A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory

Based on the modified couple stress theory (MCST), a unified higher order beam theory which contains various beam theories as special cases is proposed for buckling of a functionally graded (FG) microbeam embedded in elastic Pasternak medium. This non-classical microbeam model incorporates the material length scale parameter which can capture the size effect. The non-classical beam model reduces to the classical beam model when the material length scale parameter is set to zero. The material properties of the FG microbeam are assumed to vary in the thickness direction and are estimated through the Mori–Tanaka homogenization technique and the classical rule of mixture. The governing equations and the related boundary conditions are derived using the principal of the minimum total potential energy. The Navier-type solution is developed for simply-supported boundary conditions, and explicit expressions related to each type of beam theory are proposed for the critical buckling load. Numerical results are presented to investigate the influences the material length scale parameter, aspect ratio, different estimation method of material properties, various material compositions, and the parameters of the elastic medium on the critical buckling load. Comparison study is also performed to verify the present formulation.