Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 3, pp. 831-839, Warsaw 2012 50th Anniversary of JTAM TRANSIENT THERMOELASTIC ANALYSIS FOR A FUNCTIONALLY GRADED CIRCULAR DISK WITH PIECEWISE POWER LAW Naotake Noda Professor Emeritus, Department ofMechanical Engineering, Shizuoka University, Japan; e-mail: naotakenoda@yahoo.co.jp Yoshihiro Ootao Department of Mechanical Engineering, Graduate School of Engineering, Osaka Prefecture University, Japan; e-mail: ootao@me.osakafu-u.ac.jp Yoshinobu Tanigawa Professor Emeritus, Department ofMechanical Engineering, Graduate School of Engineering,Osaka PrefectureUniversity, Japan; e-mail: tanigawayoshinobu20070424@zeus.eonet.ne.jp The theoretical treatment of a transient thermoelastic problem involving a functionally gra- ded solid circular disk with piecewise power law due to uniform heat supply from an outer surface is studied. The solid circular disk is also cooled from the upper and lower flat sur- faces. The functionally graded circular disk consists of many thin circular layers in order to guarantee the voluntariness of material position dependency. The thermal conductivity, Young’s modulus and the coefficient of linear thermal expansion of each layer, except the first inner layer, are expressed as power functions of the radial coordinate, and their values continue on the interfaces. We obtain the exact solution for the one-dimensional tempe- rature change in a transient state, and in-plane thermoelastic response under the state of plane stress. Some numerical results for the temperature change, displacement and stress distributions are shown in figures. Key words: functionally gradedmaterial, solid circular disk, piecewise power law 1. Introduction Functionally graded materials (FGMs) are those in which two or more different material in- gredients change continuously and gradually along the certain direction. When FGMs are used under high temperature conditions or are subjected to several thermal loading, it is necessa- ry to analyze the thermal stress problems for FGMs. Because the governing equations for the temperature field and the associate thermoelastic field of FGMs become of a nonlinear form in generally, the analytical treatment is difficult. Noda and Tsuji (1991) analyzed the steady thermoelastic problem of an FGMplate. Peng and Li (2010) analyzed the steady thermal stress problem in rotating FGMhollow circular disks. On the other hand, it is well-known that thermal stress distributions in a transient state can show large values compared with the one in a steady state. Therefore, the analysis of the transient thermoelastic problem for FGMs becomes important. Obata and Noda (1995) analy- zed the transient thermal stresses in a hollow sphere of FGM by the perturbationmethod. The other exact analytical treatments are assumed that thematerial properties are given by specific functions containing the variable of the thickness coordinate. Sugano (1987) analyzed exactly one-dimensional transient thermal stresses of an FGM plate where the thermal conductivity and Young’s modulus vary exponentially, whereas Poisson’s ratio and the coefficient of linear thermal expansion vary arbitrarily in the thickness direction. Vel andBatra (2003) analyzed the three-dimensional transient thermal stresses of an FGM rectangular plate, where the material 832 N. Noda et al. properties are expressed asTaylor’s series in the thickness direction.Ootao andTanigawa (2005) discussed the three-dimensional transient thermal stress problems of an FGM rectangular plate, where the thermal conductivity, the coefficient of linear thermal expansion and Young’s modu- lus vary exponentially in the thickness direction. Ohmichi et al. (2010) analyzed the transient thermal stress problemof the stripwith boundaries oblique to the functionally graded direction. Zhao et al. (2006) treated the one-dimensional transient thermo-mechanical behavior of FGM solid cylinder, whose thermoelastic material properties vary exponentially through the thick- ness. Ootao and Tanigawa (2006) analyzed the one-dimensional solution for transient thermal stresses of an FGMhollow cylinder whosematerial properties varywith the power product form of the radial coordinate variable. Shao et al. (2007) discussed the one-dimensional transient thermo-mechanical behavior of FGM hollow cylinders, whose thermoelastic material properties are expressed as Taylor’s series. However, these studies discussed the thermoelastic problems of one-layered FGM models, which have big limitation of nonhomogeneity. Tanigawa et al. (1989) proposed the theory of la- minated compositeswhosematerial properties have constants in each layer.Ootao andTanigawa (1994, 1999), Ootao et al. (1995), and Sugano et al. (2001) analyzed the transient thermal stress problems of several analytical models using the theory of laminated composites. But the theory of laminated composites has a weak point such that the material properties are discontinuous on each interface. Guo and Noda (2007) proposed a piecewise-exponential model for the crack problems in FGMs with arbitrary material properties which are continuous on each interface in order to improve the ordinary theory of laminated composites. Ootao (2010) analyzed the transient thermoelastic problem in the FGM hollow cylinder by a piecewise-power model when the material properties can be expressed by piecewise power law. From the viewpoint of past studies, we analyze the transient thermoelastic analysis for an FGM solid circular disk whose material properties are expressed by piecewise power law to guarantee arbitrary nonhomogeneity of material properties. The FGM disk is suddenly heated from the outer surface by surrounding media, and also is cooled from the upper and lower surfaces. 2. Analysis The functionally graded solid circular disk consists ofmany layers whosematerial properties are expressed by piecewise power law of position. The thermal conductivity, Young’s modulus and the coefficient of linear thermal expansion of each layer, except the first inner layer, are expressed aspower functionsof the radial coordinate, and their values continue on each interface.Theouter radius of the solid circular disk is designated by rb. Moreover, ri is the outer radius of the ith layer. The thickness of the solid circular disk is represented by B. 2.1. Heat conduction problem TheFGMcircular disk is assumed to be initially at zero temperature and is suddenly heated from the outer surface by surrounding media of constant temperature Tb with relative heat transfer coefficient hb. TheFGMcircular disk is also cooled from theupper and lower surfaces of the ith layer by the surroundingmedia of zero temperaturewith theheat transfer coefficient γis. Theone-dimensional transientheat conduction equation for the ith layer is taken in the following form ciρi ∂Ti ∂t = 1 r ∂ ∂r [ λi(r)r ∂Ti ∂r ] − 2γsi B Ti i=1,2, . . . ,N (2.1) Transient thermoelastic analysis for a functionally graded... 833 The thermal conductivity λi and the heat capacity per unit volume ciρi in each layer are assumed to take the following forms λi(r)=      λi(const) for i=1 ciρi = const λ0i ( r ri−1 )mi for i=2, . . . ,N ciρi = const (2.2) where mi = ln(λi+1 0/λi 0) ln(ri/ri−1) λ1 =λ2 0 ciρi 6= ci+1ρi+1 (2.3) Substituting Eqs. (2.2) into Eq. (2.1), the transient heat conduction equations in dimensionless form for each layer are ∂T i ∂τ =            κi (∂2T i ∂r2 + 1 r ∂T i ∂r ) − 2Hsi ciρiB T i for i=1 λ 0 i ciρir mi i−1 ( rmi ∂2T i ∂r2 +(mi+1)r mi−1 ∂T i ∂r ) − 2Hsi ciρiB T i for i=2, . . . ,N (2.4) The initial and thermal boundary conditions in dimensionless form are τ =0 T i =0 i=1,2, . . . ,N (2.5) and r= ri T i =T i+1 i=1,2, . . . ,N −1 r= ri λi ∂T i ∂r =λi+1 ∂T i+1 ∂r i=1,2, . . . ,N −1 r=1 ∂TN ∂r +HbTN =HbTb (2.6) In Eqs. (2.4)-(2.6), we have introduced the following dimensionless values (T i,Ta,Tb)= (Ti,Ta,Tb) T0 (r,ri,B)= (r,ri,B) rb τ = λ0t c0ρ0r 2 b ciρi = ciρi c0ρ0 (λi,λ 0 i)= (λi,λ 0 i) λ0 κi = λi ciρi Hb =hbrb Hsi = γsirb λ0 (2.7) where Ti is the temperature change; t is time; and T0, λ0 and c0ρ0 are typical values of temperature, thermal conductivity and heat capacity per unit volume, respectively. Introducing theLaplace transformationwith respect to the variable τ, solutions toEqs. (2.4) canbeobtained so as to satisfy conditions (2.5) and (2.6). These solutions are shown as follows: — for i=1 T i = 1 F A ′ iI0(ω ′ ir) + ∞ ∑ j=1 { 2λ 0 2Bµ2j(2−m2) 2 [µ22jλ 0 2(2−m2) 2B+8Hs2r m2 1 ]∆ ′(µ2j) exp [ − (λ 0 2(2−m2) 2µ22j 4c2ρ2r m2 1 + 2Hs2 c2ρ2B ) τ ] ·AiJ0 ( √ M ′1iµ 2 2j +M ′ 2ir )} (2.8) 834 N. Noda et al. — fori=2, . . . ,N T i = 1 F r− mi 2 [ A ′ iIγi(ωir 1− mi 2 )+B ′ iKγi(ωir 1− mi 2 ) ] + ∞ ∑ j=1 { 2λ 0 2Bµ2j(2−m2) 2r− mi 2 [µ22jλ 0 2(2−m2) 2B+8Hs2r m2 1 ]∆ ′(µ2j) exp [ − (λ 0 2(2−m2) 2µ22j 4c2ρ2r m2 1 + 2Hs2 c2ρ2B ) τ ] · [ AiJγi ( √ M1iµ 2 2j +M2ir 1− mi 2 ) +BiYγi ( √ M1iµ 2 2j +M2ir 1− mi 2 )]} (2.9) where Iξ(·) and Kξ(·) are the modified Bessel functions of the first and second kind of the order ξ, respectively. Jξ(·) and Yξ(·) are the Bessel functions of the first and second kind of the order ξ, respectively. And ∆ and F are the determinants of (2N −1)× (2N −1) matrix [akl] and [ekl], respectively. The coefficients A1, Ai (i = 2, . . . ,N) and Bi (i = 2, . . . ,N) are defined as the determinant of the matrix similar to the coefficient matrix [akl], in which the first column, (2i−2)th column or (2i−1)th column is replaced by the constant vector {ck}, respectively. Similarly, the coefficients A ′ 1, A ′ i (i=2, . . . ,N) and B ′ i (i=2, . . . ,N) are defined as the determinant of thematrix similar to the coefficientmatrix [ekl], inwhich the first column, (2i− 2)th column or (2i− 1)th column is replaced by the constant vector {ck}, respectively. The elements of the coefficient matrices [akl], [ekl] and the constant vector {ck} are given by Eqs. (2.6). In Eqs. (2.8) and (2.9), M ′1i,M ′ 2i,M1i,M2i,∆ ′(µ2j), ω ′ i, ωi and γi are M ′1i = λ 0 2(2−m2) 2 4c2ρ2r m2 1 κi M ′2i = 2 Bκi (Hs2 c2ρ2 − Hsi ciρi ) M1i = λ 0 2(2−m2) 2ciρir mi i−1 λ 0 i(2−mi) 2c2ρ2r m2 1 M2i = 8ciρir mi i−1 λ 0 i(2−mi) 2B (Hs2 c2ρ2 − Hsi ciρi ) ∆′(µ2j)= d∆ dµ2 ∣ ∣ ∣ ∣ µ2=µ2j ω′i = √ 2Hsi λiB ωi = √ √ √ √ 8Hsir mi i−1 λ 0 iB(2−mi) 2 γi = ∣ ∣ ∣ mi 2−mi ∣ ∣ ∣ (2.10) and µ2j represent the jth positive roots of the following transcendental equation ∆(µ2)= 0 (2.11) 2.2. Thermoelastic problem The transient thermoelasticity of the FGMdisk is analyzed under the plane stress problem. The displacement-strain relations are expressed in the dimensionless form as follows εrri =uri,r εθθi = uri r (2.12) where the comma denotes partial differentiation with respect to the variable that follows. The constitutive relations are in the dimensionless form as follows { σrri σθθi } = Ei (1+νi)(1−νi) [ 1 νi νi 1 ]{ εrri εθθi } − αiEiT i 1−νi (2.13) The equilibrium equation is expressed in the dimensionless form as follows σrri,r+ 1 r (σrri−σθθi)= 0 (2.14) Transient thermoelastic analysis for a functionally graded... 835 Young’s modulus Ei, the coefficient of linear thermal expansion αi and Poisson’s ratio νi are assumed to take the following forms:— for i=1 Ei(r)=Ei(const) αi(r)=αi(const) νi = const (2.15) — for i=2, . . . ,N Ei(r)=E 0 i ( r ri−1 )li αi(r)=α 0 i ( r ri−1 )bi νi = const (2.16) where li = ln(E 0 i+1/E 0 i) ln(ri/ri−1) bi = ln(α0i+1/α 0 i) ln(ri/ri−1) E1 =E 0 2 α1 =α 0 2 νi 6= νi+1 (2.17) In Eqs. (2.12)-(2.17), the following dimensionless values are introduced σkli = σkli α0E0T0 εkli = εkli α0T0 (αi,α 0 i)= (αi,α 0 i) α0 (Ei,E 0 i)= (Ei,E 0 i ) E0 uri = uri α0T0rb (2.18) where σkli are the stress components, εkli are the strain components, uri is the displacement in the radial direction, and α0 and E0 are typical values of the coefficient of linear thermal expansion and Young’s modulus, respectively. Substitution of Eqs. (2.12), (2.13), (2.15) and (2.16) into Eq. (2.14) gives the displacement equation of equilibrium for i=2, . . . ,N uri,rr+ li+1 r uri,r+(νili−1)urir −2 = (1+νi)α 0 i r bi i−1 [(li+ bi)r bi−1T i+r biT i,r ] (2.19) If the outer surface is traction free, and the interfaces of each layer are perfectly bonded, then the boundary conditions of the outer surface and the conditions of continuity on the interfaces can be represented as follows r=1 σrrN =0 r= ri σrri =σrr,i+1 uri =ur,i+1 i=1,2, . . . ,N −1 (2.20) The solution to Eq. (2.19) can be expressed by uri =uric+urip (2.21) where uric and urip denote the homogeneous and particular solution to Eq. (2.19), respectively. We now consider the homogeneous solution, and introduce the following equation r=exp(s) (2.22) Changing a variable with the use of Eq. (2.22), the homogeneous equation of Eq. (2.19) reduces to [ d2 ds2 + li d ds − (1−νili) ] uric =0 (2.23) The homogeneous solution uric(r) for i=2, . . . ,N is given by uric =A1iexp(Λi1s)+A2iexp(Λi2s)=A1ir Λi1 +A2ir Λi2 i=2, . . . ,N (2.24) 836 N. Noda et al. where Λi1 = 1 2 [ −li+ √ l2i +4(1−νili) ] Λi2 = 1 2 [ −li− √ l2i +4(1−νili) ] (2.25) The particular solution urip for i=2, . . . ,N also can be obtained as follows urip = 1 √ l2i +4(1−νili) { C (i) a1 [−L (i) a1a(r)r Λi2 +L (i) a1b(r)r Λi1] +C (i) a2 [−L (i) a2a(r)r Λi2 +L (i) a2b(r)r Λi1] +C (i) b1 [−L (i) b1a(r)r Λi2 +L (i) b1b(r)r Λi1]+C (i) b2 [−L (i) b2a(r)r Λi2 +L (i) b2b(r)r Λi1] + ∞ ∑ j=1 { C (i) c1j[−L (i) c1ja(r)r Λi2 +L (i) c1jb(r)r Λi1]+C (i) c2j[−L (i) c2ja(r)r Λi2 +L (i) c2jb(r)r Λi1] +C (i) d1j[−L (i) d1ja(r)r Λi2 +L (i) d1jb(r)r Λi1]+C (i) d2j[−L (i) d2ja(r)r Λi2 +L (i) d2jb(r)r Λi1] } } (2.26) The expressions for C (i) a1 ,L (i) a1a(r) and so on inEq. (2.26) are omitted here for the sake of brevity. On the other hand, the solution uri for i=1 is uri =A1ir+ (1+νi)αi r r ∫ 0 rTi dr (2.27) The coefficients A1i and A2i in Eqs. (2.24) and (2.27) are unknown constants. Then, the stress components canbeevaluatedby substitutingEqs. (2.21), (2.24), (2.26) and(2.27) intoEq. (2.12), and later into Eq. (2.13). The unknown constants in Eqs. (2.24) and (2.27) can be determined so as to satisfy boundary conditions (2.20). 3. Numerical results We consider the FGMs composed of titanium alloy (Ti-6Al-4V) and zirconium oxide (ZrO2). The FGMdisk is heated from the outer surface (zirconium oxide 100%) by surroundingmedia, and is cooled from both flat surfaces. The material of the first inner layer (i = 1) is titanium alloy 100% and thematerial at the outer surface is zirconium oxide. Thematerial properties gi of the interface between the ith and (i+1)th layer are assumed as follows gi = ga+(gb−ga)fi 0¬ fi ¬ 1 i=2, . . . ,N −1 (3.1) where ga is the material property of the first inner layer, and gb is the material property of the outer surface. The heat capacity per unit volume ciρi andPoisson’s ratio νi of the ith layer use the average of values in both interfaces. The numerical parameters of heat conduction, shape and fi are presented as follows Hb =10.0 Hsi =0.1 Tb =1.0 B=0.05 (3.2) Case 1: N =3 r1 =0.1 r2 =0.55 f2 =0.1, 0.5, 0.9 (3.3) Case 2: N =4 r1 =0.1 r2 =0.4 r3 =0.7 f2 =0.1 f3 =0.2, 0.5, 0.9 (3.4) Transient thermoelastic analysis for a functionally graded... 837 Thematerial constants for titanium alloy (Ti-6Al-4V) are taken as: — for titanium alloy (Ti-6Al-4V) κ=2.61 ·10−6m2/s c=537.7J/(kg ·K) ρ=4420kg/m3 λ=6.2W/(m ·K) α=8.9 ·10−61/K E =105.8GPa ν =0.3 (3.5) — for zirconium oxide (ZrO2) κ=1.06 ·10−6m2/s c=461.4J/(kg ·K) ρ=3657kg/m3 λ=1.78W/(m ·K) α=8.7 ·10−61/K E =116.4GPa ν =0.3 (3.6) The typical values of material properties such as κ0, λ0, α0 and E0 used to normalize the numerical data, are based on those of zirconium oxide. In order to assess the influence of the material property distribution for three-layered FGM model, the numerical results forCase 1 are shown inFigs. 1 and 2. Figure 1a shows the variation of temperature change along the radial direction. Figure 1b shows the variation of the displace- ment ur along the radial direction. From Figs. 1a and 1b, it is seen that the temperature and displacement rise as time proceeds and they are the greatest in the steady state. It can be seen from Figs. 1a and 1b that the values of the temperature change at the center of the solid disk and the displacement decrease when the parameter f2 increases. Figures 2a and 2b show the variations of thermal stresses σrr and σθθ along the radial direction, respectively. From Fig. 2, the maximum tensile stress occurs in the transient state near the center of the solid circular disk. It can be seen from Fig. 2 that the maximum values of the thermal stresses σrr and σθθ decrease when the parameter f2 decreases. Fig. 1. Variation of temperature change (a) and of displacement ur (b) in the radial direction (Case 1, N =3) Fig. 2. Variation of thermal stresses in the radial direction (Case 1, N =3): (a) radial stress σrr and (b) hoop stress σθθ 838 N. Noda et al. In order to assess the influence of the material property distribution for the four-layered FGM model, the numerical results for Case 2 are shown in Fig. 3. Figures 3a and 3b show the variations of thermal stresses σrr and σθθ, respectively. It can be seen from Fig. 3 that the maximum values of thermal stresses σrr and σθθ decrease when the parameter f3 decreases. Fig. 3. Variation of thermal stresses in the radial direction (Case 2, N =4): (a) radial stress σrr and (b) hoop stress σθθ 4. Conclusion We analyzed the transient thermoelastic problem involving a functionally graded solid circular disk with piecewise power law due to uniform heat supply from the outer surface. The FGM circular disk is also cooled from the upper and lower surfaces of each layer with a constant heat transfer coefficient. The thermal conductivity, Young’s modulus and the coefficient of linear thermal expansion of each layer, except the first inner layer, are expressed as power functions of the radial coordinate in the radial direction, and their values continue on each interface. We obtained the exact solution for the transient one-dimensional temperature and transient thermoelastic response of the FGM circular disk.We carried out numerical calculations for the FGMs composed of titanium alloy (Ti-6Al-4V) and zirconium oxide (ZrO2) and examined the behavior in the transient state for the temperature change, displacement, and thermal stresses. Furthermore, the influence of the functional gradation on the temperature and thermoelastic response was investigated. References 1. GuoL.C., NodaN., 2007,Modelingmethod for a crack problemof functionally gradedmaterials with arbitrary properties – piecewise-exponential model, Int. J. Solids Struct, 44, 6768-6790 2. 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A, 418, 99-110 Analiza stanu nieustalonego termosprężystości kołowego dysku wykonanego z materiału gradientowego opisanego modelem kawałkami potęgowym Streszczenie W pracy przedstawiono analizę procesów nieustalonych termosprężystości zachodzących w kołowym dysku wykonanym z materiału gradientowego opisanego modelem matematycznym opartym na funk- cji kawałkami potęgowej przy założeniu jednorodnego ogrzewania od strony zewnętrznej. Badany dysk jest jednocześnie chłodzony na górnej i dolnej płaskiej powierzchni. Struktura dysku zawiera wiele koło- wych warstw gwarantujących dowolność kształtowania właściwości materiału w zależności od położenia. Współczynnik przewodnictwa cieplnego,moduł Younga orazwspółczynnik liniowej rozszerzalności ciepl- nej każdej warstwy, oprócz pierwszej, wyrażonowpostaci funkcji potęgowychwspółrzędnej promieniowej przy zachowaniu warunków zgodności między warstwami. Otrzymano dokładne rozwiązanie dla stanu nieustalonego termosprężystości wywołanego jednowymiarową zmianą temperatury oraz określono za- chowanie się materiału poddanego płaskiemu stanowi naprężenia. Wybrane rezultaty badań dotyczące zmian temperatury oraz rozkładów naprężeń i przemieszczeń przedstawiono graficznie. Manuscript received January 25, 2012; accepted for print February 6, 2012