Elastic-Plastic Transition of Transversely. Isotropic Thin Rotating Disc

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1 otempoy Egeeg Sceces, Vol., 9, o. 9, 4-44 Elstc-Plstc sto o svesely Isotopc h ottg Dsc Sjeev Shm d Moj Sh Deptmet o Mthemtcs JII Uvesty, -, Secto 6 Nod-7, UP, Id sjt@edml.com, moj_sh7@edml.com stct Elstc-plstc stesses hve ee deved y usg Seth s tsto theoy. esults oted hve ee dscussed umeclly d depcted gphclly. h ottg dsc mde o tsvesely sotopc mtel yelds t hghe gul speed s comped to dsc mde o sotopc mtel. ottg dsc mde o sotopc mtel equed hgh pecetge cese gul speed to ecome ully plstc om ts tl yeldg s comped to dsc mde o tsvesely sotopc mtel. ottg dsc mde o tsvesely sotopc mtel s o the se sde o desg s comped to ottg dsc mde o sotopc mtel. Mthemtcs Suject lsscto: 74, 7499 Keywods: Elstc-plstc, sto, svesely Isotopc, h ottg Dsc.. Itoducto Dsc plys mpott ole mche desg. Stess lyss o ottg dscs hs mpott ole egeeg desg. ottg dscs e the most ctcl pt o otos, tues, lywheel etc. he polem o th ottg lt dscs mde o sotopc mtel hs ee studed extesvely [-]. hkty [] d Heym [] solved the polem o the plstc stte y utlzg the soluto the elstc stte d cosde the plstc ge wth the help o esc s yeld codto. Futhe, to ot the elstc-plstc stesses, these uthos mtched the elstc d plstc stesses t the sme dus c o the dsc. Peectly elstcty d del plstcty e two exteme popetes o the mtel d the use o d-hoc ule lke yeld codto mouts to dvde the two exteme popetes y

2 44 S. Shm d M. Sh shp le, whch s ot physclly possle. he tsto theoy does ot eque these ssumptos d thus solves moe geel polem om whch cses petg to ove ssumpto c e woked out. he tsto theoy utlzes the cocept o geelzed pcpl st mesue d symptotc soluto t ctcl pots o the deetl equto deg the deomed eld d hs ee successully ppled to lge ume o polems [4-9]. he geelzed pcpl st mesue [4] s deed s e e [ e ] dej [ e ],, j,,. whee s the mesue d e j s the pcpl lms te st compoets. I ths ppe ttempt hs ee mde to study the ehvo o tsvesely sotopc th ottg dsc usg tsto theoy.. Goveg Equtos We cosde th dsc o costt desty mde o tsvesely sotopc mtel wth tel d extel d d espectvely. he dsc s ottg wth gul velocty ω out xs pepedcul to ts ple d pssg though the cete o the dsc. he dsc s th d s eectvely stte o ple stess. he dsplcemet compoets cyldcl co-odtes e gve [4-9] y, u β, v, w dz. whee β s ucto o x y oly d d s costt. he te compoets o st [4-9] e gve s, ' e [ β β ], eθθ [ β ],. ezz [ d ], eθ eθ z ez whee β d β / d Susttutg equto. equto., we get the geelzed compoets o st s e [ β β ], e e θθ zz [ β ], [ d ], eθ eθ z ez he stess-st eltos o tsvesely sotopc mtel e e eθθ e, 66 zz θθ 66 e eθθ e zz zz e eθθ ezz.

3 Elstc-plstc tsto 45 e e θθ ezz ; z θ z θ.4 whee j ' s e mtel costts. Usg equto. equto.4 we hve β P β [ { }] 66 [ β { P }] P θθ 66 β.5 θ θ z z zz whee. he equtos o equlum e ll stsed except d θθ.6 d whee ρ s the desty o the mtel. Usg equto.5 equto.6 we get o-le deetl equto β s, 66 dβ β P P β P P P P.7 dp whee β β P. he tstol pots o β equto.7 e P d P ±. he oudy codtos e t d..8. Soluto o Polem It hs ee show [4-7] tht the symptotc soluto though the pcpl stess leds om elstc to plstc stte t the tsto pot P ±. We dee the tsto ucto s, θθ [ β { P }] 66 [ β P ]. kg the logthmc deetto o equto. w..t. '', we get d dp log β P{ P } β P P d dβ. 66 dp β P P β P P dβ Susttutg the vlue o dp dβ om equto.7 equto. d tkg symptotc vlue s P ±, we get te tegto,. whee s costt o tegto d 66. Usg equto. equto., we hve θθ.4 Susttutg equto.4 equto.6, we get te tegto

4 46 S. Shm d M. Sh.5 whee s costt o tegto. Usg oudy codto.8 equto.5, we get ;.6 Susttutg the vlues o d equtos.4 d.5, we get / θθ.7 It s oud om equto.7 tht θθ s mxmum t the tel suce, theeoe yeldg o the dsc wll tke plce t the tel suce, we hve sy θθ.8 he gul speed equed o tl yeldg s gve y.9 d tstol stesses.7 ecome [ ] σ θθ θ σ. whee. d I the speed o otto s uthe cesed, the yeldg the dsc wll sped gdully; the dsc s edeed moe d moe plstc d lly ecome ully plstc t some vlue o the gul speed. Fo ully plstc stte.e., equto.7 ecome sy θθ. d the gul velocty equed o ully plstc s gve y. d stesses.7 o ully plstc stte ecome,

5 Elstc-plstc tsto 47 σ σ θ θθ. 4. Isotopc Mtel Elstc-plstc tstol stesses. o sotopc mtel ecome, σ σ θ 4. whee /. d gul speed equed o tl yeldg s gve y, 4. Fo ully plstc stte equtos 4. ecome, σ { / / } / / σ θ / 4. 6 d gul speed equed o ully plstc stte s gve y, / Equto 4. d equto 4.4 e sme s oted y Gupt d Shukl [6]. 5. Numecl Illustto s umecl exmple, elstc costts j hve ee gve tle o sotopc mtel Bss σ. d tsvesely sotopc mtels Mg d Beyl. I gue, cuves hve ee dw etwee gul speed equed o tl yeldg t the tel suce o vous d tos. It hs ee oseved tht ottg dsc mde o tsvesely sotopc mtel eque hghe gul speed to yeld s comped to dsc mde o sotopc mtel. ottg dsc mde o tsvesely sotopc mtel Mg yelds t hghe gul speed.969 o. 5 s comped to othe vlues o d tos, whle

6 48 S. Shm d M. Sh cse o sotopc mtel ottg dsc yelds t hghe gul speed. o d to. 4 s comped to othe vlues o d tos / see tle. It mes tht ottg dsc mde o tsvesely sotopc mtel eques lesse d to to yeld t the tel suce s comped to ottg dsc mde o sotopc mtel. ottg dsc mde o sotopc mtel eque hgh pecetge cese gul speed to ecome ully plstc s comped to ottg dsc mde o tsvesely sotopc mtel see tle. I gue, cuves hve ee dw etwee stesses d d to /. It s oseved tht ccumeetl stess o sotopc mtel s mxmum t the tel suce s comped to tsvesely sotopc mtel. heeoe, ottg dsc mde o tsvesely sotopc mtel s o the se sde o the desg s comped to ottg dsc mde o sotopc mtel. le : Elstc costts j used uts o N/m Mtels 44 Bss Isotopc Mtel Mgesumsvesely Isotopc Mtel Beyl svesely Isotopc Mtel le : gul speed equed o tl yeldg o vous d tos / Bss Mgesum Beyl

7 Elstc-plstc tsto 49 le : gul speed equed o tl yeldg d ully plstc stte.5 P % Mgesum Beyl Bss x s the pecetge cese gul speed whee P [ ] om tl yeldg to ully plstc stte. ckowledgmet: he utho wshes to ckowledge hs scee thks to Po. S.K. Gupt o hs ecougemet dug the pepto o ths ppe. Fg.. gul speed equed o tl yeldg o vous d tos / Fg.. Stesses o ully plstc stte wth espect to d to / eeeces [] J. hkty, ppled Plstcty, Spge Velg, Bel,. [] J. Heym, Plstc Desg o ottg Dscs, Poc. Ist. Mech. Egs., 958, [].B. Hetks d J. Igczk, Mthemtcl heoy o Elstcty, ylo d Fcs,. [4] S.K. Gupt, d Pkj, hemo Elstc-plstc sto h ottg Dsc wth Icluso, heml Scece Scetc Joul,, 7, -8.

8 44 S. Shm d M. Sh [5] S.K. Gupt d Pkj, eep sto h ottg Dsc wth gd Icluso, Deece Scece Joul, vol. 57, 7, [6] S.K. Gupt d.k. Shukl, Elstc-plstc sto h ottg Dsc, Gt, 45, 994, [7] S.K. Gupt, S. Shm d S. Pthk, eep sto h ottg Dsc o Vle Desty, Deece Scece Joul, vol. 5,, [8] S. Shm d M. Sh, eep lyss o h ottg Dsk ude Ple Stess wth o Edge Lod, WSES sctos o ppled d heoetcl Mechcs, vol. 8, 8, [9] S. Shm d M. Sh, eep sto o svesely Isotopc hck-wlled ottg ylde, dv. heo. ppl. Mech., vol. 7, 8, 5-5. eceved: Feuy, 9