本文推衍三維漸近彈性力學理論，對具簡支承邊界之功能性(Functionally Graded, FG)材料板受單軸及雙軸外壓力作用下之挫屈行為進行分析。文中假設FG板為金屬陶瓷複合材料板，楊氏係數隨厚度呈冪級數變化。彈性支承與周圍環境之相互作用以Winkler-Pasternak雙參數彈性基礎模型來模擬。將三維彈性力學理論，經過無因次化、漸近展開及漸近積分等過程，求得各階之控制方程式。文中考慮板之長寬比、寬厚比、功能性材料關係因子及彈性支承之勁度與剪切模數對臨界挫屈載重的影響。理論推衍顯示，三維漸近彈性力學理論首階解與古典板理論(Classical Plate Theory, CPT)之解相同。首階解求得後，再逐階修正漸近求得精確解。分析結果顯示本漸近理論解收斂快速且收斂解與文獻之精確解高度契合。

A three-dimensional asymptotic theory is reformulated for the buckling behavior of functionally graded plates with simply-supported under uniaxial and bi-axial compression. We assumed that the FG plate is a metal-ceramic composite plate, and the Young's modulus changes in a power series with the thickness. The interactions between the elastic support and their surrounding medium are modelled as a two-parameter Winkler-Pasternak foundation. By using a set of dimensionless variables and through the process of asymptotic expansion and asymptotic integration, the governing equations of each order are obtained. The influence of the length-width ratio, width-thickness ratio, functional material factor, stiffness and shear modulus on the critical buckling load is considered. The classical plate theory (CPT) is derived as a first-order approximation of the three-dimensional asymptotic theory. The CPT solutions can be modified order-by-order to asymptotically approach the exact elasticity solutions.

1. Pasha, A., Rajaprakash, B.M. Fabrication and mechanical properties of functionally graded materials: A review. Materials Today: Proceedings. DOI: 10.1016/j.matpr.2021.09.066.
2. Yang, S.Y., Liu, X., Cao, D.F., Mei, H., Lei, Z.T., Liu, L.S. A Study on Propagation Characteristic of One-dimensional Stress Wave in Functionally Graded Armor Composites. Journal of Physics: Conference Series. 2013; 419: 012046. DOI: 10.1088/1742-6596/419/1/012046.
3. Chin, E.S.C. Army focused research team on functionally graded armor composites. Materials Science and Engineering A. 1999; 259(2): 155-161. DOI: 10.1016/S0921-5093(98)00883-1.
4. Yılmaz, E., Çalışkan, F. A new functional graded dental implant design with biocompatible and antibacterial properties. Materials Chemistry and Physics. 2022; 277: 125481. DOI: 10.1016/j.matchemphys.2021.125481.
5. Zhang, S.Q., Zhao, Y.F., Wang, X., Chen, M., Schmidt, R. Static and dynamic analysis of functionally graded magneto-electro-elastic plates and shells. Composite Structures. 2021; 114950. DOI: 10.1016/j.compstruct.2021.114950.
6. Goudarzi, Z.M., Valefi, Z., Zamani, P. Effect of functionally graded structure design on durability and thermal insulation capacity of plasma-sprayed thick thermal barrier coating. Ceramics International. 2021; 47(24): 34361-34379. DOI: 10.1016/j.ceramint.2021.08.349.
7. Salmoria, G.V., Klauss, P., Zepon, K., Kanis, L.A., Roesler, C.R.M., Vieira, L.F. Development of functionally-graded reservoir of PCL/PG by selective laser sintering for drug delivery devices: This paper presents a selective laser sintering-fabricated drug delivery system that contains graded progesterone content. Virtual and Physical Prototyping. 2012; 7(2): 107-115. DOI: 10.1080/17452759.2012.687911
8. Jo, H., Yoon, M., Gajendiran, M., Kim, K. Recent Strategies in Fabrication of Gradient Hydrogels for Tissue Engineering Applications. Macromolecular Bioscience. 2020. 20(3); DOI: 10.1002/mabi.201900300.
9. Reddy, J.N., Wang, C.M. An overview of the relationships between solutions of the classical and shear deformation plate theories. Composites Science and Technology. 2000; 60(12-13); 2327-2335. DOI: 10.1016/S0266-3538(00)00028-2.
10. Aydogdu, M. Conditions for functionally graded plates to remain flat under in-plane loads by classical plate theory. Composite Structures. 2008; 82(1): 155-157. DOI: 10.1016/j.compstruct.2006.10.004.
11. Ebrahimi, F., Rastgo, A. An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory. Thin-Walled Structures. 2008; 46(12): 1402-1408. DOI: 10.1016/j.tws.2008.03.008.
12. Yu, T., Yin, S., Bui, T.Q., Liu, C., Wattanasakulpong, N. Buckling isogeometric analysis of functionally graded plates under combined thermal and mechanical loads. Composite Structures. 2017; 162: 54-69. DOI: 10.1016/j.compstruct.2016.11.084.
13. Wang, M., Xu, Y.G., Qiao, P., Li, Z.M. Buckling and free vibration analysis of shear deformable graphene-reinforced composite laminated plates. Composite Structures. 2022; 280: 114854. DOI: 10.1016/j.compstruct.2021.114854.
14. Kiani, Y., Mirzaei, M. Rectangular and skew shear buckling of FG-CNT reinforced composite skew plates using Ritz method. Aerospace Science and Technology. 2018; 77: 399-398. DOI: 10.1016/j.ast.2018.03.022.
15. Herskovits, J. Buckling behavior of composite and functionally graded material plates. European Journal of Mechanics - A/Solids. 2020; 80: 103921. DOI: 10.1016/j.euromechsol.2019.103921.
16. Reddy, J.N. Mechanics of Laminated Composite Plates and Shells:Theory and Analysis. 2004.
17. Zaitoun, M.W., Chikh, A., Tounsi, A., Al-Osta, M.A., Sharif, A., Al-Dulaijan, S.U., Al-Zahrani, M.M. Influence of the visco-Pasternak foundation parameters on the buckling behavior of a sandwich functional graded ceramic–metal plate in a hygrothermal environment. Thin-Walled Structures. 2022; 170: 108549. DOI: 10.1016/j.tws.2021.108549.
18. Mojahedin, A., Jabbari, M., Khorshidvand, A.R., Eslami, M.R. Buckling analysis of functionally graded circular plates made of saturated porous materials based on higher order shear deformation theory. Thin-Walled Structures. 2016; 99: 83-90. DOI: 10.1016/j.tws.2015.11.008.
19. Ansari, R., Torabi, J., Hassani, R. In-plane and shear buckling analysis of FG-CNTRC annular sector plates based on the third-order shear deformation theory using a numerical approach. Computers & Mathematics with Applications. 2018; 75(2): 486-502. DOI: 10.1016/j.camwa.2017.09.022.
20. Torabi, J., Ansari, R., Hassani, R. Numerical study on the thermal buckling analysis of CNT-reinforced composite plates with different shapes based on the higher-order shear deformation theory. European Journal of Mechanics - A/Solids. 2019; 73: 144-160. DOI: 10.1016/j.euromechsol.2018.07.009.
21. Kulkarni, K., Singh, B.N., Maiti, D.K. Analytical solution for bending and buckling analysis of functionally graded plates using inverse trigonometric shear deformation theory. Composite Structures. 2015; 134: 147-157. DOI: 10.1016/j.compstruct.2015.08.060.
22. Shimpi, R.P. Refined plate theory and its variants. AIAA JOURNAL. 2002; 40(1): 137-146. DOI: 10.2514/2.1622.
23. Shimpi, R.P., Patel, H.G. A two variable refined plate theory for orthotropic plate analysis. International Journal of Solids and Structures. 2006; 43: 6783-6799. DOI: 10.1016/j.ijsolstr.2006.02.007.
24. Mechab, I., Atmane, H.A., Tounsi, A., Belhadj, H.A., Bedia, E.A.A. A two variable refined plate theory for the bending analysis of functionally graded plates. Acta Mechanica Sinica. 2010; 26(6): 941–949. DOI: 10.1007/s10409-010-0372-1.
25. Thai, H.T., Choi, D.H. An efficient and simple refined theory for buckling analysis of functionally graded plates. Applied Mathematical Modelling. 2012; 36(3): 1008-1022. DOI: 10.1016/j.apm.2011.07.062.
26. Sobhy, M., Zenkour, A.M. Porosity and inhomogeneity effects on the buckling and vibration of double-FGM nanoplates via a quasi-3D refined theory. Composite Structures. 2019; 220: 289-303. DOI: 10.1016/j.compstruct.2019.03.096.
27. Zenkour, A.M., Aljadani, M.H. Buckling analysis of actuated functionally graded piezoelectric plates via a quasi-3D refined theory. Mechanics of Materials. 2020; 151: 103632. DOI: 10.1016/j.mechmat.2020.103632.
28. Radwan, A.F. Quasi-3D integral model for thermomechanical buckling and vibration of FG porous nanoplates embedded in an elastic medium. International Journal of Mechanical Sciences. 2019; 157-158: 320-335. DOI: 10.1016/j.ijmecsci.2019.04.031.
29. Farzam, A., Hassani, B. Thermal and mechanical buckling analysis of FG carbon nanotube reinforced composite plates using modified couple stress theory and isogeometric approach. Composite Structures. 2018; 206: 774-790. DOI: 10.1016/j.compstruct.2018.08.030.
30. Mian, M.A., Spencer, A.J.M. Exact solutions for functionally graded and laminated elastic materials. Journal of the Mechanics and Physics of Solids. 1998; 46(12): 2283-2295. DOI: 10.1016/S0022-5096(98)00048-9.
31. Asemi, K., Shariyat, M., Salehi, M., Ashrafi, H. A full compatible three-dimensional elasticity element for buckling analysis of FGM rectangular plates subjected to various combinations of biaxial normal and shear loads. Finite Elements in Analysis and Design. 2013; 74: 9-21. DOI: 10.1016/j.finel.2013.05.011.
32. Asemi, K., Salehi, M., Akhlaghi, M. Three dimensional biaxial buckling analysis of functionally graded annular sector plate fully or partially supported on Winkler elastic foundation. Aerospace Science and Technology. 2014; 39: 426-441. DOI: 10.1016/j.ast.2014.04.011.
33. Shufrin, I., Eisenberger, M. Stability of variable thickness shear deformable plates—first order and high order analyses. Thin-Walled Structures. 2005; 43: 189-207. DOI: 10.1016/j.tws.2004.07.013.
34. Thai, H.T., Vo, T.P. A new sinusoidal shear deformation theory for bending, buckling and vibration of functionally graded plates. Applied Mathematical Modelling. 2013; 37: 3269-3281. DOI: 10.1016/j.apm.2012.08.008
35. Mohammadi, M., Saidi, A.R., Jomehzadeh, E. A novel analytical approach for the buckling analysis of moderately thick functionally graded rectangular plates with two simply-supported opposite edges. Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science. 2010; 224(9). 1831–1841. DOI: 10.1243/09544062JMES1804.
36. Bodaghi, M., Saidi, A.R. Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory. Applied Mathematical Modelling. 2010; 34(11): 3659-3673. DOI: 10.1016/j.apm.2010.03.016