Elastic-Plastic Deformation of a Rotating Solid Disk of Exponentially Varying Thickness and Exponentially Varying Density

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1 Poceedings of he Inenaional MuliConfeence of Enginees Compue Scieniss 6 Vol II, IMECS 6, Mach 6-8, 6, Hong Kong Elasic-Plasic Defomaion of a Roaing Solid Dis of Exponenially Vaying hicness Exponenially Vaying Densiy Manoj Sahni Sanjeev Shama Absac In his pape he plane sae of sess in an asic-plasic oaing solid dis of exponenially vaying hicness exponenially vaying densiy has been sudied. Radial cicumfeenial sesses ae analyically deived using classical yid condiion sain hadening law. he gaphs ae ploed fo angula speed, nomalized sesses displacemen agains adii aio. I is concluded ha high angula speed is equied fo a maeial o yid fo exponenially vaiable hicness exponenially vaiable densiy ha becomes plasic as compaed o he dis wih ohe paamees, which in uns give moe significan economic design by an appiae choice of hicness densiy paamees. Index ems Roaing disc, Vaiable hicness, Vaiable densiy, Sesses. I. INRODUCION HE heoeical eseach on oaing diss has saed long ago because of is high usage in indusy. Roaing diss ae mos widy used as impoan sucual componen in engineeing. he sess analysis of pefecly plasic oaing solid dis based on esca s von-mises yid cieion has been lised in many exboos (see fo insance, [-]). he sess disibuion in an asic - plasic oaing solid dis was fis inoduced by F. Laszlo [3] in 95. Since hen a lo of eseach has been done on he oaing dis unde vaious condiions, such as plasic collapse speed, vaiable hicness, vaiable densiy, hemal effecs, ec. Game [4-5] in consideed he linea sain hadening maeial using esca s yid condiion unde diffeen bounday condiions. Game sudied he sess disibuions in oaing solid diss wih consan hicness densiy. Guven [6-7] Ocan e al. [8] exended he poblem of Game ae ino consideaion he dis wih vaiable hicness fo boh solid annula diss. A peubaion echnique is used by You e al. [9] fo solving poblem on non-linea sain hadening. Manuscip eceived Sepembe 3, 5; evised Ocobe 3, 5. Manoj Sahni is wih Depamen of Mahemaics Compue Science, School of echnology, Pi Deendayal Peoleum Univesiy, Ghinaga, Gujaa-387, INDIA. (coesponding auho phone: ; fax: ; manoj_sahani7@ediffmail.com). Sanjeev Shama is wih Depamen of Mahemaics, Jaypee Insiue of Infomaion echnology, A-, Seco 6, Noida - 37, U.P., INDIA. ( sanjeev.shama@jii.ac.in). ISBN: ISSN: (Pin); ISSN: (Online) Angula vociies unde plasic limi have been calculaed fo diffeen values of he hicness paamee by Easlan []. Easlan [] uses lipical hicness pofile fo calculaing sesses in 5. he displacemen, cicumfeenial adial sesses has been obained in ems of Whiae's funcions in he eseach pape by Zenou []. Sess analysis has been done using wo diffeen analyical mehods namy HPM ADM unde diffeen densiy hicness paamees in by Hojjai e al. [3]. Sudy of behaviou of an annula disc wih consan Poisson s aio pessue was done by Sahni e al. [4]. Sahni e al. [5] calculaed he sesses of a funcionally gaded oaing disc wih vaiable hicness exenal pessue using sess funcion. ansiion heoy [6] has been used o deive he asicplasic ansiional sesses fo a hin oaing dis of exponenially vaying hicness wih edge load inclusion. In 3, Shama e al. [7] have obained ceep sesses fo an annula dis wih vaiable hicness vaiable densiy wih edge load a he oue bounday. his heoy is used o sudy asic-plasic behaviou unde vaious condiions lie empeaue applied a he inenal suface exenal pessue in he eseach papes [8-9] using Lebesgue sain measue. In ode o obain an imal sucual design, i is necessay o esimae he angula vociy he sess disibuion of a oaing dis in fully plasic sae. he aim of his wo is o dev a useful analyical soluion fo asic - plasic oaing solid dis wih hicness densiy vaying in an exponenial fom unde he assumpion of esca s yid condiion, is associaed flow ule linea sain hadening. II. PROBLEM FORMULAION he poblem of plane sess wih he vaiaion of densiy hicness in an exponenial fom is consideed is given as hh, () Hee h ae hicness densiy a is he adius of he dis. All ohe paamees in equaion () ae geomeic paamees. he solid dis consideed is divided ino hee egions inne plasic egion, oue plasic egion asic egion. Fo each egion he yid condiion will vay. In figue, IMECS 6

2 Poceedings of he Inenaional MuliConfeence of Enginees Compue Scieniss 6 Vol II, IMECS 6, Mach 6-8, 6, Hong Kong ae he ineface adii sepaaing he wo plasic egions (inne oue) he oue asic egion, especivy. Figue : Solid dis showing he ineface adii. he plasic (inne egion) defomaion of he solid dis is govened by he yid condiion. () he equaions of equilibium ae all saisfied excep, h +h, (3) whee h is he hicness funcion, ρ is he densiy funcion ω is he angula vociy. Subsiuing equaions () () in equaion (3) inegaing, one ges n θθ A e ρ ( b) n( b) e M ω whee,. (5) he linea sain hadening law fo a linea isoic hadening maeial is +, (6) whee is he iniial ensile yid sess, η is he linea sain hadening paamee is he equivalen plasic sain. Using he equivalence of incemen of plasic wo, p p de + de θθ θ zzdeeq (7) ogehe wih he yid condiion, leads o p p e e + e η. (8) eq θ zz Consideing he incemen of plasic wo, he asic plasic sains ae added afe he inegaion, he adial displacemen is obained as ( ν ) A u + zz ( ξ) ξ dξ +. (9) η E η I is obseved ha lim zz ( ξ ) ξ dξ (4) d zz d d ξ ξ ξ lim lim. zz, () d [ ] d whee is finie a he axis. he displacemen a vanishes hence he inegaion consan. Subsiuing ino equaion (9) fom equaion (4), he displacemen becomes ( ν ) A ρω 3 η E u + M M, η whee, ξ n b M e ξ d ξ, () a n a ξ ξ n m b b b M3 e e e ξ dξada. () he plasic sain componens ae obained by subacing hei asic pas fom hei oal sains as p u ν eθ E, p du ν ν u e + d E η E η p zz ez. η (3) Hee, he asic pas of he sains ae expessed by sesses wih he hp of Hooe s law he plasic pas wih he hp of he hadening law. In he oue plasic egion he lages sess is equal o he yid sess, i.e.. Consideing he incemen of plasic wo gives ogehe wih he yid condiion one obains. Since he adial sain is puy asic, he sain-displacemen aions lead o u θθ ν E W H du ( νθθ ), (4) d E whee is he nomalized hadening paamee. Fom equaion (4) he adial sess cicumfeenial sess is calculaed as: ISBN: ISSN: (Pin); ISSN: (Online) IMECS 6

3 Poceedings of he Inenaional MuliConfeence of Enginees Compue Scieniss 6 Vol II, IMECS 6, Mach 6-8, 6, Hong Kong u N ν EN ν u + + I ( N ν ) ( N ν ) N (5) N EN u θθ + + ν u I ( N ν ) ( N ν ) (6) in which a pime denoes diffeeniaion wih espec o. Subsiuing equaions (5) (6) in he equaion of equilibium, we have d u du + n N n N ν u + d d b b N m 3 ν ν n ρ ( N ν ) ω e b +. (7) E I b o solve his diffeenial equaion, le us inoduce a new vaiable he ansfomaion u() y(z). Subsiuion of hese in equaion (7) gives a diffeenial equaion in y(z), whose homogeneous pa is d y dy z + + z + ( + N ν ) + ( N ) y. dz dz z (8) he above equaion has a soluion in ems of confluen hypegeomeic funcion given as ( + N ) ( N ) 3 4 y z A z F a, b, z + A z F a b +, b, z whee +, (9) ae abiay consans. Hee,, F (α, β, γ) is he confluen hypegeomeic funcion given as x x( x + ) x( x + )( x + ) 3 F ( x, y, z) + z + z + z +... y! y y +! y y + y + 3! () In his confluen hypegeomeic funcion y should no be zeo o a negaive inege. he geneal soluion of equaion (7) can be wien as u A P + A Q + R () whee 3 4,,, +,,. () he em in equaion () is he paicula soluion of equaion (7). can be obained using he mehod of vaiaion of paamees as: R U P + U Q (3), whee, Q ( ξ ) f ( ξ ) N ( ξ ) U d P ( ξ ) f ( ξ ) N ( ξ ) U d ξ ξ (4) is he Wonsian given by dq ( ) dp ( ) N P ( ) Q ( ). d d (5) Hee, N σ f + n ( N ) e E I b m b ν ν ρ ν ω. he adial cicumfeenial sesses ae obained as dp (6) νp A3 + N d N ν EN νq dq + + A 4 + I ( N ν ) N ν N d νr dr + + N d (7) θθ dp P A3 + ν + N EN d +. I( N ν ) N ν Q dq R dr ( ) A4 + ν + + ν d d (8) he plasic sain componens fo his egion ae given as,,. (9) In an asic egion, he sess-displacemen aions ae E ν u u + ν θθ ( ν ) E u + ν u. (3) Subsiuion of equaions (3) in equaion (3), we ge a diffeenial equaion as m 3 b d u du ν + n + n νu ρω e d d b b E (3) he geneal soluion of his equaion is given as u A5 P + A6Q + R (3) whee, P F a, b, n, b ISBN: ISSN: (Pin); ISSN: (Online) IMECS 6

4 Poceedings of he Inenaional MuliConfeence of Enginees Compue Scieniss 6 Vol II, IMECS 6, Mach 6-8, 6, Hong Kong Q F a b +, b, n b +,. he paicula soluion is obained in he fom R U P U Q (33) + (34) whee, Q f N ( ξ) ξ ξ ξ U d P f N ( ξ) ξ ξ ξ U d is he Wonsian given by dq dp N ( ) P ( ) Q d d f ( b) ( ν ) e m (35) ρ ω E (36) he adial cicumfeenial sesses ae νp dp ν R A E d ν νq dq dr ( ) A6 + + d d (37) θθ E P dp R A5 + ν + + d Q dq dr d d ν ν A6 + + ν. III. DEERMINAION OF INEGRAION CONSANS AND NUMERICAL DISCUSSIONS (38) he expessions fo sesses displacemen fo diffeen egions of defomaion conain he unnown inegaion consans,,,, he ineface adii. Fo he deeminaion of hese seven unnowns hee ae seven nonedundan condiions available:, u ae individually coninuous a, vanishes a he oue bounday of he dis, i.e. a b. hese condiions ae wien explicily as ip ip ip u u ( ) ( ), ( ) ( ), θθ θθ ( ) ( ), ( ) ( ), θθ ( ) θθ ( ). u u b,, When n m vanishes, he soluion descibes he behaviou of a oaing solid dis wih unifom hicness ISBN: ISSN: (Pin); ISSN: (Online) unifom densiy. Gaphs ae dawn fo adial cicumfeenial sesses, displacemens, plasic sains (adial cicumfeenial) wih espec o adii aio. Calculaions ae pefomed fo fou cases: ) Dis wih unifom hicness unifom densiy. ) Dis wih unifom hicness exponenially vaiable densiy. 3) Dis wih exponenially vaiable hicness unifom densiy. 4) Dis wih exponenially vaiable densiy exponenially vaiable hicness. Figue : Angula speed agains adii aio. In figue, i can seen ha fo he dis wih unifom hicness unifom densiy wih linea sain hadening paamee I /3, angula speed equied fo iniial yiding fully plasiciy is especivy. he esuls obained by Game [5] ae Ω.5499 Ω.843 fo I.5. he vaiaion beween he esuls is because of consideaion of diffeen hadening paamee, while he ends of gaphs ae same. Fo a dis wih exponenially vaiable hicness (n.5, 4), he angula speed is fo iniial yiding fully plasiciy especivy. Fo he case wih unifom hicness exponenially vaying densiy (m.5, ), angula speed equied fo iniial yiding fully plasiciy ae , which is vey high as compaed o he dis wih unifom hicness unifom densiy. Fo a dis wih exponenially vaying hicness exponenially vaying densiy (m.5,, n.5, 4), angula speed equied fo iniial yiding fully plasiciy is which is highe as compaed o ohe densiy hicness paamees. Fuhe he sesses, displacemen plasic sains ae calculaed fo he paially plasic case wih hadening paamee I /3 oue adii as.5. hese ae depiced in figue 3 fo (a) unifom hicness-vaiable densiy, (b) vaiable hicness-vaiable densiy dis. he inegaion consans in non-dimensional fom fo unifom hicness vaiable densiy ae calculaed as.6683,.88,.874,.59353, wih inne adii.53, sepaaing he inne plasic oue IMECS 6

5 Poceedings of he Inenaional MuliConfeence of Enginees Compue Scieniss 6 Vol II, IMECS 6, Mach 6-8, 6, Hong Kong plasic egion coesponding angula vociy Ω Similaly, he inegaion consans ae calculaed fo vaiable hicness-vaiable densiy dis. In non-dimensional fom hese ae.835,.58557,.5647,.44589, wih ineface adii.457 coesponding angula vociy is Ω Sesses, displacemens plasic sains ae dawn in figue 3 fo unifom hicness-vaiable densiy dis vaiable hicness vaiable densiy dis. I is seen fom figue 3 ha adial cicumfeenial sesses ae maximum a he inenal suface obseved ha upo he inne plasic egion; sesses ae same heeafe-adial sess deceases fase adially han ha of cicumfeenial sess. I has also been obseved ha a plasic sain vanishes a he ineface adii. Fom figue 3, i has been obseved ha cicumfeenial sess is maximum fo he dis whose hicness is consan densiy vaies exponenially as compaed o he dis whose hicness densiy vaies exponenially. (a) (b) Figue 4: Nomalized Sesses displacemen fo fully plasic case wih (a) unifom hicness vaiable densiy dis (m.5, ) (b) vaiable hicness - vaiable densiy dis (n.5, 4, m.5, ). (b) (a) Figue 3: Nomalized Sesses displacemen fo paially plasic case wih (a) unifom hicness vaiable densiy dis (m.5, ) (b) vaiable hicness - vaiable densiy dis (n.5, 4, m.5, ). In he fully plasic case oue ineface adii eaches he bounday, i.e...fo he dis wih unifom hicness vaiable densiy, he inegaion consans in nondimensional ae.8666,.58453,.58 wih ineface adii.4878 coesponding angula vociy is Ω.44, while fo he dis wih vaiable hicness vaiable densiy, he inegaion consans ae.8983,.7469,.444 wih ineface adii coesponding angula vociy is Ω Sesses, displacemen plasic sains ae ploed agains adii aio in figue 4. I has been obseved fom figue 4 ha plasic sains ae high a he inenal suface vanishes a he ineface adii. In addiion, cicumfeenial adial sesses ae maximum a he inenal suface. Finally he case beyond he fully plasic limi is sudied using I /3, m.5,, n.5, 4. A he angula speed Ω, he dis becomes fully plasic as he ineface adii eaches he oue bounday. Howeve his is no he collapse speed he dis can mainain angula vociies geae han Ω. he esuls fo he fully plasic case ae obained using coninuiy bounday condiions as ip ip ip u ( ) u ( ), σ ( ) σ ( ), σ θ ( ) σ θ ( ), σ ( b) ISBN: ISSN: (Pin); ISSN: (Online) IMECS 6

6 Poceedings of he Inenaional MuliConfeence of Enginees Compue Scieniss 6 Vol II, IMECS 6, Mach 6-8, 6, Hong Kong IV CONCLUSION (a) (b) Figue 5: Nomalized Sesses displacemen beyond fully plasic sae, > fo H /3, m.5,, n.5, 4 fo (a) Ω 3. (b) Ω 3.5 he inegaion consans fo he angula speed Ω 3. > Ω ae.85468,.5793,.8678 wih ineface adii.5443, while fo he dis wih angula speed Ω 3.5 > Ω, he inegaion consans ae 4.3,.47966, 3.993, wih ineface adii he sesses, displacemen plasic sains fo wo angula speeds geae han he fully plasic limi ae dawn in figue 5. I has been obseved fom figue 5 ha wih he incease in angula speed beyond he fully plasic limi, all paamees shows a significan change. Fom figues 3 4, i has also been obseved ha he magniudes of he plasic sains ae sufficienly small which jusifies he assumpion of he small defomaion heoy. An analyical soluion is obained fo asic-plasic defomaions of linea sain hadening solid dis of exponenially vaiable hicness exponenially vaiable densiy. he esuls of unifom hicness unifom densiy ae veified wih hose available in lieaue. I has been obseved ha fo exponenially vaiable hicness exponenially vaiable densiy high angula speed is equied fo a maeial o yid hen becomes plasic as compaed o he dis wih ohe paamees, which in uns give moe significan economic design by an appiae choice of hicness densiy paamees. REFERENCES [] S. P. imosheno J. N. Goodie, heoy of Elasiciy, 3 d Ed., McGaw Hill, 97. [] S. C. Ugual S. K. Fense, Advanced Sengh Applied Elasiciy, Elsevie, 987. [3] F. Laszlo, Geschleudee udehungsope im Gebie bleibende Defomaion, ZAMM, pp. 8 93, 95. [4] U. Game, esca s Yid Condiion he Roaing Solid Dis, J. Appl. Mech., vol. 5, pp , 983. [5] U. Game, Elasic-Plasic Defomaion of he Roaing Solid Dis, Ingenieu Achiv, vol. 54, pp , 984. [6] U. Guven, On he Sesses in an Elasic Plasic Annula Dis of Vaiable hicness unde Exenal Pessue, In. J. Solid Sucues, vol. 3, pp , 993. [7] U. Guven, On he Applicabiliy of esca s Yid Condiion o he Linea Hadening Roaing Solid Dis of Vaiable hicness, ZAMM, vol. 75, pp , 995. [8] Y. Ocan A. N. Easlan, Elasic-Plasic Defomaion of a Roaing Solid Dis of Exponenially Vaying hicness, Mechanics of Maeials, vol. 34, pp ,. [9] L. H. You, S. Y. Long J. J. Zhang, Peubaion Soluion of Roaing Solid Diss wih Nonlinea Sain Hadening, Mech. Res. Comm., vol. 4, pp , 997. [] A. N. Easlan, Inasic Defomaions of Roaing Vaiable hicness Solid Diss by esca von-mises Cieion, In. J. Comp. Eng. Sci., vol. 3, pp. 89-,. [] A. N. Easlan, Sess Disibuions in Elasic-Plasic Roaing Diss wih Ellipical hicness Pofiles using esca von-mises Cieion, ZAMM, vol. 85, Issue 4, pp. 5-66, 5. [] A. M. Zenou, Analyical Soluions fo Roaing Exponenially- Gaded Annula Diss wih Vaious Bounday Condiions, In. J. S. Sab. Dyn., vol. 5, pp , 5. [3] M. H. Hojjai S. Jafai, Semi-exac Soluion of Elasic nonunifom hicness Densiy Roaing Diss by Homoy Peubaion Adomain s Decomposiion Mehods, Pa Elasic Soluion, Inenaional Jounal of Pessue Vesss Piping, vol. 85, Issue, pp , 8. [4] M. Sahni M. oma, Funcionally Gaded Axisymmeic Roaing Annula Disc wih Inenal Exenal Pessue Consan Poisson s Raio, Inenaional Confeence on Chemical, Meallugical Civil Engineeing (ICCMCE; 5), Singapoe, July 9, pp. -5, 5. [5] M. Sahni R. Sahni, Roaing Funcionally Gaded Disc wih Vaiable hicness Pofile Exenal Pessue, Elsevie Pocedia Compue Science, vol. 57, pp , 5. [6] S. Shama, P. hau M. Sahni, Elasic-Plasic Defomaion of a hin Roaing Dis of Exponenially Vaying hicness Wih Edge Load Inclusion, Annals of Faculy Engineeing Hunedoaa- Inenaional Jounal of Engineeing, pp. 5-3,. [7] S. Shama M. Sahni, Ceep Analysis of hin Roaing Disc Having Vaiable hicness Vaiable Densiy wih Edge Loading, Annals of Faculy Engineeing Hunedoaa-Inenaional Jounal of Engineeing, pp , 3. [8] S. Shama, M. Sahni Sanehlaa, Elasic-Plasic ansiion of Non-Homogeneous hic-walled Cylinde unde Exenal Pessue, Applied Mahemaical Sciences, vol. 6, pp ,. [9] S. Shama M. Sahni, hemo Elasic-Plasic ansiion of a Homogeneous hic - Walled Cicula Cylinde unde Exenal Pessue, Sucual Inegiy Life, vol. 3, pp. 3 8, 3. ISBN: ISSN: (Pin); ISSN: (Online) IMECS 6

Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation

Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation Lecue 8: Kineics of Phase Gowh in a Two-componen Sysem: geneal kineics analysis based on he dilue-soluion appoximaion Today s opics: In he las Lecues, we leaned hee diffeen ways o descibe he diffusion