A Parametric Study on the Centrifugal Force-Induced Stress and Displacements in Power-Law Graded Hyperbolic Discs

Transcription

1 Oiginl Aticle A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs Abstct An extensive pmetic study on the vition of the centifugl-foce-induced stess nd displcements with the inhomogeneity indexes, pofile pmetes nd boundy conditions is conducted bsed on the utho s ecently published nlyticl fomuls fo dilly functionlly powe-lw gded otting hypebolic discs unde xisymmetic conditions. The dil vition of the thickness of the disc is chosen to obey hypebolic function defined eithe convegent o divegent. In the pesent wok, conty to the published one, it is ssumed tht both Young s modulus nd density dilly vy with the sme inhomogeneity index to enble to conduct pmetic study. Unde this dditionl ssumption, fo the vlues of the chosen powelw indexes 5, 0, 5 fo the mteil gding ule, nd the chosen pofile pmetes m, 0.75, 0.5, 0.25, 0, 0.25, 0.5, 0.75, fo hypebolic disc; the vitions of the dil stess, the hoop stess nd the dil displcement e ll illustted gphiclly fo otting disc whose both sufces e stess-fee, fo otting disc mounted igid shft t its cente nd its oute sufce is stess-fee, nd finlly fo otting disc ttched igid shft t its cente nd guided t its oute sufce ( igid csing exists t the oute sufce). Vebil Yıldıım * Univesity of Çukuov, Deptment of Mechnicl Engineeing, Adn, TURKEY. Emil: vebil@cu.edu.t *Coesponding utho Received: July 09, 207 In Revised Fom: 7Octobe 07, 207 Accepted: Octobe, 207 Avilble online: Febuy 05, 208 Keywods Elsticity solution, otting disc, functionlly gded, xisymmetic, vible thickness disk. INTRODUCTION Anlyticl nd numeicl studies on functionlly gded discs hve gined momentum since 990s. Thee e numeous studies on sttiony/otting discs with constnt/vible thickness nd mde of n isotopic nd homogeneous/non-homogeneous mteil in the vilble litetue. Some of those studies pefomed nlyticlly nd lmost diectly elevnt to this study e cited in the pesent ppe. In the litetue, especilly nlyticl studies on such stuctues subjected to only the inne pessue e eltively lge. In this section, especilly, just discs otting t constnt speed nd minly nlyticl studies bout those e cited. Güven (995) studied Tesc's yield condition nd the line hdening otting solid disk of vible thickness. Esln (2003) obtined nlyticl solutions fo the stess distibution in otting pbolic solid disks mde of n isotopic nd homogeneous mteil bsed on Tesc s yield citeion ssocited with the flow ule nd line stin hdening. Esln (2003) showed tht the defomtion behvio of the convex pbolic disk is simil to tht of the unifom thickness disk, but in the cse of concve pbolic solid disk, it is diffeent. Esln (2003) lso showed mthemticlly tht in the limiting cse the pbolic disk solution educes to the solution of otting unifom thickness solid disk. Bsed on Tesc s yield citeion, its ssocited flow ule nd line stin hdening mteil behvio, Esln (2003b) offeed nlyticl solutions fo the elstic plstic stess distibution in otting pbolic disks with fee, pessuized nd dilly constined boundy conditions. In this study it ws lso shown mthemticlly tht in the limiting cse the pbolic disk solution educes to the unifom disk solution. Apty nd Esln (2003) chieved nlyticl solutions in tems of hypegeometic functions fo the elstic defomtion of otting pbolic discs mde of isotopic nd homogeneous mteils. Cldele et l. (202) studied theoeticlly themoelstic nlysis of the Stodol's hypebolic disk mde of n isotopic nd homogeneous mteil, xisymmetic nd symmetic with espect to the mid-plne, nd subjected to dilly polynomilly vying theml lod. Vivio et l. (204) intoduced theoeticl method fo the evlution of elstic stesses nd stins in otting Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34

2 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs hypebolic disks mde of n isotopic nd homogeneous mteil. Fo otting discs mde of n isotopic nd homogeneous mteil, Esln nd Ciftci (205) used disk pofile chnging with n exponentil function. Yıldıım (207) offeed ll-in-one fomuls fo unifom discs, cylindes nd sphees subjected to the mechnicl nd theml lods. In this Refeence centifugl foce-induced, het-induced, nd pessue-induced elstic esponses hve ll been consideed fo those stuctues mde of n isotopic nd homogeneous mteil. As to otting discs mde of functionlly gded mteils, mteil gding function ws chosen s simple powe function in some Refeences (Hogn nd Chn, 999,b, You et l. (2007); Byt et l., 2008; Çllıoğlu et l., 20; Yıldıım, 206; Gng, 207), nd s n exponentil function in some studies (Zenkou, 2005, 2007; Zenkou nd Msht, 20; Esln nd Asln, 205) to get nlyticl solutions. Fom those, Hogn nd Chn (999,b) gve explicit solutions fo otting discs of constnt density nd thickness. Zenkou (2005) studied nlyticlly exponentilly gded otting nnul discs with constnt thickness. Esln nd Akış (2006) used two vints of pbolic function fo disks mde of functionlly gded mteils. Zenkou (2007) extended his study (Zenkou, 2005) fo such discs with igid csing. Byt et l. (2008), bsed on the powe-lw distibution, gve both nlyticl nd semi-nlyticl elstic solutions fo xisymmetic otting hollow discs with pbolic nd hypebolic thickness pofiles. This semi-nlytic solution ws obtined by dividing the disc with vying thickness into sub-domins with unifom thickness. By tking Young s modulus, theml expnsion coefficient nd density to be functions of the dil coodinte, closed fom solution of otting unifom cicul disks mde of powe-lw gded mteils subjected to constnt ngul velocity nd unifom tempetue is poposed by You et l. (2007). Vivio nd Vullo (2007) pesented n nlyticl pocedue bsed on the hypegeometic diffeentil eqution fo evlution of elstic stesses nd stins in otting solid o nnul conicl disks subjected to theml lod, nd hving fictitious density vition long the dius. Vivio nd Vullo (2007) lso veified thei nlyticl esults with finite element solutions. Vullo nd Vivio (2008) pesented n nlyticl pocedue fo evlution of elstic stesses nd stins in non-line vible thickness otting disks, eithe solid o nnul, subjected to theml lod, nd hving fictitious density vition long the dius. Thickness vition of disks ws descibed by mens of powe of line function by Vullo nd Vivio (2008). Peng nd Li (2009) studied themoelstic poblem of cicul nnulus mde of functionlly gded mteils with n bity gdient. Peng, nd Li (202) lso studied effects of gdient on stess distibution in otting functionlly gded solid disks. Zenkou nd Msht (20) used the modified Rung- Kutte lgoithm in thei numeicl nlysis. Çllıoğlu et l. (20) pefomed n exct stess nlysis of nnul otting discs mde of functionlly gded mteils by ssuming tht both elsticity modulus nd mteil density vy dilly s function of simple powe ule with the sme inhomogeneity pmete. Hssni et l. (20) obtined distibutions of stess nd stin components of otting hypebolic disks with non-unifom mteil popeties subjected to powe fom themo-elstic loding unde diffeent boundy conditions by semi-exct methods of Lio s homotopy nlysis method. Ageso (202) pesented nlyticl solutions fo two diffeent nnul otting disk poblems fo the elstic stess stte: The fist poblem involves n exponentilly vible pofile otting disk mde of n isotopic nd homogeneous mteil, the second is unifom disc mde of exponentilly functionlly gded mteils. Nejd et l. (203, 204) gve closed-fom nlyticl solution in tems of hype geometic functions to elstic nlysis of exponentilly functionlly gded sttiony discs subjected to intenl nd extenl pessues. Esln nd Asln (205) developed nlyticl nd numeicl solutions to otting unifom thickness exponentilly functionlly gded (FGM) solid nd nnul disks. Yıldıım (206) studied the exct elstic esponse of otting disk hving continuously vying hypebolic thickness pofile unde diffeent boundy conditions. Both convegent-hypebolic nd divegent-hypebolic disk pofiles togethe with unifom pofile e ll studied. Powe-lw gding is used fo mteil gdtion ptten. Yıldıım s (206) fomultion compises both continuously vitions of elsticity modulus nd mteil density including continuously vition of the thickness of the disc except vition of Poisson s tios. Conty to the litetue ll effects ffecting the elstic behvio of the disk with vying thickness such s intenl nd extenl pessues including ottion t constnt ngul velocity e ll studied unde fou physicl boundy conditions nd pesented in compct foms in Yıldıım s study (Yıldıım, 206). Recently, Gng (207) nlyticlly studied the stess nlysis of hypebolic simple-powe lw gded otting discs unde stess-fee conditions fo fou convegent disc pofiles nd negtive inhomogeneity indexes. Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 2/6

3 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs Figue : 3-D view of convegent/divegent hypebolic nd unifom disc pofiles In this study, unde the dditionl ssumption tht both Young s modulus nd mteil density hve vition with the sme inhomogeneity index, closed-fom fomuls deived by Yıldıım (206) e employed in the pesent wok in customized fom to study the vition of centifugl foce-induced stess nd displcements in powe-lw gded hypebolic discs with inhomogeneity pmete, pofile pmete nd boundy conditions (Fig. ). As mentioned bove, dil vition of Poisson s tio is neglected. As You et l. (2007) expessed Doubling Poisson s tio, the dil nd cicumfeentil stesses nd dil displcement hve vey few chnges. Howeve, doubling Young s modulus, the dil stess is incesed obviously, the cicumfeentil stess is ised getly, nd the dil displcement is educed noticebly. Theefoe, comped to the effects of Young s modulus, the vition of Poisson s tio cn be omitted. 2 EXPANDING YILDIRIM S (206) FORMULAS The disk whose inne dius is denoted by nd oute dius is denoted by b is ssumed to be symmetic with espect to the mid plne, nd its pofile vy dilly continuously in n hypebolicl fom h m h () whee h is the thickness of the disc t the inne sufce, is the dil coodinte, nd m is the disc pofile pmete. In Eq. (), unifom disc pofile is obtined with m=0, convegent hypebolic dick pofile is ttined with m<0 nd fo m>0 divegent hypebolic disc pofile is eched (Fig. ). Yıldıım (206) solved the following nonhomogeneous eqution govening the elstosttic behvio of otting disc mde of functionlly gded mteils fo the hypebolic discs otting t constnt ngul velocity, ω. q m m 2 2 ' '' u = 2 u u E (2) Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 3/6

4 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs In the bove eqution, the pime symbol, ( ), denotes the deivtive with espect to the dil coodinte. Poisson s tio is indicted by v; E nd e the inne sufce vlues of elsticity modulus nd the density, espectively. Repesenting Young s modulus by E, nd mteil density by, in the bove eithe E E q E Eb b (3) q b b (3b) my be pplied s mteil gding ule. In Eq. (3) β nd q e clled inhomogeneity pmetes fo both elsticity modulus nd density, espectively. If one suppose tht Mteil- is locted t the inne sufce nd Mteil-b is locted t the oute sufce, inhomogeneity pmetes in these equtions e defined s follows E E b ln ln Eb E b ln ln b (4) b In In b q b In In b (4b) Deivtion of Eq. (2) is pesented in Appendix A. The genel solution of Eq. (2) is witten in tems of unknown coefficients C nd C 2 s follows (Yıldıım, 206) m 2 u C C Whee 2 2 ρ 3 q 2 m m ξ 4 4 E 8 q 6 q 3 m 3 q Yıldıım (206) pesented explicit definitions of unknown coefficients, C nd C, fo ll possible boundy 2 conditions. Although Yıldıım s (206) fomuls e vlid fo the diffeent inhomogeneity pmetes fo both elsticity modulus nd mteil density, in the pesent study, those indexes e ssumed to be equl to ech othe, tht is q is to be used. Tht is those fomuls will be customized fo q to llow pmetic study. Let s do this. Unde this ssumption, q, the solution in Eq. (5) togethe with Eq. (3) tuns into the following ρ m 2 u C C E 8 m (5) (6) (7) Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 4/6

5 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs Boundy conditions consideed in the pesent study e pesented in Fig. 2. Fo those boundy conditions nd q, the closed-fom expessions of the dil displcement, dil nd hoop stesses e pesented in Tbles -3. Figue 2: Boundy conditions consideed in the pesent study u = Tble : Closed-fom fomuls fo fee-fee boundy conditions FREE-FREE ρ ω 2(ν )(ν + )(ν + 3) b () b () b E (β(ν + 3) + m(ν + 3) + 8) ( b )(β + m 2ν + ξ) σ = 2(ν )(ν + )(ν + 3) () () b () ( b )( β m + 2ν + ξ) + (ν ) () ( b )(β(ν + 3) + m(ν + 3) + 8) (ν + 3)ρ ω () b () + b () () b () + b () σ = ( b )(β(ν + 3) + m(ν + 3) + 8)(β + m 2ν ξ)(β + m 2ν + ξ) ρ ω () (ν + 3) () (ν(β + m ξ) 2)(β + m 2ν + ξ) b (β + m 2ν ξ)(ν(β + m + ξ) 2) + (β + m 2ν ξ) (ν + 3)b () (ν(β + m + ξ) 2) (3ν + )(β + m 2ν + ξ) () + b (β + m 2ν + ξ) (ν + 3) b () (ν(β + m ξ) 2) (3ν + )b () ( β m + 2ν + ξ) Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 5/6

6 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs σ = σ = Tble 2: Closed-fom fomuls fo fixed-guided boundy conditions FIXED-GUIDED u = E ( b )(β(ν + 3) + m(ν + 3) + 8) (ν )ρ ω () (( b ) () + (b () () ) b () + b () ρ ω () 2( b )(β(ν + 3) + m(ν + 3) + 8) () b (β + m 2ν + ξ) + ( β m + 2ν + ξ) + b () (β + m 2ν + ξ) + 2(ν + 3) () + b () (β + m 2ν ξ) + 2(ν + 3)b () ρ ω () 2( b )(β(ν + 3) + m(ν + 3) + 8) () b (ν(β + m + ξ) 2) + 2 ν(β + m ξ) + b () 2 ν(β + m + ξ) 2(3ν + ) () + b () (ν(β + m ξ) 2) + 2(3ν + )b () Tble 3: Closed-fom fomuls fo fixed-fee boundy conditions FIXED-FREE u = E (β(ν + 3) + m(ν + 3) + 8) (β + m 2ν + ξ) + b ( β m + 2ν + ξ) (ν )ρ ω () () b (β + m 2ν ξ) (β + m 2ν + ξ) + 2(ν + 3)b () + (β + m 2ν + ξ) () 2(ν + 3) b () + b ( β m + 2ν + ξ) () σ = 2(β(ν + 3) + m(ν + 3) + 8) (β + m 2ν + ξ) + b ( β m + 2ν + ξ) ρ ω () b () ((β + m 2ν) ξ ) + 2(ν + 3) (β + m 2ν + ξ) b () () + 2(ν + 3)b ( β m + 2ν + ξ) b () b () σ = 2(β(ν + 3) + m(ν + 3) + 8) (β + m 2ν + ξ) + b ( β m + 2ν + ξ) ρ ω () () b (β + m 2ν ξ)(ν(β + m + ξ) 2) (ν(β + m ξ) 2)(β + m 2ν + ξ) + 2 (ν + 3)b () (ν(β + m + ξ) 2) (3ν + )(β + m 2ν + ξ) () 2(ν + 3) b () (ν(β + m ξ) 2) 2(3ν + )b ( β m + 2ν + ξ) () Çllıoğlu et l. (20) studied the elstic esponse of powe-gded unifom stess-fee otting disks with boundy conditions: u 0 nd b 0 (Fig. 2). They ssumed tht both the Young s modulus nd the mteil density chnge with the sme inhomogeneity index, q, in thei fomultion s in the pesent wok. Howeve, they futhe ssumed tht the thickness of the disc emins constnt long the dil coodinte, tht is Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 6/6

7 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs m 0. Yıldıım (206) lso showed tht he fomuls coincides with Çllıoğlu et l. s (20) study unde those ssumptions stted in this pgph. 3 A NUMERICAL STUDY The following geometicl popeties e used in the pmetic study: 0.02 m; b 0. m. Poisson s tio is ssumed to be constnt long the dil coodinte s 0.3. Dimensionless elstic stess nd displcements e defined s σ b 2 2 ; σ b θ 2 2 ; u E u 2 3 b Fo the vlues of the chosen simple powe-lw indexes 5, 0, 5 fo the mteil gding ule (Fig. 3), nd the chosen pofile pmetes m, 0.75, 0.5, 0.25, 0, 0.25, 0.5, 0.75, fo hypebolic disc; the vitions of the dil stess, the hoop stess nd the dil displcement e ll illustted gphiclly fo otting disc whose both sufces e stess-fee, fo otting disc mounted igid shft t its cente nd its oute sufce is stess-fee, nd finlly fo otting disc ttched igid shft t its cente nd guided t its oute sufce ( igid csing exists t the oute sufce) in Figs Some numeicl esults e lso pesented in Tbles 4-6 to seve s numeicl dt fo investigtos. Accoding to Eq. (3) the positive inhomogeneity indexes suggest tht the oute sufce nd its vicinity is highly stiffe thn the middle nd the inne sufces. Howeve, the inne sufce is stiffe thn the middle nd oute sufces fo the negtive inhomogeneity indexes. The esults obtined fom pmetic study of the pesent wok my be outlined concisely s follows (see Figs 3-5): Convegent hypebolic dick pofiles, m 0, offe smlle elstic field thn divegent ones fo negtive inhomogeneity indexes including isotopic nd homogeneous mteils with 0. Howeve, fo the positive inhomogeneity pmetes some diffeences in the behvio my be obseved. Fo instnce, while the hoop stesses e smlle fo fixed-fee nd fee-fee ends of convegent disc pofiles, the dil stesses behve contily to this fo ll boundy conditions. Simil to this, fo fixed-fee nd fee-fee conditions, convegent pofiles hving positive inhomogeneity pmete pesent much smlle dil displcement vlues. As expected, fixed-guided discs hve highe elstic field thn fixed-fee nd fee-fee boundy conditions. Fo fixed-guided disc nd positive inhomogeneity index, hoop stesses e in tension-compession. Fo othe boundy conditions they e in tension. Howeve, negtive inhomogeneity pmetes offe hoop stesses in tension fo ll boundy conditions nd fo both convegent nd divegent disc pofiles. While positive inhomogeneity indexes pesent the mximum hoop stess t the oute sufce of the disc, thei loctions e t the inne sufce of the disc fo negtive inhomogeneity indexes. It my be noted tht Gng (207) nlyticlly studied convegent hypebolic discs with m, 0.75, 0.5, 0.25, nd negtive inhomogeneity indexes which my be defined ppoximtely s nd q 0.09 unde stess-fee conditions (fee-fee). He concluded tht dil nd tngentil stesses in convegent FGM hypebolic disc with negtive inhomogeneity indexes is significntly educed s comped to FGM unifom disc. This comment is in geement with the fist item of the conclusions given bove of Figs. 3-5 nd obtined fom widesped sech. (8) Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 7/6

8 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs Figue 3: Vition of the displcement, the dil nd hoop stesses with the boundy conditions nd pofile indexes fo 5 Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 8/6

9 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs Tble 4: Some numeicl esults fo 5 /b m= m=0.75 m=0.5 m=0.25 m=0 m=-0.25 m=-0.50 m=-0.75 m=- Dimensionless dil displcement (Fee-Fee) Dimensionless dil displcement (Fixed-Fee) Dimensionless dil displcement (Fixed-Fixed) Dimensionless dil stess (Fee-Fee) Dimensionless dil stess (Fixed-Fee) Dimensionless dil stess (Fixed-Fixed) Dimensionless hoop stess (Fee-Fee) Dimensionless hoop stess (Fixed-Fee) Dimensionless hoop stess (Fixed-Fixed) Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 9/6

10 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs Figue 4: Vition of the displcement, the dil nd hoop stesses with the boundy conditions nd pofile indexes fo 0 Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 0/6

11 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs Tble 5: Some numeicl esults fo 0 /b m= m=0.75 m=0.5 m=0.25 m=0 m=-0.25 m=-0.50 m=-0.75 m=- Dimensionless dil displcement (Fee-Fee) Dimensionless dil displcement (Fixed-Fee) Dimensionless dil displcement (Fixed-Fixed) Dimensionless dil stess (Fee-Fee) Dimensionless dil stess (Fixed-Fee) Dimensionless dil stess (Fixed-Fixed) Dimensionless hoop stess (Fee-Fee) Dimensionless hoop stess (Fixed-Fee) Dimensionless hoop stess (Fixed-Fixed) Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 /6

12 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs Figue 5: Vition of the displcement, the dil nd hoop stesses with the boundy conditions nd pofile indexes fo 5 Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 2/6

13 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs Tble 6: Some numeicl esults fo 5 /b m= m=0.75 m=0.5 m=0.25 m=0 m=-0.25 m=-0.50 m=-0.75 m=- Dimensionless dil displcement (Fee-Fee) Dimensionless dil displcement (Fixed-Fee) Dimensionless dil displcement (Fixed-Fixed) Dimensionless dil stess (Fee-Fee) Dimensionless dil stess (Fixed-Fee) Dimensionless dil stess (Fixed-Fixed) Dimensionless hoop stess (Fee-Fee) Dimensionless hoop stess (Fixed-Fee) Dimensionless hoop stess (Fixed-Fixed) CONCLUSIONS Afte they e customized, in this study, the closed-fom fomuls deived by Yıldıım (206) e employed to study the vition of the centifugl-foce-induced stess nd displcements in powe-lw gded hypebolic discs with inhomogeneity pmete, pofile pmete, nd boundy conditions. Conty to Yıldıım s (206) study, it is ssumed tht both Young s modulus nd mteil density chnge with the sme inhomogeneity pmete. If one suppose tht Mteil- is locted t the inne sufce nd Mteil-b is locted t the oute sufce, inhomogeneity indexes should be defined by Eq. (4). Unde this ssumption, in pctice, it is hdly confonted to get physicl metl-cemic pi to stisfy tht condition which my be defined by the following deived fom Eq. (4). E / E / (9) b b Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 3/6

14 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs On the othe hnd, in the pesent pmetic study, Eq. (4) is not used to define the inhomogeneity indexes. Eqution (3) is used by ttibuting hypotheticlly chosen vlues to the inhomogeneity indexes insted. Fo positive inhomogeneity indexes this mens tht while Mteil- is locted t the inne sufce, the mixtue of two mteils which is multiples of E exist t the othe sufces including the oute sufce. The chnge of the popeties of the mixtue is defined by Eq. (3). Tking the sme inhomogeneity index fo both Young s modulus nd density helps to conduct pmetic study by eliminting subodinte chnges in some vibles (See Eqn. (2)). Tht is, by doing so, we my cquie, t lest, bllpk estimte bout the vition of the elstic esponse of hypebolic otting discs mde of functionlly gded mteils. It my be noted tht the fomuls deived in the pesent study my be used fo both bitily chosen inhomogeneity indexes s in the pmetic study nd inhomogeneity indexes computed by Eq. (4). It is obvious tht nlyticl fomuls offeed by Yıldıım (206) should be employed to get ccute esults fo hypebolic discs mde of physiclly exist mteil-cemic pis in the lst decision stge of design pocess. Tking into considetion the bove bll-pk estimtions, the tue mteil tiloing my be done without consuming much time in the design pocess of such otting hypebolic discs mde of functionlly gded mteils. ACKNOWLEDGEMENTS The utho is gteful to the nonymous Refeees fo thei vluble suggestions, the effot nd time spent. Refeences Apty, T., nd Esln, A.N. (2003). Elstic defomtion of otting pbolic discs: nlyticl solutions (in Tukish), Jounl of the Fculty of Engineeing nd Achitectue of Gzi Univesity 8: Ageso, H. (202). Anlyticl solutions to vible thickness nd vible mteil popety otting disks fo new thee-pmete vition function, Mechnics Bsed Design of Stuctues nd Mchines 40: Byt, M., Sleem M., Shi B., Hmoud A. nd Mhdi E. (2008). Anlysis of functionlly gded otting disks with vible thickness, Mechnics Resech Communictions 35: Çllıoğlu, H., Bektþ, N.B., nd Sye, M. (20). Stess nlysis of functionlly gded otting discs: nlyticl nd numeicl solutions, Act Mechnic Sinic 27: Cldele, P.M., Vivio, F., nd Vullo, V. (202). Theml stesses of otting hypebolic disks s pticul cse of non-linely vible thickness disks, Jounl of Theml Stesses 35: Esln, A.N. (2003). Elstoplstic defomtions of otting pbolic solid disks using Tesc s yield citeion, Euopen Jounl of Mechnics A/Solids 22: Esln A.N. (2003b). Elstic plstic defomtions of otting vible thickness nnul disks with fee, pessuized nd dilly constined boundy conditions, Intentionl Jounl of Mechnicl Sciences 45: Esln, A.N., nd Akış, T. (2006). On the plne stin nd plne stess solutions of functionlly gded otting solid shft nd solid disk poblems, Act Mechnic 8/( 2): Esln, A.N., nd Asln, E., (205). Anlyticl nd numeicl solutions to otting FGM disk, Jounl of Multidiscipliny Engineeing Science nd Technology (JMEST) 2/0: Esln, A.N., nd Ciftci, B. (205). Anlyticl nd numeicl solutions to otting vible thickness disks fo new thickness pofile, Jounl of Multidiscipliny Engineeing Science nd Technology (JMEST) 2/9: Gng, M. (207). Stess nlysis of vible thickness otting FG disc, Intentionl Jounl of Pue nd Applied Physics 3/: Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 4/6

15 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs Güven, U. (995).Tesc's yield condition nd the line hdening otting solid disk of vible thickness, Zeitschift fu Angewndte Mthemtik und Mechnik 75: Hssni, A., Hojjti, M.H., Fhi, G., nd Alshti, R.A. (20). Semi-exct elstic solutions fo themomechnicl nlysis of functionlly gded otting disks, Composite Stuctues 93: Hogn, C., Chn, A. (999). The pessuized hollow cylinde o disk poblem fo functionlly gded isotopic linely elstic mteils, Jounl of Elsticity 55: Hogn, C., Chn A. (999b). The stess esponse of functionlly gded isotopic linely elstic otting disks, Jounl of Elsticity 55: Nejd, M.Z., Abedi, M., Lotfin, M.H., nd Ghnnd, M. (203) Elstic nlysis of exponentil FGM disks subjected to intenl nd extenl pessue, Centl Euopen Jounl of Engineeing 3: Nejd, M.Z., Rstgoo, A. nd Hdi, A. (204). Exct elsto-plstic nlysis of otting disks mde of functionlly gded mteils, Intentionl Jounl of Engineeing Science 85: Peng, X.L., nd Li, X.F. (2009). Themoelstic nlysis of functionlly gded nnulus with n bity gdient, Appl Mth Mech. 30: Peng, X.L. nd Li, X.F. (202). Effects of gdient on stess distibution in otting functionlly gded solid disks, J. Mech. Sci. Technol. 26: Vivio, F., nd Vullo, V. (2007). Elstic stess nlysis of otting conveging conicl disks subjected to theml lod nd hving vible density long the dius. Intentionl Jounl of Solids nd Stuctues 44: Vivio, F., Vullo, V., nd Cifni, P. (204). Theoeticl stess nlysis of otting hypebolic disk without singulities subjected to theml lod, Jounl of Theml Stesses, 37: Vullo, V., Vivio, F. (2008). Elstic stess nlysis of non-line vible thickness otting disks subjected to theml lod nd hving vible density long the dius, Inte. J. Solids Stuct. 45: Yıldıım, V. (206). Anlytic solutions to powe-lw gded hypebolic otting discs subjected to diffeent boundy conditions, Intentionl Jounl of Engineeing & Applied Sciences (IJEAS) 8/: Yıldıım, V., (207). Het-induced, pessue-induced nd centifugl-foce-induced exct xisymmetic themo-mechnicl nlyses in thick-wlled spheicl vessel, n infinite cylindicl vessel, nd unifom disc mde of n isotopic nd homogeneous mteil, Intentionl Jounl of Engineeing & Applied Sciences (IJEAS), 9 (2): You, L.H. You, X.Y. Zhng J.J. nd Li J. (2007). On otting cicul disks with vying mteil popeties, Z. Angew. Mth. Phys., ZAMP 58: Zenkou, A.M. (2005). Anlyticl solutions fo otting exponentilly-gded nnul disks with vious boundy conditions, Int. Jounl of Stuct. Stbility nd Dynmics 5: Zenkou, A.M. (2007). Elstic defomtion of the otting functionlly gded nnul disk with igid csing, Jounl of Mteils Science 42: Zenkou, A.M., nd Msht, D.S. (20). Stess function of otting vible-thickness nnul disk using exct nd numeicl methods, Engineeing 3: Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 5/6

16 A Pmetic Study on the Centifugl Foce-Induced Stess nd Displcements in Powe-Lw Gded Hypebolic Discs APPENDIX A Unde xisymmetic plne stess nd smll defomtion ssumptions, in pol coodinte system, eltions between the stin nd displcement components e s follows d u d,, the u (A.) Whee u is the dil displcement; nd e the dil nd tngentil stin components, espectively. Fo n isotopic nd non-homogeneous mteil, the stess-stin eltions (Hooke s lw) is E 2 E 2 (A.2) Whee E is the elsticity modulus; v is Poisson s tio; nd e dil stess nd cicumfeentil stess (hoop stess), espectively. The equilibium eqution of vible thickness disk otting t constnt ngul velocity,, is d h 2 2 h h 0 d Whee is the mteil density, nd h defines the pofile of the disc, h( ) h ( / ) m (A.3). Substituting Eq. (A.) in Eq. (A.2), nd then Eq. (A.2) in Eq. (A.3) yields the following non-homogeneous govening eqution in tems of dil displcement nd its deivtives. d d d d 2 E h E h 2 2 d d 2 d d d d u u u 2 d d E h E h E (A.4) In the bove eqution clled Nvie eqution, fte choosing eithe E( ) E ( / ) with ( ) ( / ) q o E( ) E ( / b) with ( ) ( / b) q b b s mteil gding ule (( de ( ) / d / E ( ) ; ( dh( ) / d) / h( ) m / ) then one my ech the following. 2 q m m d d 2 2 u 2 u u 2 = d d E (A.5) Ltin Ameicn Jounl of Solids nd Stuctues, 208, 5(4), e34 6/6