the peset study havg vaable desty ad subjected to a themal load. The e suace o the dsk s assumed to be xed to a shat. The oute suace o the dsk s ee om mechacal load. Thus, the bouday codtos o the poblem ae gve by: () = a u = () = b, T = (3) whee u ad T deote dsplacemet ad stess alog the adal decto. Fomulato o the Poblem Dsplacemet compoets cyldcal pola coodates (, θ, z), as:

3 u = β, v =, w = dz (4) whee β s ucto o = x + y oly ad d s a costat. The te sta compoets ae gve by Seth [5]: e u u = [ ( β + β ) ] [ ( d) ] u u, e θθ = [ β ] w w e zz =, eθ = eθ z = ez = (5) z z whee β = dβ d ad meag o supescpts s lmas. Substtutg eq.(5) eq. (), the geealzed compoets o sta ae gve by: [ ( β + β ) ] e = eθθ = β e zz = ( d), eθ = eθ z = ez = (6) whee β = dβ d. The stess sta elatos o themo elastc sotopc mateal ae gve by []:, [ ], [ ] Tj λδ ji µ ej ξ δj,, j =,,3 (7) whee T j s the stess compoets, λ ad µ ae Lame s costats ad I = ekk s the st sta vaat, ξ = α 3 λ + µ,α beg the coecet o themal = + Θ δ s the Koecke s delta ad j expaso ad Θ s the se o tempeatue. Futhe, Θ has to satsy Θ = (8) Eq. (7) o ths poblem becomes: λµ µξ Θ T = [ e + eθθ ] + µ e, θθ [ e + eθθ ] λ + µ ( λ + µ ) λ + µ λµ µξ Θ T = + e, θθ ( λ + µ ) T θ = Tθ z = Tz = Tzz = (9) Substtutg eq. (5) eq. (6), the sta compoets tems o stesses ae obtaed as [6]: e e θθ C [ ( β + β ) ] = T T + Θ, u u = = θθ α E C C [ β ] = T T + Θ, u u = = θθ α E C ( C) [ ] θθ w w ezz = = ( d ) = T T + α Θ, z z E C e θ = eθ z = ez =. () whee E s the Youg s modulus ad C s compessblty acto o the mateal tem o E µ 3λ µ / λ µ C = µ / λ + µ. Lame s costat, thee ae gve by = ( + ) ( + ) ad Substtutg eq. (6) eq. (9), oe get µ C T C C ( C)( P ) ξ Θ = 3 β + + +, µβ µ Cξ Θ Tθθ = 3 C β C + ( C )( P + ) +, µβ Tθ = Tθ z = Tz = Tzz =. () whee β = βp. 7

4 8 The equatos o equlbum ae all satsed except: d ( T ) T θθ + ρω = () d whee ρ vaable desty o the mateal o the otatg dsc. The tempeatue eld satsyg Laplace eq. (8) wth bouday codto Θ = Θ at = a Θ = at = b. whee Θ s costat, gve by: l Θ = Θ (3) l ( b) ( a b) Usg eqs. (), () ad (3), oe gets a o- lea deetal equato β as: ( P ) { ( )( ) } dp ρω C ( C) P( P ) + + ξ β + = + β Θ (4) dβ µ P C + C P + µ whee Θ = Θ / l ( a / b ) ad β = β P (P s ucto o β ad β s ucto o ) ad β = dβ d ( P s ucto o β ad β s ucto o oly). Soluto though the Poblem Fo dg the plastc stess, the tasto ucto s take though the pcpal stess (see SETH s [4, 5] ad PNKJ THKUR [6-]) at the tasto pot P ±. The tasto ucto τ s deed as: { } Cξ Θ τ = [ Tθθ Cξ Θ ] = ( 3 C) β C ( C)( P ) µ + + µ Takg the logathmc deetato o eq. (5) wth espect to, oe get: dp C ( C)( P ) ( P ) β d ( logτ ) β P dβ = d Cξ ( 3 C ) β { C ( C)( P ) } (6) Θ + + µ Substtutg the value dp / dβ om eq. (4) eq. (6) ad takg the asymptotc value P ± ad tegatg, oe get: ν = (7) τ whee ν = C / C ad s a costat o tegato ca be detemed by bouday codtos. Eqs. (5) ad (7) gves: µ ν Cξ Θ l ( / b) Tθθ = + (8) l ( a / b) Substtutg eq. (8) eq. () the usg eq. () ad tegatg, oe get: l ( / b) Cξ m µ ν ρω b B Cξ Θ Θ T = + + ν ( 3 k ) l a / b l a / b whee B s a costat o tegatoca be detemed by bouday codtos. (5) (9)

5 9 Substtutg eqs. (8) ad (9) secod equato o eq. (), oe get: α m m ν ρω b B α E Θ C C E Θ l b β = + + E 3 m l a / b l a b Cξ = αe C. whee Substtutg eq. () eq. (4), oe get: () α m m ν ρω b B α E Θ C C E Θ l / b u = + + () E 3 m l a / b l a / b µ 3 C / C s the Youg s modulus tem o compessblty acto ca be whee E= expessed as. Usg bouday codto (3) ad (3) eqs. (9) ad (), oe get: ( ) ( ) α ( ) m 3 m 3 m ρω b ν b a α E Θ C b a α E Θa = + ν C C C C µ 3 m b µ l a b b µ b ( ) m ρω b a α E Θ C a C E Θa B = m l a / b C Substtutg eqs. () ad (3) eqs. (8), (9), ad () espectvely, oe get the tastoal stesses ad dsplacemet as: 3 m C C b l ( b m ) a ρω νb 3 m ν a l ( a b) ( C) b T θθ = α E ( C) C C ( 3 m) b + Θ m 3 (4) ( C)( b a) + ( C) l ( a / b) b () (3) T ν a l ( / b) m ρω b ( b a ) α E ( C ) + + Θ b + C C ( 3 m) l ( a / b) ( b a) = a + b m 3 (5) C C α E ( C) a a Θ + ( C ) b ( ) m ρω b α E Θ C a a + ( 3 m) ν l a b u = m 3 (6) E ( C) α E Θ l ( / b) a + ( C) l ( a b)

6 adt ρω b m R ( 3 ) ( ν )( ) R R R + R ν R R R + R R ( ) ν ν Tθθ = Rν ( C) + αe Θ ( C) R R + l R R whee R = a / b ad R = / b o dmesoal om. m 3 (7) Ital Yeldg: The maxmum value T T θθ occus at the adus R R = (say), whch depeds upo the value o m ad C. Fo example we take C =,.5,.5 yeldg stats at the teal suace o m = -.9, -.6, -. espectvely ad o values m = -5., -4.9, -3.9 yeldg stats at the md suace. Fo the values m = 3.,.9,.6 ; T T θθ become ethe maxmum o mmum values at the exteal suace R =.e. yeldg does ot occus at the exteal suace. Fo yeldg at R = R, eq. (7) becomes: ρω b ν ( ν )( R ) R R + R ( 3 m) R T Tθθ = Y( yeldg say) R= R R R ν ν ( C) R R + αe Θ R + ( R ) R + Rν ( C) l R R whee Y s the yeldg stess. The agula speed ecessay o tal yeldg s gve by: ρ ω b Ω = = α E Θ R R + + ( 3 m) R whee S Ω Y ad ω ν ( ν )( R ) R R + R =. b ρ We toduce the ollowg o-dmesoal compoets: ( C) R ν ν RS S R ( R ) R Y Y ν ( C) l R R (8) R = / b, R = a / b, σ = T / Y, σ θ = Tθθ / Y, u = u / b, Θ = α E Θ / Y, Ω = ρ ω b / Y ad H = Y / E. Elastc-plastc tastoal stesses, agula speed ad dsplacemet om equatos (4), (5), (8) ad (6) o-dmesoal om become: 3 ν Ων R R R C R ν σθ = + Θ C R + R ( 3 m) C l R l R C ν R Ω ( R ) R Θ ( C) l R + Θ R ν R σ = + + ( R ) R ( 3 m) R R l R ν ν R + + ( R ) R ( )( ) ν l R (9) (3)

7 R R R + ( R ) R ( ) ν C Ω = RS S Θ ( C ) + ( R R ) l R ν ν Ω 3 3 Θ C R R R R R( 3 m) + R l R ad U = R R ν H ( C ) Θ l R R + ( C ) l R R (3) (3) Fully Plastc State: Stesses ad dsplacemet at the e bouday satsed the equalty T > Tθθ > Tzz ( = ) ad yeldg occus at the e adus. Fo ully-plastc state C.e. ν = /. Two plastc zoes o ully plastc state wee cosdeed as show Fg.. Thee ae two plastc zoes: () Ie-plastc zoe: T > Tθθ > Tzz ( = ) ; a o σ > σθ > σ z ( = ) ; R R R. () Oute-plastc zoe: T > T > θθ Tzz ( = ), b o σ > σ > σ ( = ) R R. θ z Whee s the adus o e plastc zoe. Fo Ie-plastc zoe, eq. (7) becomes: ( ) ρ ω b R E Θ α T Tθθ = + R R R R + Y say = ( 3 m) R R ad the agula speed equed o ully plastc state s gve by: ρω b ( 3 m) R ( 3 m) Ω = = Θ 3 4 R 3 3R + Y ( R ) ( R ) Ω Y α E wheeω = ad b ρ Y Θ = Θ. Usg equato (33) eqs. (9), (3), (3) by takg C.e. ν = /, we get the stesses ad dsplacemet o the e plastc zoe as: σ θ ( ) 3 Ω R R l R R = + Θ + ( 3 m) R R l R l R R Fgue. Two deet plastc zoes aoud the dsc o ully plastc state. (33) (34)

8 ( R ) R R ( R ) 3 Ω Θ R σ = + l R Θ ( R ) R( 3 m) R R l R R R R + Ω Θ R R l R R adu = R R H R R 4 R ( 3 m) + + Θ R l ( R ) l R R (35) (36) Fo oute plastc zoe eq. (7) becomes: ( R ) 3 ρω b T Tθθ = + αe Θ R R Y say R= + + = ( 3 m) l R ad the ad the equed agula speed s gve by: ρω b 3 m 3 m Ω = = Θ 3 R 3 ( R m m ) + + Y R R l R Ω Y α E Θ whee ω = ad b ρ Y. = Θ Usg equato (37) eqs. (9), (3), (3) by takg C.e. ν = /, oe get the stesses ad dsplacemet o the oute plastc zoe as: σ θ σ ( ) 3 Ω R R l R R = + Θ + ( 3 m) R R l R l R R Ω ( R ) R Θ R ( R ) R = + l R Θ ( R ) R ( 3 m) R R l R R R R + Ω Θ R R l R R ad U = R R H R R 4 R( 3 m) + + Θ R l R l R R (37) (38) (39) (4) RESULTS ND DISCUSSION Fo calculatg the stesses, agula speed ad dsplacemet based o the above aalyss, the ollowg values have bee take: C =.,.5,.5 ad.75, Θ = 7 F ad α = 5. 5 deg F o Methyl Methacylate [], Θ =,.75 ad.7 espectvely. I Tab., agula speed equed o tal yeldg Ω ad ully-plastc state Ω a otatg dsc havg vaable desty o deet values o m, C ad Θ has bee gve. It ca be see om the Tab. that yeldg occus at ay adus R = R o at the teal suace R =.5 o at the md suace R =.7 o the dsc depedg upo the values o m ad C. Fo example yeldg occus at the teal suace o the dsc made o compessble mateal (C =.5) at a agula speed o m = -.6 wheeas yeldg occus at the mddle suace at the agula speed o m = -3. It s also see om tab. that otatg dsc havg vaable desty ad made o compessble mateal yelds at a hghe agula speed as compae to dsc made o compessble mateal. Compessble mateal o otatg dsc wth shat havg vaable desty equed hghe pecetage ceased agula speed to become ully-

9 plastc as compae to compessble mateal. I Tab., agula speed equed o tal yeldg Ω ad ully-plastc state Ω o a otatg dsc o deet values o m=,.9 ad Θ =.,.75,.7 has bee gve. It s see that wth the eect o tempeatue, otatg dsc equed hghe pecetage agula speed to become ully plastc state wth cease tempeatue o m=.9 but evese case m=. Table. gula speed o tal yeldg Ω ad ully plastc state Θ. o a otatg dsc o deet values o m, C ad Ω 3

10 4 Table. gula speed equed o tal yeldg Ω ad ully plastc state Θ =. o a otatg dsc o m=.,.9 ad.,.75,.7 Ω I Fgs. 3(a)-3(c), cuves have bee daw betwee stesses ad adus ato R = /b o ully plastc state at deet values o m =,.9, 3.5. It s see that om gs. 3(a) ad 3(c), adal stesses s maxmum at the teal suace wheeas om g. 3(b), the ccumeetal stesses s maxmum at the oute suace o the otatg dsc. Wth the toducto o themal eect t deceases the value o adal ad ccumeetal stesses at e ad oute suace o ully-plastc state. Fgue 3(a). Stesses at ully-plastc state o deet values o tempeatue ad m=. wth espect to ad ato R=/b.

11 5 Fgue 3(b). Stesses at ully-plastc state o deet values o tempeatue ad m=.9 wth espect to ad ato R=/b. Fgue 3(c). Stesses at ully-plastc state o deet values o tempeatue ad m=3.5 wth espect to ad ato R=/b.

12 6 CONCLUSION It has bee see that compessble mateal equed hghe pecetage ceased agula speed to become ully-plastc as compae to otatg dsc made o compessble mateal. Compessble mateal o otatg dsc wth shat havg vaable desty equed hghe pecetage ceased agula speed to become ully-plastc as compae to compessble mateal. Ccumeetal stesses ae maxmal at the oute suace o the otatg dsc. Wth eect o themal load value o adal ad ccumeetal stesses at e ad oute suace o ully-plastc state must be decease. ckowledgemet The autho wshes to ackowledge hs scee thaks to Respected Po. Smeo Oka (Edto--Che o Themal Scece) ad D Vukma Bakć (Edto o Themal scece) o hs ecouagemet dug the pepaato o ths pape. Nomeclatue a,b - Ie ad oute ad o the dsc [m], ω - gula velocty o otato, [ s ] u,v,w - Dsplacemet compoets, [m] 3 ρ - Desty o mateal, [ kgm ] C - Compessblty, [ - ] T, -Stess [ kgm s ] ad Sta ate teso j e j Y - Yeld stess, [ kgm s Geek lettes R = / b; R = a / b Rad ato, [-] ] σ - Radal stess compoet ( T / Y ), [-] σ θ - Ccumeetal stess compoet ( T θθ / Y ), [-] Θ - Tempeatue, [ F ], B, d - Costats o tegato, [ - ] Reeeces: [] CHKRBRTY, J., ppled plastcty, Spge-Velag, New Yok/Bel/Hedelbeg,. [] HEYMN, J., Plastc desg o otatg dscs, Poc. Ist. Mech. Egs. 7 (958) [3] HETNRKSI, R.B., IGNCZK, J., Mathematcal Theoy o Elastcty, Taylo ad Facs, New Yok, US, 4, pp. 6. [4] SETH, B.R., Tasto theoy o elastc-plastc deomato, ceep ad elaxato, Natue, 95 (96)

13 [5] SETH, B.R., Measue cocept mechacs, It. J. No-lea Mech. I (966), [6] PNKJ, T., Some poblems elastc-plastc ad ceep tasto, Ph.D. Thess, Depatmet o Mathematcs, H.P.U. Shmla, Ida, 6. [7] GUPT, S.K., ad PNKJ, T., Ceep tasto a th otatg dsc wth gd cluso, Deece Scece Joual, Ida 57 (7), [8] PNKJ, T., ad GUPT S.K., Themo elastc-plastc tasto a th otatg dsc wth cluso, Themal Scece (7), 3-8. [9] PNKJ, T., ad GUPT, S.K., Ceep tasto a sotopc dsc havg vaable thckess subjected to teal pessue, Poceedg Natoal cademy o Scece, Ida, Secto 78 (8) I, [] PNKJ, T., Elastc-plastc tasto stesses a tasvesely sotopc thck-walled cylde subjected to teal pessue ad steady state tempeatue, Themal Scece 3 (9) 4, 7-8. [] PNKJ, T., Elastc-plastc tasto stesses a th otatg dsc wth gd cluso by tesmal deomato ude steady state tempeatue, Themal Scece 4 (), 9-9. [] PNKJ, T., Ceep tasto stesses a th otatg dsc wth shat by te deomato ude steady state tempeatue, Themal Scece 4 (), [3] PNKJ, T., Elastc-plastc tasto stesses otatg cylde by te deomato ude Steady-State tempeatue, Themal Scece5 (), [4] PNKJ, T., Ceep tasto stesses o a thck sotopc sphecal shell by tesmal deomato ude steady state o tempeatue ad teal pessue, Themal Scece 5 (), Suppl., pp. S57-S65. [5] PNKJ, T., Elastc-plastc tastoal stesses a th otatg dsc wth loadg edge, Poceedg o Iteatoal coeece o dvaces Modelg, optmzato ad Computg (MOC-), Depatmet o Mathematcs, Ida Isttute o Techology Rookee, Rookee, Dec. 5-7 () [6] PNKJ, T., Ceep tasto stesses a sphecal shell ude teal pessue by usg lebesgue measue cocept, Iteatoal joual ppled Mechacs ad Egeeg, Polad 6 () 3, [7] PNKJ, T., Steady themal stess ad sta ates a ccula cylde wth ohomogeeous compessblty subjected to themal load, Themal Scece () ole: DOI Reeece:.98/TSCI3879P. [8] PNKJ, T., Steady themal stess ad sta ates a otatg ccula cylde ude steady state tempeatue, Themal Scece () ole: DOI Reeece:.98/TSCI358P. [9] PNKj, T., Stesses a th otatg dsc o vaable thckess wth gd shat, Joual o Techology o Plastcty 37 (), -4. [] PRKUS, H., Themo-Elastcty, Spge-Velag We, New Yok, US (976) pp. 3. [] LEVITSKY, M. ad SHFFER, B.W., Resdual themal stesses a sold sphee om a themosettg mateal, J. o ppl. Mech., Tas. o SME 4 (975) 3,