Plane Thermoelastic Waves in Infinite Half-Space Caused FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 15, N o 3, 2017, pp. 545 - 563 https://doi.org/10.22190/FUME170104004F © 2017 by University of Niš, Serbia | Creative Commons Licence: CC BY-NC-ND Original scientific paper *THERMAL BUCKLING ANALYSIS OF FUNCTIONALLY GRADED CIRCULAR PLATE RESTING ON THE PASTERNAK ELASTIC FOUNDATION VIA THE DIFFERENTIAL TRANSFORM METHOD UDC 624.04 Fatemeh Farhatnia 1 , Mahsa Ghanbari-Mobarakeh 1 , Saeid Rasouli-Jazi 1 , Soheil Oveissi 2 1 Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran 2 Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran Abstract. In this paper, we propose a thermal buckling analysis of a functionally graded (FG) circular plate exhibiting polar orthotropic characteristics and resting on the Pasternak elastic foundation. The plate is assumed to be exposed to two kinds of thermal loads, namely, uniform temperature rise and linear temperature rise through thickness. The FG properties are assumed to vary continuously in the direction of thickness according to the simple power law model in terms of the volume fraction of two constituents. The governing equilibrium equations in buckling are based on the Von-Karman nonlinearity. To obtain the critical buckling temperature, we exploit a semi-numerical technique called differential transform method (DTM). This method provides fast accurate results and has a short computational calculation compared with the Taylor expansion method. Furthermore, some numerical examples are provided to consider the influence of various parameters such as volume fraction index, thickness- to-radius ratio, elastic foundation stiffness, modulus ratio of orthotropic materials and influence of boundary conditions. In order to predict the critical buckling temperature, it is observed that the critical temperature can be easily adjusted by appropriate variation of elastic foundation parameters and gradient index of FG material. Finally, the numerical results are compared with those available in the literature to confirm the accuracy and reliability of the DTM to determine the critical buckling temperature. Key Words: Thermal Buckling, Orthotropic Plate, Functionally Graded Materials (FGM), Pasternak Elastic Foundation, Differential Transform Method (DTM) Received January 4, 2017 / Accepted April 22, 2017 Corresponding author: Fatemeh Farhatnia Affiliation: Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran E-mail: farhatnia@iaukhsh.ac.ir mailto:farhatnia@iaukhsh.ac.ir 546 F. FARHATNIA, M. GHANBARI-MOBARAKEH, S.R. JAZI, S. OVEISSI 1. INTRODUCTION One of the most important undesired phenomena in a mechanical structure as observed in plates is thermal buckling. The response of the plate to buckling depends on its mechanical properties. Functionally graded materials (FGM) are made of ceramic and metal constituents, including high mechanical strength, good machinery ability, and high thermal resistance. FGMs are a good choice to be employed as the material constituents in a plate exposed to high thermal gradients. They are less likely to delaminate at high temperatures due to the continuity of their physical and mechanical properties. In addition, the orthotropic properties of FGMs have received great attention because they increase the tolerance of plates to various types of loading. Numerous studies have been conducted on thermal plate buckling. In what follows, we mention the studies related to the subject of this paper. Dewey and Costello [1] proposed an analytical and experimental method to investigate the thermal buckling of flat plates where the modulus of elasticity changes due to the thermal gradient. Najafizadeh and Eslami [2] discussed the thermo-mechanical response of plates based on the first-order shear deformation theory. They derived nonlinear and linear governing equations from the energy method and by using the calculus of variations. In another work, Najafizadeh and Eslami [3] discussed the thermoelastic buckling of the circular orthotropic composite plates under various kinds of thermal loading. They obtained the governing equations based on the Love-Kirchhoff hypothesis and the Sanders’ nonlinear strain-displacement relation. Li et al. [4] examined the nonlinear vibration and thermal buckling of orthotropic annular plates with a centric rigid mass. They employed the Hamilton’s principle to derive the governing equations based on the Von- Karman nonlinearity. Najafizadeh and Heydari [5] assessed the thermal buckling of circular plates in functionally graded materials under uniform radial compression subject to various types of thermal loads. They established the governing equations in buckling, using variational method and solved them via Bessel functions. Prakash and Ganapathy [6] applied the finite element method to analyze the vibrations and thermal buckling of circular FGM plates. Zhao et al. [7] studied the thermal and mechanical buckling behavior of plates with arbitrary geometry, including plates containing square and circular holes in the center. Zenkour and Sobhy [8] investigated the thermal buckling of FG sandwich plates, using sinusoidal shear deformation plate theory. The thermal loads are assumed to have uniform, linear, and non-linear distribution through thickness. Jalali et al. [9] assessed the thermal buckling of circular FGM plates with varying thickness, using the pseudo-spectral method (PSM). Evaluating the reaction of plates resting on elastic foundations and subject to different types of loads has a great scientific importance, particularly in modern engineering structures. Kiani and Eslami [10] studied the exact solution of thermal buckling in annular FGM plates resting on the Pasternak elastic foundation. They examined the effects of geometrical parameters, power-law index, and coefficients of the elastic foundation on the critical buckling temperature. Jabbari et al. [11] studied the buckling of a solid porous circular plate subjected to thermal loading. They derived the governing equations based on the Sanders’ nonlinear strain-displacement relation and determined the pre-buckling temperatures and the critical buckling temperatures. Yaghoobi and Fereidoon [12] proposed the thermal and mechanical buckling of functionally graded (FG) plates resting on elastic foundation, using nth-order shear deformation theory. They obtained the governing equations via exploiting the minimum total potential energy method. Mansouri and Shariat [13] predicted the thermal buckling response of heterogeneous orthotropic plates based on Thermal Buckling Analysis of Functionally Graded Circular Plate ... 547 the high-order theories. They employed a new version of the DQM (Differential Quadrature Method) to solve the governing differential equations. Mirzaei and Kiani [14] investigated the thermally induced bifurcation buckling of rectangular composite plates reinforced with single-walled carbon nanotubes. They revealed that in most cases, the FG-X pattern of CNTs is the most influential case since it results in higher critical buckling temperatures. Yu et al. [15] evaluated the new numerical results of thermal buckling of functionally graded plates (FGPs) with internal defects (for example, crack or cutout), using an effective numerical method. They employed the new formulation of the first-order shear deformation plate theory associated with extended isogeometric analysis (XIGA) and level sets. Moreover, they investigated the influences of various aspect ratios, including gradient index, crack length, plate thickness, cutout size, and boundary conditions on the critical buckling temperature rise (CBTR). Tung [16] studied the nonlinear bending and post-buckling behavior of functionally graded sandwich plates resting on elastic foundations and subject to uniform external pressure, thermal loading, and uniaxial compression in the thermal environment. Sun et al. [17] numerically investigated the thermomechanical buckling and post-buckling of a functionally graded material (FGM) Timoshenko beam resting on a two-parameter non-linear elastic foundation and subject to only a temperature rise, using the shooting method. In the present research, we accomplish the thermal buckling of an orthotropic FG circular plate. To that end, the Differential Transformation Method (DTM) is applied to solve the governing equation of thermal buckling. The literature review indicates that the present work is the first attempt to exploit DTM to evaluate the critical buckling temperature. A number of studies have been carried out by this method [18-24]. The results are presented in four categories based on linear and uniform thermal load and simply supported and clamped edge conditions in the proceeding sections. The effects of parameters such as volume fraction index, stiffness of the Pasternak elastic foundation, thickness-to-radius ratio, and modulus ratio of orthotropic material to critical temperature are also investigated. 2. MATHEMATICAL FORMULATION OF CONSTITUTIVE EQUATIONS Consider a functionally graded circular solid supported on the Pasternak elastic foundation (Fig. 1). A polar coordinate system (r, θ, z) is employed to label the material points of the plate in radial, circumferential, and thickness directions. Fig. 1 Schematic of problem Based on the Sanders’ kinematic relations and the Von-Karman nonlinear assumption, the strain-displacement relationships are written in polar coordinates system as follows: 548 F. FARHATNIA, M. GHANBARI-MOBARAKEH, S.R. JAZI, S. OVEISSI , , 2 rr rr rr r r r zk zk zk                (1) where rr  ,  , and r are mid-surface strains and krr, kθθ, and krθ are the curvatures defined as follows: 2 2 2 1 , 2 rr rr u w w k r r r               2 2 2 2 2 1 1 1 1 , 2 v u w w w k r r r rr r                       (2) 2 2 1 v 1 1 1 , r r u v w w w w k r r r r r r rr                                where u, v, and w represent the middle-plane displacements in the polar coordinates. Stress-strain relationships in polar coordinates are expressed as follows: 11 12 21 22 66 ( ) ( ) ( ( ) ( ) ( ) ( )) ( ) rr rr T rr T rr T rr T r r a a a a a                             (3) in which, 11 21 12 22 66 0 ( ) ( ) ( )( ) , ( ) , , , 1 1 1 1 , , ( ) , ( ) , ( ) r rr r r r r r r r r r r r r r T rr r T E E EE a a a a a G E E z z zz T T T T z z T                                                        (4) where εT is the thermal strain, αr and αθ are the thermal expansion coefficients in radial and circumferential directions, respectively. Moreover, T and T0 are the current and reference temperatures, respectively. The readers interested in the thermo-elastic stress- strain relations in a symmetrical case of deformation can find more details in the study conducted by Kiani et al. [10]. Upon substituting Eq. (4) into Eq. (3), the constitutive relations for orthotropic FG plate can be re-written as follows: 11 12 11 12 21 22 21 22 66 ( ) ( ) rr rr r rr r r r a a a a T a a a a T a                               (5) In addition, Grθ is shear modulus, and Er and Eθ are Young’s modulus of the plate in radial and circumferential directions, respectively. Based on the power-law model in polar coordinates, the material properties are as follows: 1 ( ) , ( ) 2 n r r r r r rcm m cm c m z E E E E E E h z      (6) 1 ( ) , 2 ( ) n r r r r r rcm m cm c m z z h           (7) Thermal Buckling Analysis of Functionally Graded Circular Plate ... 549 1 ( ) , ) 2 ( n cm m cm c m z h z               (8) 1 ( ) , 2 ( ) n r r r r r rcm m cm c m z G G G G G G h z          (9) where h, n and subscripts m and c represent the thickness, FG power index, the metal and ceramic properties, respectively. By assuming the polar orthotropic characteristics of plate, Young’s modulus in the circumferential coordinate is defined as follows: 1 (( ) ( ) ( ) ) 2 n r cm m z E E E E E h z z z             (10) where μ is the orthotropic modulus ratio. 3. GOVERNING EQUATIONS IN THERMAL BUCKLING Stress resultants and stress couples are obtained as follows: 2 2 ( , , ) ( , , ) h rr r rr r h N N N dz         (11) 2 2 ( , , ) ( , , ) h rr r rr r h M M M z dz         (12) Substituting Eq. (5) into Eqs. (11) and (12) yields the stress resultants and stress couples as follows: 1 1 2 2 1rr rr rr N A B k A B k C T        (13) 2 2 3 3 2rr rr N A B k A B k C T         (14) 1 1 2 2 3rr rr rr M B D k B D k C T        (15) 2 2 3 3 4rr rr M B D k B D k C T         (16) where,   1 2 3 2 1 2 3 11 12 22 2 1 2 3 2 2 2 1 3 11 12 2 4 21 22 2 2 1 ( , ) (1, )( ) , ( , ) (1, )( ) h h h h r r h h A A A B B B z a a a dz D D D z C C z a a dz C C z a a dz                                   550 F. FARHATNIA, M. GHANBARI-MOBARAKEH, S.R. JAZI, S. OVEISSI When the plate is subjected to the mechanical loading, the total energy is given by: V U  (17) where Ω and U are the potential energy of the external loading and the strain energy, respectively. Ω is the summation of the potential energy of mechanical loading and two parameters elastic foundation reaction. The elastic foundation is exerted on the lower surface of the plate, as shown in Fig. 1. In this research, the mechanical loading is absent. U is the summation of thermal strain energy, membrane strain energy, bending strain energy, coupled bending-membrane strain energy, and elastic foundation strain energy: 2 / 2 0 0 / 2 ( ) 1 2 R h h rr rr rr rr r U T rdzdrT d                            (18) By integrating through thickness, U can be expressed as follows: 2 0 0 2 2 2 2 1 ( 2 ) 2 1 R rr rr rr rr r r r r w g g U N k M N k M N rdrd k M w w k w k k r r                                      (19) where kw and kg denote the stiffness of Pasternak elastic foundation (tension and shear foundation parameters, respectively). By substituting Eqs. (13-16) into Eq. (19), setting the resultant expression into the expression of the total potential energy function, Eq. (17), and employing Euler equations [5], the governing equilibrium equations in the buckling of the plate resting on the Pasternak foundation based on the von-Karman nonlinearity are obtained as follows [10]: , , , , , , , , , , ,2 2 , , , ,2 2 1 0 1 2 0 2 1 1 2 2 1 1 1 ( ( ) 1 , 0) r r r r r r r r rr r r r r r r r rr r w r rg r N N N N r r N N N r r M M M M M M N w r r r r r N w w k w k w w w r r r r                                     (20) The governing equations of equilibrium can be expressed in terms of displacement components by assuming the symmetric state of buckling which gives the variation with respect to the circumferential direction set to zero, substituting Eqs. (13-16) into Eqs. (20), and utilizing Eq. (2):   22 2 12 2 2 2 3 2 2 3 2 2 11 1 2(1 ) 1 0 r r T T r ku u u w w w E r r r r rr r rk N Nw w w E r r rr r r                                               (21) Thermal Buckling Analysis of Functionally Graded Circular Plate ... 551 2 3 2 3 2 2 22 3 2 2 3 3 2 2 4 3 2 3 , ,4 3 2 2 3 , , (2 )1 2 (1 ) 2 1 1 0 r r T T r rr r w g r rr ku u u u w w w w w E r r r r rr r r r r r rk w w w w E N w N w r r rr r r r r k w k w w r                                                                    (22) where, 2 1 2 3 3 1 1 , , 1 2 2 2 1 1 1 12 3 4 4 2 cm m cm m cm E E h E E h E n n n E E h E n n n                             3.1. Types of thermal loading In this study, two cases of temperature rise are considered: uniform and linear temperature differences. When the plate is not thick enough, the assumption of linear distribution is reasonable [10]. But when the temperature on the top and bottom surfaces of the plate is constant and no source of heat is available in it, the temperature variation can be defined as a linear function of thickness coordinate; for instance, the temperature distribution in aircraft window, walls of the building, or furnace. By assuming that the reference temperature of plate is T0 and the displacement in the radial direction is prevented due to restraint on the edge of the plate, N T r and N T θ as the representatives of stress resulting from the thermal gradients can be determined in two categories due to the temperature rise across thickness as follows:  Uniform thermal loading: 1 2 (1 ) 1 2 1(1 ) T cm m m cm cm cmr r m m r E E E N C T h E T n n                    (23) 2 2 ( ) 1 2 1(1 ) T cm m m cm cm cmr m m r E E E N C T h E T n n                      (24)  U Linear thermal loading: 1 ( ) ( ) 2 m c m z T z T T T h          (25) 02 2 (1 ) ( ) 1 2 1(1 ) (1 ) 2 2 2 2 (1 ) T cm m m cm cm cmr r m m m r m m cm m m cm cm cmr r E E E N h T T E n n E E E E h T n n                                   (26) 552 F. FARHATNIA, M. GHANBARI-MOBARAKEH, S.R. JAZI, S. OVEISSI 02 2 ( ) ( ) 1 2 1(1 ) ( ) 2 2 2 2(1 ) T cm m m cm cm cmr m m m r m m cm m m cm cm cmr r E E E N h T T E n n E E E Eh T n n                                      (27) where ∆T=T(z)-T0. By eliminating the radial displacement components of u and after performing some mathematical manipulations, the equations of the equilibrium can be summarized in one equation in terms of out-plane displacement component of w as follows: 4 3 2 2 4 3 2 2 3 2 2 1 )( ) ( 0 T T E r g g w w w w w w w D N k N k k w r r r rr r r r r r                          (28) where, 2 2 2 1 3 1 ( ) / E (1 ) E r E E ED    . Based on the adjacent equilibrium criteria, the state of stable equilibrium may be designated by w0; in addition, it is w0+w1 in the neighborhood of stability state when w1 can be represented to any small increment of displacement. Similar to out-plane displacement, the stress resultants are divided into two terms representing the stable equilibrium and the neighboring state [10]. Upon substituting w0+w1 and stress resultants in two terms into the governing equation (28) and performing some mathematical manipulations, the stability equation is obtained as follows: 4 3 2 2 1 1 1 1 1 04 3 2 2 3 2 1 0 1 2 ( ) 1 ( ) 0 T E r g T g w w w w w w D N k r rr r r r r r w N k k w r r                         (29) where N T r0 and N T θ0 are the pre-buckling thermal loading, and the following relation can be established: N T r0= N T θ0= -N T . The following dimensionless quantities are defined to deal with the problem under consideration in the dimensionless forms: 22 42 1Φ , , ,  , , TT gT T wr r g w E E E E k RN R k Rw N Rr n n K K h R D D D D        4 3 2 2 3 2 3 4 3 2 2 2 3 Φ Φ Φ Φ 2 ( ) Φ ( ) Φ 0 T r g T g w d d d d d n K dd d d d d n K K d                        (30) 4. SOLVING THE GOVERNING EQUATION BY DTM The differential transform method (DTM) is a numerical method based on the Taylor series expansion that proposes the solution in the form of polynomials [24]. This method is a fast convergent method in comparison with the Taylor series in order to solve the differential equations. The advantage of this method is its low computational manipulation and its applicability to handle linear and non-linear ordinary and partial differential equations. By Thermal Buckling Analysis of Functionally Graded Circular Plate ... 553 exploiting the DTM, the differential equations are reduced to the recurrence relations and convert the boundary conditions into a set of algebraic equations. The differential transform of the k-th derivative of function f(r) is defined as follows: 0 1 ( ) [ ] ! k k r r d f r F k k dr         (31) where f(r) and F[k] are the original function and transform function, respectively. The inverse differential transform of F[k] is defined as follows [5]: 0 0 ( ) [ ]( ) k k f r F k r r     (32) In the domain of R, original function f(r) is considered to be analytical, and r=r0 represents any point in R. f(r) is represented by power series whose center is located at r0. From Eqs. (31) and (32), it can be concluded that: 0 0 0 ( ) ( ) ( ) ! k k k k r r r r d f r f r k dr            (33) Table 1 demonstrates the fundamental mathematical properties of the differential transform method (DTM): Table 1 Fundamental theorems of DTM [22] Original Function Transformed Function ( ) ( ) ( )f r y r z r  [ ] [ ] [ ]F k Y k Z k  ( ) ( ) f r y r  [ ] [ ]F k Y k  ( ) ( ). ( )f r y r z r 1 0 1 1 [ ] [ ] [ ] k k F k Y ZK K K    ( ( )) ( ) ( ) m m d y r f r dr  ! [ ] [ ] ! ( )m k F k Y k k m      n f r r  0 [ ] ( ) 1 if k n F k if k n k n      Hence, function Ф[ξ] is obtained as follows: 0 1 2 0 Φ[ ] [ ] [0] [1] [2]   N k k k               (34) To solve Eq. (35), we use the differential transform relationships of the k-th derivative of the function of non-dimensional out-plane displacement Ф; moreover, DTM theorems are listed in Table 1: 554 F. FARHATNIA, M. GHANBARI-MOBARAKEH, S.R. JAZI, S. OVEISSI 1 4 3 1 1 1 1 1 14 0 Φ ( 3)( 4)( 3)( 2)( 1) [ 4] k k d k k k k k k k k k k k d                  1 3 2 1 1 1 1 13 0 Φ ( 2)( 3)( 2)( 1) [ 3] k k d k k k k k k k k k d                1 2 1 1 1 12 0 Φ ( 1)( 2)( 1) [ 2] k k d k k k k k k k d              Φ ( 1) [ 1] d k k d      (35) 1 2 3 1 1 1 12 0 Φ ( 3)( 2)( 1) [ 2] k k d k k k k k k k d              1 2 1 1 1 0 Φ ( 2)( 1) [ 1] k k d k k k k k d            1 3 1 1 0 Φ ( 3) [ ] k k k k k       Substituting the aforementioned relationships into the governing equation in thermal buckling Eq. (28) yields: 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 2 ( 3)( 4)( 3)( 2)( 1) [ 4] 2 ( 2)( 3)( 2)( 1) [ 3] ( 1)( 2)( 1) [ 2] ( 1) [ 1] ( 3)( 2)( 1) [ 2] k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k A k k k k k k k A                                                             1 1 1 1 1 1 1 0 0 ( 2)( 1) [ 1] ( 3) [ ] 0 k k w k k k k k k K k k k                (36) By utilizing the appropriate theorems of the DT method (see Table 1 and the simplified form of Eq. 36), we have the following recurrence relation: 1 2 2 2 ( 2 )( 1) [ 1] [ 3] [ 1] , 3 ( )( 1 ) ) ( w A k A k k K k k k k k                 (37) where A1=n T r+Kg and A2=n T θ+Kg. The following boundary conditions are imposed as clamped and simply supported boundary conditions on the edges of the plate. However, by applying differential transformation method to boundary conditions, we can obtain: Thermal Buckling Analysis of Functionally Graded Circular Plate ... 555  Non-dimensional clamped edge: 1 1 0 , 0 d d        (38)  Differential transform of clamped edge condition with DTM: 0 0 [ ] 0, [ ] 0 N N k k k k k       (39)  Non-dimensional simply supported boundary condition: 2 21 1 0, 0 E d d D r dd                  (40)  Differential transform of simply supported edge condition with DTM: 0 0 [ ] 0, ( 1 ) [ ] 0 N N k k k k k k          (41) In this study, the symmetrical thermal buckling behavior of a plate is considered, and the regularity condition is imposed besides the boundary conditions. The non-dimensional form of regularity condition and its differential transform are as follows:   0 0, 1 0 d d        (42) Upon substituting k=3,5,7,… into the recurrence Eq. (37), we get: 2 1 2 1 2 [1] φ[3] 0 3 12 (6 3 ) [3] [1] (20 5 ) [5] [3] φ[5] 0,  φ[7] 0 ,   15 240 35 1260 w w A A A K A A K                       (43) It can be concluded that for odd values of k in Eq. (34), φ[k] equals zero [23-24]. Therefore, by using recurrence Eq. (37), we can find that φ[k] can be determined in terms of φ[0] and φ[2]. By using recurrence relation (37) for k=2, 4, 6,…, we can obtain the following equations: 1 2 1 2 1 2 2( ) [2] [0] [4] 8 72 (12 4 ) [4] [2] (30 6 ) [6] [4] [6] , [8] , 24 600 48 2352 w w w A A K A A K A A K                         (44) Hence, all the φ[k] with even values of k in Eq. (34) can be expressed in terms of φ[2], φ[0]. 556 F. FARHATNIA, M. GHANBARI-MOBARAKEH, S.R. JAZI, S. OVEISSI In order to determine the critical buckling temperature, recurrence relation (42) and imposed boundary conditions of Eq. (39) are simultaneously employed for clamped edges. Therefore, a set of two homogenous equations are established in terms of φ[0] and φ[2], as follows: 11 12 21 22 [0] [2] 0 [0] [2] 0 H H H H          (45) where H11, H12, H21, and H22 are the coefficients of polynomials of N-th order. 1 2 11 1 2 1 2 1 2 12 1 2 21 1 2 1 2 1 2 22 (3 ) 1 8 72 (6 150)(8 72) (3 )( ) 1 4 36 24 600 (6 150)(4 36) 6(3 ) 2 18 (6 150)(2 36) (12 4 )( ) 2 9 4 100 (4 100)(4 36) w w w w w w K A A K H KA A A A A A H K A A K H KA A A A A A H                                                       (46) For a non-trivial solution of Eq. (45), the determinant of coefficients must vanish, leading to the eigenvalue problem. Hence, we have: 11 12 21 22 0 H H H H  (47) In a similar manner, we can obtain the solution for the simply supported plate by utilizing the recurrence relation (37) and the imposed boundary conditions of Eq. (41). 5. NUMERICAL RESULTS In this section, some numerical results are presented for the thermal buckling of orthotropic FG circular plates of two categories of uniform and linear temperature rise under clamped and simply supported boundary conditions. Due to the high volume of the calculations required, the system of algebraic equations presented in the preceding section are implemented in a computer code in MATLAB software, and the numerical results are presented in a tabulated form. For the numerical results, an FG plate composed of aluminum (as metal) and alumina (as ceramic) is considered. Young’s modulus of aluminum (Al) and alumina are Em=70 Gpa and Ec=380 Gpa, respectively. The thermal expansion coefficients are αm=23×10 -6 K -1 and αc=7.4×10 -6 K -1 for metal and ceramic constituents, respectively. Poisson’s ratio remains constant at ν=0.3. It is assumed that the material properties are assumed to be temperature-independent. To examine the convergence rate of DT method, we obtain the results for a clamped isotropic homogeneous circular plate with the thickness-to- radius ratio of h/R=0.01. We observe that the number of terms (k=20) is sufficient to get precise values of the critical temperature. This trend remains constant in other numerical results as presented in the following section. Thermal Buckling Analysis of Functionally Graded Circular Plate ... 557 5.1. Uniform loading - clamped boundary condition Table 2 shows the variation of the critical buckling temperature with respect to FG power-law index, thickness-to-radius ratio, and orthotropic ratio. As it is observed, the critical temperature increases as the value of μ rises. According to Table 2, for a specific h/R ratio, the buckling critical temperature is reduced by increasing the value of n; however, increasing the h/R ratio increases the values of the critical temperature. Table 2 Variation of the critical temperature ( o K) of the clamped plate versus FG power-law index, orthotropic ratio, and thickness-to-radius ratio in case of uniform thermal loading n μ h/R 0 0.5 1 2 5 10 0.50 0.010 0.150 0.020 0.030 0.040 7.070 15.911 28.280 63.641 113.132 4.081 9.104 16.081 36.153 64.214 3.163 7.390 12.851 28.934 52.560 2.914 6.552 11.652 26.214 46.603 2.151 6.060 11.141 26.042 46.142 2.002 5.950 10.860 25.851 45.823 0.70 0.010 0.150 0.020 0.030 0.040 13.780 31.020 55.131 124.061 220.552 7.813 17.570 31.232 70.280 124.944 6.422 14.414 25.611 55.412 98.503 5.684 12.780 22.714 51.091 90.831 5.164 11.181 21.421 52.732 93.721 5.010 10.551 21.090 50.261 92.361 0.9 0.010 0.150 0.020 0.030 0.040 21.381 48.110 85.541 192.462 342.152 12.112 27.260 48.460 109.033 193.842 9.5404 21.480 38.881 89.412 152.872 8.812 19.823 35.234 79.270 140.920 9.150 20.444 36.341 81.841 145.382 9.040 20.121 33.370 80.124 142.534 1 0.010 0.150 0.020 0.030 0.040 24.950 56.150 99.824 224.581 399.270 14.142 31.813 56.552 127.240 226.214 11.594 25.083 46.370 104.343 185.480 10.280 23.132 41.113 92.504 164.451 10.001 21.863 38.411 89.432 161.652 8.901 20.520 38.604 88.104 160.412 1.5 0.010 0.150 0.020 0.030 0.040 51.633 114.280 203.141 456.602 807.412 29.262 64.744 115.081 258.660 457.411 23.480 53.091 94.421 213.192 360.590 21.260 47.120 83.662 188.150 332.542 21.941 48.562 80.311 194.121 343.150 22.561 49.933 78.760 199.480 352.761 2 0.010 0.150 0.020 0.030 0.040 87.150 200.051 349.374 794.632 1408.221 49.371 113.361 197.942 450.160 797.630 40.634 89.381 162.401 355.991 651.631 35.890 82.393 143.881 327.270 579.880 33.151 78.050 141.450 322.644 558.262 31.134 77.412 140.661 320.180 555.180 558 F. FARHATNIA, M. GHANBARI-MOBARAKEH, S.R. JAZI, S. OVEISSI Table 3 exhibits the critical buckling temperature with respect to elastic foundation coefficient, orthotropic ratio, and FG power-law index. As elastic foundation coefficients increase by increasing the orthotropic ratio, the critical temperature increases. On the other hand, when the FG power-law index increases, the critical temperature decreases. This shows that the pure ceramic plate, compared to the metal-ceramic plate, is more stable at the elevated working temperature. Table 3 Variation of the critical temperature ( o K) of the clamped plate versus FG power-law index, orthotropic ratio, and elastic foundation on the critical temperature in case of uniform thermal loading, h/R=0.020 kw, kg μ n (0,0) (100,0) (200,0) (500,0) (100,10) (200,20) 0 55.131 63.121 67.341 81.470 77.470 96.911 0.50 31.232 35.690 38.150 46.151 43.880 54.880 0.7 1 25.611 29.341 31.280 37.851 35.903 44.841 2 22.714 26.030 27.741 33.581 31.960 39.801 5 21.422 23.822 24.633 31.610 30.841 37.250 0 99.824 114.293 121.780 147.531 140.050 175.360 0.5 56.552 64.520 69.270 83.812 79.512 99.5281 1 1 46.370 53.141 56.343 68.301 65.381 81.334 2 41.113 47.282 50.480 61.012 57.722 73.982 5 38.411 44.690 47.742 59.553 53.801 71.552 0 203.142 232.570 248.150 300.222 285.450 357.153 0.5 115.081 131.764 140.580 170.244 161.711 202.300 1.5 1 94.421 108.111 115.342 139.544 132.680 165.823 2 83.662 95.781 102.233 123.644 117.583 147.221 5 80.313 92.812 100.433 120.810 115.283 145.342 5.2. Linear loading - clamped boundary conditions Table (4) represents the effect of increasing the thickness-to-radius ratio on increasing the critical temperature in case of linear thermal loading condition. This table clearly shows that the trend of variation of the critical temperature is confirmed in Table 2 as well. The comparison of the two tables indicates that the rate of increase is higher for linear thermal loading as compared to the uniform one. Nevertheless, for small h/R and non-zero quantities of n, this trend is reversed. Thermal Buckling Analysis of Functionally Graded Circular Plate ... 559 Table 4 Variation of the critical buckling temperature ( o K) for the clamped plate in case of linear thermal loading, (kw, kg)=0 μ h/r n 0 0.5 1 2 5 0.50 0.010 0.150 0.020 0.030 0.040 8.480 26.531 50.740 67.592 220.670 2.652 12.780 26.521 37.052 122.801 0.843 8.673 19.272 27.860 93.292 0.351 6.640 15.472 22.873 77.060 0.305 6.062 14.763 20.160 73.893 0.70 0.010 0.150 0.020 0.030 0.040 15.173 43.640 93.142 234.551 432.540 4.750 21.021 48.682 128.581 240.714 1.511 14.272 35.372 96.700 182.851 0.523 10.920 28.403 79.370 151.231 0.451 10.284 27.932 77.370 150.681 0.9 0.010 0.150 0.020 0.030 0.040 22.772 76.230 151.082 364.921 664.300 7.123 36.710 78.962 200.153 369.6821 2.274 24.922 57.371 150.444 280.832 0.650 19.110 46.122 123.480 231.901 0.563 18.713 44.920 122.212 231.493 1 0.010 0.150 0.020 0.030 0.040 29.912 92.291 179.632 429.171 778.532 8.273 43.632 93.131 234.563 432.552 2.980 30.171 68.214 176.932 329.124 0.731 23.101 54.772 145.231 271.880 0.660 21.860 53.790 143.053 264.810 1.5 0.010 0.150 0.020 0.030 0.040 62.510 197.501 355.670 931.304 1572.631 19.562 95.113 185.882 510.520 875.182 15.032 71.954 151.512 390.711 671.040 10.080 57.064 118.450 322.131 555.614 9.840 55.860 115.042 315.793 551.260 2 0.010 0.150 0.020 0.030 0.040 101.694 325.971 617.394 1511.632 2266.591 31.813 156.981 322.672 828.650 1261.371 30.290 124.653 251.290 639.903 971.882 22.394 100.291 205.614 528.782 805.652 21.350 98.140 205.380 524.433 803.570 Table (5) shows the effect of the presence of elastic foundation on the critical buckling temperature for linear loading condition. As it is observed, the critical temperature decreases when the elastic foundation coefficient gets more values. In order to validate the present solutions, we compare them with the results obtained from the study conducted by Ghiasian et al. [25]. Table (6) shows the critical temperature for the clamped plate without elastic foundation. The values are Em=201 Gpa, αm=12.33×10 -6 K -1 , Ec=350 Gpa, and αc=5.87×10 -6 K -1 , and the Poisson’s ratio is ν=0.3. 560 F. FARHATNIA, M. GHANBARI-MOBARAKEH, S.R. JAZI, S. OVEISSI Table 5 Effect of elastic foundation on the critical buckling temperature for linear temperature rise, h/R=0.020 (kw, kg) μ n (0,0) (100,0) (200,0) (500,0) (100,10) (200,20) 0 93.140 107.621 114.160 136.523 142.211 163.120 0.50 48.681 56.560 60.163 71.362 74.610 54.442 1 35.371 41.212 43.363 51.851 54.661 61.681 0.7 2 28.401 33.280 35.222 41.634 43.362 49.742 5 27.931 33.111 35.022 41.411 43.331 48.040 0 179.632 207.291 219.152 264.062 274.664 312.511 0.5 93.131 105.611 111.761 134.290 141.562 158.322 1 68.210 79.390 83.900 100.812 104.361 118.692 1 2 54.772 64.361 66.274 80.2380 85.331 117.322 5 54.390 65.830 65.182 79.012 83.030 98.190 0 355.670 416.954 439.520 521.322 545.123 622.433 0.5 185.880 217.913 229.722 272.253 284.620 325.342 1 151.510 177.622 187.244 222.363 234.163 265.253 1.5 2 118.450 138.863 146.370 173.632 180.853 207.283 5 117.841 140.721 145.342 172.884 180.052 206.220 Table 6 Comparison of the results of the present study with those of Ref. [25] h R p.e. * 0.040 p.e. * 0.030 p.e. * 0.020 p.e. * 0.010 n 0.857 199.644 0.884 112.291 0.893 49.910 1.11 12.731 Present 201.369 113.293 50.360 12.591 0 Ref. [25] 0.015 141.149 0.264 83.142 0.146 37.001 2.88 9.532 Present 141.170 83.362 37.055 9.265 0.5 Ref. [25] 0.038 133.358 0.105 74.921 0.654 33.120 2.41 8.536 Present 133.307 75.000 33.338 8.335 1 Ref. [25] 0.676 122.314 0.768 68.752 0.792 30.554 0.792 7.761 Present 123.147 69.284 30.798 7.700 2 Ref. [25] 0.253 114.452 0.356 64.326 0.732 28.486 2.718 7.370 Present 114.742 64.556 28.696 7.175 5 Ref. [25] * The percentage of error Thermal Buckling Analysis of Functionally Graded Circular Plate ... 561 5.3. Effect of boundary conditions on the critical buckling temperature In circular FG plates, due to the stretching-bending coupling exposed to uniform thermal loading, the asymmetric material distribution induces the pre-buckling thermal moments. Therefore, the bifurcation buckling may not occur, and the buckling critical temperature is not available [26]. However, the plates with clamped edges can tolerate the bending moments and remain in an un-deformed configuration. In this study, we consider that the variation of the critical buckling temperature is tabulated for a homogeneous orthotropic circular plate under simply supported boundary condition, as shown in Table 7. Table 7 Variation of the critical buckling temperature for a homogeneous orthotropic circular plate under simply supported boundary condition and linear and uniform temperature rises μ h R 0.50 0.70 0.90 Uniform Linear Uniform Linear Uniform Linear 0.010 2.06 4.12 3.72 7.45 5.37 10.74 0.150 4.63 9.26 8.37 16.74 12.08 24.17 0.020 8.23 16.46 14.88 29.76 21.48 42.96 0.030 18.52 37.04 33.49 66.97 48.34 96.67 0.040 32.92 65.84 59.53 119.04 85.93 171.86 h R 1.0 1.50 2.00 Uniform Linear Uniform Linear Uniform Linear 0.010 6.03 12.07 10.31 20.76 15.80 34.28 0.150 13.58 27.15 23.29 46.24 39.64 79.55 0.020 24.13 48.27 41.46 73.41 70.46 126.31 0.030 54.30 108.60 93.10 187.88 159.56 318.12 0.040 96.54 193.08 165.95 333.14 253.41 569.59 6. CONCLUSIONS In this paper, we have analyzed the thermal buckling of a circular FGM plate resting on the Pasternak elastic foundation and subjected to uniform and linear thermal loading by employing the differential transform method (DTM) to obtain the solutions. Some remarkable conclusions obtained in this study are as follows:  By utilizing the differential transform method (DTM), the governing differential equation in the thermal buckling can be transformed to algebraic equations in the sub-domains. DTM is capable of deriving the analytical solution to determine the critical buckling temperature for FG orthotropic plate resting on two-parameter (Pasternak) foundations.  Comparing the results with those existing in the literature indicates that DTM is a fast convergent, precise, and cost-efficient tool to analyze the thermal buckling behavior of functionally graded plates. 562 F. FARHATNIA, M. GHANBARI-MOBARAKEH, S.R. JAZI, S. OVEISSI  By increasing volume fraction n, the flexural rigidity is reduced as the plate becomes more metal-rich. Consequently, the critical buckling temperature is reduced. This leads to the conclusion that for a ceramic-rich plate, the value of the critical temperature is the maximum.  The present study has investigated the effect of the presence of elastic foundation, as a controlling parameter, on the critical temperature for two boundary conditions, namely clamped and simply supported edges. The numerical results indicate that increasing the parameters of elastic foundation increases the critical temperature. Hence, the critical temperature in buckling can be adjusted effectively.  According to the results, when the temperature rises linearly, the critical temperature gets higher values in comparison with the uniform temperature rise.  As the ratio of Young’s modulus in the circumferential direction to that in the radial one increases, the plate demonstrates a more resistant behavior. Therefore, the critical buckling temperature increases. REFERENCES 1. Dewey, B.R, Costello, G.A., 1968, Thermal buckling of nonhomogeneous plates, Nuclear Engineering and Design, 7(3), pp. 249-261. 2. Najafizadeh, M.M., Eslami, M.R., 2002, First-order-theory-based thermoelastic stability of functionally graded material circular plate, AIAA J, 40(7), pp.1444–1450. 3. Najafizadeh, M.M., Eslami, M.M., 2002 Thermoelastic stability of orthotropic circular plates, Journal of Thermal Stresses, 25(10), pp. 985-1005. 4. Li, S.R., Zhou, Y.H., Song, X., 2002, Non-linear vibration and thermal buckling of an orthotropic annular plate with a centric rigid mass, Journal of Sound and Vibration, 251(1), pp. 141-152. 5. Najafizadeh, M.M., Heydari, H.R., 2004, Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory, European Journal of Mechanics A/Solids., 23, pp. 1085– 1100. 6. Prakash, T., Ganapathi, M., 2006, Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method, Composites: Part B. Engineering, 37(7–8), pp.642–649. 7. Zhao, X., Lee, K.M., Liew, K.M., 2009, Mechanical and thermal buckling analysis of functionally graded plates, Composite Structure, 90(2), pp.161-17. 8. Zenkour, A.M., Sobhy, M., 2010, Thermal buckling of various types of FGM sandwich plates, Composite Structure, 93(1), pp. 93-102. 9. Jalali, S.K, Naei, M.H., Poorsolhjouy, A., 2010, Thermal stability analysis of circular functionally graded sandwich plates of variable thickness using pseudo-spectral method, Mater. Des., 31(10), pp.4755–63. 10. Kiani, Y., Eslami, M.R., 2013, An exact solution for thermal buckling of annular FGM plates on an elastic medium, Composites Part B: Engineering, 45(1), pp.101-110. 11. Jabbari, M., Hashemitaheri, M., Mojahedin, M.R., 2014, Thermal Buckling Analysis of Functionally Graded Thin Circular Plate Made of Saturated Porous Materials, Journal of Thermal Stresses, 37(2), pp. 202-220. 12. Yaghoobi, H., Fereidooni, A., 2014, Mechanical and thermal buckling analysis of functionally graded plates resting on elastic foundations: An assessment of a simple refined nth-order shear deformation theory, Composites Part B: Engineering, 62, pp. 11-26. 13. Mansouri, M.H., Shariyat, M., 2014, Thermal buckling predictions of three types of high-order theories for the heterogeneous orthotropic plates, using the new version of DQM, Composite Structure, 113, pp. 40-55. 14. Mirzaei, M., Kiani, Y., 2016, Thermal buckling of temperature-dependent FG-CNT-reinforced composite plates, Meccanica, 51(9), pp. 2185–2201 15. Yu, T., Bui, T.Q., Yin, S., Doan, D.H., Wu, C.T., Do, T.V., Tanaka, S., 2016, On the thermal buckling analysis of functionally graded plates with internal defects using extended isogeometric analysis, Composite Structures, 136, pp. 684–695. https://link.springer.com/journal/11012/51/9/page/1 http://www.sciencedirect.com/science/journal/02638223/136/supp/C Thermal Buckling Analysis of Functionally Graded Circular Plate ... 563 16. Tung, H.V., 2015, Thermal and thermomechanical post-buckling of FGM sandwich plates resting on elastic foundations with tangential edge constraints and temperature-dependent properties, Composite Structures, 131(1), pp. 1028–1039. 17. Sun, Y., Li, S.R., Batra, R.C., 2016, Thermal buckling and post-buckling of FG Timoshenko beams on nonlinear elastic foundation, Journal of Thermal Stresses, 39(1), pp. 11-26. 18. Attarinejad, R., Semnani, Sh.J., Shahba, A., 2006, Basic displacement functions for free vibration analysis of non-prismatic Timoshenko beams, Journal of Finite Elements in Analysis and Design, 46 (10), pp. 916–929. 19. Ozdemir, O., Kaya, M.O., 2006, Flap wise bending vibration analysis of a rotating tapered cantilever Bernoulli–Euler beam by differential transform method, Journal of Sound and Vibration, 289, pp. 413–420. 20. Yalcin, H.S., Arikoglu, A., Ozkol, I., 2009, Free vibration analysis of circular plates by differential transformation method, Computational and Applied Mathematics, 212, pp.377–386. 21. Yeh, Y.L., Wang, C.C., Jang, M.J., 2007, Using finite difference and differential transformation method to analyze of large deflections of orthotropic rectangular plate problem, Applied Mathematics and Computation, 190(2), pp.1146-1156. 22. Abbasi, S., Farhatnia, F., Jazi, S. R., 2013, Application of Differential Transformation Method (DTM) for bending Analysis of Functionally Graded Circular Plates, Caspian Journal of Applied Sciences Research, 2(4), pp. 17-23. 23. Abbasi, S., Farhatnia, F., Jazi, S.R., 2014, A semi-analytical solution on static analysis of circular plate exposed to non-uniform axisymmetric transverse loading resting on Winkler elastic foundation, Archives of Civil and Mechanical engineering, 14, pp. 476-488. 24. Lai, R., Ahlawat, N., 2015, Axisymmetric vibrations and buckling analysis of functionally graded circular plates via differential transform method, European Journal of Mechanics A/Solids, 52, pp. 85-94. 25. Ghiasian, SE., Kiani, Y., Sadighi, M., Eslami, M.R., 2014, Thermal buckling of shear deformable temperature dependent circular/annular FGM plates, International journal of Mechanical Sciences, 81, pp.137-148. 26. Li, S., Zhang, J., Zhao, Y., 2007, Nonlinear thermomechanical post-buckling of circular FGM plate with geometric imperfection, Thin-Walled Structures, 45(5), pp. 528-536. http://www.sciencedirect.com/science/journal/02638223/131/supp/C