In this paper, buckling and post-buckling analysis of functionally graded (FG) micro-beams subjected to the thermal gradient are presented. The FG micro-beam is embedded on an elastic substrate medium modeled by Winkler-Pasternak foundation. To capture the size effect, a modified couple stress theory is applied. Based on the minimum potential energy principle, and using Timoshenko beam hypothesis and von Kármán strains, governing differential equations are derived. Both Fourier series solution and exact analytical method are presented for solving the system of coupled differential equations. Because Fourier series solution cannot satisfy all boundary conditions, it is not able to predict the post-buckling response of FG micro-beams, correctly. Moreover, in numerical result section, the effects of length-scale parameter, power-law index, transverse and axial load, temperature gradient, and elastic foundation constants on the post-buckling behavior of simply supported FG micro-beams are investigated. Obtained responses demonstrated that the buckling of the FG micro-beams are significantly sensitive to the variation of mentioned parameters.

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