Dynamics of a non-circular-shaped nanorod with deformable boundaries based on second-order strain gradient theory

In this study, a general method is developed for the torsional vibration of non-circular-shaped nanorods with varying boundary conditions using second-order strain gradient theory. In most of the studies in the literature, the cross section of the rods is considered to be circular. The reason for this is that the use of warping function is inevitable when the cross section geometry is not circular. For circular cross sections after torsion, the warping is very small and is considered to be non-existent. For non-circular sections, cross section warping should be taken into account in mathematical calculations. The cross section geometry is different from circular in this study, and the boundary conditions are not rigid, contrary to most studies in the literature. In this paper, the second-order strain gradient theory and the most general solution method are discussed. In some specific cases, it is possible to transform the problem into many studies found in the literature. The correctness of the algorithm is tested by comparing the resulting solutions with closed solutions found in the literature. The influence of some variables on the torsional frequencies is illustrated by a series of graphical figures, and the superiority of the applied method is summarized.