Size-dependent longitudinal vibration of embedded fg bishop nanotubes by incorporating deformable boundary conditions

Abstract

In this study, the axial vibration analysis of radially functionally graded composite nanotubes in an elastic medium based on Bishop's theory is investigated. The material parameters used change functionally from the inside to the outside for the hollow section, as described in the literature. Eringen's nonlocal elasticity theory in differential form is applied to study higher-order effects in the nano-scale. First, the equation of motion and boundary conditions for the problem in the axial direction are established. The Fourier coefficient is determined by combining the Fourier sine series with the equation of motion. With higher-order boundary conditions, an eigenvalue problem is formulated using the Stokes' transform. In this eigenvalue problem, the elastic medium parameter, rigidities of axial springs at the boundaries, nonlocal parameter, and the volume fraction index are incorporated. From this perspective, the study of these parameters in compact form is possible. The results obtained are compared with those of similar studies in the literature, showing excellent agreement. The effects of several parameters are detailed through various graphs and tables.