Stokes’ transform solution method for dynamics of a short-fiber-reinforced nanorod via second-order strain gradient theory

This study focuses on a study of free vibration of non-circular composite nanorods using second-order strain gradient theory. Short-fibers are used to strengthen the mechanical behavior of the cross-section. The main objective of this study is to present a general solution that can calculate the torsional vibration frequencies of non-circular and short-fiber-reinforced nanorods on the basis of second-order strain gradient theory for arbitrary (rigid and non-rigid) boundary conditions. For this purpose, elastic springs are placed in the torsional direction to provide flexibility in boundary conditions and spring constants are included in the problem. A Fourier sine series is utilized as the rotation function. At the boundaries, constant coefficients are used as the rotation function and preparation is made to exclude these constant coefficients for the construction of the eigenvalue problem. Using the torsional moment function equation, i.e., the force boundary condition and the Fourier coefficient, a set of equations consisting of two equations of infinite series are formed and the most general eigenvalue problem is established with the help of the coefficients matrix of this set of equations. In order to investigate the influence of various parameters, solutions are performed and the results obtained are shown in the form of a series of graphs and tables.