Free Vibration Analysis of Thin-Walled Beams Using Two-Phase Local–Nonlocal Constitutive Model

A mathematical model is developed based on the thin-walled beams theory for free vibration analysis of nano/micro scale beams having nonlocal properties and arbitrary cross sections. Constitutive relations are defined by using two-phase local–nonlocal constitutive formulation. Equations of motion are derived by use of Hamilton‘s principle. Both the local and nonlocal part of the model is solved by the displacement-based finite element method. Numerical results are obtained and examined for nonlocal box beams and collapsed carbon nanotubes. In general, it is observed that the natural frequency decreases by increasing the nonlocal parameter or the volume fraction of the nonlocal part.