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The aim of the current study is to investigate torsional vibrations of bi-directional functionally graded (FG) small-scale tubes. The size effect is captured using the modified couple stress theory (MCST). Through-the-thickness distribution profile of material properties are characterized by employing a power-law function, which incorporates a gradation index,  β , and longitudinal varation in material properties is featured by an exponential function which employs a gradient parameter,  α . Similar to other material properties, continuous spatial variation of the length scale parameter is also incorporated into the analysis of two-dimensionally FG tubes. The model, including governing equations of motion and boundary conditions, is developed through the utilization of Hamilton’s principle. By employing differential quadrature method (DQM) numerical results regarding torsional vibrations are provided. Accuracy of the proposed model and procedures are verified through verification comparisons made to limiting cases available in the literature. Numerical results reveal effects of various material and geometric parameters on natural frequencies and indicate that torsional free vibration characteristics of small-sized tubes are sensitive to distribution profile of constituents, geometric aspect ratios and effective length scale parameter. Results also demonstrate that length scale parameter variation has significant influence on natural frequencies which justifies the necessity of its consideration in analyses.

Torsional dynamics of bi-directional functionally graded small-scale tubes possessing a variable length scale parameter