DETERMINE VARIATION OF POISSON RATIOS AND THERMAL CREEP STRESSES AND STRAIN RATES IN AN ISOTROPIC DISC

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1 Kagujevac J. Sci. 38 (6) -8. UDC DETERMINE VARIATION OF POISSON RATIOS AND THERMAL CREEP STRESSES AND STRAIN RATES IN AN ISOTROPIC DISC Nishi Gupta, Satya Bi Sigh, Pakaj Thaku 3*, Depatmet of Mathematics, Pujabi Uivesity Patiala, Pujab 47, Idia 3 Depatmet of Mathematics, IEC Uivesity, Baddi, Sola, Himachal Padesh 743, Idia * Coespodig autho; pakaj_thaku@yahoo.co.i (Received Apil, 6) ABSTRACT. Seth s tasitio theoy is applied to the pom of themal ceep tasitio stesses ad stai ates i a thi otatig disc with shaft havig vaia desity by fiite defomatio. Neithe the yield citeio o the associated flow ule is assumed hee. The esults obtaied hee ae applica to compessi mateials. If the additioal coditio of icompessibility is imposed, the the expessio fo stesses coespods to those aisig fom Tesca yield coditio. Themal effect deceased value of adial stess at the iteal suface of the otatig isotopic disc made of compessi mateial as well as icompessi mateial ad this value of adial stess futhe much iceases with the icease i agula speed. With the itoductio of themal effects, the maximum value of stai ates futhe iceases at the iteal suface fo compessi mateials as compae to icompessi mateial. Keywods: stai ates, displacemet, agula speed, disc, themal stesses. INTRODUCTION Rotatig discs povide a aea of eseach ad study due to thei vast utilizatio i otatig machiey such as compesso, tubo geeatos, pumps, compessos, flywheels, shik fits, automotive bakig systems, ship popelles, compute disc dives, steam ad gas tubie otos. Theoetical ivestigatio of the stesses ad stai ates i aula discs otatig at high speeds have eceived widespead attetio due to a lage umbe of applicatios i mechaical ad stuctual egieeig. They ae usually opeated at elatively highe agula speed ad high tempeatue. Theefoe the pedictio of log tem steady state ceep defomatio is vey impotat fo these applicatios. The classical theoies of ceep stat with the assumptios of costitutive equatios fo ceep ad the classical theoies of plasticity eed a exta elatio called the yield coditio i additio to the flow ules. The desciptio of the defomatios i a solid subjected to exteal foces is thus give by a diffeet set of equatios fo elastic, plastic ad ceep defomatios. Solutios fo thi isotopic discs ca be foud i most of the stadad ceep text books (LUBHAN, 96; ODUIST, 974; KARAUS, 98; BOYLE, 983; NABARRO, 99; PENNY, 99; HOFFMAN, ). I most of egieeig applicatio, the disc has to opeate ude elevated tempeatue ad is simultaeously subjected to high stesses caused by disc otatio at high

2 6 speed (LASKAJ, 999). As a esult of sevee mechaical ad themal loadig, the mateial of disc udegoes ceep defomatio, theeby affectig pefomace of the system (FARHI et al. 4, 8). I ecet yeas, the pom of ceep i otatig discs made of fuctioally gaded mateials, subjected to sevee mechaical ad themal load, has attacted the iteest to may eseaches. GUPTA et al. (7) study etwok modelig of ceep behavio i otatig composite disc. DEEPAK et al. () ivestigated the pom ceep modelig i fuctioally gaded otatig disc of vaia thickess. DEEPAK et al. () discuss the pom ceep behavio of otatig FGM disc with liea ad hypebolic thickess pofiles. SETH (96) ivestigated Tasitio theoy of elastic-plastic defomatio, ceep ad elaxatio. GUPTA et al. (979, 98,, 7, 8) aalyzed ceep tasitio i thi otatig disc ad cylide havig vaious coditios. SHKULA (996) ivestigated ceep tasitio i a thi otatig ohomogeeous disc by usig Seth theoy. THAKUR () aalyzed ceep tasitio stesses i a thi otatig disc with shaft by fiite defomatio ude steady state tempeatue. SHARMA Sajeev et al., (, 3) aalyzed ceep defomatio i a thi otatig disk of expoetially vayig thickess with iclusio ad edge load by usig Seth tasitio theoy. THAKUR et. al. () study themal ceep stesses ad stai ates i a cicula disc with shaft havig vaia desity by usig Seth tasitio theoy. WAHL (96) has ivestigated ceep defomatio i otatig discs assumig small defomatio, icompessibility coditio, Tesc a yield citeio, its associated flow ule ad a powe stai law. The ecessity of iceasig use of ad-hoc semi-empiical laws i the classical theoy of elastic-plastic ad ceep tasitio lies i the fact that the latte does ot ecogize the existece of the tasitio state betwee elastic ad plastic oes ad the ceep. We have show i this eseach pape that assumptios of yield coditios i such poms become uecessay oce we ecogize that the tasitio fom plastic state to ceep, as explaied by Seth, is a asymptotic pocess ad that tasitio state is a sepaate state which caot be eplaced by a yield suface as has always bee doe i the cuet liteatue. This teatmet i the classical theoy amouts to divide two exteme popeties of a mateial by a shap lie which is physically impossi. It has bee clea fom ou wok that idetificatio of the tasitio state is basically impotat. Thee ae, at peset, thee ways to idetify the tasitio state. The most geeal oe amog all is the vaishig of the Jacobia of tasfomatio fom elastic state to plastic state ad plastic to ceep. A ivaiat elatio amog the stai (stess) ivaiats is obtaied fom this coditio ad it is foud that most of the yields coditios peset i cuet liteatue ae obtaia fom it as special cases. Also ou esults iclude the Bauschige's effect while the classical yield coditios fail to accout fo it. The classical theoy of elasticity, plasticity ad ceep makes use of liea stai measue. But we have show that tasitio fields ae sub-hamoic (supe-hamoic) fields ad that they ae o-liea ad o-cosevative i chaacte ad hece it is vey impotat that a o-liea stai measue such as the Almasi measue should be used i the costitutive equatio. The ecogitio of tasitio state o mid-zoe as a sepaate state ecessitates showig the existece of the costitutive equatio fo that state. I this cotext, we have used Seth's tasitio theoy to obtai the stesses ad stais i the tasitio state c c / v ; ad the same may be obtaied fo the plastic state whe a cetai paamete whee ν is the Poisso's atio of the mateial, is made to appoach zeo. Fom these solutios the costitutive equatios fo both tasitio ad plastic states ae obtaied, the latte takes the fom of the Levy-vo-Mises equatio. I atue tasitios do occu fequetly ad the existig classical theoy fails to explai them successfully. Thus the tasitio theoy, as it stads, ow ca be fuitfully exploited to explai a vaiety of physical pheomea ad hece

3 has a vey wide applicatio i all applied scieces. Seth s tasitio theoy does ot acquie ay assumptios like a yield coditio, icompessibility coditio ad thus poses ad solves a moe geeal pom 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 citical poits o tuig poits of the diffeetial equatios defiig the defomed field ad has bee successfully applied to a lage umbe of poms (SETH, 96, 966, 97, 97, 974; GUPTA et al., 979, 98,, 7, 8; SHKULA, 996; DEEPAK et. al.,, ; SHARMA et al.,, 3; THAKUR,, ). I this pape we discuss umeical study of possio atios ad themal stess ad stai ates i a isotopic disc by usig Seth tasitio theoy. 7 GOVERNING EQUATIONS OF THE PROBLEM Coside a cicula disc with cetal boe of adius a ad exteal adius b ad havig uifom thickess otatig with a agula velocity ω of gadually iceasig magitude about a axis pepedicula to its plae ad passig though the cete. The thickess of the disc is assumed sufficietly small so that it is effectively i a state of plae stess ( T zz =). The tempeatue at the cetal boe of the disc is Θ as show i Figue. Figue. Geomety of disk Displacemet Coodiate: The displacemet compoets i cylidical pola co-odiate ae give by (SETH, 96, 966, 97, 97): u = ( β),v =, w = dz () whee u, v, w (displacemet compoets); β is positio fuctio, depedig o = x y oly, ad d is a costat. The geealized compoets of stai ae give by Seth s (97): [ ( β β ) ] e = e = β e zz = ( d), e = e z = ez = () whee,, z be pola co-odiates ad β = dβ / d., [ ], [ ] Stess-Stai Relatio: The stess stai elatios fo themo elastic isotopic mateial ae (Pakus, 976): T = λδ I µ e ξθ δ, (i, j =,, 3) (3) ij i j ij ij whee T ij ae the stess compoets, λ ad µ ae Lame s costats, stai ivaiat, ξ α 3 λ µ δ is the Koecke s delta, ij I = ekk is the fist =, α beig the coefficiet of themal expasio, ad Θ is the tempeatue. Futhe, Θ has to satisfy: Θ = d Θ dθ d dθ d = o Θ k = ; which has solutios: Θ = k log k (4) d d d d d

4 8 whee k ad k ae costat of itegatio ad ca detemied fom the bouday coditio. Substitutig (Equatio ()) i (Equatio (3)), the stesses ae obtaied as: µ cξθ T = 3 c β c ( c)( P ), µβ µ cξθ T = 3 c β c ( c)( P ), T = T z = Tz = Tzz = () µβ whee c is the compessibility facto of the mateial i tem of Lame s costat, ad ae give by c = µ / λ µ. Equatio of equilibium: The adial equilibium of a elemet of the otatig disc equies: d ( T ) T ρω = (6) d whee T ad T ae the adial ad cicumfeetial stesses, ρ is the mateial desity of the disk ad ω is the costat agula velocity. Bouday coditios: The tempeatue satisfyig Laplace (Equatio (4)) with bouday coditio: Θ = Θ ad u = at = a, Θ = ad whee Θ is costat, give by (Pakus, 976): k ad k fom (Equatio (4)), we get: T = at = b, (7) k = Θ log a / b ad k = log b. Substitutig ( b) ( a b) Θ Θ = log (8) log Citical poits o Tuig poits: Usig (Equatios () & (8)) i (Equatio (6)), we get a o- liea diffeetial equatio i β as: { } dp ρω cξθ ( c) β P ( P ) = β ( P ) P c ( c)( P ) dβ µ µ whee Θ = Θ / log( a / b) ad β = βp (P is fuctio of β ad β is fuctio of ). Tasitio poits of β i (Equatio (9)) ae P ad P ±. (9) SOLUTION OF THE PROBLEM Fo fidig the themal ceep stesses ad stai ates, the tasitio fuctio is take though picipal stess diffeece (see SETH, 96, 966, 97, 97, 974; GUPTA et al. 979, 98,, 7, 8; SHKULA, 996; SANJEEV et al.,, 3; DEEPAK et. al., ; THAKUR,,, 6) at the tasitio poit P - we defie the tasitio fuctio as: µβ = T T = ( P ) () whee is fuctio of oly ad is dimesio.

5 Takig the logaithmic diffeetiatig of eq. () with espect to ad substitutig eq. (9) ad takig asymptotic value P -, we get: d ρw cξθ ( l ) = ( 3 c) () d ( c) µ D µβ ( c) Asymptotic value of β as P - is D/; D beig a costat. Itegatig equatio (Equatio ()) with espect to ad usig(equatio ()), we get = T T = A exp φ ψ, () whee A, f ad g ae costat of itegatios,which ca be detemie by bouday coditio ad c ν = be Poisso atio s, cξ = αe / ( ν ), c = [ ( ) ν ( ) ], αθ ν ω ρ ( ν ) ω ρ ( ν ) ψ =, φ = =. Fom D µ D ED (Equatios () ad ()), we have T T = A exp ( φ ψ ) (3) Substitutig (Equatio (3)) i (Equatio (6)), we get: ρω T = B A exp( φ ψ ) d (4) whee B is a costat of itegatio, which ca be detemie by bouday coditio. Usig bouday coditio (Equatio (7)) i (Equatio (4)), we get ρω B = A exp( ξ ψ ) d. = b Substitutig the value of B i (Equatio (4)), we get: ( b ) ρω () b T = A exp( ξ ψ ) d Substitutig (Equatio () i (Equatio (3)), we get: k ( ξ ψ ) exp( ξ ψ ) ( b ) b ρω 3 T = A exp d (6) Compaig (Equatios () ad (3)) ad the takig asymptotic value P, we get: ( ν ) β = A exp ( ξ ψ ) (7) E µ = E / ν is the Youg s modulus i tem of Poisso s atio. Usig (Equatio whee (7)) i (Equatio ()), we get 9 ( ν ) k 3 u = A exp ( ξ ψ ) E whee u be the displacemet compoet. (8)

6 Usig bouday coditio (Equatio (7)) i (Equatio (8)), we get; A E =. Substitutig the value of costat A i (Equatios (), (6) ν a exp φ a ξ a ad (8)), we get: T b E exp φ ψ d ρω b = ( ν ) a exp( φa ψ a ) (9) exp φ ψ b E exp k a a a 3 ( ν ) exp ( φ ψ ) ( b ) ρω T = ( φ ψ ) d () ( ξ ψ ) ( ξ ψ ) exp u = a exp a a. () (Equatios. (9) ()) give ceep stesses ad displacemet fo a thi otatig disc with shaft at tempeatue Θ. We itoduce the followig o-dimesioal compoets as: R = / b, R = a / b, σ = T / E, σ = T / E, Ω = ρω b / E, u = u / b ad αθ = Θ. (Equatios (9) - ()) i o-dimesioal fom become: R exp( ξr ψ R ) Ω ( R ) σ = dr ( ν ) R exp ( ξr ψ R ) exp( ) R R ξ R ψ R () 3 ( ν ) exp ( ξ ψ ) ( R ) Ω σ = R exp k ( ξr ψ R ) dr R R R (3) ( ξ ξ ) ( ξ ψ ) ( ν ) b R R exp R R u = R R (4) R exp R R Ω Θ ( ν ) b wheeξ = ; ψ = (Costats); σ (Tagetial stesses); D D l R (Radial stess); R = / b ad R = a / b (Radii atios). Fully-Plastic state: Fo a disc made of icompessi mateial ( ν / o c = ) (Equatios () to (4)) become: k4 R exp( ξr ψ R ) Ω ( R ) σ = dr k4 k4 3R exp( ξr ψ R ) () exp( ) R R ξ R ψ R Ω ( R k ) 4 σ = R exp k ( ξr ψ R ) dr 4 3R exp ξ R ψ R (6) k4 R ( ξ ψ ) ( ξ ψ ) R exp R R u = R R R exp R R k4 σ (7)

7 3Ω b whee ξ = 4D 3 Θb ; k 4 = ad ψ =. D l R DISTRIBUTION VARIATION PARAMETER IN POISSON RATIOS: Figue 3(a). Pecetage decease i adial. Vesus mateials Poisso atios. Fo umeically distibutio of Poisso atios vaious pecetage decease i adial as well as cicumfeetial stesses as show i Fig. 3(a) ad Fig. 3(b). It ca be calculated fom equatios (), (3), () ad (6) by takig values of ν =.33,. 48,., Ω =,, = /, /3, /7 ad tempeatue Θ =,. It has bee see fom Fig. 3(a) that value of pecetage decease i adial stess must be iceased fo icompessi mateial (i.e. ν =.) as compae to compessi mateials (i.e. ν =.33,.48) fo measue N ( =/7). Fom Fig. 3(b), It has bee obseved that value of pecetage decease i cicumfeetial stesses must be icease fo measue = /, /7 at agula speed Ω = fo icompessi mateial as compae to compessi mateial but evese i case fo measue = /3. With the iceased i agula speed the value of pecetage decease i adial as well as cicumfeetial stesses must be decease fo icompessi as well as compessi mateials. Figue 3(b). Pecetage decease i Cicumfeetial stesses. Vesus mateials Poisso atios. ESTIMATION OF CREEP PARAMETERS Whe the ceep sets i, the stais should be eplaced by stai ates ad the stess-stai elatios (Equatio (3)) become: ν ν e& ij = σ ij δijt αθ (8) E E whee e& ij is the stai ate teso with espect to flow paamete t. Diffeetiatig (Equatio (4)) with espect to time t, we get: e & = β & β (9) Fo SWAINGER measue (i.e. = ), (equatio (9)) become : & = & β. (3) ε

8 whee ε& is the SWAINGER stai measue. Fom (Equatio ()) the tasitio value β is give by: / [ ] β = µ σ σ (3) Usig (Equatios (9)-(3)) i (Equatio (8)), we get: & ε & ε = = [ ( σ σ )( ν )] [ σ νσ αθ] [ ( σ σ )( ν )] [ σ νσ αθ] [ ( σ σ )( ν )] [ ν ( σ σ ) αθ] & ε = (3) zz whee ε& ε& ϑad ε& zz ae stai ates teso. These ae the costitutive equatios used by ODQUIST (974) fo fidig the ceep stesses ad stai ates povided we put = /N. NUMERICAL RESULTS AND DISCUSSION Fo calculatig stesses, stai-ates ad displacemet based o the above aalysis, the followig values have bee take Ω = ρω b / E =, ν =. (icompessi mateial), ν =.487 ad.333 (compessi mateials), = /3, /, /7 (i.e N =3,, 7), α =. deg F (fo Methyl Methacylate; LEVITSKY et. al., 97 ), Θ = ad,, F, Θ = αθ =. ad ad D =. I classical theoy measue N is equal to /. Defiite itegals i the equatios () ad (3) have bee solved by usig Simpso s ule. It has bee see fom Fig. 4 ad, cuve have bee betwee adial stesses vesus tempeatue Θ =, fo measue = /7, /, /3, Ω =,. With the themal effect stesses must be deceases. Fo measue =/7, decease pecetage chage ae -.4%, -3.%, -4.3% at agula speed Ω = ad -.9%, -.4%, -.8% (-ve sig idicates decease value) at agula speed Ω = havig possio atios ν =.33,.48,.. Fo measue =/, decease pecetage chage ae -.6 %, -.%, -.3% at agula speed Ω = ad -.8%, -%, -.3% (-ve sig idicates decease value) at agula speed Ω = as show havig possio atios ν =.33,.48,.. Fo measue =/3, decease i pecetage chage ae -.7 %, -6.4%, -7% at agula speed Ω = ad -. %, -.%, -.% (-ve sig idicates decease value) at agula speed Ω = havig possio atios ν =.33,.48,.. Fom Ta, it has bee see that themal effect deceases the value of adial stesses as well as cicumfeetial stesses at the iteal suface fo compessi mateial as compae to icompessi mateial fo measue = /7, / ad /3. Pecetage chage i adial ad cicumfeetial stesses should be deceased with effect of tempeatue. Cuves ae poduced betwee stesses ad displacemet alog the adii atio R = /b (see Figues (a) ad (b)) fo otatig disc made of compessi as well as icompessi mateials with agula speed Ω = ad. It is also obseved fom (Figues (a)-(b)) that the adial stess has maximum value at the iteal suface of the otatig disc made of

9 compessi mateial (i.e. ν =.33 say Coppe;.48 say satuated clay) as compae to icompessi mateials (i.e. ν =. say ubbe) fo measue = /7 (i.e. N = 7) at agula speed ( Ω =). The values of adial stess futhe iceases at the iteal suface with icease value of agula speed ( Ω =) fo measue = /7 (i.e. N = 7), = / (i.e. N =) ad =/3 (i.e. N = 3) espectively. Themal effect deceases the values of adial stess at the iteal suface. 3 Figue 4(a). Ω = Figue 4(b). Ω = Figues 4(a)-4(b). Pecetage chage i adial stess with ad without tempeatue at the iitial yieldig to become fully plastic.

10 4 Ta. Paetage chage stesses fo iitial yieldig ad fully plastic state Iteal suface of the Disc Agula Speed Ω Tempeatue Θ Measue / / /3 /3 /7 /7 / / /3 /3 /7 /7 / / /3 /3 /7 /7 / / /3 /3 /7 /7 / / /3 /3 /7 /7 / / /3 /3 /7 /7 Radial stesses σ ν =.33 (compessi mateial) ν =.48 (compessi mateial) ν =. (Icompes si mateial) ν =.33 (compessi mateial) ν =.48 (compessi mateial) ν =. (Icompes si mateial) Pecetage chage (Decease) Cicumfeetial stess σ.6% % % % % % % % % % % % % % % % % % ν =.33 (compessi mateial) ν =.48 (compessi mateial) ν =. (Icompes si mateial) ν =.33 (compessi mateial) ν =.48 (compessi mateial) ν =. (Icompes si mateial) Pecetage chage (Decease) 48% 77.4% 3.33% 78.43% 47.76% 43.96% 9.99% 4.% 7.4 %.7 %.7 %.9 % 4.87 %.4 % 4.7 %. %.33 %.4 % Figue (a). Stesses ad displacemet distibutio alog the adii atio R = /b at agula speed Ω =.

11 Figue (b). Stesses ad displacemet distibutio alog the adii atio R = /b at agula speed Ω =. Cuve ae poduced fo stai ates alog the adii atio R = /b (see Figues 6(a) ad 6(b)) fo otatig disc made of compessi mateials (i.e. satuated clay o coppe) as well as icompessi mateial (i.e. ubbe) with agula speed Ω = ad 7 fo measue = /7, /, /3 ( i.e. N = 7,, 3). It has bee see (Figues 6(a)-6(b)) that otatig disc made of compessi mateials has maximum value of stai at the iteal suface as compaed to disc made of icompessi mateial fo measue = /7, /, /3 (i.e. N =7,, 3) at agula speed Ω =. Sice the values of stai ates futhe iceases at the iteal suface with icease value of agula speed say Ω = espectively. With the itoductio of themal effects the maximum value of stai ates futhe iceases at the iteal suface fo compessi mateials (i.e. satuated clay o coppe) as compae to icompessi mateial. Measue decease value of stai ates at the iteal suface. Rotatig disc is likely to factue by cleavage close to the shaft at the boe. CONCLUSION Themal effect deceased value of adial stess at the iteal suface of the otatig isotopic disc made of compessi mateial as well as icompessi mateial ad this value of adial stess futhe much iceases with the icease i agula speed. With the itoductio of themal effects the maximum value of stai ates futhe iceases at the iteal suface fo compessi mateials as compae to icompessi mateial. Ackowledgmet 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-3-463).

12 6 Figue 6(a). Stai ates distibutio i disc alog the adii atio R = /b at agula speed at agula speed Ω =. Figue 6(b). Stai ates distibutio i disc alog the adii atio R = /b at agula speed at agula speed Ω =.

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