THERMOELASTIC ANALYSIS OF THICK-WALLED FINITE-LENGTH CYLINDERS OF FUNCTIONALLY GRADED MATERIALS

Journal of Thermal Stresses, 28: 391 408, 2005 Copyright # Taylor & Francis Inc. ISSN: 0149-5739 print/1521-074x online DOI: 10.1080/01495730590916623 THERMOELASTIC ANALYSIS OF THICK-WALLED FINITE-LENGTH CYLINDERS OF FUNCTIONALLY GRADED MATERIALS M. Ruhi, A. Angoshtari, and R. Naghdabadi Department of Mechanical Engineering, Sharif University of Technology Tehran, Iran A semianalytical thermoelasticity solution for thick-walled finite-length cylinders made of functionally graded (FG) materials is presented. The governing partial differential equations are reduced to ordinary differential equations using Fourier expansion series in the axial coordinate. The radial domain is divided into some virtual subdomains in which the power-law distribution is used for the thermomechanical properties of the constituent components. Imposing the necessary continuity conditions between adjacent subdomains, together with the global boundary conditions, a set of linear algebraic equations are obtained. Solution of the linear algebraic equations yields the thermoelastic responses for each subdomain as exponential functions of the radial coordinate. Some results for the stress, strain, and displacement components through the thickness and along the length are presented due to uniform internal pressure and thermal loading. Based on the results, the gradation of the constitutive components is a significant parameter in the thermomechanical responses of FG cylinders. Keywords: Finite cylinder; Functionally graded materials; Thermoelasticity Functionally graded materials (FGMs) are a new class of materials constructed to operate in high-temperature environments. Functionally graded structures are typically those in which the volume fraction of two or more materials is varied continuously as a function of position along certain dimension(s) of the structure to achieve a required function [1,2]. FGMs are made of a mixture of ceramic and a combination of different metals. The composition is varied from a ceramic-rich surface to a metalrich surface, with a desired variation of the volume fractions of the two materials between two surfaces. The gradual change of material properties makes FGMs preferable in many applications. The continuous changes in the microstructure of FGMs distinguish them from fiber-reinforced laminated composite materials, which have a mismatch of mechanical properties across an interface of two discrete materials bonded together. In addition, fiber-reinforced laminated composite materials, in comparison with FGMs, suffer from the presence of residual stresses due to the difference in coefficients of Received 29 July 2004; accepted 11 November 2004. Address correspondence to R. Naghdabadi, Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9567, Tehran, Iran. E-mail: naghdabd@sharif.edu 391

392 M. RUHI ET AL. thermal expansion of the fiber and matrix. In FGMs, such limitations are avoided or reduced by gradually changing the volume fraction of the constituents rather than abruptly changing them across an interface. Thus, the study of thermoelastic responses of FGMs to different loading conditions is essential and valuable. There have been some studies that deal with thermal stresses in the basic structural components of FGMs. Suresh and Mortensen [2] have provided a very good introduction to the fundamentals of FGMs. Noda [3] has presented an extensive review that covers a wide range of topics from thermoelastic to thermoinelastic problems. He has discussed the importance of temperature-dependent properties on stresses and suggested that those properties of the material should be taken into account to perform more accurate analysis. He has also presented analytical methods to handle transient heat conduction in FGMs. Tanigawa [4] has compiled a comprehensive review on thermoelastic analysis of FGMs. Fukui and Yamanaka [5] have examined the effects of the gradation of components on the strength and deformation of thick-walled functionally graded (FG) tubes under internal pressure in the case of plain strain. Fukui and his coworkers [6] further extended their previous work by considering a thick-walled FG tube under uniform thermal loading. They also investigated the effect of graded components on residual stresses and estimated the optimum composition gradient generated by compressive circumferential stress at the inner surface. Zimmerman and Lutz [7] have presented solutions for the problem of the uniform heating of an FG circular cylinder using the Frobenius series method. Using a perturbation approach, Obata and Noda [8] have investigated the thermal stresses in an FG hollow sphere and an FG hollow circular cylinder. Ootao and Tanigawa [9] have conducted an approximate analysis of three-dimensional thermal stresses in an FG rectangular plate. Liew et al. [10] have presented an analysis of the thermomechanical behavior of hollow circular cylinders of FGM. They used the exponential volume fraction function and assumed the state of plain strain and developed an analytical model to deal with FG hollow circular cylinders that are subjected to the action of an arbitrary steady-state or transient temperature field. Ma and Wang [11] have employed the third-order shear deformation plate theory (TSDPT) to solve the axisymmetric bending and buckling problems of FG circular plates. They have also presented comparisons of the TSDPT solutions to the first-order shear deformation plate theory (FSDPT) and the classical plate theory. They concluded that the FSDPT is enough to consider the effect of shear deformation on axisymmetric bending and buckling of FG circular plates. Shahsiah and Eslami [12] have studied thermal buckling of FG cylindrical shells based on the first-order shear deformation shell theory (FSDST) and Sanders kinematic equations. They continued their study on thermal instability of FG cylindrical shells based on the FSDST and improved Donnell equations [13]. Ozturk and Tutuncu [14] have provided closed-form solutions for stresses and displacements in FG cylindrical vessels subjected to internal pressure alone using the infinitesimal theory of elasticity. They assumed material stiffness obeying a power law through the wall thickness with the Poisson ratio held constant. They approximated a radially varying elastic modulus by E(r) ¼ E 0 r b, where r is the radius of the cylinder. Since r is greater than zero, by adjusting the constants E 0 and b it is possible to obtain physically meaningful results. Different values of the inhomogeneity

THERMOELASTIC ANALYSIS OF THICK-WALLED CYLINDERS 393 constant b for a given geometry renders FG cylinders with different base materials; however, the various b values demonstrate the effect of inhomogeneity on the stress distribution. Ootao and Tanigawa [15] have studied the theoretical treatment of a transient thermoelastic problem involving an FG thick strip due to a nonuniform heat supply in the width direction. They obtained the exact solution for the two-dimensional temperature change in a transient state and thermal stresses of a simply supported strip under the state of plane strain. In this study, we consider a finite-length, thick-walled FG circular cylinder subjected to internal pressure and uniform thermal loading. The cylinder is radially graded and is simply supported at two longitudinal edges. The equilibrium equations are derived based on the thermoelasticity theory. The governing thermoelasticity equations are reduced to ordinary differential equations (ODEs) using Fourier s expansion series in the longitudinal (axial) coordinate. The radial domain is divided into finite subdomains in which the thermomechanical properties are assumed to be constant. Thus, the governing equations in each subdomain are a set of ODEs with constant coefficients. Imposing the continuity conditions at the interface of the adjacent subdomains, together with the global boundary conditions, a set of linear algebraic equations are derived. More subdomains (divisions) considered for the cylinder in the radial direction results in greater accuracy in the obtained solution. Solving the linear algebraic equations, the thermoelastic responses for a finite-length thick-walled FG cylinder are obtained and the normalized stress, strain, and displacement components in the cylinder length and thickness are presented. GRADIATION RELATIONS The material property gradation through the thickness is represented in terms of the volume fraction by the following expression [1]: pðrþ ¼ðp o p i ÞV þ p i V ¼ f þ 1 n f ¼ r R 2 t ð1þ where p(r) denotes a generic material property and p o and p i denote the property of the outer and inner faces of the cylinder, respectively, such as elastic moduli. Also, t is the total thickness of the cylinder, R is the mean radius of the cylinder ½ðR i þ R o Þ=2Š and n is a parameter that dictates the material variation profile through the thickness. Here we assume that the elastic modulus E and thermal coefficient of expansion, a, vary according to Eq. (1), and the Poisson ratio t is a constant. THERMOELASTIC EQUATIONS Consider a thick-walled finite-length FG cylinder with inner radius R i and outer radius R o, as shown in Figure 1, subjected to internal pressure P and a uniform thermal loading DT from a stress-free state.

394 M. RUHI ET AL. Figure 1 Configuration of the thick-walled finite-length cylinder. We use cylindrical coordinate system (r, h, z) and assume axial symmetry. The linear relations between the strain and the displacement components are e r ¼ @u e h ¼ u @r r c rz ¼ @u @z þ @w @r e z ¼ @w @z ð2þ where u and w are the radial and longitudinal displacements, respectively. The linear constitutive thermoelastic equations in the cylindrical coordinate system are used in the form of te r r ¼ ð1 þ tþð1 2tÞ e þ E 1 þ t e r EaDT 1 2t te r h ¼ ð1 þ tþð1 2tÞ e þ E 1 þ t e h EaDT 1 2t te r z ¼ ð1 þ tþð1 2tÞ e þ E 1 þ t e z EaDT 1 2t E s rz ¼ 21þ ð tþ c rz ð3þ where t, E, and a are the Poisson ratio, elastic modulus, and thermal expansion coefficient, respectively, and e ¼ e r þ e h þ e z. The equilibrium equations in the absence of body forces have the following forms: @r r @r þ r r r h þ @s rz r @z ¼ 0 @s rz @r þ s rz r þ @r ð4þ z @z ¼ 0

THERMOELASTIC ANALYSIS OF THICK-WALLED CYLINDERS 395 Substituting Eq. (2) into Eq. (3) and then into Eq. (4), we obtain the Navier thermoelastic equations: te @E @u ð1 þ tþð1 2tÞ @r @r þ u r þ @w þ E @2 u @r @r 2 þ 1 @u r @r u r 2 þ @2 w @z@r 1 @E @u 1 þ t @r @r þ E @2 u @r 2 @E adt @a DT E @r 1 2t @r 1 2t þ 1 E @u r 1 þ t @r u E @ 2 u þ r 2ð1 þ tþ @z 2 þ @2 w ¼ 0 ð5þ @r@z 1 @E @u 2ð1 þ tþ @r @z þ @w @r þ E @2 u @r@z þ @2 w @r 2 þ 1 E @u r 2ð1 þ tþ @z þ @w @r te @ 2 u þ ð1 þ tþð1 2tÞ @r@z þ 1 @u r @z þ @2 w @z 2 þ E @ 2 w 1 þ t @z 2 Ea @DT 1 2t @z ¼ 0 It is noted that, in Navier equation (5), the displacement components and thermomechanical properties are functions of r and z only (because of the axial symmetry). BOUNDARY CONDITIONS Let the boundary conditions of the finite-length cylinder be simply supported at the two longitudinal edges, u ¼ 0 r z ¼ 0 at z ¼ 0; L ð6þ and the following traction conditions on the inner and outer surfaces must be satisfied: r r ¼ P s rz ¼ 0 at r ¼ R i ð7aþ where P is the internal pressure of the cylinder. r r ¼ 0 s rz ¼ 0 at r ¼ R o ð7bþ METHOD OF ANALYSIS The boundary conditions in Eqs. (6) and (7) are satisfied by the following expansions: u ¼ X1 / r sinðb m zþ m¼1 w ¼ X1 / z cosðb m zþ m¼1 ð8þ

396 M. RUHI ET AL. where / r and / z are functions of r only and b m ¼ mp=l. We also use the following Fourier expansions for the internal pressure P and the temperature increment DT: P ¼ X1 P m sinðb m zþ m¼1 DT ¼ X1 T m sinðb m zþ m¼1 ð9þ Substituting Eqs. (8) and (9) into Navier equations (5) yields " # c m d 2 1 dr 2 þ d cm 2 dr þ cm 3 / r þ c m d 4 dr þ cm 5 / z þ c m 6 ¼ 0 " ð10þ d1 m d dr þ dm 2 / r þ d3 m d 2 dr 2 þ d dm 4 dr þ dm 5 #/ z d m6 ¼ 0 where and c m 1 ¼ te ð1 þ tþð1 2tÞ þ E 1 þ t c m 2 ¼ t @E ð1 þ tþð1 2tÞ @r þ t E ð1 þ tþð1 2tÞ r þ 1 @E 1 þ t @r þ E rð1 þ tþ c m 3 ¼ t 1 @E ð1 þ tþð1 2tÞr @r t E ð1 þ tþð1 2tÞr 2 1 E ð1 þ tþr 2 E 21þ ð t c m 4 ¼ t ð1 þ tþð1 2tÞ Eb E m 21þ ð tþ b m c m 5 ¼ t @E ð1 þ tþð1 2tÞ @r b m c m 6 ¼ @E at m @a T m E @r 1 2t @r 1 2t d1 m ¼ E 21þ ð tþ b t m þ ð1 þ tþð1 2tÞ Eb m d2 m ¼ 1 @E 21þ ð tþ @r b m þ 1 E r 21þ ð tþ b t E m þ ð1 þ tþð1 2tÞ r b m d3 m ¼ E 21þ ð tþ d4 m ¼ 1 @E 21þ ð tþ @r þ 1 E r 21þ ð tþ d5 m ¼ t ð1 þ tþð1 2tÞ Eb2 m E ð1 þ tþ b2 m d6 m Ea ¼ 1 2t b mt m Þ b2 m ð11þ ð12þ

SOLUTION ALGORITHM The ordinary differential equations (10) can be solved by different methods, but it is difficult to provide an analytical solution to them. Hence, in this study, an analytical solution is obtained by dividing the radial domain into finite subdomains with thickness t(k), as shown in Figure 2. Evaluating coefficients of Eq. (10) at r=r(k), mean radius of kth division, and using them instead of the variable coefficients in the original equations, two ODEs with constant coefficients are obtained that are valid in the kth subdomain. That is, ( ) THERMOELASTIC ANALYSIS OF THICK-WALLED CYLINDERS 397 c k;m 1 d k;m 1 dr 2 þ d ck;m 2 dr þ ck;m 3 / r þ c k;m d 4 dr þ ck;m 5 / z þ c k;m 6 ¼ 0 ( ) d dr þ dk;m 2 / r þ d k;m d 2 3 dr þ d dk;m 4 dr þ dk;m 5 / z þ d k;m 6 ¼ 0 d 2 where c k;m i and c k;m i are constant coefficients of Eq. (10) evaluated at r ¼ r(k). These coefficients represent the effect of the continuously changing material properties of the FG cylinder in the radial direction. Using the preceding technique, two ordinary differential equations (10) with variable (r-dependent) coefficients are changed into 2n ODEs with constant coefficients in which n is the number of virtual subdomains. Therefore, the solution for Eqs. (13) can be written in the form k;m / r ¼ / z A k;m B k;m A k;m exp X k;m r where A k,m, B k,m, and X k,m are constants for the kth division. It is noted that the preceding solution is valid for ð13þ ð14þ rðkþ tðkþ 2 r rðkþþ tðkþ 2 ð15þ Figure 2 Dividing radial domain into finite subdomains.

398 M. RUHI ET AL. where r(k) and t(k) are the mean radius and thickness of the kth subdomain, respectively. Substituting Eq. (14) into Eq. (13), we obtain a fourth-order polynomial for the kth radial subdomain. Solving the polynomial for X k,m and using Eqs. (14), we are left with four unknowns A k,m. It is noted that B k,m is calculated from the linear algebraic equations obtained by substituting X k,m into Eq. (13). The unknowns A k,m are determined by applying the necessary boundary conditions between each of two adjacent subdomains. For this purpose, the continuity of the radial and longitudinal displacements, u and w, as well as radial stress r r and shear stress s rz, are imposed at the interfaces of the adjacent subdomains. According to the Fourier expansion used in Eq. (8), the continuity conditions at the interfaces are / k tðkþ r r¼rðkþþ 2 r k r tðkþ r¼rðkþþ 2 / k z tðkþ r¼rðkþþ 2 s k tðkþ rz r¼rðkþþ 2 ¼ / kþ1 r ¼ r kþ1 r ¼ / kþ1 z ¼ s kþ1 rz tðkþ1þ r¼rðkþ1þ 2 tðkþ1þ r¼rðkþ1þ 2 tðkþ1þ r¼rðkþ1þ 2 tðkþ1þ r¼rðkþ1þ 2 Continuity conditions (16) together with global boundary conditions (7) yield a set of linear algebraic equations in terms of A k,m. Solving the resultant linear algebraic equations for A k,m, the unknown coefficients of Eq. (14) are calculated. Then the displacement components u and w are determined in each radial subdomain using Eq. (8). By increasing the number of subdivisions, the accuracy of the results is improved. ð16þ NUMERICAL RESULTS To verify the Navier solution obtained for the simply supported finite-length thick-walled FG cylinder under uniform internal pressure and uniform thermal loading, two case studies are investigated. The analysis is conducted using aluminum as the outer surface metal and zirconia as the inner surface ceramic. According to Eq. (1), the following material properties are used in computing the numerical results: E out ¼ E Al ¼ 70 GPa a out ¼ a Al ¼ 23 10 6 K 1 t ¼ 0:3 E in ¼ E Cer ¼ 151 GPa a in ¼ a Cer ¼ 10 10 6 K 1 ð17þ where E, a, and t are the modulus of elasticity, thermal coefficient of expansion, and Poisson ratio, respectively. Two cases of loading are studied: uniform internal pressure (P in ) and uniform thermal loading (DT). In both cases, the following geometric properties (Figure 1) are considered: t ¼ 20 mm R in ¼ 2:5t L ¼ 10R in ð18þ where t, R in,andl are the cylinder thickness, internal radius, and length, respectively.

THERMOELASTIC ANALYSIS OF THICK-WALLED CYLINDERS 399 Figure 3 Metal volume fraction (V m ) through the thickness of the cylinder. Figure 3 shows the distribution of the metal volume fraction V m through the cylinder thickness for various values of the power-law index n. Results for the Uniform Internal Pressure Figures 4 and 5 show the nondimensionalized radial displacement U r ¼ u r E Cer t=p in R 2 in through the thickness and along the length (at the inner surface of the cylinder) for different values of the power-law index n. As expected, the radial displacement values for the full metal (aluminum) cylinder are greater than those for the full ceramic (zirconia) cylinder due to the higher elasticity modulus of ceramic. It is also observed that the radial displacement increases with the increase of the powerlaw index n from zero (homogenized aluminum cylinder) up to its maximum value for n ¼ 0.2 and then decreases to its minimum value for n ¼1 (homogenized zirconia cylinder). Thus, for a given pair of materials, there is a particular volume fraction for which the radial displacement attains its maximum value under a given internal pressure. Figure 6 shows the variation of the normalized radial and angular strains (e r ¼ e r E Cer =P in, e h ¼ e h E Cer =P in ) through the thickness for different values of the power-law index n. As expected, e h is positive whereas e r is negative for a prescribed internal pressure. Also, as observed in the figure, the extreme values for the strains occur at about n ¼ 0.1, which is near the state of a homogenized full metal cylinder. This is because of the lower stiffness of the metal in comparison with the ceramic. It is noted that these extreme values do not occur in the homogenized state.

400 M. RUHI ET AL. Figure 4 Normalized radial displacements through the thickness at the middle length of the cylinder due to uniform internal pressure for different values of the power-law index n. Figure 5 Normalized radial displacements along the length at the inner surface of the cylinder due to uniform internal pressure for different values of the power law index n.

THERMOELASTIC ANALYSIS OF THICK-WALLED CYLINDERS 401 Figure 6 Normalized strains through the thickness at the middle length of the cylinder due to uniform internal pressure for different values of the power law index n. Figure 7 Normalized radial stress through the thickness at the middle length of the cylinder due to uniform internal pressure for different values of the power law index n.

402 M. RUHI ET AL. Figure 8 Normalized hoop stress through the thickness at the middle length of the cylinder due to uniform internal pressure for different values of the power law index n. Figure 7 illustrates the through-the-thickness variation of the normalized radial stress (r r ¼ r r =P in ) for different values of the power-law index n. As shown in the figure, for n < 1 the radial stress has a minimum value that is greater than the internal pressure and does not occur at the inner surface of the cylinder. In Figures 8 and 9 the through-the-thickness and along-the-length distributions of the normalized hoop stress (r h ¼ r h =P in ) are presented. As shown in the figures, the hoop stresses at the inner surface are always greater than the hoop stresses at the outer surface for all values of n. The maximum hoop stress at the inner surface occurs for the power-law index n of about 0.2. Also, the curve peaks in Figure 9 reveal the longitudinal boundary effects of the cylinder under uniform internal pressure. Results for Uniform Thermal Loading Figures 10 and 11 illustrate the nondimensionalized radial displacements due to uniform thermal loading (U r th ¼ u r th =ta Cer DT) through the thickness and along the length (at the inner surface) of the cylinder, respectively. As shown in these figures, the radial displacements for the homogenized states are positive, whereas these values are negative for the FG cylinder with the selected values of n. It is also noted that the maximum radial displacement occurs at about n ¼ 0.2.

THERMOELASTIC ANALYSIS OF THICK-WALLED CYLINDERS 403 Figure 9 Normalized hoop stress along the length at the inner surface of the cylinder due to uniform internal pressure for different values of the power law index n. The through-the-thickness variation of the normalized radial and angular strains (e r ¼ e r =a Cer DT, e h ¼ e h =a Cer DT) for different values of the power-law index n are presented in Figure 12. Because the radial displacements in the homogenized states (full metal or full ceramic) have a linear behavior, as shown in Figure 10, e r and e h are constant (with the same values) in the homogenized states [see Figure 12 and Eq. (2)]. Also, despite the positive values of e h in the homogenized states (full metal or full ceramic), the angular strains e h are negative for the selected values of the power-law index n; it is the same for e r for some specific values of n. It is also noted that, for different values of n, e h always attains its maximum value at the inner surface as shown in Figure 12. Figure 13 shows the variation of the normalized thermal radial stress (r r th ¼ r r th =a Cer DT) through the thickness at the middle length of the cylinder due to the uniform thermal loading. As shown in the figure, both the maximum value of the thermal radial stress and the position at which it occurs depend on the value of the power-law index n. It is also noted that the thermal radial stress is positive for small values of n and negative for large values of n. Thus, it is concluded that, for a particular value of n, the thermal radial stress distribution through the thickness of the cylinder is close to that of the homogenized states, which is zero, as shown in the figure.

404 M. RUHI ET AL. Figure 10 Normalized radial displacements through the thickness at the middle length of the cylinder due to uniform thermal loading for different values of the power law index n. Figure 11 Normalized radial displacements along the length at the inner surface of the cylinder due to uniform thermal loading for different values of the power law index n.

THERMOELASTIC ANALYSIS OF THICK-WALLED CYLINDERS 405 Figure 12 Normalized strains through the thickness at the middle length of the cylinder due to uniform thermal loading for different values of the power law index n. Figure 13 Normalized radial stress through the thickness at the middle length of the cylinder due to uniform thermal loading for different values of the power law index n.

406 M. RUHI ET AL. Figure 14 Normalized hoop stress through the length at the inner surface of the cylinder due to uniform thermal loading for different values of the power law index n. Figure 15 Normalized hoop stress through the thickness at the middle length of the cylinder due to uniform thermal loading for different values of the power law index n.

THERMOELASTIC ANALYSIS OF THICK-WALLED CYLINDERS 407 Figure 14 shows the along-the-length variation of thermal hoop stresses for different values of the power-law index n. The boundary effects due to the longitudinal supports are shown in both homogenized and FG states. Figure 15 illustrates the effect of the power-law index n on the normalized thermal hoop stress (r h th ¼ r h th =a Cer DT) through the thickness at the middle length of the cylinder. It is observed that the response of the FG cylinder to thermal loading is complicated and it is noted that the maximum values of the thermal hoop stresses occur at the inner surface of the cylinder. CONCLUSIONS A semianalytical solution for thermoelasticity equilibrium equations of a thickwalled finite circular cylinder made of FGMs is presented. The effects of the radial gradation of constitutive components on stress, strain, and displacement components of the FG cylinder have been investigated for both mechanical and uniform thermal loadings. It is seen that, for a given pair of materials, there is a particular volume fraction that elicits a specified mechanical response under a given thermomechanical loading. It is additionally observed that, for some particular values of the power-law index n, the radial stress due to internal pressure attains its minimum value at a surface close to the internal surface of the cylinder and not at the boundary surface (internal surface). This minimum value is a little greater than the internal pressure; thus, the gradation of the constitutive components plays an important role in determining the thermomechanical responses of FG thick-walled cylinders. REFERENCES 1. J. N. Reddy, Analysis of Functionally Graded Plates, Int. J. Numer. Method, Eng., vol. 47, pp. 663 684, 2000. 2. S. Suresh and A. Mortensen, Fundamentals of Functionally Graded Materials, IOM Communications Limited, London, 1998. 3. N. Noda, Thermal Stress in Materials with Temperature-Dependent Properties, Appl. Mech. Rev., vol. 44, pp. 383 397, 1991. 4. Y. Tanigawa, Some Basic Thermoplastic Problems for Nonhomogeneous Structural Materials, Appl. Mech. Rev., vol. 48, pp. 377 389, 1995. 5. Y. Fukui and N. Yamanaka, Elastic Analysis for Thick-Walled Tubes of Functionally Graded Material Subjected to Internal Pressure, JSME Int. J. I, vol. 35, no. 4, pp. 379 385, 1992. 6. Y. Fukui, N. Yamanaka, and K. Wakashima, The Stresses and Strains in a Thick-Walled Tube for Functionally Graded Material under Uniform Thermal Loading, JSME Int. J. A, vol. 36, no. 2, pp. 156 162, 1993. 7. R. W. Zimmerman and M. P. Lutz, Thermal Stresses and Thermal Expansion in a Uniformly Heated Functionally Graded Cylinder, J. Thermal Stresses, vol. 22, pp. 177 188, 1999. 8. Y. Obata and N. Noda, Steady Thermal Stresses in a Hollow Circular Cylinder and a Hollow Sphere of a Functionally Gradient Material, J. Thermal Stresses, vol. 17, pp. 471 488, 1994. 9. Y. Ootao and Y. Tanigawa, Three-Dimensional Transient Thermal Stress of Functionally Graded Rectangular Plate Due to Partial Heating, J. Thermal Stresses, vol. 22, pp. 35 55, 1999.

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