Vibration characteristics of two-dimensional FGM nanobeams with couple stress and surface energy under general boundary conditions

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This paper presents a comprehensive investigation of the size-dependent vibration response of two-dimensional functionally graded (2D-FG) micro/nanoscale beams with various boundary conditions in the context of a microstructure-surface energy-based model, for the first time. The size-dependent 2D-FG beam model is developed based on Euler-Bernoulli beam theory with the aid of the modified couple stress and surface elasticity theories to capture the influence of microstructure and surface energy, respectively. Power-law function is adopted to describe the distribution of the material properties through thickness and length directions, simultaneously. Continuous spatial variation of the single material length-scale parameter and the three surface constants are accounted into the analysis of 2D-FG beams. Moreover, the present model considers the axial and bending displacements, the deviation of the physical neutral plane from the midplane counterpart, and the Poisson's effect. The size-dependent equations of motion are derived by means of Lagrange equations.To this end, the natural frequencies of 2D-FG nanobeams with various boundary conditions are obtained using Ritz method. The accuracy and stability of the developed variational formulation and solution procedure are verified through a convergence and comparative study with some limiting cases available in the literature. An extensive parametric study is carried out to examine the influence of the effects of material distributions, variable material length-scale parameter, variable surface residual stress, variable surface mass density, slenderness ratio, and boundary conditions on the natural frequencies of 2D-FG nanobeams in the context of microstructure-surface energy-dependent model. It is noticeable that the free vibration response is highly dependent on the material properties and nonclassical microstructure and surface energy parameters.