Variable cross section nanorod model for longitudinal vibration analysis of cracked nanorods

This paper investigates the dynamic response of cracked nanorods under longitudinal vibration by proposing a new variable cross-section nanorod model applicable to different boundary conditions. The nonlocal theory has been used to derive the governing differential equation; a shape function has been proposed to model the cross sections of the cracked nanorods under the axial vibrations. The generalized differential quadrature method in conjunction with the fourth-order Runge-Kutta method has been used to solve the governing differential equation. The generalized differential quadrature method shows that increasing the number of grid points from N = 15 to N = 25 resulted in under 1% change in the computed natural frequencies, hence demonstrating spatial convergence. A parametric study is carried out to study the local-to-nonlocal frequencies ratio considering nonlocal modulus, nanorod length, and nondimensional crack position. The results show that an increase of nonlocal modulus ( μ ) from 0.1 to 0.3 leads to approximately 8.3 % reduction in frequency. The proposed model successfully avoids dealing with the continuity condition or the crack spring model for the cracked nanorod problem. For various boundary conditions, the proposed model shows good agreement with previous studies that represented cracked nanorods using an equivalent axial spring to connect two segments, with the closest correspondence observed for the fixed–fixed case.