# Closed Form Solutions for Thermal Buckling of Functionally Graded Rectangular Thin Plates
## 1 Introduction  
Designable Functionally Graded Materials (FCMs) are increasingly widely used in the aerospace, nuclear, and ship industries and other various industry departments, especially in thermal environments with ultra-high temperatures and a large temperature gradient. FCMs were proposed as thermal barrier materials by Japanese scientist [PERSON] [1; 2] to solve many problems in the new generation of aerospace airplane thermal protection systems. The volume content of each material in FCMs is continuously distributed in the spatial position on the macroscopic scale, and the physical properties change smoothly, so that the stress concentration of FCMs can be better avoided or reduced. Since FCMs usually work in extremely high temperature environments, in order to ensure structural stability, the critical buckling temperature of FGM structures is very important in theory and application. Many researchers have studied the thermal buckling of FGM plates based on different theories in the past two decades. [PERSON] and [PERSON] [3] developed closed form solutions for the thermal buckling of simply supported FGM thin plates based on the Classical thin Plate Theory (CPT). [PERSON] and [PERSON] [4] obtained closed form solutions for the thermal buckling of simply supported imperfect FGM thin plates. [PERSON] et al. [5] investigated the critical buckling temperature of fully clamped FGM thin plates resting on elastic foundations; they used the Bubnov-Galerkin Solution (BGS), Power Series Solution (PSS) and Semi-Levy Solution (SLS) respectively to represent the lateral displacement which cannot be obtained by the Navier solution, and compared the solutions using these different approaches. [PERSON] et al. [6] studied the critical buckling load and buckling model of FGM thin plate with in-plane material inhomogeneity by using radial basis function, but the thermal environment was not taken into account in their work. Shear deformation plays a more prominent role for thick plates, so shear deformation theories are more accurate than CPT in this case. [PERSON] [7] and [PERSON] et al. [8] obtained a closed form solution forthermal buckling of simply supported FGM thick plates based on the first-order shear deformation theory (FSDT), and compared the results with those of CPT. [PERSON] and [PERSON] [9] developed closed form solutions for thermal buckling of imperfect simply supported FGM thick plates based on FSDT. [PERSON] et al. [10] obtained a finite element solution for the critical buckling temperature based on neutral surface and FSDT. [PERSON] and [PERSON] [11] presented thermal postbuckling and vibration analyses based on FSDT by using the nonlinear finite element method. [PERSON] and [PERSON] [12] studied the thermal stability of eccentrically stiffened sigmoid FGM thick plate with metal-ceramic-metal layers by the Galerkin method based on FSDT. There are also some researches based on high-order shear deformation theories. [PERSON] and [PERSON] [13] developed closed form solutions for thermal buckling of simply supported FGM thick plates based on the third-order shear deformation theory (TSDT). [PERSON] and [PERSON] [14] investigated the critical buckling temperature of clamped FGM circular thick plates based on TSDT, and compared it with results based on CPT and FSDT. [PERSON] [15] investigated the critical buckling temperature of simply supported FGM thick plates based on the 2D higher-order deformation theory that he developed, and compared the results with CPT and other lower-order shear deformation theories such as FSDT and TSDT. [PERSON] and [PERSON] [16] studied thermal buckling of simply supported FGM thick plates based on sinusoidal shear deformation plate theory (SPT) and obtained closed form solutions. Some researchers have used three-dimensional linear elasticity theory to analyze the thermal buckling of FGM plates.