A novel functional and mixed finite element analysis of functionally graded micro-beams based on modified couple stress theory

Abstract Static bending, free vibration and buckling analyses of functionally graded (FG) Timoshenko micro-beams based on modified couple stress theory are presented under the framework of a new functional and mixed finite element method (MFEM). The new functional has been constituted for FG Timoshenko micro-beams through a scientific procedure based on the Gâteaux differential. Detailed derivation of mixed finite element formulation which allows C0 type continuous shape functions is given. The results of MFEM formulation for static bending, fundamental vibration frequency and critical buckling load of a straight FG micro-beam under various boundary conditions are obtained and compared with the existing literature. It is shown that, with simple C0 type shape functions it is possible to effectively capture variations of dimensionless fundamental frequencies and critical buckling load of moderately thick FG micro-beams for various boundary conditions and material parameters by completely avoiding the shear locking. Also FG tapered micro-beams are analyzed as a demonstration of versatility. Another distinctive feature of the new formulation is its capability of identification of the separate contributions of the coupled and the classical parts of the section moment. The new functional and the mixed finite element formulation are the original contributions of this study.