New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method

Abstract

New analytic bending, buckling, and free vibration solutions of rectangular nanoplates with combinations of clamped and simply supported edges are obtained by an up-to-date symplectic superposition method. The problems are reformulated in the Hamiltonian system and symplectic space, where the mathematical solution framework involves the construction of symplectic eigenvalue problems and symplectic eigen expansion. The analytic symplectic solutions are derived for several elaborated fundamental subproblems, the superposition of which yields the final analytic solutions. Besides Lévy-type solutions, non-Lévy-type solutions are also obtained, which cannot be achieved by conventional analytic methods. Comprehensive numerical results can provide benchmarks for other solution methods.

A review of size-dependent continuum mechanics models for micro- and nano-structures

“…Likewise, they proved that the hardening effects of the cantilever nanobeam by employing the small-scale influences were obvious, where the obtained results for C-C and S-S nanobeams were different. Novel analytical bending/buckling/free vibration behavior of rectangular nanoplates using the symplectic superposition scheme with consideration of the clamped as well as S-S edges was proposed by Zheng et al [386]. By employing Hamilton's principle as well as the symplectic space the governing equations were derived based on the mathematical solution comprised of two significant eigenvalue problems which were the construction and the expansion of them.…”

Symplectic superposition solutions for free in-plane vibration of orthotropic rectangular plates with general boundary conditions

“…From the open literature, analytic solutions for free in-plane vibration of orthotropic rectangular plates with non-Lévy-type boundary conditions, i.e., those without two opposite edges simply supported, are still quite deficient, which motivates the present work. It is noteworthy that a symplectic superposition method (SSM) has been developed recently to deal with plate and shell problems involving out-of-plane deformation, including bending 17,18 , buckling 19,20 , and transverse vibration [21][22][23] . Such an analytic method is developed based on an elegant integration of the superposition technique and the symplectic approach 24,25 ; it is not conducted in the Euclidean space but in the symplectic space where several important mathematical treatments, such as the separation of variables and the symplectic eigen expansion, prove to be valid.…”