Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 4, pp. 1053-1065, Warsaw 2015 DOI: 10.15632/jtam-pl.53.4.1053 TIME-DEPENDENT THERMO-ELASTIC CREEP ANALYSIS OF THICK-WALLED SPHERICAL PRESSURE VESSELS MADE OF FUNCTIONALLY GRADED MATERIALS Mosayeb Davoudi Kashkoli, Mohammad Zamani Nejad Mechanical Engineering Department, Yasouj University, Yasouj, Iran e-mail: m.zamani.n@gmail.com; m zamani@yu.ac.ir Assuming that the thermo-elastic creep response of thematerial is governedbyNorton’s law and material properties, except Poisson’s ratio, are considered as a function of the radius of the spherical vessel, an analytical solution is presented for calculation of stresses and displacements of axisymmetric thick-walled spherical pressure vessels made of functionally gradedmaterials. This analytical solution could be used to study the time and temperature dependence of stresses in spherical vessels made of functionally graded materials. Creep stresses and displacements are plotted against dimensionless radius and time for different values of the powers of the material properties. Keywords: sphericalpressurevessels, creep, time-dependent, thermo-elastic, functionallygra- ded material 1. Introduction Composites are commonly employed invarious structural andengineering applications.Recently, a new class of composite materials known as functionally graded materials (FGMs) has drawn considerable attention. From viewpoints of solidmechanics, FGMs are non-homogeneous elastic mediums (Ghannad and Nejad, 2013). FGMs are those in which two or more different material ingredients change continuously and gradually along a certain direction (Noda et al., 2012). The mechanical and thermal responses of materials with spatial gradients in composition are of considerable interests in numerous industrial applications such as tribology, biomechanics, nanotechnology and high temperature technologies. These materials are usually mixtures of metal and ceramic which exhibit excellent thermal resistance with low levels of thermal stresses (Alashti et al., 2013). Extensive studieshavebeencarriedout, both theoretically andnumerically, on thermo-elastic creep stress distribution in functionally gradient materials. Time-dependent creep analysis of FGM spheres and cylinders has been an active area of research over the past decade. Assuming the infinitesimal strain theory, Finnie and Heller (1959) studied creep problems in engineering materials and a steady-state creep solution for a spherical vessel under internal pressure. Johnson and Khan (1963) obtained a theoretical analysis of the distribution of stress and strain in metallic thick-walled spherical pressure vessels subject to internal and external pressuresat elevated temperatures.Penny (1967) investigated creepof spherical shells containing discontinuities. In this study, the solution procedure has been applied to pressurized spherical shells conta- ining discontinuities with a view to discovering, in broad terms, how stresses change with time and how strains accumulate during the creep process. Bhatnagar andArya (1975) obtained cre- ep analysis of a pressurized thick-walled spherical vessel made of a homogeneous and isotropic material bymaking use of the finite strain theory, and with considering large strains. Miyazaki 1054 M.D. Kashkoli, M.Z. Nejad et al. (1977) presented a parametric analysis of the creep buckling of a shallow spherical shell subjected to uniform external pressure using the finite element incremental method. Xiroucha- kis and Jones (1979) investigated creep buckling behavior of a geometrically imperfect complete spherical shell subjected to a uniform external pressure using Sanders’ equilibrium and kinema- tic equations appropriately modified to include the influence of initial stress-free imperfections in the radius. In this study, the Norton-Bailey constitutive equations are used to describe the secondary creep behavior, and elastic effects are retained. Arya et al. (1980) studied effect of material ani- sotropy on creep of pressurized thick-walled spherical vessel considering the large strain theory. Kao (1981) obtained creepdeformations and creepbuckling times for axisymmetric shallow sphe- rical shells with and without initial imperfections. For nonlinear creeps, both strain-hardening and time-hardening rules are employed in this study. Creep of a sphere subjected to inner and outer pressures, and also thermal stress, was discussed by Sakaki et al. (1990) by using internal stress arising from a spherically symmetric, finite plastic strain. Assuming that the elastic beha- vior of the material is undergoing both creep and dimensional changes, Miller (1995) presented a solution for stresses and displacements in a thick spherical shell subjected to internal and external pressure loads. Based on basic equations of steady-state creep of spherically symme- tric problems, You et al. (2008) proposed a simple and efficient iterative method to determine creep deformations and stresses in thick-walled spherical vessels with varying creep properties subjected to internal pressure. Using a long-term material creep constitutive model defined by the Θ projection concept, Loghman and Shokouhi (2009) evaluated the damage histories of a thick-walled sphere subjected to an internal pressure and a thermal gradient. They studied the creep stress and damage histories of thick-walled spheres using the material constant creep and creep rupture properties defined by the theta projection concept. Aleayoub and Loghman (2010) studied time-dependent creep stress redistribution analysis of thick-walled FGM spheres subjected to internal pressure and a uniform temperature field. In this study, using equations of equilibrium, compatibility and stress-strain relations, a dif- ferential equation, containing creep strains, for radial stress is obtained. Ignoring creep strains in this differential equation, a closed-form solution for initial thermo-elastic stresses at zero time is presented. Pankaj (2011) investigated creep stresses for a thick isotropic spherical shell by finitesimal deformation under steady-state temperature and internal pressure by using Seth’s transition theory. Marcadon (2011) presented mechanical modelling of the creep behavior of Hollow-Sphere Structures. Based on basic equations of steady-state creep of spherically symme- tric problems,Nejad et al. (2011) presented a new exact closed form solution for creep stresses in isotropic and homogeneous thick spherical pressure vessels. Loghman et al. (2011) investigated time-dependent creep stress redistributionanalysis of thick-walled spheresmadeof a functionally gradedmaterial (FGM) subjected to internal pressure. In another study, Loghman et al. (2012) investigatedmagneto-thermo-elastic creep behavior of thick-walled spheresmade of functionally graded materials (FGM) placed in uniform magnetic and distributed temperature fields and subjected to internal pressure using the method of successive elastic solution. They developed a semi-analytical method in conjunction withMendelson’s method of successive elastic solution to obtain history of stresses and strains. Assuming that the creep response of the material is governed by Norton’s law, Nejad et al. (2013) presented a new exact solution for steady state creep stresses of hollow thick-walled spherical shells subjected to internal and external pressure, made of functionally graded materials (FGMs). By using the method of successive elastic solu- tion, Fesharaki et al. (2014) presented a semi-analytical solution for the time-dependent creep behavior of hollow spheres under thermomechanical loads. In this study, assuming that the thermo-elastic creep response of thematerial is governed by Norton’s law, an analytical solution is presented for the calculation of stresses anddisplacements of FGM thick-walled spherical pressure vessels. For the creep material behavior, the solution is Time-dependent thermo-elastic creep analysis of thick-walled spherical pressure... 1055 asymptotic. For the stress analysis after creeping for a long time, an iterative procedure is necessary. 2. Geometry and loading condition, material properties and creep constitutive model 2.1. Geometry and loading condition A thick-walled spherical vessel made of a functionally graded material with inner radius a, and outer radius b, subjected to internal pressure Pi and external pressure Po and a distributed temperature field due to steady-state heat conduction from the inner surface to the outer surface of the vessel is considered. 2.2. Material properties Thematerial properties are assumed to be radially dependent E(r)= Ei (r a )n1 α(r)= αi (r a )n2 λ(r)= λi (r a )n3 (2.1) Here Ei, αi and λi are the modulus of elasticity, linear expansion and thermal conductivity on the linear surface, r = a and n1, n2 and n3 are inhomogeneity constants determined empirically. Fig. 1. Geometry and boundary conditions of the sphere 2.3. Creep constitutive model For materials with creep behavior, Norton’s law (1956) is used to describe the relations between the rates of stress σ̇ij and strain ε̇ij in the multi-axial form ε̇ij = 1+ν E σ̇ij − ν E σ̇kkδij + 3 2 Dσ (N−1) eff Sij (2.2) and Sij = σij − 1 3 σkkδij σeff = √ 3 2 SijSij = 1√ 2 √ (σrr −σθθ)2+(σrr −σφφ)2+(σφφ −σθθ)2 = σrr −σθθ (2.3) where D and N are material constants for creep. σeff is the effective stress, Sij is the deviator stress tensor and σrr and σθθ = σφφ are respectively the radial and circumferential stresses. 1056 M.D. Kashkoli, M.Z. Nejad 3. Heat conduction formulation In the steady-state case, the heat conduction equation for the one-dimensional problem in sphe- rical coordinates simplifies to 1 r2 ∂ ∂r ( r2λ ∂T ∂r ) =0 (3.1) We can determine the temperature distribution in the spherical vessel by solving Eq. (3.1) and applying appropriate boundary conditions. Equation (3.1) may be integrated twice to obtain the general solution T(r)= A1r −n3−1+A2 (3.2) It is assumed that the inner surface is exposed to uniform heat flux, whereas the outer surface is exposed to airstream. To obtain the constants of integration A1 and A2, we introduce the following boundary conditions −λT ′ = { qa for r = a h∞(T −T∞) for r = b (3.3) where T ′ = dT/dr. Applying these conditions to the general solution, we obtain A1 = an3+2qa (n3+1)λi A2 = T∞+ qa h∞ (a b )2 − a n3+2qa λi(n3+1)bn3+1 (3.4) Substituting the constants of integration A1 and A2 into the general solution, we obtain the temperature distribution T(r)= T∞+ qa h∞ (a b )2 + an3+2qa (n3+1)λi (r−n3−1− b−n3−1) (3.5) 4. Formulation of the thermo-elastic creep analysis 4.1. Solution for linear elastic behavior of FGM thick spherical pressure vessels For the stress analysis inan FGM thick spherical pressure vessel, having material creep be- havior, solutions of the stresses at a time equal to zero (i.e. the initial stress state) are needed, which correspond to the solution of materials with linear elastic behavior. In this Section, equ- ations to calculate such linear stresses in FGM thick spherical pressure vessel analytically will be given. The elastic stress-strain relations in each material read σrr = E (1+ν)(1−2ν) [(1−ν)εrr +2νεθθ − (1+ν)αT ] σθθ = σφφ = E (1+ν)(1−2ν) [εθθ +νεrr − (1+ν)αT ] (4.1) whereσrr andσθθ = σφφ are radial and circumferential stresses, respectively.HereE, ν andα are Young’s modulus, Poisson’s ratio and thermal expansion coefficient, respectively, and T = T(r) is the temperature distribution in the sphere. The strain displacement relation is written as εrr = dur dr εθθ = ur r (4.2) Time-dependent thermo-elastic creep analysis of thick-walled spherical pressure... 1057 where εrr and εθθ = εφφ are radial and circumferential strains and ur is the displacement in the r-direction. The equation of the stress equilibrium inside the FGM spherical pressure vessel is dσrr dr + 2 r (σrr −σθθ)= 0 (4.3) UsingEqs. (4.1)-(4.3), the essential differential equation for the displacement ur can be obtained as d2ur dr2 + n1+2 r dur dr + ν′′n1−2 r2 ur = ν ′(Arn2−1+Brn2−n3−2) ν′ = 1+ν 1−ν ν′′ = 2ν 1−ν (4.4) where A =(n1+n2) [T∞αi an2 + qaαi h∞an2 (a b )2] − a n3−n2+2qaαi(n1+1)b−n3−1 λi(n3+1) B = (n1+1)an3−n2+2qaαi λi(n3+1) − an3−n2+2qaαi λi (4.5) It is obvious that the homogeneous solution for Eq. (4.4) can be obtained by assuming ur = Cr x (4.6) Substituting Eq. (4.6) into Eq. (4.4) one can obtain the following characteristic equation x2+(n1+1)x+(ν ′′n1−2)= 0 (4.7) The roots of Eq. (4.7) are x1 = −(n1+1)+ √ (n1+1)2−4(ν′′n1−2) 2 x2 = −(n1+1)− √ (n1+1)2−4(ν′′n1−2) 2 (4.8) For selected common values of n1 =±0.4,±0.8 and ν =0.292 in this study, the discriminant of Eqs. (4.8) is always greater thanzero; therefore,x1 andx2 are real anddistinct.Thehomogeneous solution to Eq. (4.4) is then as follows uh = C1r x1 +C2r x2 (4.9) The particular solution to differential equation (4.4) can be obtained as up = u1r x1 +u2r x2 (4.10) where u1 = ∫ −rx2P(r) W(rx1,rx2) dr u2 = ∫ rx1P(r) W(rx1,rx2) dr (4.11) in which P(r)= ν′(Arn2−1+Brn2−n3−2) (4.12) is the expression on the right-hand side of Eq. (4.4), and W is defined as W(rx1,rx2)= ∣∣∣∣∣ rx1 rx2 x1r x1−1 x2r x2−1 ∣∣∣∣∣ =(x2−x1)r x1+x2−1 (4.13) 1058 M.D. Kashkoli, M.Z. Nejad Therefore, u1 and u2 can be obtained by the following integration u1 = ν ′ [∫ −rx2Arn2−1 (x2−x1)rx1+x2−1 dr+ ∫ −rx2Brn2−n3−2 (x2−x1)rx1+x2−1 dr ] =−ν′ [ Arn2−x1+1 (x2−x1)(n2−x1+1) + Brn2−n3−x1 (x2−x1)(n2−n3−x1) ] u2 = ν ′ [∫ rx1Arn2−1 (x2−x1)rx1+x2−1 dr+ ∫ rx1Brn2−n3−2 (x2−x1)rx1+x2−1 dr ] = ν′ [ Arn2−x2+1 (x2−x1)(n2−x2+1) + Brn2−n3−x2 (x2−x1)(n2−n3−x2) ] (4.14) Substituting Eqs. (4.14) into Eq. (4.10), one can obtain the particular solution as up = ν ′ [ Arn2+1 (n2−x2+1)(n2−x1+1) + Brn2−n3 (n2−n3−x2)(n2−n3−x1) ] (4.15) The complete solution to Eq. (4.4) can be written as ur(r)= C1r x1 +C2r x2+ν′ [ Arn2+1 (n2−x2+1)(n2−x1+1) + Brn2−n3 (n2−n3−x2)(n2−n3−x1) ] (4.16) The corresponding stresses are σrr = Ei ( r a )n1 (1+ν)(1−2ν) { C1r x1−1[2ν +(1−ν)x1]+C2rx2−1[2ν +(1−ν)x2] + Aν′[(n2+1)(1−ν)+2ν]rn2 (n2−x2+1)(n2−x1+1) + Bν′[(n2−n3)(1−ν)+2ν]rn2−n3−1 (n2−n3−x2)(n2−n3−x1) − (1+ν) [(T∞αi an2 + qaαi h∞an2 (a b )2 − an3−n2+2qaαib −n3−1 λi(n3+1) ) rn2 + an3−n2+2qaαi λi(n3+1) rn2−n3−1 ]} σθθ = Ei ( r a )n1 (1+ν)(1−2ν) { C1r x1−1(1+x1ν)+C2r x2−1(1+νx2) + Aν′[(n2+1)ν +1]rn2 (n2−x2+1)(n2−x1+1) + Bν′[(n2−n3)ν +1]rn2−n3−1 (n2−n3−x2)(n2−n3−x1) − (1+ν) [(T∞αi an2 + qaαi h∞an2 (a b )2 − a n3−n2+2qaαib −n3−1 λi(n3+1) ) rn2 + an3−n2+2qaαi λi(n3+1) rn2−n3−1 ]} (4.17) To determine the unknown constants C1 and C2 in each material, boundary conditions have to be used, which are σrr = { −Pi for r = a −Po for r = b (4.18) The unknown constants C1 and C2 are given in Appendix. Time-dependent thermo-elastic creep analysis of thick-walled spherical pressure... 1059 4.2. Solution for creep behavior of FGM thick spherical pressure vessel The relations between the rates of strain and displacement are ε̇rr = du̇r dr ε̇θθ = u̇r r (4.19) And the equilibrium equation of the stress rate is dσ̇rr dr + 2 r (σ̇rr − σ̇θθ)= 0 (4.20) The relations between the rates of stress and strain are σ̇rr = E (1+ν)(1−2ν) [(1−ν)ε̇rr +2νε̇θθ]− 3 2 E (1−ν)(1−2ν) Dσ (N−1) eff S ′ rr σ̇θθ = E (1+ν)(1−2ν) [ε̇θθ +νε̇rr]− 3 2 E (1−ν)(1−2ν) Dσ (N−1) eff S ′ θθ (4.21) where S′rr =(1−ν)Srr +2νSθθ S′θθ = Sθθ +νSrr (4.22) Using Eqs. (4.19)-(4.22), the essential differential equation for the displacement rate u̇r in FGM spherical vessel can be obtained as d2u̇r dr2 + du̇r dr (2 r + d lnE dr ) + u̇r r ( 2ν 1−ν d lnE dr − 2 r ) = d lnE dr 3 2(1−ν) Dσ (N−1) eff S ′ rr + 1 1−ν d dr (3 2 Dσ (N−1) eff S ′ rr ) + 3 r(1−ν) Dσ (N−1) eff (S ′ rr −S′θθ) (4.23) In general, the quantities σeff , S′rr and S ′ θθ are very complicated functions of the coordinate r, even in an implicit function form. Therefore, it is almost impossible to find an exact analytical solution to Eq. (4.23).We can alternatively find an asymptotical solution to Eq. (4.23). At first, we assume that σeff , S′rr and S ′ θθ are constant, i.e. they are independent of the coordinate r d2u̇r dr2 + 1+n1 r du̇r dr − νn1−1 r2 u̇r = 3 2 D r σ (N−1) eff [S′rr(1+n1−v′)+S′θθ(n1ν′−1+ν′)] (4.24) The homogeneous solution to Eq. (4.24) is then uh = D1r x1 +D2r x2 (4.25) The particular solution to differential equation (4.24) can be obtained as up = u ′ 1r x1 +u′2r x2 (4.26) where u′1 = ∫ −rx2Hr−1 (x2−x1)rx1+x2−1 dr u′2 = ∫ rx1Hr−1 (x2−x1)rx1+x2−1 dr H = 3 2 D (1−ν) σ (N−1) eff [S ′ rr(2+n1)−2S′θθ] (4.27) The complete solution to Eq. (4.24) can be written as u̇r(r)= D1r x1 +D2r x2 + 3 2 Drσ (N−1) eff [(n1+2)S ′ rr −2S′θθ] (1−ν)[n1(ν′′−1)−2] (4.28) 1060 M.D. Kashkoli, M.Z. Nejad where the unknown constants D1 and D2 can be determined from the boundary conditions. The corresponding stress rates are σ̇rr = Ei ( r a )n1 (1−2ν)(1+ν) { D1r x1−1[2ν +(1−ν)x1]+D2rx2−1[2ν +(1−ν)x2] + 3 2 ν′ n1(ν′′−1)−2 Dσ (N−1) eff [(n1+2)S ′ rr −2S′θθ]− 3 2 Dσ (N−1) eff S′rr } σ̇θθ = Ei ( r a )n1 (1−2ν)(1+ν) { D1r x1−1(1+νx1)+D2r x2−1(1+νx2) + 3 2 1+ν n1(ν′′−1)−2 Dσ (N−1) eff [(n1+2)S ′ rr −2S′θθ]− 3 2 Dσ (N−1) eff S′θθ } (4.29) To determine the unknown constants D1 and D2 in each material, boundary conditions have to beused. Since the inside andoutside pressuresdonot changewith time, theboundaryconditions for stress rates on the inner and outer surfaces may be written as σ̇rr = { 0 for r = a 0 for r = b (4.30) The unknown constants D1 and D2 are given in Appendix.When the stress rate is known, the calculation of stresses at any time ti should be performed iteratively σ (i) ij (r,ti)= σ (i−1) ij (r,ti−1)+ σ̇ (i) ij (r,ti)dt (i) ti = i∑ k=0 dt(k) (4.31) To obtain a generally useful solution, a higher-order approximation of σeff , S′rr and S ′ θθ should bemade σeff (r)=σeff(r)+ d dr [σeff (r)] ∣∣∣ r=r 1! (r−r)+ d2 dr2 [σeff (r)] ∣∣∣ r=r 2! (r−r)2+ d3 dr3 [σeff (r)] ∣∣∣ r=r 3! (r−r)3 S′rr(r)= S ′ rr(r)+ d dr [S′rr(r)] ∣∣∣ r=r 1! (r−r)+ d2 dr2 [S′rr(r)] ∣∣∣ r=r 2! (r−r)2+ d3 dr3 [S′rr(r)] ∣∣∣ r=r 3! (r−r)3 (4.32) S′θθ(r)= S ′ θθ(r)+ d dr [S′θθ(r)] ∣∣∣ r=r 1! (r−r)+ d2 dr2 [S′θθ(r)] ∣∣∣ r=r 2! (r−r)2+ d3 dr3 [S′θθ(r)] ∣∣∣ r=r 3! (r−r)3 where r is the center point of the wall thickness in the following analysis. 5. Numerical results and discussion In the previous Sections, the analytical solution of creep stresses for FGM thick-walled spheri- cal vessels subjected to uniform pressures on the inner and outer surfaces has been obtained. In this Section, some profiles are plotted for the radial displacement, radial stress and circum- ferential stress as a function of the radial direction and time. An FGM thick-walled spherical vessel with creep behavior under internal and external pressure is considered.Radii of the sphere are a = 20mm, b = 40mm. Mechanical properties of the sphere such as modulus of elastici- ty, linear expansion and thermal conductivity are assumed to be varying through the radius. The inhomogeneity constants n1 = n2 = n3 = n, and n ranges from −0.8 to +0.8. The fol- lowing data for loading and material properties are used in this investigation: Ei = 207GPa, Time-dependent thermo-elastic creep analysis of thick-walled spherical pressure... 1061 ν =0.292, αi =10.8 ·10−6K−1, Pi =80MPa, Po =0MPa, qa =500W/m2, λi =43W/(m◦C), h∞ =6.5W/(m2 ◦C), T∞ =25◦C, D =1.4 ·10−8, N =2.25. The distributions of creep stress components σrr and σθθ after 10h of creeping for values of n = ±0.4, ±0.8 are plotted in Fig. 2. It must be noted from Fig. 2a that the radial stress increases as n decreases, and that the radial stress for different values of n is compressive. The absolutemaximums of radial stress occur at the outer edge. Itmeans themaximum shear stress, which is τmax = (σθθ −σrr)/2 for each value of n will be very high on the outer surface of the vessel. Fig. 2. Normalized radial and circumferential stresses versus dimensionless radius after 10h of creeping It is also clear from Fig. 2a that the maximum changes in the radial stresses with time take place for the material n = 0.8 and the minimum changes occur for n = ±0.4, −0.8. The circumferential stress shown in Fig. 2b remains compressive throughout and is observed to decrease with the increasing radius for n = −0.4, −0.8, and reaches the minimum value somewhere towards the inner radius followed by an increase with a further increase in the radius. It also can be seen from Fig. 2b that the circumferential stress remains compressive throughout the cylinder for n =+0.8 with the maximum value at the inner radius and zero at the outer radius under the imposed boundary conditions, and that theminimum changes occur for n =0.4. Time dependent stress redistributions at the point r = 30mm are shown in Fig. 3. It can be seen in Fig. 3a that the radial stress increases as time increases. It must be noted from Fig. 3b that, for n =+0.4, +0.8 the circumferential stress decreases as time increases, whereas for n =−0.4,−0.8 the circumferential stress increases as time increases. Fig. 3. Time-dependent radial and circumferential stresses at the point r =30mm The radial displacement along the radius is plotted in Fig. 4a. There is an increase in the value of the radial displacement as n increases and the maximum value of radial displacement occurs at the outer edge. The time-dependent radial displacement at the point r = 30mm is shown in Fig. 4b. Figure 4b shows that the radial displacement redistribution at the point 1062 M.D. Kashkoli, M.Z. Nejad r = 30mm increases as time increases for n =−0.4, −0.8, while for n =+0.4, +0.8 the radial displacement decreases as time increases. Figure 5 shows the effect of adding external pressure to the radial and circumferential stresses. It can be seen inFig. 5 that the radial stress decreases as the external pressure increases while the circumferential stress increases as the external pressure increases. Fig. 4. (a) Normalized radial displacement versus dimensionless radius after 10h of creeping, (b) time-dependent radial displacement at the point r =30mm Fig. 5. The effect of adding external pressure to the radial and circumferential stresses Temperature distribution of four different values of n is shown in Fig. 6. It can be seen in Fig. 6 that the maximum values of temperature occur at the inner radius for n = −0.8 and that the minimum values of temperature occur at the outer radius for all values of n under the imposed boundary conditions. Fig. 6. Temperature distribution of FGM thick-walled spherical vessel for values of n =±0.4,±0.8 6. Conclusions In this paper, assuming that the thermo-creep response of thematerial is governed byNorton’s law, an analytical solution is presented for the calculation of stresses and displacements of FGM thick-walled spherical pressure vessels. For the stress analysis inasphere, having material creep Time-dependent thermo-elastic creep analysis of thick-walled spherical pressure... 1063 behavior, the solutions of the stresses at a time equal to zero (i.e. the initial stress state) are needed,which corresponds to the solution ofmaterials with linear elastic behavior. It is assumed that the material properties change as graded in the radial direction to a power law function. To show the effect of inhomogeneity on the stress distributions, different values are considered for inhomogeneity constants. The pressure, inner radius and outer radius are considered con- stant. The heat conduction equation for the one-dimensional problem in spherical coordinates is used to obtain temperature distribution in the sphere. For the creep material behavior, the solution is asymptotic. For the stress analysis after creeping for a long time, the iterative proce- dure is necessary. It could be seen that the inhomogeneity constants have significant influence on the distributions of the creep stresses and radial displacement. By increasing the grading parameter n, the normalized radial stress increases due to internal pressure and temperature distribution while the normalized circumferential stress decreases (Fig. 2). The absolute maxi- mums of radial and circumferential stresses occur at the outer edge. It must be noted that the radial and circumferential stresses at the point r = 30mm for different values of n are com- pressive. As can be seen, the absolute maximum of radial stress at the point r =30mm occurs at a time equal to 10 hours for different values of n, whereas for n = +0.4, +0.8 the absolute maximum of circumferential stress occurs at a time equal to zero, and for n =−0.4, −0.8 the absolute maximum of circumferential stress occurs at a time equal to 10 hours. Appendix The unknown constants in Eqs. (4.17) are C1 = −Pi(1+ν)(1−2ν) Ei[2ν +(1−ν)x1]ax1−1 − C2[2ν +(1−ν)x2]ax2−1 [2ν +(1−ν)x1]ax1−1 − Aν′[(n2+1)(1−ν)+2ν]an2 (n2−x2+1)(n2−x1+1)[2ν +(1−ν)x1]ax1−1 − Bν ′[(n2−n3)(1−ν)+2ν]an2−n3−1 (n2−n3−x2)(n2−n3−x1)[2ν +(1−ν)x1]ax1−1 + 1+ν [2ν +(1−ν)x1]ax1−1 {[T∞αi an2 + qaαi h∞an2 (a b )2 − an3−n2+2qaαib −n3−1 λi(n3+1) ] an2 + aqaαi λi(n3+1) } (A.1) C2 = ( Pib x1−1− Poan1+x1−1 bn1 ) (1+ν)(1−2ν) Ei[2ν +(1−ν)x2](bx2−1ax1−1− bx1−1ax2−1) − Aν′[(n2+1)(1−ν)+2ν](n2+ν′+1)(bn2ax1−1− bx1−1an2) (n2−x2+1)(n2−x1+1)[2ν +(1−ν)x2](bx2−1ax1−1−bx1−1ax2−1) − Bν ′[(n2−n3)(1−ν)+2ν](bn2−n3−1ax1−1− bx1−1an2−n3−1) (n2−n3−x2)(n2−n3−x1)[2ν +(1−ν)x2](bx2−1ax1−1− bx1−1ax2−1) + 1+ν [2ν +(1−ν)x2](bx2−1ax1−1− bx1−1ax2−1) {[T∞αi an2 + qaαi h∞an2 (a b )2 − an3−n2+2qaαib −n3−1 λi(n3+1) ] (bn2ax1−1− bx1−1an2) + qaαi λi(n3+1) (bn2−n3+1an3−n2+x1+1− bx1−1a) } (A.2) 1064 M.D. Kashkoli, M.Z. Nejad The unknown constants in Eqs. (4.29) are D1 = 3 2 ν′Dσ (N−1) eff (ax2−1− bx2−1)[(n1+2)S′rr −2S′θθ] [2ν +(1−ν)x1][n1(ν′′−1)−2](ax1−1bx2−1− bx1−1ax2−1) + 3 2 Dσ (N−1) eff S′rr(b x2−1−ax2−1) [2ν +(1−ν)x1](ax1−1bx2−1− bx1−1ax2−1) (A.3) D2 = 3 2 ν′Dσ (N−1) eff (ax1−1− bx1−1)[(n1+2)S′rr −2S′θθ] [2ν +(1−ν)x2][n1(ν′′−1)−2](ax2−1bx1−1− bx2−1ax1−1) + 3 2 Dσ (N−1) eff S′rr(b x1−1−ax1−1) [2ν +(1−ν)x2](ax2−1bx1−1− bx2−1ax1−1) (A.4) References 1. AlashtiR.A.,KhorsandM., TarahhomiM.H., 2013,Three-dimensional asymmetric thermo- elastic analysis of a functionally gradedrotatingcylindrical shell,Journal ofTheoretical andApplied Mechanics, 51, 1, 143-158 2. Aleayoub S.M.A., LoghmanA., 2010,Creep stress redistribution analysis of thick-walledFGM spheres, Journal of Solid Mechanics, 2, 10, 115-128 3. AryaV.K., Debnath K.K, Bhatnagar N.S., 1980, The spherical vessel with anisotropic creep properties considering large strains, International Journal of Non-Linear Mechanics, 15, 185-193 4. Bhatnagar N.S., Arya V.K., 1975, Creep of thick-walled spherical vessels under internal pres- sure considering large strains, Indian Journal of Pure and Applied Mathematics, 6, 10, 1080-1089 5. Bhatnagar N.S., Pradnya K., Arya V.K., 1987, Primary creep analysis of an anisotropic thick-walled spherical shell, Journal of Pressure Vessel Technology, 109, 347-351 6. Finnie I., HellerW.R., 1959,Creep of EngineeringMaterials, McGraw-Hill BookCo. Inc., New York 7. Ghannad M., Nejad M.Z., 2013, Elastic solution of pressurized clamped-clamped thick cylin- drical shells made of functionally gradedmaterials, Journal of Theoretical and Applied Mechanics, 51, 4, 1067-1079 8. Fesharaki J., LoghmanA.,YazdipoorM.,Golabi S., 2014, Semi-analytical solution of time- dependent thermomechanical creepbehaviorofFGMhollowspheres,Mechanics ofTime-Dependent Materials, 18, 41-53 9. JohnsonA.E.,KhanB., 1963,Creepofmetallic thick-walled spherical vessels subject to pressure andradial thermalgradientat elevated temperatures, International Journal ofMechanical Sciences, 5, 507-532 10. Kao R., 1981, Nonlinear creep buckling analysis of initially imperfect shallow spherical shells, Computers and Structures, 14, 1/2, 111-122 11. Loghman A., Aleayoub S.M.A., Hasani Sadi M., 2012, Time-dependent magnetothermoela- stic creepmodeling of FGM spheres usingmethod of successive elastic solution,Applied Mathema- tical Modelling, 36, 836-845 12. Loghman A., Ghorbanpourarani A., Aleayoub S.M.A., 2011, Time-dependent creep stress redistribution analysis of thick-walled functionally graded spheres, Mechanics of Time-Dependent Materials, 15, 353-365 13. LoghmanA., Shokouhi N., 2009,Creep damage evaluation of thick-walled spheres using a long- -term creep constitutive model, Journal of Mechanical Science and Technology, 23, 2577-2582 Time-dependent thermo-elastic creep analysis of thick-walled spherical pressure... 1065 14. Marcadon V., 2011, Mechanical modelling of the creep behaviour of hollow-sphere structures, Computational Materials Science, 50, 3005-3015 15. Miller G.K., 1995, Stresses in a spherical pressure vessel undergoing creep and dimensional changes, International Journal of Solids and Structures, 32, 14, 2077-2093 16. Miyazaki N., Yagawa G., Ando Y., 1977, A parametric analysis of creep buckling of a shallow spherical shell by the finite elementmethod, Nuclear Engineering and Design, 41, 257-263 17. NejadM.Z.,Hoseini Z., NiknejadA.,GhannadM., 2011,Anewanalytical solution for creep stresses in thick-walled spherical pressure vessels,Journal of Basic andApplied Scientific Research, 1, 11, 2162-2166 18. Nejad M.Z., Hoseini Z., Taghizadeh T., Niknejad A., 2013, Closed-form analytical solution for creep stressesofpressurized functionallygradedmaterial thick spherical shells,AdvancedScience Letters, 19, 2, 464-467 19. Noda N., Ootao Y., Tanigawa Y., 2012, Transient thermoelastic analysis for a functionally graded circular disk with piecewise power law, Journal of Theoretical and Applied Mechanics, 50, 3, 831-839 20. Pankaj T., 2011, Creep transition stresses of a thick isotropic spherical shell by finitesimal defor- mation under steady-state of temperature and internal pressure,Thermal Science, 15, 2, 157-165 21. Penny R.K., 1967, The creep of spherical shells containing discontinuities, International Journal of Mechanical Sciences, 5, 373-388 22. Sakaki T., Kuroki T., Sugimoto K., 1990, Creep of a hollow sphere, Journal of Applied Me- chanics, Transactions ASME, 57, 276-281 23. Xirouchakis P.C., Jones N., 1979, Axisymmetric and bifurcation creep buckling of externally pressurized spherical shells, International Journal of Solids and Structures, 16, 131-148 24. You L.H., Ou H., Zheng Z.Y., 2008, Steady-state creep analysis of thick-walled spherical pres- sure vessels with varying creep properties, Journal of Pressure Vessel Technology, 130, 0145011- -0145015 Manuscript received March 8, 2014; accepted for print July 15, 2015