Nonlocal strain gradient approach for axial vibration analysis of arbitrary restrained nanorod

Axial free vibration analysis of small size-dependent nanorod subjected to deformable restrained boundary conditions is carried out in the present work. Unlike previous works, the formulation is rewritten without resorting to any un-deformable boundary conditions neither clamped ends with Navier approximation nor considering nanorod as a compact form without any discontinuities, and the boundary conditions are assumed to be gradually deformable in the axial direction. Within the framework of Fourier sine series and Stokes' transformation, an eigenvalue problem is constructed to obtain the axial vibration frequencies. In addition, the higher-order elasticity model contains a material scale parameter considering the prominence of strain gradient stress field and a nonlocal coefficient considering the prominence of nonlocal elastic stress field. The validity of the presented procedure is checked by comparing the obtained results by giving proper values to elastic spring parameters, and good agreement is achieved. Numerical results and graphical representation are presented to demonstrate the applicability of the presented eigenvalue solution to examine the free axial response of nanorods with arbitrary boundary conditions. Effects of small-scale parameters on the dynamic response of nanorods are discussed in detail.