COMPARATIVE THERMAL BUCKLING ANALYSIS OF FUNCTIONALLY GRADED PLATE

HERMAL SCIENCE: Year 7, Vol., No. 6B, pp. 957-969 957 Introduction COMPARAIVE HERMAL BUCKLING ANALYSIS OF FUNCIONALLY GRADED PLAE by Dragan V. ČUKANOVIĆ a*, Gordana M. BOGDANOVIĆ b, Aleksandar B. RADAKOVIĆ c, Dragan I. MILOSAVLJEVIĆ b, Ljiljana V. VELJOVIĆ b, and Igor M. BALAĆ d a Faculty of ecnical Sciences, University of Pristina, Kosovska Mitrovica, Serbia b Faculty of Engineering, University of Kragujevac, Kragujevac, Serbia c State University of Novi Pazar, Novi Pazar, Serbia d Faculty of Mecanical Engineering, University of Belgrade, Belgrade, Serbia Original scientific paper ttps://doi.org/.98/sci668c A termal buckling analysis of functionally graded tick rectangular plates according to von Karman non-linear teory is presented. e material properties of te functionally graded plate, except for te Poisson s ratio, were assumed to be graded in te tickness direction, according to a power-law distribution, in terms of te volume fractions of te metal and ceramic constituents. Formulations of equilibrium and stability equations are derived using te ig order sear deformation teory based on different types of sape functions. Analytical metod for determination of te critical buckling temperature for uniform increase of temperature, linear and non-linear cange of temperature across tickness of a plate is developed. Numerical results were obtained in МALAB software using combinations of symbolic and numeric values. e paper presents comparative results of critical buckling temperature for different types of sape functions. e accuracy of te formulation presented is verified by comparing to results available from te literature. Key words: termal buckling, von Karman non-linear teory, sape function, iger-order sear deformation teory, power-law distribution Functionally graded materials (FGM) are composite materials in wic tere is a continuous and a discontinuous variation of teir cemical composition and/or micro-structure troug defined geometric distance. Mecanical properties suc as Young s modulus of elasticity, Poisson s ratio, sear modulus, as well as material tickness, are graded in recommended directions, and a gradient property can be stepwise or continuous, []. Delamination between layers is te biggest and te most frequently analyzed problem concerning conventional composite laminates. Most frequently used FGM is metal/ceramics, were ceramics ave a good temperature resistance, wile metal is superior in terms of tougness. Functionally graded materials, wic contain metal and ceramic constituents, improve termo mecanical properties between layers because of wic delamination of layers sould be avoided due to continuous cange between properties of te constituents. By varying a percentage of volume fraction content of te two or more materials, FGM can be formed so tat it acieves a desired gradient property in specific directions. * Corresponding autor; e-mail: dragan.cukanovic@pr.ac.rs

958 HERMAL SCIENCE: Year 7, Vol., No. 6B, pp. 957-969 e termoelastic beavior of a FG rectangular ceramic-metal plate was presented by Praveen and Reddy [] by using a four node rectangular finite element based on te first order sear deformation teory (FSD), including von Karman s non-linear effect. Using and expanding te adopted formulation by Praveen and Reddy, Reddy [3] studied te static analysis of te FG rectangular plates using te tird order sear deformation teory (SD). Using te SD, e defined displacement field based on te finite element of te plate wit te eigt-degrees of freedom per node. is formulation explains te termo mecanical coupling and von Karman s geometrical non-linearity. Woo and Meguid [] studied non-linear deformations of tin FG plates and sells using von Karman s classical non-linear plate teory under termo mecanical loads. e autors compared te stresses and displacements for ceramic, metal and FG plates and tey concluded tat displacements of te FG plate were, even wit a small ceramic volume fraction, significantly smaller tan displacements of te metal plate. Ma and Wang [5] researced large deformations by bending and buckling of an axisymmetrical simply supported and fixed circular FG plate using te von Karman s non-linear plate teory. e autors of te paper made an assumption tat mecanical and termal properties of FG materials vary continuously according to te power law of te volume fraction of te constituents. Lane [6] derived equilibrium and stability equations of a moderately tick rectangular simply supported FG plate under termal loads using FSD. Buckling temperature is derived for two types of termal loading uniform temperature increase and gradient increase troug te tickness of te plate. Ci and Cung [7, 8] obtained a closed form solution of a simply supported FG rectangular moderately tick plate under transverse load using te classical plate teory and Fourier series expansion. ey assumed tat te elastic modulus varies in te tickness direction of te plate, depending on te variation of te volume fraction of te constituents. Poisson s ratio remains constant. Closed form analytical solution is proven by comparing to numerical results wit finite element metod. Cung and Ce [9] analyzed a simply supported, elastic, moderately tick, rectangular FG plate under linear temperature canges in te tickness direction of te plate. ey assumed tat Young s modulus of elasticity and Poisson s ratio are constant trougout te plate. However, te termal expansion coefficient varies depending on te variation of te volume fraction of te constituents, based on te power law or exponential function in te tickness direction. Description of te problem Functionally graded rectangular plates of a b dimensions, were te z-axis is in direction of tickness,, are studied in tis paper. Young s modulus of elasticity, termal expansion coefficient and canges in temperature are defined according to te power law []: z E( z) = E + E +, m cm p z α( z) = α + α +, m cm E = E E, α = α α, cm c m cm c m = cr c m p z ( z) = + + m cr respectively, were subscripts c and m refer to ceramics and metal, respectively. By using substitution ( / + z / ) = Λ, eac of te previously mentioned laws represents a function of L. If te product is furter defined as E(z) α(z) (z), it is not difficult to conclude tat for defined properties of materials and defined values of p and s (p defines te percentage of ceramic or metal volume, < s < []), cr remains te only unknown value in te product. Analytical procedure for determining te critical buckling temperature for uniform increase of temperature, linear and non-linear cange of temperature across te tickness of a plate, is developed ere. s ()

HERMAL SCIENCE: Year 7, Vol., No. 6B, pp. 957-969 959 Displacement field and constitutive equations Disadvantages of te classical lamination teory, and te first FSD wic require correctional factors, are eliminated by many autors by introducing te sear deformation sape functions (SF), tab.. Many of tose SF are introduced in order to give te better results for specific kinds of loads and specific static or dynamical problems. It sould be empasized tat te following SF are not generally applicable to all kinds of problems. Таble. Sear deformation SF defined by different autors Number of sape function Name of autors Sape function f (z) SF Ambartsumain [] ( z/ )( / z /3) SF Reissner and Stavsky [] ( 5 z/)( z ) /3 SF 3 Stein [3] ( / π) sin ( πz / ) SF Mantari, et al. [] ( π ) cos ( π z/ ) / sin z / e + πz/ SF 5-6 Mantari et al. [5] tan ( mz) zm sec ( m/ ), m = { /5, π/} SF 7 Karama, et al. [6] Aydogdu [7] SF 8 Mantari, et al. [8] z exp [ ( z / ) ], z exp [ ( z / ) /ln α], α > ( z/ ) z.85 +.8z SF 9 Meice, et al. [9] ξ[ ( / π) sin ( πz / ) z], ξ = {,/cos( π/) } SF Soldatos [] sin ( z / ) z cos ( / ) [ ] SF Akavci [] zsec ( z / ) zsec ( π/) ( π/) tan ( π/) SF Akavci [] ( 3 π/ ) tan ( z / ) ( 3 π/ ) zsec ( / ) SF 3 Mecab, et al. [] e displacement field is ere taken as follows, [5]: (,,, ) = (,, ) zcos / sin z / + cos / + cos / w u( xyzt,,, ) = u( xyt,, ) z ( xyt,, ) + f( z) θ x x w v( xyzt,,, ) = v ( xyt,, ) z ( xyt,, ) + f( z) θ y y wxyzt w xyt were f (z) is a SF. In order to define components of unit loads, it is necessary to apply te relations between displacements and strains in accordance wit te von Karman s non-linear teory of elasticity []. Using a generalized Hooke s law, as well as te stiffness matrix [], and taking into account te effect of te cange in temperature eq. () and termal expansion, wic cause a strain αδ [3], te following components of unit loads are obtained: ()

96 HERMAL SCIENCE: Year 7, Vol., No. 6B, pp. 957-969 + + + + + {N} = {σ} d z = [Q]{k }d z+ [Q]{k } zd z+ [Q]{k } f( z)d z [Q ] α( z) ( z)dz + + + + + {M} = {σ} zd z = [Q]{k } zd z+ [Q]{k } z d z+ [Q]{k } zf( z)d z [Q ] zα ( z) ( z)dz + + + + + [ ] {P} = {σ} f( z)d z = [Q]{k } f( z)d z+ [Q]{k } zf( z)d z+ [Q]{k } f( z) d z [Q ] zα ( z) ( z)dz were + + ' ' S S {R} = {τ} f ( z)d z = [Q ]{k }[ f ( z)] dz Q Q Q 6 Q + Q Q Q5 [Q] = Q Q Q 6, [Q ], [Q ] Q Q S = Q5 Q = + 55 Q6 Q6 C 66 {σ}= { σxx σ yy σxy}, { τ}= { τxz τ yz} { Nxx Nyy Nxy} { Mxx M yy Mxy} { Pxx Pyy Pxy } { Ry Rx} {N} =, {M} = {P} =, {R} = {k } u x = + w v w u v w w + + + x y y y x x y w w w {k } = x y xy θ θ x y θ θ x y S {k } = +, {k } = x y y x { θx θy} (3) In eq. (3), by grouping te terms wit te elements of constitutive matrix, it is possible to define new matrices: + { } =,,,,, ij ij ij ij ij ij z f z z zf z f ij ( z) A,B, D, E, F, G Q d z, for i, j=,,6 + ( z) d, (, ) (,5) ' H = Q f z lr = lr lr In order to get an equilibrium equation, it is necessary to define te deformation energy in te following form []: ()

HERMAL SCIENCE: Year 7, Vol., No. 6B, pp. 957-969 96 + { σxx εxx α( z ) ( z) + σ yy εyy α( z ) ( z) + σzzεzz + τxyγ xy + τxzγ xz + τ yzγ yz}d d U= A Using te principles of minimum potential energy, equilibrium equations become: δu : N + N =, δv : N + N = xx, x xy, y yy, y xy, x δ w : M + M + M + N w + N w + N w = xx, xx xy, xy yy, yy xx, xx xy, xy yy, yy δθ : P + P R =, δθ : P + P R = x xx, x xy, y x y xy, x yy, y y Stability equation for a tick FG plate is derived based on te equilibrium eq. (6). e stability equation of te plate under termal load can be defined using te displacement components u, ν, w, θ x, and θ y. Displacement components of te next stable configuration are: Az (5) (6) u = u + u, v= v + v, w= w + w θ = θ + θ, θ = θ + θ x x x y y y (7) were u, ν, w, θ x, and θ y are te displacement components of arbitrarily small deviation from te stable configuration. Assuming tat temperature is constant in te xy-plane of te plate and tat it is canging only in te tickness direction of te plate, te stability equation can be derived by substituting eq. (7) and eq. (3) into eq. (6). In suc obtained equation, terms u, ν, w, θ x, and θ y do not exist because tey vanis due to satisfying te equilibrium condition eq. (6). erefore, te stability equations of te functionally graded rectangular plate are: δu : N + N =, δv : N + N = xx, x xy, y yy, y xy, x δ w : M + M + M + N w + N w + N w = xx, xx xy, xy yy, yy xx, xx xy, xy yy, yy δθ : P + P R =, δθ : P + P R = x xx, x xy, y x y xy, x yy, y y (8) were N xx, N yy, and N xy, are te resultants of te pre-buckling forces: / α ( z) ( z) / E z N = N = d z, N. xx yy = xy υ Equation (8) can be solved by using te analytical and numerical metods. In order to obtain analytical solutions, assumed solution forms and boundary conditions are adopted in accordance to Navier s metods applied in [-6]. Procedure for obtaining te results by combining te symbolic and numerical coefficient values, wic occur in tese kinds of problems, is implemented. Boundary conditions along edges of te simply supported rectangular plate, according to [6], are te following: v = w = θ = N = M = P = along edges x = and x = a u = w = θ = N = M = P = along edges y = and y = b y xx xx xx x yy yy yy aking into account te previously defined boundary conditions, based on [], it is possible to assume tat Navier s solution is in te following form: (9) ()

96 HERMAL SCIENCE: Year 7, Vol., No. 6B, pp. 957-969 mπx nπy mπx nπy u( xy, ) = U cos sin, v( xy, ) = V sin cos mn a b a b mn m= n= m= n= mπx nπy mπx nπy w( xy, ) = W sin sin, θ ( xy, ) = cos sin a b a b mn x xmn m= n= m= n= θ y mπx nπy, = sin cos ymn a b ( xy) m= n= () were U mn, V mn, W mn, xmn, ymn are arbitrary parameters, wic are to be determined. Furtermore, using te Navier s solution, equilibrium equation becomes: [ ] L Ω I { U} = () were U = {U mn V mn W mn xmn ymn } and Ω is te buckling parameter. Coefficients L ij, (i, j = -5) are defined in te following way:,,,,,, L = α A + β A L = αβ A + A L = L = α D + β D 66 66 3 66 L = αβ D + D L = α A + β A L = L = αβ D + D 5 6 66 3 6 L = α C + β C, L = α E + α β E + α β E + β E 5 66 33 66 L = α F αβ F αβ E, L = α β F α β F 3 3 66 35 β F 3 66 L = H + α G + α G, L = αβ G + G, L = H + α G + α G 66 5 66 55 55 66 wile te matrix I ij, (i, j = -5) is defined:, {I} = α N + β N, i, j = 3 xx yy were α = mπ / α, β = nπ / b. o obtain te non-trivial solutions, it is necessary for te determinant in eq. () to be equal to zero: Numerical results (3) () L Ω I = (5) e aim of tis section is to ceck te accuracy and te effectiveness of te given teory in determining te critical buckling temperature of FG plates for uniform increase of temperature, linear and non-linear cange temperature across tickness. In order to do tat, different numerical examples are sown, and te obtained results were compared to te results available from te literature. e teory presented in tis paper is verified by te examples of te square plate a/b=, wic was considered in [7, 8, ]. Unlike [8], were SD is applied, and [7, ] were iger order sear deformation teory (HSD) is applied, based on just one SF, comparative analysis of all SF (tab. ) is done ere. Besides te results, wic are available in te literature, tis paper sows results for a/ (5 and ) and a/b ( and 5) ratios. Material properties, used in te numerical examples, were: Metal (Aluminum): E M =.7 5 [MPa], n =.3, α M = 3 6 [ o C ], Ceramics (Alumina): E C = 3.8 5 [MPa], n =.3, α C = 7. 6 [ o C ].

HERMAL SCIENCE: Year 7, Vol., No. 6B, pp. 957-969 963 able. Comparison of critical buckling temperatures ( cr ) of rectangular FGM plates under a uniform increase of temperature (a / = 5, a / = and m = n = ) p Source a / = 5 a / = a / b = a / b = a / b = 5 a / b = a / b = a / b = 5 [7] --- --- --- 68.68 --- --- [8] --- --- --- 67.8 --- --- [] --- --- --- 68.75 --- --- SF 5583. 959.8 39.8 68.68 377.3 87.65 SF 5583. 959.8 39.8 68.68 377.3 87.65 SF 3 5585.559 97.95 335. 68.8 378.59 89.38 SF 56.88 8.93 55.5 6.68 376. 39.689 SF 5 5583.6 959.86 389.5 68.68 377.37 87.36 SF 6 5587.88 97.575 365.33 69. 379.87 883.95 SF 7 559.9 995.856 358.38 69.5 375.5 98.8 SF 8 559.659 999.86 368.379 69.83 375.77 93.98 SF 9 559.659 999.86 368.379 69.83 375.77 93.98 SF 5583. 958.98 38.7 68.68 377.3 87.799 SF 559.9 983.3 386.89 69.9 375.9 9.39 SF 558.65 966.6 397.57 68.75 377.766 88.6 SF 3 5583.5 96. 3.66 68.68 377.333 873.66 [7] --- --- --- 758.39 --- --- [8] --- --- --- 757.89 --- --- [] --- --- --- 758. --- --- SF 67.53 5398.66.8 758.395 775.555 66.86 SF 67.53 5398.66.8 758.395 775.555 66.86 SF 3 67.9 53.3 8.777 758.5 775.899 65.35 SF 687.6 57.68 88.3 759.588 78.3 657.5 SF 5 67.5 5397.995 99.578 758.395 775.553 66.73 SF 6 673.373 5.958 37.5 758.57 776. 6.98 SF 7 67.69 5.66 395.63 758.6 776.83 63.733 SF 8 67.939 55.59 9.68 758.63 776.963 633.93 SF 9 67.939 55.59 9.68 758.63 776.963 633.93 SF 67.5 5397.86 95.8 758.395 775.55 66.79 SF 67.835 58.33 7.98 758.69 777.99 69.78 SF 67.7 5.9 53.939 758.3 775.7 6.967 SF 3 67.559 5398.3 7.357 758.396 775.563 67.36 [7] --- --- --- 69.69 --- --- [8] --- --- --- 69.59 --- --- [] --- --- --- 69.57 --- --- SF 76.788 5.58 796.373 69.69 56.3 8.685 SF 76.788 5.58 796.373 69.69 56.3 8.685 SF 3 75.59 3.6 8.79 69.5 56.35 839.65 SF 9.659 65.87 877.768 693.799 568.5 99.58 SF 5 76.857 5.76 7963.55 69.7 56.66 8.86 SF 6 85.7 3.9 796.675 693.59 566. 87.95 SF 7 76.87 9.5 89.78 69.597 56.7 89.98 SF 8 76.7.599 83.8 69.65 56.8 85.8 SF 9 76.7.599 83.8 69.65 56.8 85.8 SF 77.6 6.87 796.6 69.77 56. 8.65 SF 86.66 3.65 796.595 693.66 566.85 876.9 SF 75.69 3.57 799.68 69.57 56.57 838.988 SF 3 76.58 5.6 7967.39 69.67 56.98 8.75

Čukanović, D. V., et al.: Comparative ermal Buckling Analysis of Functionally Graded Plate HERMAL SCIENCE: Year 7, Vol., No. 6B, pp. 957-969 96 In te case of graded cange in temperature, te temperature at te metal surface is M = 5 oc. able sows te values of critical buckling temperatures of rectangular FG plates under a uniform increase of temperature. ick and moderately tick plates were considered in ratios a / = 5 and a / =. Using te ratios, a / =, a / m = and p =, tere was a good matc of results for all te given SF wit te results from te papers [, 7, 8]. An insignificant deviation was noticed in te SF marked as SF. During te analytical procedure in MALAB, it is noticed tat only SF, SF,, SF, could give solutions for integrals defined in eq. () in closed form, wile in te case of oter SF, a numerical integration ad to be conducted. It is noticed tat te difference between results obtained by different SF increases wit te increase of te (a / b) ratio, and it decreases wit te increase of (a / ) ratio, proving te fact tat te effect of SF is inversely proportional to te plate tickness. Figure (a) a sows te decrease of te difference between te obtained results wit te increase of te value p, so wen p > 5, te constant ratio a / = and a variable ratio (a / b), te curves merge. e SF do not ave a significant effect on tis beavior because te curves obtained by te use of, SF, SF and completely overlap, wic can be clearly seen in te fig. (a). In fig. (b), it can be clearly seen tat te increase of (a / ) ratio, regardless of te p value, causes te curves to asymptotically approac zero, wic is in accordance wit te fact tat tin plates ave lower resistance to temperature cange. SF SF Δcr 5 8 3 p= 6 p = 5, 6 p= (a) a/ = 3 a/b 5 SF SF Δcr p= p= p = 5, 3 5 6 a/ 7 (b) a/b = Figure. Effect of te aspect ratio a / b and a / on te critical buckling temperature Δcr under an uniform increase of temperature Similar situation can be noticed under te linear cange of temperature. All te presented SF gave te results, wic matc te ones from te papers [, 7, 8]. e greatest deviation is noticed in te SF. able 3 sows te obtained values of te critical buckling temperature, Δcr, as well as te matcing wit values given in te literature. e curves in fig. (a)-(c) are of te same nature as in te previous case. Figure (d) clearly sows tat wit te ratio values a / = 5, a / =, a / b = and te increase of te value p, te curves asymptotically approac a specific value. e value of te orizontal asymptote for a rectangular plate depends mainly on te a / ratio. able sows numerous values of critical buckling temperatures of rectangular FG plates under a non-linear cange of temperature across teir tickness. ere was a perfect matc of te results for square plates under ratio values of a / = 5, a / = and s =, s = 5 for all SF wit te results given in te references [, 7, 9]. A deviation of te results for all SF is witin te permitted limits, wic can be clearly seen on te diagram sowing a complete overlap of curves wic correspond to te different SF.

HERMAL SCIENCE: Year 7, Vol., No. 6B, pp. 957-969 965 able 3. Comparison of critical buckling temperatures (Δ cr ) of rectangular FGM plates under a linear cange of temperature across teir tickness (a / = 5, a / =, m = n = and m = 5 C) p Source a / = 5 a / = a / b = a / b = a / b = 5 a / b = a / b = a / b = 5 [7] --- --- --- 37.36 --- --- [8] --- --- --- 3.968 --- --- [] --- --- --- 37.5 --- --- SF 56.885 98.895 6375.6 37.36 78.6 5735.33 SF 56.885 98.895 6375.6 37.36 78.6 5735.33 SF 3 6.7 93.39 669.5 37.6 786.39 577.67 SF 33.76 6.586 998.9 333.365 758.3 69.378 SF 5 56.85 98.57 6368.3 37.363 78.66 573.7 SF 6 65.765 93.5 6.67 38. 788.976 5757.83 SF 7 7.8 98.7 75.76 38.5 79.9 586.67 SF 8 73.39 988.37 76.759 38.567 79.55 5855.963 SF 9 73.39 988.37 76.759 38.567 79.55 5855.963 SF 56.8 97.96 6353. 37.36 78.65 5733.597 SF 7.89 956.69 66.578 38.859 79.9 579.637 SF 59.9 9.9 6585.39 37.56 785.533 5758.5 SF 3 57. 9.9 6399.38 37.368 78.667 5737.3 [7] --- --- ---.96 --- --- [8] --- --- ---.3 --- --- [] --- --- --- 3. --- --- SF 5.989.5 873.95.968 33.65 6.76 SF 5.989.5 873.95.968 33.65 6.76 SF 3 5.635.98 3.85 3.7 33.6.8 SF 53.867 53.9 5.5 5.5 3333.38.9 SF 5 5.976.38 87.58.967 33.6 6.3 SF 6 5.3 3.687 75.3 3.95 33.75 5.958 SF 7 56.798.5 338.7 3.373 333.9 53. SF 8 57.38 7.6 36.99 3.6 333.57 57. SF 9 57.38 7.6 36.99 3.6 333.57 57. SF 5.956.9 863.5.968 33.68 5.758 SF 57.86 33.956 773.75 3.55 333.5 9.76 SF 5.9.95 97.7 3. 33.96 6.5 SF 3 5..989 885.9.969 33.63 7.37 [7] --- --- --- 8.63 --- --- [8] --- --- --- 8.38 --- --- [] --- --- --- 8. --- --- SF 5.755 73.6.59 8.639 759.6 8569.3 SF 5.755 73.6.59 8.639 759.6 8569.3 SF 3 3.5 7.89 88.59 8.37 757.95 8567.36 SF 5.33 755.535 5.89.596 769.93 875.76 SF 5 5.876 73.938.89 8.65 759. 8569.6 SF 6.99 788.936 99.738.5 766.6 86.86 SF 7 5. 75.68 33.938 8.67 758.597 858.9 SF 8 5.63 75.59 35.3 8.98 758.77 8587.89 SF 9 5.63 75.59 35.3 8.98 758.77 8587.89 SF 6. 7.595 99.6 8.68 759.35 857.7 SF 3.88 795.8.369.358 767.635 863.39 SF 3.8 739.935 58.6 8.8 758. 8566.37 SF 3 5.39 7.776 9.953 8.6 758.976 8568.

966 HERMAL SCIENCE: Year 7, Vol., No. 6B, pp. 957-969 able. Comparision of critical buckling temperatures (Δ cr ) of rectangular FGM plates under a non-linear cange of temperature across teir tickness (a / = 5, a / =, m = n = and m = 5 C) p Source a / = 5 a / = a / b = a / b = 5 a / b = a / b = 5 s = s = 5 s = s = 5 s = s = 5 s = s = 5 [7] 673. 337. --- --- 8. 968. --- --- [9] 67.6 3383.3 --- --- 8. 968.9 --- --- [] 6738.8 3377.7 --- --- 8. 968.5 --- --- SF 6735.3 337.6 69563. 396.8 8. 968. 386.9 775.9 SF 6735.3 337.6 69563. 396.8 8. 968. 386.9 775.9 SF 3 67.6 3383.3 738.3 76.7 8. 968.9 38658.9 7737.8 SF 685.6 337. 7368.3 796.7 85. 97. 393. 78688. SF 5 6735. 337.5 6955. 39.9 8. 968. 386. 77. SF 6 678.6 3397. 698.9 3836.8 8.3 968.7 38636.7 7773. SF 7 6757.7 3355. 776. 5.3 8.6 9685.3 38769. 77538.5 SF 8 6759.9 3359.9 78. 6.3 8.8 9685.7 38783.9 77567.8 SF 9 6759.9 3359.9 78. 6.3 8.8 9685.7 38783.9 77567.8 SF 6735. 337. 6953. 396. 8. 968. 386. 77.7 SF 6759. 3358. 693.8 3887.7 83. 9686.5 38685.9 7737.9 SF 6738.8 3377.6 69877.7 39755. 8. 968.5 38637. 777. SF 3 6735.5 337. 69598.9 3998. 8. 968. 3865.9 77.9 [7] 75. 58. --- ---. 3. --- --- [9] 758.6 587.8 --- --- 6.8 38. --- --- [] 757.5 585.6 --- --- 6.7 38. --- --- SF 756. 58.8 33.5 699.7 6.6 37.9 79.8 3669. SF 756. 58.8 33.5 699.7 6.6 37.9 79.8 3669. SF 3 758.5 587.8 335.5 7356.5 6.7 38. 79. 36739. SF 75.6 537. 367.7 738.7 9.9 3.7 8.3 3735.5 SF 5 756. 58.7 398. 6989. 6.6 37.9 79. 3669. SF 6 76. 593.3 339.7 69535. 7. 38.9 79.9 367. SF 7 76.8 53.5 366.6 7.8 7. 39. 797.3 36833.7 SF 8 765.6 53.3 3685.8 795. 7.3 39.3 7976.5 3686.3 SF 9 765.6 53.3 3685.8 795. 7.3 39.3 7976.5 3686.3 SF 756. 58.7 388. 69869.7 6.6 37.9 7899.7 36689. SF 765.3 53.7 3395. 69595. 7. 39.6 7935.5 3676.3 SF 757.5 585.6 35. 7.7 6.7 38. 795. 367. SF 3 756. 58.9 3. 69935.8 6.6 37.9 79. 36693.7 [7] 55. 953. --- --- 67. 88. --- --- [9] 5536.9 955.5 --- --- 676.6 88. --- --- [] 5537.3 956. --- --- 676.7 88.5 --- --- SF 55. 953.8 99.8 3339. 677. 885. 79.3 87. SF 55. 953.8 99.8 3339. 677. 885. 79.3 87. SF 3 5536.9 955.5 955.5 3359. 676.6 88. 789.8 8.9 SF 5573.8 9589. 66.8 3555.9 679.7 889.7 7.9 658. SF 5 55. 953. 97.6 33388. 677. 885. 79.8 87.9 SF 6 556. 9567. 93. 3338.7 679. 888.6 868.5 8. SF 7 5539. 959.6 97.8 3393.5 676.7 88.6 83. 3.9 SF 8 5539.8 953.5 978. 3397. 676.8 88.7 87. 39.7 SF 9 5539.8 953.5 978. 3397. 676.8 88.7 87. 39.7 SF 55.5 953.7 93. 3338.5 677. 885. 793.8 89.6 SF 556. 957. 95.5 3338.6 679.3 889. 879. 36.6 SF 5537.3 956. 983.6 3359. 676.7 88.5 788. 8. SF 3 5539.5 953. 97.3 33.9 676.9 885. 79. 85.

Čukanović, D. V., et al.: Comparative ermal Buckling Analysis of Functionally Graded Plate HERMAL SCIENCE: Year 7, Vol., No. 6B, pp. 957-969 3 5 5 SF SF 3 Δcr p= p= p = 5, 5 8 6 (a) a/ = Δcr 3 a/b 5 p= p= p = 5, p = 5, (b) a/ = 5 Δcr 3 a/b 5 SF SF 3 8 6 a/ = 5 p= SF SF 3 p= SF SF 3 3 5 Δcr 967 (c) a/b = 3 5 6 a/ 7 a/ = 5 (d) a/ = 5 and a/ =, a/b = 5 a/ Figure. Effect of te aspect ratio a / b and a / on te critical buckling temperature Δcr under a linear cange of temperature Figure 3(c) sows tat te increase of non-linearity of temperature cange causes te fastest increase of Δcr for te value p =. For larger p values, curves of te temperature cange increase somewat more slowly, but comparing to te previous cases, tere is no complete matcing of curves wen p = 5 or p =. For te value s > 6, tese curves start to go in te different directions and teir separation is clearly seen. 6 Δcr 3 5 SF SF p= p= p = 5, 6 p= p = 5, a/b 3 (a) a/ =, s = 3 5 SF SF p= 8 3 Δcr 3 a/ 3 p= p=5 6 p= 8 (b) a/b =, s = 3 Δcr 3 5 6 7 p = SF SF S 6 8 (c) a/ =, a/b = Figure 3. Effect of te aspect ratio a / b, a / and parameter s on te critical buckling temperature Δcr under a non-linear cange of temperature Conclusion Based on te presented results, it can be concluded tat SF given in tab. are acceptable for te termo mecanical analysis of functionally graded plates. e results obtained

968 HERMAL SCIENCE: Year 7, Vol., No. 6B, pp. 957-969 troug te developed analytical procedure matced te results given in te literature. aking into account te fact tat SF ave a significantly larger effect on tick and moderately tick plates, results are limited to ratios a / = 5 and a / =. is paper proves tat te volume fraction of metal//ceramic constituents and te sape of te plate (a / b ratio), ave a significantly larger effect on te temperature resistance tan te cosen deformation teory. It is also sown tat te biggest deviations occur wen te value of te parameter p is low, wile te increase of te value p causes te differences between te critical temperatures to be reduced or completely vanis. Separation of te curves, wic correspond to te greater values of te parameter p, occurs wit te increase of non-linearity of a temperature cange, namely wit te increase of te s parameter. Nomenclature a, b, plate lengt, widt, tickness, [m] E c, E m elasticity modulus of ceramics, metals, [Pa] f (z) sape function, [ ] {k } membrane strain components, [ ] {k } strain components due to bending, [ ] {k } strain components due to plane sear, [ ] {k s } strain components due to transversal sear, [ ] {N} force resultant per unit lengt, [Nm ] {M} moment resultant per unit lengt, [N] {P}, {R} ig order moment resultants per unit lengt, [Nm] {Q} constitutive matrix, [Pa] c, m temperature of ceramic, metal, [ o C] D cr critical buckling temperature, [ o C] u, v, w displacement components at a general point of plate, [m] u, v, w displacement components at mid-plane of plate, [m] References Greek symbols a c, a m coefficient of termal expansion of ceramics, metals, [ o C ] q x, q y rotation of transversal normal due to transverse sear, [ ] {s} plane stresses components, [Pa] {t} transversal sear stresses components, [Pa] {W} buckling parametar, [ ] Acronyms FGM functionally graded material CP classical plate teory FSD first order sear deformation teory SD tird order sear deformation teory HSD ig order sear deformation teory FEM finite element metod MALAB Matrix laboratory Superscripts p volume fraction index, [ ] s degree of power law function, [ ] [] Markovic, S., Sinteza i karakterizacija Bai -x Sn x O 3 praova i viseslojni keramicki materijala (Syntesis and Caracterization of Bai -x Sn x O 3 Powders and Multilayer Ceramic Materials in Serbian), P. D. tesis, University of Belgrade, Belgrade, Serbia, 8 [] Praveen, G. N., Reddy, J. N., Nonlinear ransient ermoelastic Analysis of Functionlly Graded Ceramic Metal Plates, International Journal of Solids and Structures, 35 (998), 33, pp. 57-76 [3] Reddy, J. N., Analysis of Functionally Graded Plates, International Journal for Numerical Metods in Engineering, 7 (), -3, pp. 663-68 [] Woo, J., Meguid, S. A., Nonlinear Analysis of Functionally Graded Plates and Sallow Sells, International Journal of Solids and Structures, 38 (), -3, pp. 79-7 [5] Ma, L. S., Wang,. J., Nonlinear Bending and Post-Buckling of a Functionally Graded Circular Plate under Mecanical and ermal Loadings, International Journal of Solids and Structures, (3), 3-, pp. 33-333 [6] Lane, W., ermal Buckling of a Simply Supported Moderately ick Rectangular FGM Plate, Composite Structures, 6, (), pp. -8 [7] Ci, S.-H., Cung, Y.-L., Mecanical Beavior of Functionally Graded Material Plates under ransverse Load-Part I: Analysis, International Journal of Solids and Structures, 3 (6), 3, pp. 3657-367 [8] Ci, S.-H., Cung, Y.-L., Mecanical Beavior of Functionally Graded Material Plates under ransverse Load-Part II: Numerical Results, International Journal of Solids and Structures, 3 (6), 3, pp. 3675-369

HERMAL SCIENCE: Year 7, Vol., No. 6B, pp. 957-969 969 [9] Cung, Y.-L., Ce, W.-., e flexibility of Functionally Graded Material Plates Subjected to Uniform Loads, Journal of Mecanics of Materials and Structures, (7),, pp. 63-86 [] Akavci, S. S., ermal Buckling Analysis of Functionally Graded Plates on an Elastic Foundation According to a Hyperbolic Sear Deformation eory, Mecanics of Composite Materials, 5 (),, pp. 97- [] Ambartsumyan, S. A., On te eory of Anisotropic Sells and Plates, Proceedings, Non-Homogeniety in Elasticity and Plasticity, IUAM, Symposium, (Ed. W. Olszak), Warsaw, 958 [] Reissner, E., Stavsky, Y., Bending and Stretcing of Certain ypes of Heterogeneous Aeolotropic Elastic Plates, ASME Journal of Applied Mecanics, 8 (96), 3, pp. -8 [3] Stein, M., Nonlinear eory for Plates and Sells Including te Effects of ransverse Searing, AIAA Journals, (986), 9, pp. 537-5 [] Mantari, J. L., et al., Bending and Free Vibration Analysis of Isotropic and Multilayered Plates and Sells by Using a New Accurate Higerorder Sear Deformation eory, Composites Part B: Engineering, 3 (), 8, pp. 338-336 [5] Mantari, J. L., et al., A New rigonometric Sear Deformation eory for Isotropic, Laminated Composite and Sandwic Plates, International Journal of Solids and Structures, 9 (),, pp. 3-53 [6] Karama, M., et al., Mecanical Beaviour of Laminated Composite Beam by te New Multi-Layered Laminated Composite Structures Model wit ransverse Sear Stress Continuity, International Journal of Solids and Structures, (3), 6, pp. 55-56 [7] Aydogdu, M., A New Sear Deformation eory for Laminated Composite Plates, Composite Structures, 89 (9),, pp. 9- [8] Mantari, J. L., et al., A New angential-exponential Higer Order Sear Deformation eory for Advanced Composite Plates, Composites Part B: Engineering, 6 (), Apr., pp. 39-38 [9] Meice, N. E. et al., A New Hyperbolic Sear Deformation eory for Buckling and Vibration of Functionally Graded Sandwic Plate, International Journal of Mecanical Sciences, 53 (),, pp. 37-7 [] Soldatos, K., A ransverse Sear Deformation eory for Homogeneous Monoclinic Plates, Acta Mecanica, 9 (99), 3, pp. 95- [] Akavci, S. S., wo New Hyperbolic Sear Displacement Models for Ortotropic Laminated Composite Plates, Mecanics of Composite Materials, 6 (),, pp. 5-6 [] Mecab, B., et al., Analysis of ick Ortotropic Laminated Composite Plates Based on Higer Order Sear Deformation eory by te New Function Under ermo-mecanical Loading, Composites Part B: Engineering, 3 (), 3, pp. 53-58 [3] Milosavljević D. I., at al., Failure Criteria of Fibre Reinforced Composites in Homogenous emperature Field, ermal Science, (), Suppl., pp. S85-S97 [] Mantari, J. L., et al., Bending Response of Functionally Graded Plates by Using a new Higer Order Sear Deformation eory, Composite Structures, 9 (),, pp. 7-73 [5] Grover, N., et al., Flexural Beavior of General Laminated Composite and Sandwic Plates Using a Secant Function Based Sear Deformation eory, Latin American Journal of Solids and Structures, (), 7, pp. 75-97 [6] Reddy. J. N., Mecanics of Laminated Composite Plates and Sells: eory and Analysis, CRC Press, Boca Raton, Fla., USA, [7] Bouiadjra, M. B., et al., ermal Buckling of Functionally Graded Plates According to a Four-Variable Refined Plate eory, Journal of ermal Stresses, 35 (), 8, pp. 677-69 [8] Javaeri, R., Eslami, M. R., ermal Buckling of Functionally Graded Plates Based on a Higer Order eory, Journal of ermal Stresses, 5 (), 7, pp. 63-65 [9] Zenkour, A. M., Masat, D. S., ermal Buckling Analysis of Ceramic-Metal Functionally Graded Plates, Natural Science, (), 9, pp. 968-978 Paper submitted: June, 6 Paper revised: July 8, 6 Paper accepted: July, 6 7 Society of ermal Engineers of Serbia Publised by te Vinča Institute of Nuclear Sciences, Belgrade, Serbia. is is an open access article distributed under te CC BY-NC-ND. terms and conditions