Creep modeling in functionally graded rotating disc of variable thickness

Transcription

1 Jounl of Mechnicl Science nd Technology 4 () () ~3 DOI.7/s Ceep modeling in functionlly gded otting disc of vile thickness D. Deepk, V. K. Gupt,* nd A. K. Dhm 3 Deptment of Mechnicl Engg., R.I.E.I.T.,Rilmj, 44533, Indi Mech. Engg., UCoE, Punji Univesity, Ptil-47, Indi 3 Deptment of Physics, Punji Univesity, Ptil-47, Indi (Mnuscipt Received July, 9; Revised My 3, ; Accepted July 7, ) Astct Ceep ehvio of otting discs mde of functionlly gded mteils with linely vying thickness hs een investigted. The discs unde investigtion e mde of composite contining silicon cide pticles in mtix of pue luminum. The ceep ehvio of the composite hs een descied y theshold stess sed ceep lw y ssuming stess exponent of 5. The effect of imposing line pticle gdient on the distiution of stesses nd stin tes in the composite disc hs een investigted. The study indictes tht with incese in pticle gdient in the disc, the dil stess inceses thoughout the disc, whees the tngentil nd effective stesses incese ne the inne dius ut decese ne the oute dius. The stedy stte stin tes in the composite disc, hving gdient in the distiution of einfocement, e significntly lowe thn tht oseved in disc hving unifom distiution of einfocement. Keywods: Functionlly gded mteil; Modeling; Rotting disc; Stedy stte ceep Intoduction Rotting discs povide n e of esech nd study due to thei vst utiliztion in otting mchiney such s stem nd gs tuine otos, tuo genetos, pumps, compessos, flywheels, utomotive king systems, ship popelles nd compute disc dives [-4]. In most of these pplictions, the disc hs to opete unde elevted tempetue nd is simultneously sujected to high stesses cused y disc ottion t high speed [5]. As esult of sevee mechnicl nd theml lodings, the mteil of disc undegoes ceep defomtions, theey ffecting pefomnce of the system [, 6-8]. As n exmple, in the tuine oto, thee is lwys possiility tht the het fom the extenl sufce is tnsmitted to the shft nd then to the eings, which hs dvese effects on the functioning nd efficiency of the oto [9]. Unde high tempetue, otting disc mde of monolithic mteil my not pefom well. The excellent mechnicl popeties like high specific stength nd stiffness, nd high tempetue stility offeed y luminum mtix composites einfoced with SiC pticles, whiskes o fies mke them suitle fo otting disc pplictions exposed to elevted tempetue [-]. In ecent yes, the polem of ceep in otting discs This ppe ws ecommended fo puliction in evised fom y Associte Edito Seong Beom Lee * Coesponding utho. Tel.: , Fx: E-mil ddess: guptvk_7@yhoo.co.in KSME & Spinge mde of functionlly gded mteils (FGMs), sujected to sevee mechnicl nd theml lods, hs ttcted the inteest of mny eseches. In FGMs the volume fction of one o moe constituent mteil is vied continuously s function of position long cetin dimension(s) [, 3]. These mteils e designed nd developed to opete in high tempetue envionments. The stedy stte ceep ehvio of otting disc mde of functionlly gded (FG) isotopic Al- SiCp (suscipt p fo pticle) hving constnt thickness nd opeting unde constnt tempetue ws studied y Singh nd Ry [4]. The study ssumed linely decesing distiution of SiCp fom the inne to the oute dius of disc. It is eveled tht the stedy stte ceep esponse of FGM disc is significntly supeio s comped to simil disc ut contining unifom distiution of SiCp. Gupt et l. [, 5] nlyzed ceep ehvio of otting disc mde of FG Al-SiCp hving unifom thickness nd opeting unde dil theml gdient. The study indictes tht fo the ssumed line distiution of SiCp, the stedy-stte stin tes in n FGM disc e significntly lowe thn tht oseved in n isotopic disc hving unifom distiution of SiCp. Byt et l. [6] cied out themo-elstic nlysis of n FG otting disc with smll nd lge deflections nd oseved tht fo pticul vlues of gding index (n) of mteil popeties mechnicl esponses in FG disc cn e smlle thn oseved in homogeneous disc. Byt et l. [9] otined elstic solutions fo xisymmetic otting discs mde of FGM with vile thickness. Kodkheili nd Nghddi [7] otined semi-nlyticl

2 D. Deepk et l. / Jounl of Mechnicl Science nd Technology 4 () () ~3 themo-elstic solution fo hollow nd solid otting xisymmetic disc mde of FGM unde plne stess condition. The litetue evels tht the stesses in otting discs (nnul o solid) hving vile thickness, with thickness decesing fom the cente towds the peiphey, e much lowe thn those oseved in unifom-thickness disc opeting t the sme ngul velocity [8-]. By employing vile thickness disc the plstic limit ngul velocity inceses nd the mgnitude of stesses nd defomtions in the disc educes []. Jhed et l. [] oseved tht the use of vile thickness disc helps in minimizing the weight of disc in eospce pplictions. Gupt et l. [] lso oseved tht otting disc whose density nd thickness decese dilly is on the sfe side of design in compison to flt disc hving vile density. Deepk et l. [3] noticed tht the stesses nd stin tes in otting disc mde of isotopic composite cn e significntly educed y selecting linely vying thickness pofile s comped to hypeolic o constnt thickness disc pofile. The litetue consulted so f evels tht the studies petining to ceep in otting disc mde of FGM, tht too of vile thickness, e the scnt. Futhe, the ceep pefomnce of composite disc with linely vying thickness is oseved to e supeio s comped to discs hving othe thickness pofiles. With these foethoughts, it is decided to investigte the stedy stte ceep, which occupies 3-4% of totl ceep life, in otting disc mde of functionlly gded Al-SiCp nd hving linely vying thickness. The content of SiCp in Al mtix is ssumed to vy linely with mximum SiCp content t the inne dius nd minimum SiCp content t the oute dius. A mthemticl model hs een developed to descie the stedy ceep ehvio of the composite disc. The model is used to investigte the effect of imposing vious kinds of line pticle (SiCp) gdients on the stedy stte ceep esponse of vile thickness disc.. Disc pofile nd ssumptions In this study, the ceep esponse hs een clculted fo otting discs hving linely vying thickness nd constnt thickness while keeping equl volume of oth the disc. The inne () nd oute () dii of oth the discs e ssumed, espectively, s 3.75 mm nd 5.4 mm while the thickness (t) of unifom thickness disc is tken s 5.4 mm. The dimensions of the disc selected in this study e simil to those epoted elie [8]. In cse of linely vying thickness disc, the thickness h( t ny dius is given y h( = h + c(, () ( h h ) whee the constnt c =, nd h nd h e, espectively, the disc thickness t = nd = ( ). Since the volume of linely vying thickness disc is equl to tht of constnt thickness disc, π h ( d = π ( ) t. () Sustituting h( fom Eq. () into the ove eqution nd simplifying, we get, 3( + ) t h ( + ) h =. (3) ( + ) Assuming h =3.97 mm nd sustituting = 3.75 mm, = 5.4 mm nd t = 5.4 mm in Eq. (3), we get h = 43. mm. The nlysis cied out in this study is sed on the following ssumptions: () Mteil of the disc is incompessile nd loclly isotopic, i.e., the popeties of the disc emin constnt t given dius in ll the diections ut chnge with the chnge in dius. () Stesses t ny point in the disc emin constnt with time, i.e., stedy stte condition of stess is ssumed. (3) Elstic defomtions in the disc e smll nd neglected s comped to ceep defomtions. (4) The xil stess ( z ) thoughout the disc emins zeo. 3. Distiution of einfocement The distiution of SiCp in the FGM disc deceses linely fom the inne to oute dius; theefoe, the density nd the ceep constnts will lso vy with dil distnce. The mount (vol%) of SiCp, V(, t ny dius, is given y ( ) ( = Vmx ( Vmx Vmin ) = δ δ, (4) ( ) V whee δ = ( V mx + δ ), ( Vmx Vmin ) δ =. ( ) Vmx nd V min e the mximum nd minimum SiCp contents, espectively, t the inne () nd the oute () dius of the disc. Accoding to ule of mixtue, the density ρ( of the FGM disc t ny dius my e witten s, [ V( ] ρm + V( ρd ( ρd ρm) V( ρ( = = ρm +, (5) whee, ρ m ( = kg/m 3 ) nd ρ d ( = 3 kg/m 3 ) e,

3 D. Deepk et l. / Jounl of Mechnicl Science nd Technology 4 () () ~3 3 espectively, the densities of pue luminum mtix nd silicon cide pticles [4, 5]. Sustituting V ( fom Eq. (4) into Eq. (5), ( ρd ρm )( δ δ. ρ( = ρm + = Aρ Bρ., (6) whee, A nd, B ρ ρ δ = ρm + ( ρd ρm ). δ ( ρd ρm ) =. The vege pticle content in the disc (V v ) cn e expessed s, V v = πh( V ( d π ( ) t. (7) Sustituting h( nd V (, espectively, fom Eq. () nd Eq. (4) into Eq. (7), the minimum SiCp content (V min ) in the disc my e otined s, V min V = mx 3 ( h h + c 5 c+ c+ h + 3 c) 3V 3 3 ( h + c+ h + c c h c) 4. Selection of ceep lw 3 v t( ) In luminum-sed composites, undegoing stedy stte ceep, the effective ceep te (ε & ) is elted to the effective stess ( ) though well documented ceep lw given y [6, 7]: n & ' Q = A ( ) ε exp, (9) E RT whee the symols A, n, Q, E, R, T nd ( ) denote, espectively, the stuctue dependent pmete, tue stess exponent, tue ctivtion enegy, tempetue-dependent Young s modulus, gs constnt, opeting tempetue nd theshold stess (i.e., the minimum stess equied to initite ceep). The tue stess exponent (n) in Eq. (9) is usully selected s 3, 5 nd 8, which coespond to thee well-documented ceep cses fo metls nd lloys: (i) n = 3 fo ceep contolled y viscous glide pocesses of disloction, (ii) n = 5 fo ceep contolled y high tempetue disloction clim (lttice diffusion), nd (iii) n = 8 fo lttice diffusion-contolled ceep with (8) constnt stuctue [7]. Though, some of the investigtos [8-3] hve used tue stess exponent of 8 to descie stedy stte ceep in Al-SiCp,w (suscipt p fo pticle nd w fo whiske composites ut lge nume of othe investigtos [3-39] hve suggested tht stess exponent of eithe ~3 o ~5, the thn 8, povides ette desciption of expeimentl stedy stte ceep dt oseved fo discontinuously einfoced Al-SiC composites. As consequence, stess exponent of 5 is used in this study to descie stedy stte ceep ehvio of the composite discs. 5. Estimtion of ceep pmetes The ceep lw given in Eq. (9) my ltentively e witten s [ M { ( }] n & ε =, () ( / n Q whee M A ' ( ) = exp is ceep pmete. E RT The ceep pmetes M ( nd ( given in Eq. () e dependent on the type of mteil nd e lso ffected y the tempetue (T) of ppliction. In composite, the dispesoid size (P) nd the content of dispesoid (V) e the pimy mteil viles ffecting these pmetes. In this study, the vlues of M nd hve een extcted fom the expeimentl unixil ceep esults epoted fo Al-SiC P [3]. Fo this pupose the individul set of ceep dt epoted in [3] e / 5 epesented on ε& vesus line plots s shown in Figs. ()-(c). The slopes nd intecepts of these gphs yield the vlues of ceep pmetes M nd s epoted in Tle. This ppoch of detemining the theshold stess ( ) is known s line extpoltion technique [4]. To void vition due to systemtic eo, if ny, in the expeimentl esults, the ceep esults fom single souce [3] hve een used. /5 The ε& vesus plots, coesponding to the oseved expeimentl dt points of Al- SiC P [3] fo vious comintions of SiC P size, SiC P content nd opeting tempetue, exhiit n excellent lineity, s evident fom Figs. ()-(c). The coefficient of coeltion fo these plots is oseved in excess to.96 s evident fom Tle. In the light of these fcts, the choice of stess exponent n = 5, to descie the stedy stte ceep ehvio of Al-SiC P, is justified. Fo the ceep pmetes given in Tle, the egession nlysis hs een pefomed in DATAFIT softwe to estimte the vlues of M nd t ny dius of the FGM disc in tems of P, V nd T. The developed egession equtions e, M ( =.88, () P T V ( ( =.84P.3T +.85V (.7 () + To veify the ccucy of the egession equtions developed, the ceep pmetes coesponding to oseved expeimentl

4 4 D. Deepk et l. / Jounl of Mechnicl Science nd Technology 4 () () ~3 Tle. Ceep pmetes used fo Al-SiCp in the pesent study. P (µm) T (K) V (vol%) M (s -/5 / MP) 4.35E-3 8.7E E E-3.63E-3.7E-3.63E-3 4.4E-3 5.9E-3 (MP) Coefficient of Coeltion () () () () (c) Fig.. Vition of /5 ε& vesus in Al-SiCp fo diffeent: () SiCp sizes, () vol % of SiCp (c) Opeting tempetues [3]. dt points, s given in Tle, hve een clculted fom Eqs. () nd (). The ceep pmetes thus otined hve een sustituted in ceep lw given y Eq. () to estimte the stin tes coesponding to the epoted expeimentl stess levels [3] fo vious comintions of P, V nd T. The stin tes thus estimted hve een comped with the expeimentlly oseved stin tes [3] s shown in Figs. ()-(c). An excellent geement is oseved etween the theoeticl nd the expeimentl stin tes. In otting disc mde of FGM, with SiC P content vying long dius s V(, oth the ceep pmetes M nd will lso vy long the dil diection. In the pesent study, the size of SiC P (P) is tken s.7 µm nd the opeting tempetue (T) is ssumed s 63 K ove the entie disc. Thus, fo given FGM disc with known pticle gdient, oth the ceep pmetes will e functions of dil distnce only. The vlues of M ( nd ( t ny dius, could e estimted (c) Fig.. Compison of expeimentl [3] nd estimted stin tes in Al-SiCp fo diffeent: () sizes of SiCp, () vol% of SiCp nd (c) Opeting tempetues. y sustituting the pticle content V( t the coesponding loctions into Eqs. () nd () espectively. 6. Mthemticl fomultion It is ssumed in this study tht the disc mteil is isotopic nd yield ccoding to von-mises yield citeion [4] given y, ( ij ) = =, (3) f J k whee f ( ij ) is the potentil function, k is constnt nd J is given y J = [( 3 ) + ( 3 ) + ( ) ]. (4) 6 The stin incement ( ij ) is elted to the potentil function, f ), though the ssocited flow ule given y, ( ij f ( ij ) ij = dλ ij, (5)

5 D. Deepk et l. / Jounl of Mechnicl Science nd Technology 4 () () ~3 5 whee d λ is the popotionlity fcto tht depends upon ij, d ij nd ε ij, pt fom stin histoy ecuse of stin hdening. Sustituting the yield citeion given y Eq. (3) into Eq. (5), the following constitutive equtions e otined in tems of pincipl stin incements d ε, d ε nd d ε 33 nd pincipl stesses, nd 33 : ( + 33) = dλ, 3 ( + 33) = dλ, (6) 3 ( + ) 33 = 33 dλ. 3 The effective stess ( ) nd stin incement ( d ε ) e given y, / = [( 33) + ( 33 ) + ( ) ], (7) = [( 3 / ) + ( 33) + ( 33 ) ] (8) Sustituting the vlues of d ε, d ε nd d ε 33 fom Eqs. (6) into Eq. (8) nd using Eq. (7), we otin, = dλ. (9) 3 Fom Eqs. (6) nd (9) we get the stin incements ( + d ε = 33), ( + d ε = 33), () ( + ) d ε 33 = 33. On integting the ove set of equtions (Eqs. ()), one gets the stin tes, ε & ε = & [ 33], ε & ε = & [ 33], () ε & ε 33 = & [ 33 ]. Let A nd A denote the es of tnsvese section of the disc element with oute dii nd, espectively, ut with the sme inne dius. The vlues of A nd A my e expessed s, A = hd. () A = hd The pol moment of e I nd I of these disc elements, with oute dii espectively s nd ut with the sme inne dius, is given y I = h d. (3) I = h d The vege tngentil stess in the disc ) is ( θv θ v = h d A θ. (4) The set of constitutive equtions (Eq. ()) fo ceep in n isotopic composite mteil unde ixil stte of stess (i.e. Z =) tkes the following fom when the efeence fme is long the pincipl diections, θ nd z [5]: & ε & ε = [ θ ], & ε & εθ = [ θ ], (5) & ε & ε z = [ θ ], whee & ε, & ε & θ, ε z nd, θ, z e, espectively, the stin tes nd the stesses in, θ nd z diections, s indicted y espective suscipts. The effective stess ( ), given y Eq. (7), in otting disc unde ixil stte of stess my e expessed s, [ ( ) ] θ + + θ =. (6) Sustituting the vlue of ε & fom Eq. () nd fom Eq. (6) into the fist eqution mongst set of equtions in Eq. (5), the dil stin te ecomes

6 6 D. Deepk et l. / Jounl of Mechnicl Science nd Technology 4 () () ~3 du & ε = & = d [{ x( } [x( ] x( + )] [ M ( { ( } ] n / (7) whee x = θ is the tio of dil nd tngentil stesses, u is the dil defomtion nd u& = du / dt is the dil defomtion te. Similly, the second eqution of Eq. (5), yields the tngentil stin te, u = & [ x( ] [{ x( } x( + )] [ M ( { ( } ] n & ε θ = /. (8) Dividing Eq. (7) y Eq. (8) nd integting the esulting eqution etween limits to, we get = φ( u& u& exp d, (9) whee u& is the dil defomtion te t the inne dius nd [x( ] φ ( =. [ x( ] Sustituting u& fom Eq. (9) into Eq. (8), we get, [ M ( { ( } ] n u& φ( [ x( ] exp d = [{ x( } x( + )] / Simplifying the ove eqution, the tngentil stess ( θ ) is otined s / u n ψ ( θ = & + ψ (, (3) M ( whee / n ψ ( ψ ( =, / [{ x( } x( + )] ( ψ ( =, (3) / nd, [{ x( } x( + )] / [{ x( } x( + )] = φ( ψ ( exp d. (3) [ x( ] Consideing the equiliium of foces cting on n element of the disc hving vying thickness, the foce equiliium eqution my e witten s [8, 4]. d [ h( ] h( θ + ρ( ω h( = (33) d whee ρ( is the density of composite disc. It is ssumed tht the disc is connected to the shft y mens of splines whee smll xil movement is pemitted. Theefoe, fee-fee condition pplies [6], i.e. = t nd = t. (34) = = Integting Eq. (33) etween limits to unde the imposed oundy conditions given in Eq. (34), we get, 4 4 ( h + c)( ) h( = 4 θ d ω Aρ I Bρ 5 5 c( ). 5 (35) Multiplying Eq. (3) y h(d nd integting the esulting eqution etween limits to, we get / n u& = A h d θv ( ) ψ ( ). (36) h( ψ ( d M ( Dividing Eq. (35) y A nd noting Eq. (4), we get θv = h( θ d A ω = Aρ I A ( h + c)( B ρ ) c( ) 5 Knowing θv, the tngentil stess ( θ ) is otined fte n sustituting the vlue of u & / fom Eq. (36) into Eq. (3) s follows: ψ ( A θv h( ψ ( d = h( ψ ( M ( d M ( θ + ψ (. (37) Integting Eq. (33) etween limits to, the dil stess is otined s h( θd ω Aρ I =. (38) h( ( + )( ) ( ) h c c B ω ρ 4 5

7 D. Deepk et l. / Jounl of Mechnicl Science nd Technology 4 () () ~3 7 The tngentil stess ( θ ) nd the dil stess ( ) t ny point in the composite disc e detemined fom Eqs. (37) nd (38) espectively. Using these vlues, the stin tes ε& nd ε& e clculted, espectively, fom Eqs. (7) nd (8). θ 7. Numeicl computtions Following the pocedue descied in Section 6, the stesses nd stin tes in the disc e estimted though n itetive numeicl scheme shown in Fig. 3. The itetion is continued untill the pocess conveges nd yields the vlues of stesses t diffeent points of the dius gid. Fo pid convegence 75% of the vlue of θ otined in the cuent itetion hs een mixed with 5% of the vlue of θ otined in the pevious itetion nd this modified vlue is used in the next itetion. ITER = Itetion no. [ θ] ITER [ θ] ITER- E RR = [ ] ITER- h = Limiting Vlue of ERR (=.) ITM = Mximum no. of itetion = 5 8. Results nd discussions A compute code, sed on the mthemticl fomultion pesented in Section 6, hs een developed to otin the distiution of stesses nd stedy stte ceep tes in vious FGM discs hving the sme vege SiCp content. Fo compison, the esults hve lso een otined fo non-fgm disc hving vege SiCp content equl to tht in FGM discs ut distiuted unifomly. 8. Vlidtion Befoe discussing the esults otined in this study, it is necessy to check the vlidity of nlysis cied out nd the softwe developed. Fo this pupose, the dil nd tngentil ceep stins hve een computed in otting steel disc y following the cuent nlyticl pocedue. The esults otined e comped with the ville expeimentl esults fo steel disc [43]. The opeting conditions, dimensions nd ceep pmetes used fo the steel disc e epoted in Tle. To estimte the vlues of pmetes M nd fo steel disc, the ceep lw given y Eq. () is integted etween limits of t fom to t to give t n ε = [ M ( )] f ( t dt, (39) ) Fig. 3. Numeicl scheme of computtion. whee ε is the effective stin nd f(t) is function of time t. Simil to the wok of Whl et l. [43], the function f(t) is ssumed to e unity duing the vlidtion pocess. Whl et l. [43] oseved tht fo steel disc the vege stin t the end of 8 hs nd men stess ( ) = 5,5 psi is.9 in/in, whees t men stess ( ) = 9,45 psi, the stin oseved is.9 in/in. Using the ove epoted vlues of stess nd stin in Eq. (39), the ceep pmetes M nd fo steel disc e estimted s.5 X -4 s -/ 5 /MP nd MP. These pmetes hve een used in the developed compute code to otin the distiution of tngentil nd dil ceep stins in the steel disc. Fig. 4 shows good geement etween the esults otined y the pocedue outlined in this study nd the ville expeimentl esults fo steel disc [43], which vlidtes the nlysis cied out nd the softwe developed in this study.

8 8 D. Deepk et l. / Jounl of Mechnicl Science nd Technology 4 () () ~3 Tle. Pmetes nd opeting conditions fo steel disc. Tle 3. Desciption of otting discs. Pmetes fo steel disc Opeting conditions Density of disc mteil (ρ) = 7,83.8 kg/m 3 Disc dius: = 3.75 mm, = 5.4 mm Disc Thickness: t = 5.4 mm Stess exponent: n = 5 Ceep pmetes: M =.5-4 s -/ 5 / MP = MP Disc pm = 5, Opeting tempetue = 8.78 K Ceep dution = 8 hs Disc (Nottion) Pticle (SiCp) Content (vol%) V mx V min V v Pticle Gdient (PG) = V mx - V min (vol%) Unifom/Non-FGM (D). FGM (D) FGM (D3) FGM (D4) M ( x -3 ), s -/5 /MP Non-FGM Disc D (PG = %) FGM Disc D (PG = 9.48%) FGM Disc D3 (PG = 8.97%) FGM Disc D4 (PG = 8.46%) Rdius (mm) () Fig. 4. Compison of theoeticl (Pesent Study) nd expeimentl [43] stins in the steel disc (Disc pm = 5, Ceep dution 8 hs) Non-FGM Disc D (PG = %) FGM Disc D (PG = 9.48%) FGM Disc D3 (PG = 8.97%) FGM Disc D4 (PG = 8.46%) (MP) Non-FGM Disc D (PG = %) FGM Disc D (PG = 9.48%) FGM Disc D3 (PG = 8.97%) FGM Disc D4 (PG = 8.46%) V (Vol %) Rdius (mm) 5 () Fig. 6. Vition of ceep pmetes in composite discs Rdius (mm) Fig. 5. Vition of pticle content in composite discs. 8. Vition of ceep pmetes Fig. 5 shows the distiution of einfocement (SiCp) in vious discs used in this study (efe to Tle 3). The SiCp content deceses linely fom the inne to the oute dius in FGM discs (D-D4) while in unifom disc D (non-fgm disc) the content of SiCp emins vol % ove the entie dius. Figs. 6() nd 6() show, espectively, the vition of ceep pmetes M nd with dil distnce in the composite discs. The vlue of pmete M oseved in FGM discs (D-D4) inceses with incesing dil distnce. The incese oseved in M is due to decese in pticle content V( in the FGM discs (D-D4) on moving fom the inne to the oute dius, s evident fom Eq. (). With the incese in pticle gdient (PG), defined s the diffeence of mxi-

9 D. Deepk et l. / Jounl of Mechnicl Science nd Technology 4 () () ~3 9 mum nd minimum pticle content in the disc, the distiution of M ecomes much steepe. On the othe hnd, the theshold stess ( ) shown in Fig. 6(), deceses linely with incesing dil distnce s oseved fo the FGM disc (D-D4). The theshold stess is highe in egions hving moe mount of SiCp comped to those hving eltively lesse mount of SiCp, which is lso eveled fom Eq. (). Simil to the ceep pmete M, the vition of lso ecomes steepe with the incese in pticle gdient in the FGM discs. Both the ceep pmetes, M nd, oseved in unifom disc (D) emins constnt due to unifom distiution of SiCp einfocement (i.e. vol% SiCp) ove the entie disc. 8.3 Distiution of stesses nd stin tes Rdil Stess (MP) Non-FGM Disc D (PG = %) FGM Disc D (PG = 9.48%) FGM Disc D3 (PG = 8.97%) FGM Disc D4 (PG = 8.46%) Rdius (mm) The effect of pticle gdient (PG) on ceep ehvio of unifom nd FGM discs is shown in Figs. (7)-(8). The dil stess in ll the discs, Fig. 7(), inceses fom zeo t inne dius, eches mximum efoe dopping to zeo gin t the oute dius, unde the imposed oundy conditions given in Eq. (34). By incesing PG in the disc, the dil stess is oseved to incese ove the entie disc. The dil stess is highest in FGM disc D4 nd lowest in unifom disc D. The mximum vlue of dil stess in FGM discs D, D3 nd D4 is highe, espectively, y out %, 3% nd 33%, when comped with those oseved in unifom disc D. As comped to unifom thickness disc D, the dil stess in FGM discs D, D3 nd D4 is highe ne the inne dius due to highe density, which cuses the centifugl foce to incese. But towd the oute dius, in spite of lowe density of FGM discs, the dil stess is highe in FGM discs thn the unifom thickness disc D due to eltively lowe thickness. Fig. 7() shows the vition of tngentil stess in diffeent composite discs. By incesing PG in the FGM disc the tngentil stess inceses ne the inne dius ut deceses towds the oute dius when comped with distiution of tngentil stess in unifom disc D. The distiution of tngentil stess exhiits cossove t dil distnce of ound mm, Fig. 7(). In the FGM discs (D- D4), the highe nd lowe density of the disc, espectively, ne the inne nd oute dii, comped to the density of unifom disc D, e esponsile fo eltively highe nd lowe tngentil stesses ne the inne nd oute dii, s eveled fom equiliium Eq. (38). As comped to unifom disc D, the mximum vlues of tngentil stess in FGM discs D, D3 nd D4 e highe y out %, % nd 3% espectively. The vition of effective stess with dil distnce fo diffeent discs, s shown in Fig. 7(c), is simil those oseved fo tngentil stess in Fig. 7(). The stin tes, tngentil s well s dil, decese significntly with incesing PG in the disc s evident fom Figs. 8() nd 8(). Though, ne the inne dius the tngentil stess in the FGM discs is highe thn the unifom disc D due to highe density, ut the FGM discs exhiit lowe tngentil Fig. 7() Effect of pticle gdient on dil stess. Tngentil Stess (MP) Non-FGM Disc D (PG = %) FGM Disc D (PG = 9.48%) FGM Disc D3 (PG = 8.97%) FGM Disc D4 (PG = 8.46%) Rdius (mm) Fig. 7() Effect of pticle gdient on tngentil stess. Effective Stess (MP) Non-FGM Disc D (PG = %) FGM Disc D (PG = 9.48%) FGM Disc D3 (PG = 8.97%) FGM Disc D4 (PG = 8.46%) Rdius (mm) Fig. 7(c) Effect of pticle gdient on effective stess.

10 3 D. Deepk et l. / Jounl of Mechnicl Science nd Technology 4 () () ~3 5e-7 6 Tngentil Stin (s - ) 4e-7 3e-7 e-7 Non-FGM Disc D (PG = %) FGM Disc D (PG = 9.48%) FGM Disc D3 (PG = 8.97%) FGM Disc D4 (PG = 8.46%) Stess Inhomogeneity (MP) Rdil Stess Tngentil Stess Effective Stess e-7 e Rdius (mm) Fig. 8() Effect of pticle gdient on tngentil stin te Pticle Gdient (Vol % ) Fig. 9() Effect of pticle gdient on stess inhomogeneity. 5e-8 e+ 4.e-7 3.5e-7 Rdil Stin Rte Tngentil Stin Rte Rdil Stin (s - ) -5e-8 -e-7 -e-7 Non-FGM Disc D (PG = %) FGM Disc D (PG = 9.48%) FGM Disc D3 (PG = 8.97%) FGM Disc D4 (PG = 8.46%) Stin Rte Inhomogeneity ( s - ) 3.e-7.5e-7.e-7.5e-7.e-7 -e-7 5.e-8-3e Rdius (mm).e Pticle Gdient (Vol % ) Fig. 8() Effect of pticle gdient on dil stin te. Fig. 9() Effect of pticle gdient on stin te inhomogeneity. ceep tes ne the inne dius thn the unifom disc D. It is ttiuted to lowe nd highe vlues of pmetes M nd, Figs. 6()-(), ne the inne dius of FGM discs (D-D4) thn the unifom disc D. On the othe hnd, the tngentil stin te ne the oute dius of the FGM discs (D-D4) is lowe thn the unifom disc D, due to lowe tngentil stess, Fig. 7(), cused y the lowe density of FGM discs towds the oute dius. At the oute dius, the incese nd decese in pmetes M nd, espectively, with incesing PG, Figs. 6()-(), could not dominte the decese in tngentil stess. The decese in tngentil stin te is moe t the inne dius thn tht oseved towds the oute dius. The effect of incesing the PG on the dil stin te, Fig. 8(), is simil to tht oseved fo tngentil stin te in Fig. 8(). By incesing the PG eyond 9.48%, the ntue of dil stin te, which is genelly compessive, ecomes tensile in the middle of the disc, s noticed fo FGM discs D3 nd D4. The mximum tensile dil stin te is oseved somewhee in the middle of FGM disc D4, which possesses mximum PG. It is lso evident fom Figs. 8() nd 8() tht esides decese in stin tes with incesing PG, the distiution of stin tes ecomes eltively moe unifom. Theefoe, the FGM disc hving highe PG will hve less chnces of distotion. The stess inhomogeneity, defined s the diffeence of mximum nd minimum vlues of stess, inceses with incesing pticle gdient s shown in Fig. 9(). Fo % incese in pticle gdient in the FGM disc, the inhomogeneity in dil, tngentil nd effective stess inceses y out.3 MP,.37 MP nd.47 MP espectively. Unlike stess inhomogeneity, the tngentil s well s dil stin te inhomogeneity in the disc deceses significntly with incesing pticle gdient s shown in Fig. 9(). The decese is steepe up to out 8% pticle gdient followed y eltively lowe decese with futhe incese in pticle gdient. The decese oseved in tngentil stin te inhomogeneity is eltively highe thn tht oseved in dil stin te. 9. Conclusions The study cied out hs led to the following conclusions: In the pesence of pticle gdient in otting disc, the

11 D. Deepk et l. / Jounl of Mechnicl Science nd Technology 4 () () ~3 3 tngentil stess inceses ne the inne dius ut deceses ne the oute dius. The dil stess in otting disc inceses eveywhee with the incese in pticle gdient. In the FGM discs with linely decesing pticle content fom the inne to the oute dius, the stedy stte ceep esponse in tems of stin tes is supeio to tht oseved in disc contining unifom distiution of einfocement, when oth the discs hve the sme vege mount of einfocement. By employing highe pticle gdient in the FGM disc, the distiution of stin tes in the disc ecomes eltively moe unifom. Refeences [] V. K. Gupt, S. B. Singh, H. N. Chndwt nd S. Ry, Modeling of ceep ehvio of otting disc in the pesence of oth composition nd theml gdients, Jounl of Engineeing Mteils nd Technology, 7 (5) [] M. H. Hojjti nd A. Hssni, Theoeticl nd numeicl nlysis of otting discs of non-unifom thickness nd density, Intentionl Jounl of Pessue Vessels nd Piping 85 () (8) [3] S. B. Singh nd S. Ry, Modeling the nisotopy nd ceep in othotopic luminum-silicon cide composite otting disc, Mechnics of Mteils 34 () [4] L. H. You, X. Y. You, J. J. Zhng nd J. Li, On otting cicul disks with vying mteil popeties, Zeitschift fu ngewndte Mthemtik und Physik 58 (7) [5] B. Fshi nd J. Biddi, Optimum design of inhomogeneous otting discs unde secondy ceep, Intentionl Jounl of Pessue Vessels nd Piping 85 (8) [6] V. K. Gupt, N. Kwt nd S. Ry, Atificil neul netwok modeling of ceep ehvio in otting composite disc, Engineeing Computtions: Intentionl Jounl fo Compute-Aided Engineeing nd Softwe 4 () (7) [7] B. Fshi, H. Jhed nd A. Mehin, Optimum design of inhomogeneous non-unifom otting discs, Computes nd Stuctues 8 (4) [8] M. Lskj, B. Muphy nd K. Houngn, Impoving the efficiency of cooling the font disc ke on V8 cing c, Poject epot, Monsh Univesity, Meloune (999). [9] M. Byt, M. Sleem, B. B. Shi, A. M. S. Hkoud nd E. Mhdi, Anlysis of functionlly gded otting disks with vile thickness, Mechnics Resech Communictions 35 (8) [] T. G. Nieh, Ceep uptue of silicon cide einfoced luminum composite, Metllugicl Tnsctions 5A (984) [] T. G. Nieh, K. Xi nd T. G. Lngdon, Mechnicl popeties of discontinuous SiC einfoced luminium composites t elevted tempetues, Jounl of Engineeing Mteils nd Technology (988) [] J. N. Reddy, Anlysis of functionlly gded pltes, Intentionl Jounl of Numeicl Method Engineeing 47 () [3] S. Suesh nd A. Motensen, Fundmentls of Functionlly Gded Mteils, pocessing nd themomechnicl ehvio of gded metls nd metls-cemic composites, IOM Communictions Limited, London (998). [4] S. B. Singh nd S. Ry, Stedy stte ceep ehvio in n isotopic functionlly gded mteil otting disc of Al-SiC composite, Metllugicl nd Mteils Tnsctions 3 (7) () [5] V. K. Gupt, S. B. Singh, H. N. Chndwt nd S. Ry, Stedy stte ceep nd mteil pmetes in otting disc of Al-SiC p composite, Euopen Jounl of Mechnics A/Solids 3 (4) [6] M. N. Byt, M. Sleem, B. B. Shi, A. M. S. Hmoud nd E. Mhdi, Themo elstic nlysis of functionlly gded otting disk with smll nd lge deflections, Thin- Wlled Stuctues 45 (7) [7] S. A. H. Kodkheili nd R. Nghddi, Themo elstic nlysis of functionlly gded otting disk, Composite Stuctues 79 (4) (7) [8] N. S. Bhtng, P. S. Kulkni nd V. K. Ay, Stedystte ceep of othotopic otting disks of vile thickness, Nucle Engineeing nd Design 9 () (986) -44. [9] A. N. Esln nd Y. Ocn, Elstic-plstic defomtion of otting disk of exponentilly vying thickness, Mechnics of Mteils 34 () [] Y. Ocn nd A. N. Esln, Elstic-plstic stesses in linely hdening otting solid disks of vile thickness, Mechnics Resech Communictions 9 () [] H. Jhed, B. Fshi nd J. Biddi, Minimum weight design of inhomogeneous otting discs, Intentionl Jounl of Pessue Vessels nd Piping 8 (5) [] S. K Gupt, S. Shm nd S. Pthk, Ceep tnsition in thin otting disc hving vile thickness nd vile density, Indin Jounl of Pue Applied Mthemtics 3 () () [3] D. Deepk, V. K. Gupt nd A. K. Dhm, Stedy stte ceep in otting composite disc of vile thickness, Intentionl Jounl of Mteils Resech () [4] T. W. Clyne nd P. J. Withes, An Intoduction to Metl Mtix Composites, Cmidge Univ. Pess, Cmidge, UK, (993) 479. [5] Metls Hndook (vol. ), Ameicn Society of Metls, 9 th Ed., Metls Pk, Ohio, USA, (978) 74. [6] Z. Y. M nd S. C. Tjong, Ceep defomtion chcteistics of discontinuously einfoced luminium mtix composites, Composites Science nd Technology 6 (5) () [7] S. C. Tjong nd Z. Y. M, Micostuctul nd mechnicl chcteistics in situ metl mtix composites, Mteils Science nd Engineeing 9 () [8] G. Gonzlez-Doncel nd O. D. Shey, High tempetue

12 3 D. Deepk et l. / Jounl of Mechnicl Science nd Technology 4 () () ~3 ceep ehvio of metl mtix luminium-sic composites, Act Metllugic et Mteili 4 () (993) [9] R. S. Mish nd A. B. Pndey, Some osevtion on the high-tempetue ceep ehvio of 66 Al-SiC composites, Metllugicl Tnsctions A (99) [3] A. B. Pndey, R. S. Mish nd Y. R. Mhjn, Stedy stte ceep ehvio of silicon cide pticulte einfoced luminium composites, Act Metllugic et Mteili 4 (99) [3] A. B. Pndey, R. S. Mish nd Y. R. Mhjn, Hightempetue ceep of Al-TiB pticulte composites, Mteils Science nd Engineeing 89A (994) [3] J. Cdek, H. Oikw nd V. Sustek, Theshold ceep ehvio of discontinuous luminium nd luminium lloy mtix composites: An Oveview, Mteils Science nd Engineeing A9 (995) 9-. [33] Y. Li nd T. G. Lngdon, Ceep ehvio of n Al-66 metl mtix composite einfoced with lumin pticultes, Act Mteili 45 () (997) [34] Y. Li nd T. G. Lngdon, An exmintion of sustuctue-invint model fo the ceep of metl mtix composites, Mteils Science nd Engineeing A65 (999) [35] Y. Li nd F. A. Mohmed, An investigtion of ceep ehvio in n SiC-4 Al composite, Act Mteili 45 () (997) [36] F. A. Mohmed, K. T. Pk nd E. J. Lveni, Ceep ehvio of discontinuous SiC Al composites, Mteils Science nd Engineeing 5A (99) -35. [37] K. T. Pk, E. J. Lveni nd F. A. Mohmed, High tempetue ceep of silicon cide pticulte einfoced luminum, Act Metllugic et Mteili, 38 () (99) [38] K. T. Pk nd F. A. Mohmed, Ceep stengthening in discontinuous SiC Al composite, Metllugicl nd Mteils Tnsctions 6A (995) [39] H. Yoshiok, Y. Suzumu, J. Cdek, S. J. Zhu nd K. Milick, Ceep ehvio of ODS luminium einfoced y silicon cide pticultes: ODS Al 3 SiC P composite, Mteils Science nd Engineeing A48 () (998) [4] R. Lgneog nd B. Begmn, The stess/ceep ehvio of pecipittion-hdened lloys, Metl Science () (976) -8. [4] V. Mises, Tnsltion of Mechnik de festen koepe im plstisch-defomlem Zustnd Nchichten von de koniglichen Gsellschft de Wissenschften, Technicl Memondum 88488, NASA, Wshington DC (986) [4] S. P. Timoshenko nd J. N.Goodie, Theoy of Elsticity, McGw-Hill, Singpoe (97) 8. [43] A. M. Whl, G. O. Snkey, M. J. Mnjoine nd E. Shoemke, Ceep tests of otting disks t elevted tempetue nd compison with theoy, Jounl of Applied Mechnics (954) Dhmpl Deepk holds Bchelos Degee in Mechnicl Engineeing fom SLIET, Longowl (Indi) nd Mstes in Poduction Engineeing fom GNDEC, Ludhin (Indi). Pesently, he is woking s n Assistnt Pofesso in the Mechnicl Engineeing Deptment t RIEIT, Nwnsheh, Indi. He hs nume of pulictions in efeeed Jounls nd Confeences. His min esech e is Mechnics of Composite Mteils. V. K. Gupt eceived his Bchelos Degee in Mechnicl Engineeing fom HBTI, Knpu, (Indi) nd Mstes in Industil Metllugy fom IIT Rookee (Indi). He then otined his Ph.D. in Mechnicl Engineeing fom TIET, Ptil (Indi). D. Gupt is cuently woking s Rede in Mechnicl Engineeing t UCoE, Punji Univesity, Ptil, Indi. He hs pulished thity five plus esech ppes in efeeed Jounls nd Confeences. His min esech inteests e Mechnics of Composite Mteils, Ftigue Behvio of PVD Coted Steels Components nd Hd Pt Tuning. D. Ashok K. Dhm holds Ph.D. in Theoeticl Physics fom Punji Univesity, Ptil, Indi. Pesently, he is woking s Pofesso of Physics t Punji Univesity, Ptil (Indi). His min es of esech e Intemolecul Foces, Atomic nd Molecul Physics, nd Computtionl Physics. He hs pulished thity plus esech ppes in the leding elted Intentionl/Ntionl Jounls of epute, nd hs pesented ppes in out twelve Intentionl Confeences on Intemolecul Foces/Tnspot Popeties. He hs spent six yes t Univesity of Westen Ontio (Cnd) nd ye nd hlf t Univesity of Wteloo (Cnd) on sticls.