The effects of multi-directional functionally graded materials on the natural frequency of the doubly-curved nanoshells

Up to now, no studies have been yet reported to study the mechanical behaviors of three dimensionally functionally graded (FG) shell structures. In this paper, the free vibration of three dimensionally FG nanoplates and nanoshells are investigated for the first time. All of the mechanical properties expect passion’s ratio are assumed to be changed along the length, width and thickness directions, which can vary according to an arbitrary function. The small scale effect due to nanostructures is considered based on nonlocal Eringen’s theory where Hamilton’s principle is adopted to derive the equations of motion. Galerkin solution method is adopted to obtain the natural frequencies of the FG nanostructures. Dynamic results are reported for nanoplates, spherical nanoshells, and cylindrical nanoshells for simply supported boundary conditions. The influences of several parameters, such as nonlocal parameter and functionally graded indexes are investigated on the natural frequency of the nanostructures. The results of the presented study can be served as benchmarks for future mechanical analysis of three-dimensionally FG shell structures.