Elastic limit angular speed of solid and annular disks under thermomechanical

Transcription

1 MultiCft Intentionl Jounl of Engineeing, Science nd Technology Vol. 8, No., 016, pp INTERNATIONAL JOURNAL OF ENGINEERING, SCIENCE AND TECHNOLOGY MultiCft Limited. All ights eseved Elstic limit ngul speed of solid nd nnul disks unde themomechnicl loding Piymd Nyk 1, Kshinth Sh * 1, Deptment of Mechnicl Engineeing, Jdvpu Univesity, Kolkt-70003, INDIA * Coesponding Autho: e-mil: kshinthsh@gmil.com, Tel , Fx Astct In this study, the influence of themo-mechnicl loding on stesses nd defomtion sttes in otting disk with vying thicknesses hs een evluted sed on vitionl pinciple consideing the dil displcement field s unknown. The solutions till the elstic limit yield stess of the mteil e ttined with the ssumption of plne stess condition. MATLAB computtionl simultion softwe is used to implement the solution lgoithm. The nlysis is ccomplished fo diffeent disk geometies s well s tempetue distiution pofiles. Limit ngul speed of the disks is clculted unde themo-mechnicl loding nd ccounted fo in dimensionl mode s limit peiphel speed nd dimensionless fom s nomlized limit ngul speed. The influence of tempetue on yield stess nd susequently on the limit peiphel speed is lso studied fo ny given tempetue distiution nd esults e funished in dimensionl fom. The effects of tempetue field on othe mteil popeties e lso studied. The esults of some educed polems e vlidted with those ville in litetue nd vey good geement is oseved. The new esults, funished gphiclly s design monogms, might pove helpful fo the pcticing enginees. Keywods: Rotting disks, Vitionl method, Themo-elstic stesses. DOI: 1. Intoduction Rotting disks e vitl pts of divese mechnicl pplictions, fo instnce disk kes, flywheels, ges, cicul sws, hd disks, gs s well s stem tuine otos, intenl comustion engines, centifugl compessos, nd in eospce industies. Mechnicl design of disks entils the ssessment of centifugl s well s theml stesses nd they need to e designed fo ppoximte unifom stess distiutions. The nlyses of stesses nd stins of these otting elements hve een studied y sevel uthos nd investigtos. Vious types of theoeticl, semi-nlyticl o numeicl ppoches e employed to solve complicted polems involving divese thickness pofiles, vition in mechnicl s well s physicl popeties nd oundy conditions, etc. The ehvio of otting disks unde high tempetue hs investigted y Thompson (1946) in which he gve numeicl method to the tuine disk polem y ccounting fo point-to-point vitions in disk thickness nd in ll othe physicl popeties pt fom Poisson s tio. In stuctul design pocess, it is expected tht the estimtion of the stess distiution nd the ngul velocity of otting disk in fully-plstic stte is impotnt nd found momentum in 1980s whee sevel eseches employed the Tesc yield citeion. Gme (1983) pointed out tht the displcement field clculted ccoding to Tesc s citeion s well s its elted flow ule t the elsto-plstic intefce of otting disk, ws discontinuous nd negtive plstic stin cused y tensile stess. To wok out this polem, the utho Gme (1984, 1985) dvnced n ext line stin hdening fo otting disk of constnt thickness of elstic plstic mteil hving stte of plne stess. Rees (1999) pplied von Mises citei nd its ssocited flow ule to void the insufficiency of Tesc solution in the elsto-plstic stess distiution of otting disk y using numeicl itetion method.

2 31 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp Tesc s yield condition hs een found to pedict slightly lowe limit ngul velocities thn tht of von Mises. The ide ws extended y Esln (003) in pesenting n nlyticl solution fo elstic-plstic defomtion of otting vile thickness nnul disks with fee, pessuized s well s dilly-constined oundy conditions. Anlysis of elstic stess stte in otting disks up to the point of yielding with the ojective to detemine limit ngul speed hs lso een cied out ecently. Vivio nd Vullo (007) s well s Vullo nd Vivio (008) hve studied the stesses nd stins in vile thickness nnul nd solid otting elstic disks unde the influence of theml lods s well s contining vile density ll long the dius. Hojjti nd Jfi (008) epoted ppoximte solutions of simil polems ppeing s n infinite powe seies fo nonline equtions, y using Adomin's decomposition method (ADM). Byt et l. (008) otined solutions fo the elstic nd themo-elstic polems fo functionlly gded (FG) otting disks unde the ssumption tht the mteil popeties s well s disk thickness vies following powe-lw functions of dius. The xisymmetic displcements s well s stesses in functionlly gded hollow cylindes, disks nd sphees unde the influence of unifom intenl pessue, using plne elsticity theoy nd Complementy Functions method, hs een poposed y Tutuncu nd Temel (009). Bhowmick et l. (010) studied the limit ngul speed of extenlly loded otting disks suject to shink-fit y using vitionl method. In susequent ppe, Bhowmick et l. (010) lso investigted the development of elstic-plstic font in otting solid disks of nonunifom thickness with exponentil s well s polic geomety vition y extending vitionl method in elstoplstic egime. Axisymmetic themo-elstic nlysis of otting dilly-gded hollow cicul disk unde the influence of dil tempetue distiution ws conducted y Peng nd Li (010) using one-dimensionl elsticity theoy. Nie nd Bt (010) consideed stesses in isotopic, line themo-elstic incompessile functionlly gded otting disks of vile thickness. Closed-fom solutions e pesented wheeve possile; othewise the govening equtions e solved y diffeentil qudtue method. Rttn et l. (010) investigted stedy-stte ceep esponse of n isotopic FGM otting disc of luminum silicon cide pticulte composites nd consideed ceep in the pesence of theml esidul stess s well. Wue nd Schweize (010) studied the dynmic themo-elstic polems of otting disks with sttiony souces of het nd foces t smll, odeed es of the sufce. A thoough litetue eview in this esech e evels tht nlyses of polems e cied out mostly though nlyticl methods nd no dedicted wok on limit ngul speed of otting disks unde themo-mechnicl loding hs een cied out. The pincipl gol of this esech is to study the effect of themo-mechnicl loding on stesses nd defomtion sttes in otting disk with vying thicknesses. Fo this pupose, numeicl method sed on vitionl pinciple hs een poposed to fomulte the polem nd to chieve n estimted solution of the unknown displcement field fom the govening eqution. MATLAB computtionl simultion softwe is used to implement the solution lgoithm. The esults epoted y vious othe eseches e vlidted nd good geement is otined. The esults funished in the pesent study would povide significnt undestnding of the ehvio of otting disks unde themo-mechnicl loding. Futhemoe, the effect of disk geomety nd tempetue field vition on the pefomnce of otting disks is consideed nd nomlized vlue of limit ngul speed is funished.. Mthemticl Fomultion In the development of the mthemticl model, it is ssumed tht the disk mteil is homogeneous, isotopic nd line elstic. Fo the polem t hnd, esponse nlysis of themo-elstic stesses is studied unde the stuctue of smll defomtion. The disk is thin, symmetic with espect to the mid-plne nd plne stess ( σ z = 0 ) ssumption is justified. Rdil displcements will occu in otting disk, due to the oth centifugl lod nd theml lod. Besides the mgnitude of loding, the dil displcement field is lso govened y the oundy conditions of the disk. The esult fo the displcement field is found fom the minimum totl potentil enegy pinciple, δ ( U + V ) = 0, whee U is the stin enegy stoed in the disk nd V is the potentil enegy evolving oth fom centifugl foce s well s theml lod. It is ledy stted tht the mteil of the disk is isotopic, so the theml stin t ny loction of the disk is sme in ll the diections. If α is theml expnsion coefficient nd * T ( ) is the chnge in tempetue t ny dius, then the theml stin is given y αt ( ) ε =. The totl stin is otined y dding the elstic stin nd the theml stin. Thus, the components of the totl stin e given y ε = e + ε nd εθ = e θ + ε, whee ε nd ε θ denote the dil nd cicumfeentil components, espectively, with espect to the totl stin nd e nd e θ e the dil nd cicumfeentil components of the elstic stin. The elstic stin components ε = σ νσθ E + αt nd e elted to stesses y Hooke s lw nd thus the expession of totl stin components ecome, ( ) ε ( σ νσ ) E αt( ), = whee E nd ν e elstic modulus nd Poisson s tio. Fo xisymmetic polems with smll stins, θ θ + the eltions etween stins nd dil displcement e given y ε = du d nd ε θ = u. The totl stin enegy tht comes fom the stess nd stin field of the disk, nd expessed s

3 3 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp Π = ( σε ) dv = ( σ ε + σ θ εθ )dv (1) Vol Vol Sustituting the eltions etween stess-stin nd stin-displcement, eqution (1) ecomes, πe u du du u du Π = + ν u + ( 1+ ν ) αt + hd () ( 1 ν ) d d d At this point, the totl enegy Π is decomposed into two pts: A stin enegy U nd the potentil enegy V. The fist pt of potentil enegy V centifugl comes fom centifugl foce field, while the second pt V theml, comes fom the theml lod. The expessions fo U, V centifugl nd V theml e given elow. E π u U = + νu ( 1 ν ) du d du + d hd V = πρω u hd (4) centifugl πe u du Vtheml = α ( 1+ ν ) + 1 d Thd ν So, the expession fo potentil enegy V unde comined loding ecomes, E π u du V = πρω u hd α (6) ( 1+ ν ) 1 d Thd + ν On sustitution of equtions (3) nd (6) in the enegy pinciple ( U + V ) = 0, δ the govening equiliium eqution is otined s, ( 1 + ν ) E π u du du E π α u du δ u hd u hd Thd = 0 ( 1 ) + ν + πρω d d + (7) ν ( 1 ) d ν Eqution (7) is stted in nomlized co-odinte, ξ ( = ( ) ( )) nd using the nottion =, it tkes the following fom, (3) (5) δ ( 1 ν ) 0 ( ξ + ) E E 1 ( 1 + ν ) ( 1 ν ) α 1 0 u ν u du + + d ξ u ξ + ( ξ + ) du dξ du d ξ ( ξ + ) + Thdξ = 0 1 hdξ ρω + 0 ( ξ ) uhdξ (8) u ( 0 ) = 0 nd σ = 0, whees fo n nnul disk, these conditions e σ = 0 nd σ = 0. The necessy displcement function employed to poduce the highe ode othogonl functions tht stisfies ove oundy conditions e given elow, fo solid nd nnul disks. The oundy conditions of the displcement function u ( ξ ) fo solid disk e ( 3 + ν ) ρω ( ) α 1 ν φ o = + 1+ ν T d + T d (9) 8E 3 + ν ( 1 ν ) 1 ν

4 33 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp φ o ( ) α + ρω = 8E 3 + ν ( 1 ν ) ( 1 ν ) + ( 1+ ν ) 1 + ν T d + T ( ) d ( 1 ν ) ν + ( 3 + ν ) Fo computtionl pupose, the set of othogonl functions e employed to estimte the displcement function ( ξ ) Function ( ) o c φ,i 1,,...,n. (9) u s follows, u ξ (10) i i = φ is nomlized nd the govening eqution in mtix fom is otined y sustituting eqution (10) in eqution (8). δ E ( 1 ν ) 1 ρω ( ciφi ) ( ξ + ) ν + {( ξ + ) ( c φ )} j j ( c φ ) i i d Eα 1 hdξ ( 1 ν ) 0 ( c jφ j ) ( ξ + ) d( c φ ) dξ + ( ξ + ) d ( c jφ j ) c φ + j j i dξ i dξ hdξ Thdξ = 0 (11) Replcing opeto δ in eqution (11) y minimiztion pinciple s, c j, j = 1,,..., n, the govening yields in geement with Glekin s eo E n n 1 φiφ j ν ' ' ( ξ + ) ' ' c φ φ φ φ φ φ hdξ i + i ν ( ξ ) j + i j + i j 1 i= 1 j= n 1 ρω { ( ξ ) φ } E hdξ α n 1 ' = + j + { φ j + ( ξ + ) φ j } Thdξ j= 1 1 ν j= (1) ' whee indictes diffeentition with espect to nomlized coodinte ξ. Eqution (1) cn e expessed s [ K ]{ c} = { f }, which yields the solution vecto { c i } though single step mtix invesion pocess. The polem consideed in the pesent ppe is von-mises yield citeion unde plne stess cn e witten in the fom σ vm θ y = σ σ σ + σ σ (13) θ Fom the solution vecto of eqution (1), the esulting displcement field is post-pocessed to detemine the von-mises stess coesponding to the theml nd centifugl loding. The loding is initited t low vlues nd incements e povided until the condition of yielding is eched. 3. Results nd Discussions Figue 1. Thickness pofiles of disk coesponding to geometic pmetes n, k. The effect of tempetue on the limit ngul speed fo ny given tempetue distiution nd oundy conditions e estlished in the pesent study. The nlysis is cied out fo fou diffeent pofiles of disk, unifom, tpe, exponentil nd

5 34 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp k [ ] policlly vying thickness. The expession fo n exponentilly vying disk given y h( ξ ) h0 exp n( ξ ) k policlly vying disk the expession ecomes, h( ξ ) h 1 ( ξ ) [ ] =, whees fo = 0 n, whee h 0 indictes disk thickness t the inne dius of the disk. With the fom of disk pofile function fo polic thickness vition, unifom thickness disk is otined y setting n = 0 nd linely vying thickness (tpe) is otined y setting k = 1. The pofiles consideed in this study e deived y using constnt volume citei, which would help to chcteize the pefomnce of the disk. The thickness of the unifom disk is tken s 5% of its oute dius nd coesponding to this volume, othe disk pofiles e clculted. It is futhe ssumed tht fo disks of vying thickness, the tip thickness is 1% of oute dius. This pticul ssumption is indeed necessy to detemine the geomety pmetes n nd k, s shown in Figue 1. Figue. Rotting disk unde vious tempetue distiution pofiles fo () specified oute sufce tempetue (T ), nd () specified inne sufce tempetue (T ). The dimensionless ngul speed, ω y ρ σ y coesponding to the onset of yielding is defined s nomlized limit ngul speed ( ω y ) nd consideed s the design pmete fo disks unde unifom tempetue envionment. The nlysis is lso cied out fo vious vying tempetue pofiles following unifom, line, exponentil nd polic tempetue distiution, s expessed in eqution (14) nd shown in Figue () nd (). It my e noted tht the mthemticl eltions fo the fou types of tempetue distiutions e not identicl with tht of the thickness distiution eltions. Unifom: T( ) = T o T (14) Line: T ( ) = T + ( T T )ξ (14) Exponentil: T ( ) ( T T ) ln( ) ln( ) Polic: T ( ) T + ( T T ) ξ = (14c) = (14d) The tempetue field ptten is function of tempetue oundy conditions lso. Two diffeent, incesing nd decesing, tempetue oundy conditions e ssumed i) T = 0 C, T = 1000 C nd ii) T = 1000 C, T = 0 C, whee T nd T e the inne nd oute sufce tempetue of the isotopic disk t = nd = espectively. The tempetue distiution pofiles, shown in Figue () nd (), coespond to these two oundy conditions. In the definition of dimensionless ngul speed ω y ρ σ y, it is ssumed tht the disk mteil popeties E, ρ, σ y, α nd ν emin constnt in the vile theml field T ( ). The peliminy pt of the study is cied out sed on this ssumption ut in the lte pt, when the disk mteil popeties e ssumed to e function of theml field T ( ), dimensionl vlue of limit ngul tip speed, ω y in (m/s) is used s the design pmete. The numeicl nlysis of the pesent study is cied out y consideing system pmete vlues s, E = 10 GP, ν = 0.3, ρ = 7800 kg m, 3 6 α = nd initil yield stess σ 350 MP. The nume of coodinte functions used in the C 1 y =

6 35 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp pesent study to ppoximte the displcement field, s expessed in eqution (10), is 11. This pticul nume hs een ived t fte cying out the necessy convegence tests on nomlized limit ngul speed fo unifom disk. The vlidtion plots fo the displcement nd stess (dil nd tngentil) field e pesented in Figue 3() to (e) which shows tht the geement of the esults of computtion with the sme using the esults of Vivio nd Vullo (007) is vey good. Detil discussions on these figues e epoted in Vivio nd Vullo (007) nd hence they e not epeted hee to mintin evity. Figue 3. Compison plots fo displcement (u), dil σ nd tngentil stesses σ t with the esults of Vivio nd Vullo (007) fo: () solid tpe disk, () nnul tpe disk, (c) nnul tpe disk sujected to tempetue gdient nd (d) nnul tpe disk with density vition (e) nnul tpe disk sujected to oth density nd tempetue gdient. Figue 4. Plots fo nomlized limit ngul speed ( ω y ) with oute sufce tempetue (T ) hving diffeent tempetue distiutions: () unifom, () line, (c) exponentil nd (d) polic. 3.1 Effect of tempetue distiutions: The effect of tempetue distiutions (T ) on the nomlized limit ngul speed y ω is pesented in Figue 4() to (d) fo nnul disks nd in ech figue, vition is shown fo fou diffeent disk geometies. The inne sufce tempetue T is set to 0 C nd the spect tio of disk (/) is tken s 0.8. It is ssumed tht mteil popeties E, α, ρ nd ν emin constnt in the vile theml field of the disk. It is oseved tht with incese in oute sufce tempetue T, the nomlized limit ngul speed deceses fo ll type of disk geometies. Howeve, fo unifom tempetue distiution the nomlized limit ngul speed emins constnt. Amongst the diffeent geometies, nomlized limit ngul speed is mximum fo exponentil disk geomety. Agin, in cse of exponentil tempetue distiution, the decese of nomlized limit ngul

7 36 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp speed is pominent fo diffeent disk geometies. It is oseved in the nlyticl study tht mong ll the physicl pmetes, coefficient of theml expnsion α hs the pedominnt effect on the vition in ω y. A simil study on the nomlized limit ngul speed vition with tempetue is indicted in Figue 5() to (d), fo the pescied disk geometies nd tempetue distiutions. Hee, the inne sufce tempetue T is vied ut the oute sufce tempetue T is kept fixed t 0 C. In Figue 5(), the nomlized limit ngul speed is constnt fo unifom tempetue distiution s tht of the pevious cse while in Figue 5() to (d), the nomlized limit ngul speed inceses with incese in inne sufce tempetue T fo diffeent disk geometies fo ll types of tempetue distiutions. These figues show tht the nomlized limit ngul speed is mximum fo exponentil disk geomety. The esults lso indicte tht fo exponentil tempetue distiution; the incese of nomlized limit ngul speed is pominent fo diffeent disk geometies. Figue 5. Plots fo nomlized limit ngul speed ( ω y ) with oute sufce tempetue (T ) hving diffeent tempetue distiutions: () unifom, () line, (c) exponentil nd (d) polic. Figue 6. Distiution of dil ( σ ) nd tngentil ( t ) to () pue centifugl loding due to ottion only, () comined loding with highe inne sufce tempetue ( ) σ stess field coesponding to yield limit stte in n nnul disk sujected comined loding with highe oute sufce tempetue ( T ). To exploe the phenomenon in gete detil dil ( σ ) nd tngentil ( t ) T nd (c) σ stess fields of n nnul disk (/=0.8) is plotted in Figue 6() to (c) coesponding to the yield limit stte, ttined y centifugl nd the two types of theml loding. Figue 6() indictes the stess sttes coesponding to limit ngul speed, nd Figues 6(, c) consides tempetue field effect coesponding to highe vlue of inne nd oute sufce tempetue, espectively. The dil stess ( σ ) field is lmost simil in ll the cses ut thee is vition in tngentil stess ( σ t ). The eduction in tngentil stess field towds the oute dius is most pominent in Figue 6(c), which coespond to the cse of highe oute sufce tempetue ( T ), whees in Figue 6() the decement is minimum. The tngentil stess field coming fom pue centifugl loding, s shown in Figue 6(), ppes to e in n intemedite stte. Hence the ntue of vition in ( ω y ), s oseved in Figues 4 nd 5 is quite justified fo comined centifugl nd theml loding. The effect of disk geometies on ehvio of themlly loded disks is lso investigted fo the nomlized limit ngul speed vition with tempetue nd shown gin in Figue 7() to (d). The individul figues e fo pticul disk geomety nd in ech figue the inne sufce tempetue T is set to 0 C nd oute sufce tempetue T is vied up to 1500 C. Fo ll

8 37 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp tempetue distiutions the nomlized limit ngul speed ( ω y ) initite fom the sme initil vlue nd deceses with incese in the sufce tempetue of the othe oundy. Howeve, fo unifom tempetue distiution, the cuve emins constnt. It is lso seen tht the decese of nomlized limit ngul speed is moe fo exponentil tempetue distiution, in ll the cses of diffeent thickness pofiles. Agin, when the fou figues e comped, it is oseved tht the initil vlue of nomlized limit ω is mximum fo exponentil disk geomety nd minimum fo unifom disk geomety. ngul speed y In Figues 8() to (d), the vitions in nomlized limit ngul speed with incesing inne sufce tempetue T e lso pesented fo the pescied disk geometies nd tempetue distiutions. Now, the oute sufce tempetue T is set to 0 C while the inne sufce tempetue T is vied up to 1500 C. In these figues lso, the initil vlue of nomlized limit ngul speed is sme fo ll tempetue distiutions nd it inceses with incese in the inne sufce tempetue, except fo unifom tempetue distiution. Figue 7. Effect of oute sufce tempetue distiutions T on nomlized limit ngul speed geometies: () unifom, () tpe, (c) exponentil nd (d) polic. ω fo diffeent disk y Figue 8. Effect of inne sufce tempetue distiutions T on nomlized limit ngul speed geometies:. () unifom, () tpe, (c) exponentil nd (d) polic. ω fo diffeent disk y Figue 9. Vition of disk mteil popeties ρ, σ y nd E with tempetue: () density, σ y nd (c) Elsticity modulus (E) fo stuctul steel with gde S350GD+Z t diffeent specified tempetues. ρ () yield Stess

9 38 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp In the next pt of the study it is ssumed tht the disk mteil popeties ρ, σ nd E e functions of theml field T ( ), nd hence limit ngul tip speed, ω y in (m/s) is used s the design pmete. The vitions in popeties e shown in Figue 9 fo stuctul steel with gde S350GD+Z, following Outinen nd Mkelinen (00). The effect of density vition with ρ = ρo 1 3αT, gphicl epesenttion of which is shown in Figue 9(). tempetue is otined y using the eltion [ { }] The figue coesponds to diffeent. Vition in yield stess y 6 α = , ut fo othe vlues of theml expnsion coefficient, the effect would e σ nd elsticity modulus (E) with tempetue field is otined y using est fit cuves on C 1 the tulted vlues of Outinen nd Mkelinen (00). 3. Effect of tempetue on the vition of density: Figue 10() to (d) show the effect of vious disk geometies nd tempetue distiutions on the limit ngul tip speed ( ω y ), tking the tempetue effect of density vition in to considetion. The inne sufce tempetue T is set to 0 C nd oute sufce tempetue T is vied up to 1600 C. As cn e seen, with incese in T, the limit ngul tip speed deceses fo ll the tempetue distiutions. But in cse of unifom tempetue distiution, the cuve emins constnt. The tempetue effect on density is ppently not much ponounced in Figues 10() to (d), ut fo highe vlues of theml expnsion coefficient, the effect is found to e moe. The effect of density vition with tempetue fo the pescied disk geometies nd tempetue distiution e lso depicted in Figue 11() to (d). In this cse, the oute sufce tempetue T is set to 0 C nd the limit ngul tip speed vition with incesing inne sufce tempetue T is plotted. The vition of density is govened y the sme eltion of the pevious cse, s shown in Figue 9(). As the inne sufce tempetue inceses the limit ngul tip speed inceses except fo unifom tempetue distiution just like in the pevious cse. y Figue 10. Effect of density vition on limit ngul tip speed fo diffeent tempetue distiutions of oute sufce T : () unifom, () line, (c) exponentil nd (d) polic tempetue. Figue 11. Effect of density vition on limit ngul tip speed fo diffeent tempetue distiutions of inne sufce T : () unifom, () line, (c) exponentil nd (d) polic tempetue. 3.3 Effect of tempetue on yield stess: The effect of tempetue on yield stess is studied consideing othe mteil popety vlues E, α, ρ nd s constnt. The nlysis is cied out fo fou diffeent tempetue distiution pofiles nd disk pofiles. Fo ech of the non-unifom tempetue distiution pofiles s pesented gphiclly in Figue (), the tempetue oundy condition is ssumed s T = 0 C nd T = 1000 C. The vition of yield stess with tempetue is ledy epoted in Figue 9(). Figues 1() nd () shows the vition of yield stess with the disk dius using T s pmete fo disk with line nd

10 39 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp polic tempetue distiutions. The figues e deived fom the tempetue vition with dius nd yield stess vition with tempetue, i.e., fom the comintion of Figue () nd Figue 9(). It cn e seen tht in oth the cses the yield stess deceses s dius of the disk inceses nd oute sufce tempetues inceses. Figue 1. Yield Stess vition in the disk fo () line tempetue distiution nd () polic tempetue distiution. Figue 13. Vition of von Mises stess ( σ vm ) fo polic tempetue distiution pofile fo unifom disk t () ω = 0 d s nd () ω = 1000 d s, t fou diffeent oute sufce tempetues ( T ). Figue 14. Vition of von Mises stess ) ( vm σ fo polic tempetue distiution pofile fo () unifom disk nd () polic disk, t fou diffeent ngul speeds.

11 40 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp The effect of tempetue vition on von Mises stess is studied fist nd the stess distiution with dius is plotted in Figue 13() nd () fo unifom disk with polic tempetue distiution pofile. The study is cied out fo two diffeent ngul speeds ω = 0 d s nd ω = 1000 d s espectively. It is oseved tht von Mises stess ( σ vm ) fist deceses towds the inne dius of the disk nd then emins constnt fo ll the specified oute sufce tempetues nd gin thee is n incese in von Mises stess ( σ vm ) towds the oute dius of the disk. It is lso oseved tht the induced von Mises stess pofile incese with oute sufce tempetue vlues. Fo the two diffeent cses of sttic nd otting disk, the von Mises stess is highe fo the otting one, eing ppent in Figue 13(). In the next study, the vition of von Mises stess fo diffeent vlues of ngul speeds is consideed nd shown in Figue 14() nd (). In this cse, tempetue distiution is ssumed to e polic nd the two figues pesent esults fo unifom nd polic thickness disks. It is oseved tht von Mises stess ( σ vm ) inceses with incese in ngul speeds ut the incese is significntly lowe in polic disk geomety s comped to unifom disk geomety. Figue 15. Plots of limit peiphel speed with oute sufce tempetue ( T ) hving diffeent tempetue distiutions: () unifom, () line, (c) exponentil nd (d) polic tempetue. A compison of the induced stess pofile of von Mises stess ( σ vm ) due to themo-mechnicl loding with the llowle yield stess pofile (s shown in Figue 1) would estlish the limit ngul speed fo ny given tempetue distiution nd oundy conditions. This ngul speed is not menle fo nomliztion y the pmete ω y ρ σ y, s σ y is field vile hee. Hence this speed is clled s limit peiphel speed nd is pesented in dimensionl fom. Figue 15() to (d) show the vition of limit peiphel speed with T fo diffeent disk geometies nd tempetue distiutions. Fom these figues it is illustted tht, thee is fll of limit peiphel speed with incese in oute sufce tempetue nd this eduction is moe pominent fte 400 C. The osevtion is suppoted y the fct tht eyond this tempetue the yield stess lso flls pidly with tempetue, s shown in Figue 1. Figue 16. Vition of Elsticity modulus long the dius of the disk hving diffeent tempetue distiutions () line nd () polic tempetue.

12 41 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp Effect of tempetue on elsticity modulus: The effect of tempetue on elsticity modulus is studied consideing the numeicl vlues of the system pmete s σ y = 350 MP, ν = 0. 3, ρ = kg m nd α = C 1. Fo ech of the non-unifom tempetue distiution pofile s pesented gphiclly in Figue (), the tempetue oundy condition is ssumed s T = 0 C, T = 1000 C. The vition of elsticity modulus with tempetue is otined fom the expeimentl vlues epoted y Outinen nd Mkelinen (00) nd pesented elie in Figue 9(c). The vition of elsticity modulus with dius is shown in Figue 16() nd () fo disk hving line nd polic tempetue distiutions t vious specified oute sufce tempetues. It could e oseved tht fo oth cses the elsticity modulus declined with dius nd this tend ecomes moe pedominnt with incese in the vlues of oute sufce tempetues. Figue 17() to (d) indictes the vitions of nomlized limit ngul speed ω ρ σ ) ( y y, with oute sufce tempetue T fo diffeent disk geometies nd in ech figue the effect is shown fo fou diffeent tempetue distiutions. These figues indicte fll of nomlized limit ngul speed with incesing oute sufce tempetues except fo unifom tempetue distiution, whee it emins constnt. The eduction is moe pominent in exponentil tempetue distiution. It my e noted tht ω hs een used s design pmete ecuse density nd yield stess is ssumed to e constnt in this nlysis. y Figue 17. Plots of nomlized limit ngul speed with oute sufce tempetue ( T ) hving diffeent tempetue distiutions: () unifom, () line, (c) exponentil nd (d) polic tempetue. Figue 18. Simultneous vition of density nd elsticity modulus (E) with tempetue. 3.5 Effect of tempetue on simultneous vition of density nd elsticity modulus: The influence of tempetue on oth density nd elsticity modulus is consideed in this section with the sme vition of density nd elsticity modulus with tempetue, s consideed peviously. The simultneous vition of density nd elsticity modulus with tempetue is pesented in Figue 18 once gin, eing the simultneous plots of Figues 9() nd 9(c) up to 1000 C. The influence of tempetue on oth density nd elsticity modulus with tempetue fo the pescied disk geometies nd tempetue distiution e depicted in Figues 19() to (d). In this cse, the tempetue oundy condition is estimted t T = 0 C, nd oute sufce tempetue T is vied up to 1000 C. The vition of limit peiphel tip speed ω y with incesing oute sufce tempetue T is plotted, fo ech of the fou tempetue distiution pofiles. The effects of fou diffeent disk geometies e pesented in ech of these figues. Thee is decese in limit peiphel speed when oute sufce

13 4 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp tempetue inceses ut exception exists fo unifom tempetue distiution. The decese of limit peiphel speed is moe fo exponentil tempetue distiution, in line with the osevtions of pevious cses, ut the effect is slightly moe pominent due to the comined influence of tempetue on oth density nd elsticity modulus. Figue 19. Plots of limit peiphel speed with oute sufce tempetue ( T ) hving diffeent tempetue distiutions: () unifom, () line, (c) exponentil nd (d) polic tempetue. Finlly, 3D nd contou plots of von Mises stess showing its simultneous vition with ottionl speed nd tempetue fo fou diffeent disk geometies is pesented in Figues 0 () to (d). In these figues, tempetue field is ssumed to hve line vition, with the tempetue oundy conditions: T = 0 C nd T =pescied. Howeve, the physicl pmetes of the disk mteil e ssumed to e constnt, in these plots. It is oseved tht with incese in T s well s ottionl speed, von Mises stess inceses, ut theml effect is the insignificnt. Anothe set of plots with high vlue of theml co-efficient would evel the theml stess effect, ut such plot is omitted to mintin evity. () () (c) (d) Figue 0. 3D nd contou plots of von Mises stess showing it s vition with ottionl speed (d/sec) nd T ( o C) fo vious disk geometies: () unifom, () tpe, (c) exponentil nd (d) polic.

14 43 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp Conclusions This ppe investigtes the otting disk ehvio fo comined theml lod nd ottionl ineti effects y using n ppoximte solution. The estimted solution of the unknown displcement field fom the govening eqution is otined y supposing seies estimtion following Glekin s pinciple. The poposed method hs een vlidted successfully with pulished mteil nd eltively good geement is documented. The influence of disk geomety nd tempetue field vition on the pefomnce of otting disks is consideed nd nomlized vlue of limit ngul speed is funished. Tempetue effect on yield stess is lso studied nd the limit peiphel speed fo ny given tempetue distiution nd oundy conditions is estlished unde themo-mechnicl loding in dimensionl fom. Also the influence of tempetue field on the mteil popeties, including density, elsticity modulus s well s comintion of oth density nd elsticity modulus is consideed in the pesent ppe nd the numeicl esults e funished. A futue study in this diection would give moe undestnding of the design of otting disks using vious industil pplictions. These esults e funished gphiclly s design monogms which might pove helpful fo the pcticing enginees. Acknowledgement The fist utho is gteful to Council of Scientific & Industil Resech (CSIR) of the Govenment of Indi fo the fellowship gnt (File No. 09/096(0810)/014-EMR-I) to cy out the esech wok. Nomencltue, Disk s inne nd oute dii c Unknown coefficients vecto { } i e,e θ Elstic stin in dil nd tngentil diections E Mteil s elsticity modulus f Lod vecto { } h 0,h Thickness t the oot nd thickness t ny dius of the disk [ K] Stiffness mtix n, k Pmetes tht contol the disk s thickness vition, θ, z Rdil, tngentil s well s xil diections Pmete ( ) T ( ) Tempetue distiution with dius T, T Disk s inne s well s oute sufce tempetue u Rdil displcement field U, V Stin enegy nd disk s wok potentil α Coefficient of theml expnsion δ Vitionl opeto ε,ε θ Totl stin in dil nd tngentil diections ε Theml stin φ Goup of othogonl functions employed fo the i ppoximtion of displcement field Stt function of othogonl set of functions φo ν Poisson s tio ρ Mteil s density σ Yield stess of the disk mteil y σ,σ θ Rdil nd tngentil stesses σ von Mises stess vm ω Disk s dimensionl ngul speed ω Dimensionl vlue of limit ngul speed y

15 44 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp ω y Nomlized limit ngul speed, ω ρ σ y ω y Limit peiphel speed ξ Nomlized co-odinte in dil diection, ( ) ' Diffeentition w..t. nomlized coodinte ξ y Refeences Byt M., Sleem M., Shi B.B., Hmoud A.M.S nd Mhdi E., 008. Anlysis of functionlly gded otting disks with vile thickness. Mechnics Resech Communictions. Elsevie, Vol. 35, pp Bhowmick S., Mis D. nd Sh K.N., 010. A pmetic study of extenlly loded otting disks unde shink fit. Poceedings of the Institution of Mechnicl Enginees, Pt C: Jounl of Mechnicl Engineeing Science. Pofessionl Engineeing Pulishing, vol. 4, pp Bhowmick S., Mis D. nd Sh K.N., 010. Vitionl fomultion sed nlysis on gowth of yield font in high speed otting solid disks. Intentionl Jounl of Engineeing, Science nd Technology. Multicft, Vol., pp Esln A.N., 003. Elstic plstic defomtions of otting vile thickness nnul disks with fee, pessuized nd dillyconstined oundy conditions. Intentionl Jounl of Mechnicl Sciences. Elsevie, Vol. 45, pp Gme U., Tesc s yield condition nd the otting disk. Jounl of Applied Mechnics. Tns ASME, Vol. 50, pp Gme U., Elstic plstic defomtion of the otting solid disk. Ingenieu-Achiv. Spinge, Vol. 54, pp Gme U., Stess distiution in the otting elstic plstic disk. ZAMM- Jounl of Applied Mthemtics nd Mechnics. Wiley, Vol. 65, pp Hojjti M.H. nd Jfi S., 008. Semi-exct solution of elstic non-unifom thickness nd density otting disks y homotopy petution nd Adomin s decomposition method Pt I: elstic solution. Intentionl Jounl of Pessue Vessels nd Piping. Elsevie, Vol. 85, pp Nie G.J. nd Bt R.C., 010. Stess nlysis nd mteil tiloing in isotopic line themoelstic incompessile functionlly gded otting disks of vile thickness. Composite Stuctues. Elsevie, Vol. 9, pp Outinen J. nd Mkelinen P., 00. Mechnicl popeties of stuctul steel t elevted tempetues nd fte cooling down. Second Intentionl Wokshop (Stuctues in Fie), Chistchuch. Peng X.L. nd Li X.F., 010. Theml stess in otting functionlly gded hollow cicul disks. Composite Stuctues. Elsevie, Vol. 9, pp Rttn M., Chmoli N. nd Singh S.B., 010. Ceep nlysis of n isotopic functionlly gded otting disk. Intentionl Jounl of Contempoy Mthemticl Sciences, Vol. 5, pp Rees D. W. A., Elstic plstic stesses in otting disks y von-mises nd Tesc. ZAMM- Jounl of Applied Mthemtics nd Mechnics. Wiley, Vol. 79, pp Thompson A.S., Stesses in otting discs t high tempetue. Jounl of Applied Mechnics. Tns ASME, Vol. 13, pp Tutuncu N. nd Temel B., 009. A novel ppoch to stess nlysis of pessuized FGM cylindes, disks nd sphees. Composite Stuctues. Elsevie, Vol. 91, pp Vivio F. nd Vullo V., 007. Elstic stess nlysis of otting conveging conicl disks sujected to theml lod nd hving vile density long the dius. Intentionl Jounl of Solids nd Stuctues. Elsevie, Vol. 44, pp Vivio F. nd Vullo V., 008. Elstic stess nlysis of non-line vile thickness otting disks sujected to theml lod nd hving vile density long the dius.. Intentionl Jounl of Solids nd Stuctues. Elsevie, Vol. 45, pp Wue J. nd Schweize B., 010. Dynmics of otting themoelstic disks with sttiony het souce. Applied Mthemtics nd Computtion. Elsevie, Vol. 15, pp

16 45 Nyk nd Sh / Intentionl Jounl of Engineeing, Science nd Technology, Vol. 8, No., 016, pp Biogphicl notes Piymd Nyk completed he B.Tech in Mechnicl Engineeing fom College of Engineeing nd Technology, Bhunesw, Odish in 007 nd Mste in Mechnicl Engineeing in Mchine Design speciliztion fom Jdvpu Univesity in 013. She hs woked fo n IT industy fo two yes nd lso engged in teching fo two yes. Pesently she is woking s senio esech fellow unde CSIR Fellowship. He cuent e of esech is effect of theml loding on post elstic ehviou of otting disk. Kshinth Sh did his gdution in 198 fom the Mechnicl Engineeing Deptment of Jdvpu Univesity. He joined cdemi s lectue in 1987, fte seving industy fo five yes. M. Sh teches mchine elements nd system design in unde-gdute level nd in the post-gdute level he dels with stess, defomtion nd dynmic nlysis of stuctul elements. Pesently he is woking with polems involving geometic nd mteil nonlineities. He hs supevised mny esech students nd hs pulished sevel jounl nd confeence ppes in his cdemic cee. Received July 015 Accepted Mch 016 Finl cceptnce in evised fom June 016