Size-dependent free vibration analysis of multidirectional functionally graded nanobeams via a nonlocal strain gradient theory

The free vibration behavior of a new advanced functionally graded (FG) nanobeam is presented in this work using the recently proposed nonlocal higher-order shear deformation theory. In the present theory, the stress tensor can satisfy the parabolic variation of the shear stress distribution throughout the thickness direction and also fulfill the requirement that the shear stress on the top and bottom surfaces of the FG nanobeam is zero. Two common types of FG structures, namely, FG hardcore and FG softcore, are considered here for analysis with three schemes. The material properties of the FG nanobeam are assumed to vary continuously in both the longitudinal and transversal directions according to a combined simple power-law distribution in terms of the volume fractions of the constituents. The governing equations of the FG nanobeam with simply supported boundary conditions are derived using the proposed higher-order shear deformation plate theory. The nonlocal strain gradient theory is employed to capture the microstructure-dependent effect. The influence of the structural geometry, the gradient index, and the nonlocal and length scale parameters on the vibration frequency is investigated. Finally, many new results are also reported in the current study, which will serve as a benchmark for future research.