On buckling and post-buckling behavior of functionally graded micro-beams in thermal environment

This study provides an exact solution for the size dependent buckling and post-buckling behavior of functionally graded (FG) micro-beams with arbitrary boundary conditions which are subjected to combined thermo-mechanical loading. To this end, a theoretical formulation including the effects of size dependency, elastic foundation and uniform temperature distribution is first derived using the modified couple stress theory and through the principle of minimum total potential energy. Next, the nonlinear equations governing bending and stretching behavior of FG micro-beams are uncoupled to a fourth-order ordinary differential equation. Finally, the differential operator method is utilized to exactly solve the decoupled equation. Also, a Fourier series solution is presented for doubly-simply supported FG micro-beams to show the importance of exact solution. In the numerical results section, the effects of the geometric ratios, material distribution, temperature variation, and material length scale parameter on the post-buckling behavior are discussed in detail. Findings show that the Fourier series solution is not able to correctly predict the post-buckling behavior of FG micro-beams, since the effect of flexural-extensional coupling stiffness term appearing in the natural boundary condition is ignored. Also, it is seen that critical values of axial traction obtained from the post-buckling analysis are significantly varied with the transverse force per unit length and temperature variation, while the buckling analysis predicts that buckling values of beam remain constant. Therefore, it can be concluded that the buckling analysis is inadequate for analyzing FG micro-beams under thermo-mechanical loadings.