Transverse Free Vibration of Axisymmetric Functionally Graded Circular Nanoplates with Radial Loads

Purpose The functionally graded circular nanoplate is a commonly seen component in the nano-electromechanical system. It is indispensable to examine free vibration behaviors of such an axisymmetric nanostructure subjected to uniformly distributed loads in the radial direction. Although the vibration engineering and technology have been fully studied at the macro-scale, there are still many unsolved problems at the micro-scale. The present research aims to promote the theoretical characterization of vibration behaviors at micro/nano-scale and further provide a basis for the development of vibration testing technologies. Methods Using the nonlocal strain gradient approach and Mindlin plate theory, we develop the theoretical model describing the transverse free vibration of the axisymmetric functionally graded circular nanoplate. First, by considering the nonlocal strain gradient constitutive relation, we derive the equation of motion via Hamilton's principle in polar coordinate system. Subsequently, the differential quadrature method is employed to solve the equation of motion numerically. Results The natural frequencies of circular nanoplates decrease with an increase of a radial compression while increase with an increase of a radial tension. The first-mode natural frequency reduces to zero under a certain radial compression, bringing about the dynamical instability. The natural frequencies are sensitive to the radial compression, and the clamped boundary constraint is more resistant to external loads than the simply supported one. An increase in the nonlocal parameter results in lower natural frequencies, while an increase in the strain gradient characteristic parameter results in higher ones. Conclusions It is demonstrated that the strain gradient characteristic parameter has a threshold in the present model for functionally graded circular nanoplates. In the circumstance of a lower nonlocal parameter than the strain gradient characteristic parameter, the circular nanoplate shows hardening behaviors. In the circumstance of a greater nonlocal parameter, the circular nanoplate shows softening behaviors. When the two internal characteristic parameters are equal, the stiffness of nanoplates remains unchanged and degenerates into its classical counterpart.

Automated summary

This study established a theoretical model for the transverse free vibration of axisymmetric functionally graded circular nanoplates using the nonlocal strain gradient method and Mindlin plate theory. The results showed that the natural frequencies of the circular nanoplates are affected by radial compression, boundary constraints, and internal characteristic parameters. The study also concluded that the nanoplates exhibit different behaviors based on the relationship between the nonlocal parameter and strain gradient characteristic parameter.