Elastic-Plastic Analysis of a Thin Rotating Disk of Exponentially Variable Thickness with Inclusion

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1 WSS NSIONS o MHMIS Sjv Shm, Moj Sh, vd Kum lstc-plstc lyss o h ottg Ds o xpotlly Vl hcss wth Icluso SNJV SHM, MNOJ SHNI & VIND KUM 3, Dptmt o Mthmtcs, JII Dmd Uvsty, -, Scto 6, Nod-7, INDI 3 B.I.. xtso t, Nod-3, INDI sjt@dml.com, moj_sh7@dml.com, vdchuh@yhoo.co. stct: - sto thoy hs usd to dv th lstc-plstc d tstol stsss. sults otd hv dscussd umclly d dpctd gphclly. It hs osvd tht o dsc wth xpotlly vyg thcss, hgh gul spd s qud o tl yldg t tl suc s compd to th lt dsc d xpotlly vyg thcss o 4 owds. hus w c coclud tht lt dsc.75 s o th s sd o th dsg s t qus hgh pctg cs gul spd to com ully plstc s compd to th lt dsc wth th compssl ctos,.5,tc.. Kywods: lstc, plstc, compsslty, tstol stsss, sotopc, ottg ds. Nomcltu o Symols:- d : Itl d xtl d o dsc u,v,w : Dsplcmt compots x, y, z : ts co-odts,θ, z : yldcl pol co-odts d : St d stss tso : Fst st vt β : Fucto o oly : ld stss P : Fucto o β oly : Mtl costts λ d μ : Lm s costts : dl dstc /, / : ompsslty σ / - dl stss compots θ σ / - cumtl stss compots Itoducto hs pp s cocd wth th lyss o ottg sold ds md o sotopc mtl wth xpotlly vyg thcss. h my pplctos o such typ o ottg dss, such s tus, otos, lywhls d wth th dvt o computs, ds dvs. h us o ottg ds mchy d stuctul pplctos hs gtd cosdl tst my polms dom o sold mchcs. h lyss o stss dstuto ccul ds ottg t hgh spd s mpott o tt udstdg o th hvo d optmum dsg o stuctus. h lyss o th ottg dscs md o sotopc mtl hs dscussd xtsvly y mosho d Good []. I th clsscl thoy, solutos o such typ o dscs md o sotopc mtl c oud most o stdd txt oos [-5]. hty [] d Hym [6] solvd th polm o th plstc stt y utlzg th soluto th lstc g d cosdg th plstc stt wth th hlp o sc s, Vo-Mss o y oth clsscl yld codto. H [7] hs vstgtd lstc d plstc stsss o sotopc mtls wth vl thcss. sl [8] hs clcultd lstc d plstc stsss hvg vl thcss usg sc s yld cto, ts ssoctd low ul d l st hdg. Wg [9] hs vstgtd domto o lstc hl gs. scu [] gv som umcl mthod o dtmg stsss ollg gs whl Mh [] clcultd stsss y usg t lmt mthod o wd tu otos. sto s tul phomo d th s hdly y ch o scc o tchology whch w do ot com coss tsto om o stt to oth. t tsto, th udmtl stuctu o th mdum udgos chg. h ptcls costtutg mdum -g thmslvs d gv s to sp, otto, votcty d oth o-l cts. hs suggsts tht t tsto, o-l tms vy mpott d glcto o whch my ot pst th l physcl phomo. ho tsto ISSN: Issu 5, Volum 9, My

2 WSS NSIONS o MHMIS lds o-l, o-cosvtv d vsl tu d should ot ttd s supposto o cts. lstcty-plstcty, vsco-lstc, cp, tgu, lxto som o th xmpls o tsto. t pst, such polms l lstc-plstc, cp d tgu ttd y ssumg d-hoc, smmpcl lws wth th sult tht dscotuts, sgul sucs, o-dtl gos hv to toducd ov whch two succssv stts o mdum mtchd togth. I ss o pps, Boh [7] hs gv tly dt otto to ths tstg polm o tsto. H hs dscussd out tsto thoy o lstc-plstc d cp domto. sto thoy th qus th yld cto o th ssoctd low uls to dv th tstol d plstc stsss. h tsto thoy utlzs th cocpt o glzd pcpl st msu d symptotc soluto t ctcl pots o tug pots o th dtl quto dg th domd ld d hs succssully ppld to lg um o polms [4-6]. h glzd pcpl st msu [] s dd s, [ ] d [ ],, j,,3 wh s th msu d s th pcpl lms t st compots. Fo -, -,,, t gvs uchy, G, Hcy, Swg d lms msus spctvly. I ths pp ttmpt hs md to study th hvo o sotopc th ottg ds wth xpotlly vl thcss d dg lod usg tsto thoy. h thcss o th dsc s ssumd to vy log th dus th om h h wh h s th costt thcss, s th gomtc pmt d s th dus o th ds. Ojctv o th Pst Study I od to xpl th lstc-plstc domto, t s st cssy to cogz th tsto stt s symptotc o d ths wo; t s ou m m to lmt th d o ssumg sm-mpcl lws, yld codto. W lso ot th costtutv quto cospodg to th tsto stt. Boh [7] dtd th tsto stt whch th govg dtl systm shows som ctclty. h gl yld codto o tsto s dtd om th vshg o th Jco o X,, Z tsomto,, wh X,,Z, x,y,z x, y, z th coodts o pot th udomd d domd stt spctvly. 3 Govg qutos W cosd th ds o costt dsty wth ctl o o dus d xtl dus. h dsc s ottg wth gul spd ω out xs ppdcul to ts pl d pssd though th ct o th dsc. h thcss o th dsc s ssumd to costt d s t to suctly smll so tht t s ctvly stt o pl stss, tht s, th xl stss ZZ s zo. h ds s ssumd to symmtc wth spct to th md pl, d th gomty o th cosdto s pstd gu. Fg. Isotopc dsc hvg xpotlly vl thcss. h dsplcmt compots cyldcl pol coodts gv y []. u β ; v ; w dz wh β s ucto o x y oly d d s costt. h t st compots gv s, [ β β ' ] u u u u β zz θ wh Sjv Shm, Moj Sh, vd Kum w w z z θz dβ β d '. z [ d ] 3 O susttuto o quto 3, th glzd compots o st gv s ISSN: Issu 5, Volum 9, My

3 [ β β ' ] [ β ] zz [ d ] θ θz z 4 h stss-st ltos o sotopc mtl gv s, λδ I μ,, j,,3 5 wh d th stss d st compots spctvly, λ d μ th Lm s costts d I s th st st vt, δ s th Koc s dlt. quto 5 o ths polm coms λμ λ μ λμ μ λ μ [ ] μ [ ] θ θz z zz 6 Susttutg quto 3 5, th st compots tms o stsss otd s [ β ' β ] [ β ] zz WSS NSIONS o MHMIS [ d ] θ θz z 7 wh s th oug s modulus d s th compsslty cto o th mtl. I tms o Lm s costt thy gv s 3λ μ μ μ d λ μ λ μ Susttutg quto 4 6, w gt th stsss s μ β ' 3 β β μ β ' 3 β β θ z z zz θ 8 qutos o qulum ll stsd xcpt d h h ρ ω h d 9 wh ρ s dsty o th mtl d h s th xpotlly vl thcss o th dsc. Usg quto 8 9, w gt o-l dtl quto β s dp Pβ P β P dβ μ { P } β { P } 3 wh β ' βp P s ucto o β d β s ucto o. sto o tug pots o P quto P d P ±. h oudy codtos : u t t Sjv Shm, Moj Sh, vd Kum 4 Soluto hough th Pcpl Stss It hs show lg um o css tht th symptotc soluto though th pcpl stss lds om lstc to plstc stt t th tsto pot P ±, w d th tsto ucto s μ 3 β P [ ] g th logthmc dtto o quto wth spct to d usg quto, w gt ISSN: Issu 5, Volum 9, My

4 [ ] P P P P P d d 3 3 log β β β μβ β 3 g th symptotc vlu o quto 3 s ± P d tgtg, w gt 4 wh s costt o tgto, whch c dtmd y th oudy codto. Fom quto d 4, w hv μ 5 Susttutg quto 5 9 d tgtg, w gt h B μ 6 wh B s costt o tgto d d. Susttutg qutos 5 d 6 scod quto o 7, w gt h B β 7 Susttutg quto 7, w gt h B u 8 wh 3 μ s th oug s modulus tms o compsslty cto. Usg oudy codto quto 6 d 8, w gt h B, μ 9 Susttutg th vlus o costt o tgto d B om quto 9 qutos 5, 6 d 8 spctvly, w gt th tstol stsss d dsplcmt s u Fom quto d, w gt 3 4. Itl ldg Fom quto 3, t s s tht s mxmum t th tl suc.. t, tho yldg wll t plc t th tl suc o th dsc d quto 3 com,. sy d th gul spd cssy o tl yldg s gv y 3 4 WSS NSIONS o MHMIS Sjv Shm, Moj Sh, vd Kum ISSN: Issu 5, Volum 9, My

5 4. Fully Plstc Stt h dsc coms ully plstc t th xtl suc.. t d quto 3 com { } * gul spd qud o th dsc to com ully plstc s gv y wh * w * ρ. 3 [ ] 5 W toduc th ollowg o-dmsol compots s u,, σ, σ θ, d u stol stsss, gul spd d dsplcmt c otd om quto - d 4 o-dmsol om s, σ θ σ WSS NSIONS o MHMIS d u d d o.. d 6 7 d 8 9 Stsss, dsplcmt d gul spd o ullyplstc stt otd om quto 6-8 d 5 s σ θ Sjv Shm, Moj Sh, vd Kum d σ d u d d 3 d Numcl Illustto d Dscusso I gu, cuvs hv dw tw gul spd d vous d tos o o dt compsslty ctos,.5,.5,.75. It hs osvd tht o lt dsc, hgh gul spd s qud o tl yldg t tl suc o compssl mtl s compd to compssl mtl. Fo compssl mtl, hgh pctg cs gul spd s qud o mtl to com ully plstc. s thcss o th ds vs xpotlly dcsg dlly, hgh gul spd s qud o tl yldg t tl suc o compssl mtl s compd to compssl mtl. It hs s tht wth th cs o om owds, lss gul spd s qud o ully plstcty t tl suc s compd to th ds wth thcss t. It c s om tl, s thcss o dsc dcss dlly, hgh gul spd s qud o th compssl mtl.75 to com ully plstc. s th thcss o dsc dcss dlly, gul spd qud o ully plstc stt s much lss s compd to th lt dsc. I gus 3-6, cuvs hv dw o tstol stsss d dsplcmt. Fo lt dsc- dl, ccumtl tstol stsss mxmum t tl suc. Fo ds md o compssl mtl, dl s wll s ccumtl stsss, s mxmum t tl suc s compd to ds md o compssl mtl. h dsplcmt s mxmum t xtl suc o compssl/ compssl mtl. h dsplcmt s lg o compssl mtl s compd to compssl mtl. Fo ds wth xpotlly vyg thcss, ISSN: Issu 5, Volum 9, My

6 dl stss s mxmum t tl suc whl ccumtl stss d dsplcmt s mxmum t xtl suc. cumtl stss s gog o csg s th thcss o dsc dcsg dlly. cumtl stss s dcsg s compsslty o dsc css..,.5,.5. Fo lt dsc, dl d ccumtl plstc stsss mxmum t tl suc s c s om gu 7. Fo dsc whos thcss vyg xpotlly dcsg dlly, dl stss s mxmum t tl suc whl ccumtl stss s mxmum t xtl suc. s th thcss o th ds dcss dlly, th ccumtl stss css. l gul spd qud o Itl ldg d Fully Plstc stt wth Vl thcss.5 WSS NSIONS o MHMIS 4 Pctg cs gul spd oclusos: Fo ds wth xpotlly vyg thcss, hgh gul spd s qud o tl yldg t tl suc s compd to th lt ds d xpotlly vyg thcss o 4 owds. hus w c coclud tht lt dsc.75 s o th s sd o th dsg s t qus hgh pctg cs gul spd to com ully plstc s compd to th lt dsc wth th compssl ctos,..,.5,.5, tc. d th dsc whch thcss dcss dlly.. o 3, 4, 5, tc.. cowldgmt: h uthos wshs to cowldg th sc ths to Po. S.K. Gupt o hs cougmt dug th ppto o th pp. cs: [] S.P. mosho d J.N. Good, hoy o lstcty, 3 d dto, Nw o: McGw-Hll Boo oy., Lodo, 95. [] J. hty, ppld Plstcty, Spg Vlg, Bl,. Sjv Shm, Moj Sh, vd Kum [3] W. H d B.D. ddy, Plstcty, Mthmtcl hoy d Numcl lyss, Spg Vlg, Bl, 999. [4].B. Hts d J. Igcz, Mthmtcl hoy o lstcty, ylo d Fcs, 3. [5] I.S. Soolo, Mthmtcl thoy o lstcty, d dto, Nw o: McGw-Hll Boo o., 95. [6] J. Hym, Plstc dsg o ottg dscs, Poc. Ist. Mch. gs., 958, pp [7].P.S. H d h K-u, lyss o Hgh- Spd ottg Dss wth Vl hcss d Ihomogty, sctos o th SM, 6, 994, pp [8].N. sl d.oc, lstc-plstc Domto o ottg sold ds o xpotlly vyg thcss, Mchcs o Mtls, vol. 34,, pp [9] Xu Wg d Xu, O Lg Domtos o lstc Hl gs, WSS sctos o ppld d hotcl Mchcs, vol., 7, pp. 4. [] Io scu, D Lpdtscu d Hotu odoscu, Numcl Mthods o Dtmto th lstc Stsss ollg Bgs, WSS sctos o ppld d hotcl Mchcs, vol., Nov. 6, pp [] Z.I. Mh d M.S. ouh, lculto o Dymc Stsss usg Ft lmt Mthod d Pdcto o Ftgu Flu o Wd u oto, WSS sctos o ppld d hotcl Mchcs, vol. 3, 8, pp [] B.. Sth, sto lyss o ollps o hc-wlld yld, ZMM, 5, 97, pp [3] B.. Sth, p sto, J. Mth. Phys. Sc., vol. 8, 97, pp. -. [4] S. Huls, sto hoy o p Shlls ud Uom Pssu, ZMM, vol. 46, 966, pp [5] S.K. Gupt d V.D., hmo lstc- Plstc d p sto ottg yld, J. Mth. Phy. Sc., 3, 989, pp [6] S.K. Gupt d Pj, hmo lstc-plstc sto th ottg Dsc wth cluso, hml Scc sctc joul,, 7, pp.3-8. [7] B. N. Boh, hmo lstc-plstc tsto, otmpoy Mthmtcs, vol. 379, 5, pp [8] S. Shm, lstc-plstc sto o Nohomogous hc-wlld cul yld ud Itl Pssu, D. Sc. Joul, vol. 54, No., 4. [9] S. Shm d M. Sh, p sto o svsly Isotopc hc-wlld ottg ISSN: Issu 5, Volum 9, My

7 WSS NSIONS o MHMIS yld, dv. ho. ppl. Mch., vol. 7, 8, pp [] S.K. Gupt, S. Shm & S. Pth, p tsto th ottg dsc o vl dsty, Dc Scc Joul, 5,, pp. -7. [] S.K. Gupt & Pj, p sto h ottg Dsc wth gd cluso, D. Sc. Joul, 57, 7, [] S.K.Gupt,.L.Dhm & V.D., p sto oso, It. J Nol Mchcs, 3, 979, [3] S.K. Gupt &.L. Dhm, p sto thc-wlld cyld Sjv Shm, Moj Sh, vd Kum ud tl pssu, ZMM, 59, 979, pp [4] B.N.Boh, hmo lstc-plstc sto o us ud Uom Pssu d Stdy stt tmptu, Id J. o Pu d ppld Mthmtcs Vol. 3, 97, 8-9. [5] S. Shm d M. Sh, p lyss o h ottg Dsc ud Pl Stss wth o dg lod, WSS sctos o ppld d hotcl Mchcs, 8. [6] U. G u& v, lstc-plstc ottg ds wth gd cluso, Mch. Stuct. d Mch., vol. 7, 999, pp Fg. gul spd qud o tl yldg t th tl suc o th ottg dsc wth vl thcss,, 4. ISSN: Issu 5, Volum 9, My

8 WSS NSIONS o MHMIS Sjv Shm, Moj Sh, vd Kum Fg.3 stol stsss d dsplcmt th ottg dsc log vous d to / wth compsslty o vl thcss K,, 4. Fg.4 stol stsss d dsplcmt th ottg dsc log vous d to / wth compsslty.5 o vl thcss K,, 4. ISSN: Issu 5, Volum 9, My

9 WSS NSIONS o MHMIS Sjv Shm, Moj Sh, vd Kum Fg.5 stol stsss d dsplcmt th ottg dsc log vous d to / wth compsslty.5 o vl thcss K,, 4. Fg.6 stol stsss d dsplcmt th ottg dsc log vous d to / wth compsslty.75 o vl thcss K,,4. ISSN: Issu 5, Volum 9, My

10 WSS NSIONS o MHMIS Sjv Shm, Moj Sh, vd Kum Fg.7 Plstc stsss d dsplcmt th ottg dsc log vous d to / o vl thcss K,, 4. ISSN: Issu 5, Volum 9, My