Longitudinal Vibration of Nanobeams by Two-Phase Local/Nonlocal Elasticity, Rayleigh Theory, and Generalized Differential Quadrature Method

To solve a differential equation of motion via more reliable procedures, it is essential to realize their efficiency. Whether Rayleigh's theory can be a compatible platform with two-phase local/nonlocal elasticity to render more reliable results compared to other theories or not is the main question that will be answered by this paper. Thus, nanobeam modeled by Rayleigh beam theory is analyzed by two-phase local/nonlocal elasticity. Governing equation in presence of the axial and transverse displacements is derived by means of Hamilton’s principle and differential law of two-phase elasticity. Next, fourth-order Generalized Differential Quadrature Method (GDQM) is utilized to attain the discretized two-phase formulation. In order to confirm, the method and the results are compared with the exact solution prepared and presented in the literature. Moreover, the effects of various parameters such as geometrical properties like thickness, mode shape number, Local phase fraction coefficient, and nonlocal factor on the natural frequency are investigated to clarify that utilizing these theories with a common goal how ends with more accurate results, and how affects the natural frequencies.