Multi-objective optimization of functionally graded materials, thickness and aspect ratio in micro-beams embedded in an elastic medium

Optimal design of micron-scale beams as a general case is an important problem for development of micro-electromechanical devices. For various applications, the mechanical parameters such as mass, maximum deflection and stress, natural frequency and buckling load are considered in strategies of micro-manufacturing technologies. However, all parameters are not of equal importance in each operating condition but multi-objective optimization is able to select optimal states of micro-beams which have desirable performances in various micro-electromechanical devices. This paper provides optimal states of design variables including thickness, distribution parameter of functionally graded materials, and aspect ratio in simply supported FG micro-beams resting on the elastic foundation using analytical solutions. The elastic medium is assumed to be as a two-layered foundation including a shear layer and a linear normal layer. Also, the size effect on the mechanical parameters is considered using the modified strain gradient theory and non-dominated sorting genetic algorithm-II is employed to optimization procedure. The target functions are defined such that the maximum deflection, maximum stress and mass must be minimized while natural frequency and critical buckling load must be maximized. The optimum patterns of FG micro-beams are presented for exponential and power-law FGMs and the effect of theory type and elastic foundation discussed in details. Findings indicate that the elastic foundation coefficients and internal length scale parameters of materials have the significant influences on the distribution of design variables. It is seen that the optimum values of inhomogeneity parameter and aspect ratio for E-FG micro-beams predicted by the modified strain gradient theory are larger than those of the classical continuum theory. Also, the multi-objective optimization is able to improve the normalized values of mass, maximum deflection, buckling load and natural frequency of P-FG micro-beams.