On the symplectic superposition method for new analytic bending, buckling, and free vibration solutions of rectangular nanoplates with all edges free

Abstract

This paper extends an up-to-date symplectic superposition method for analytic bending, buckling, and free vibration solutions of rectangular nanoplates based on the nonlocal theory. By transferring the problems to the Hamiltonian system expression, the symplectic approach becomes applicable, which involves generation of symplectic eigenvalue problems and symplectic eigen expansion. An original problem is then converted into superposition of several elaborated subproblems that are solved by the symplectic approach. The analytic bending deflection, natural frequency, buckling load, and associated mode shape solutions are obtained by the equivalence between the original problem and the superposition. As a representative class of non-Levy-type nanoplates that were not solved by conventional analytic methods, the nanoplates with all edges free are focused on the rapid convergence of the present solutions confirming comprehensive results tabulated with the accuracy of five significant figures, which can serve as benchmarks for future comparison. Quantitative parameter analyses are performed with the analytic solutions to investigate the effects of the nonlocal parameter and plate dimension on the mechanical behaviors of free nanoplates. The present method has merits on tackling complex boundary-value problems of higher-order partial differential equations represented by those for nanoplates and may be further developed for more analytic solutions in future studies.