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This work presents the analytical analysis for free linear vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates in the framework of nonlocal strain gradient theory (NSGT) and the first-order shear deformation plate theory (FSDPT). The nanoplate is considered made of a mixture of an isotropic polymer matrix and reinforced carbon nanotubes (CNTs). Four different distributions of CNTs are examined, including uniformly distributed and FG reinforcements (FG-O, FG-X, and FG-V). The governing equations of motion are established based on Hamilton’s principle. The closed-form analytical solution for the natural frequency of FG-CNTRC nanoplates with simply supported all edges is carried out by using the Navier-type solution. The impact of some key parameters on the natural frequencies of FG-CNTRC nanoplates is also studied and discussed. The result shows that FG-CNTRC nanoplates reveal the softening- or hardening-stiffness effects depending on the relationship between the nonlocal parameter and the material length scale parameter. The aspect ratios of FG-CNTRC nanoplates, the volume fraction, and the distribution pattern of CNTs also have an important impact on the vibration behavior of FG-CNTRC nanoplates.
Hieu et al. (Mon,) studied this question.