New Analytic Solutions For Nanoplates Based On The Nonlocal Theory

As a typical two-dimensional nanostructure,rectangular nanoplates are widely used in biosensors,resonators,storage devices,micro switches,etc.,and are used in micro/nanoelectro-mechanical systems as an important structural component.However,for nanoscale structures,the small size effect cannot be ignored,and classical continuum mechanics does not consider this problem.In order to overcome this limitation,various generalized continuum mechanics theories represented by the nonlocal theory provide a new idea for studying such structures.Because the governing differential equations of nanoplates under the nonlocal theory are difficult to solve mathematically,the existing researches mostly use numerical approximation methods or traditional analytic methods to deal with some simply boundary conditions.The analytic solutions to complex boundary conditions have always been a difficult problem.Based on Eringen's nonlocal elastic theory in combination with Kirchhoff's classical thin plate theory,this dissertation establishes a nonlocal rectangular nanoplate model,and introduces its bending,free vibration and buckling problems into the Hamiltonian system.The symplectic superposition method is reasonably used to solve rectangular nanoplates' problems with various boundary conditions.For the bending problems of rectangular nanoplates,the Hamiltonian system is established by constructing generalized displacements and generalized forces as dual variables,and the symplectic analytic solutions of two basic problems are obtained rationally by the methods of separating variable and symplectic eigen expansion.On this basis,the bending analytic solutions of nanoplates with various boundary conditions under uniform/concentrated loads are obtained by using the idea of superposition method.The consistency between the analytic results and the finite element results degraded to the classical plate theory proves the validity and accuracy of the symplectic superposition method.For the free vibration and buckling problems of rectangular nanoplates,this dissertation directly introduces dual variables of basic displacement variables,establishes the Hamiltonian system for such problems,and rationally yields the symplectic analytic solutions of two types of basic problems: those with two opposite edges simply supported or slidingly clamped.The analytic solutions for natural frequencies and critical buckling loads are obtained directly for nanoplates with two opposite edges simply supported,while for other complex boundaries,the analytic solutions of the original problems are obtained by superposing the solutions of the basic problems.The numerical results are compared with those from the published literature and the finite element results degraded to the classical plate theory,and their consistency proves the accuracy of the symplectic superposition method.Based on the above results,the effects of nonlocal parameter,boundary conditions,modal order,and plate length,etc.on the vibration and buckling behaviors of nanoplates are also investigated.In this dissertation,combined with the nonlocal theory,the symplectic superposition method is extended from the traditional plate and shell problems to nanoplates,and the analytic solutions for bending,free vibration and buckling problems of rectangular nanoplate are obtained,which provides a reference for the study of the mechanical behaviors of nanoplates,and can be used as a benchmark for comparison with numerical methods and other analytical methods.This method divides an original problem with complex boundaries into several basic sub-problems with simple boundaries,uses the symplectic geometry method to solve the subproblems in the Hamiltonian system,and uses the superposition method to make each physical quantity satisfy the actual boundary conditions to establish an equation,and then yields the analytic solution of the original problem.In this process,there is no need to assume a trial function,and the mathematical derivation is full analytic and rational,thus the generality is strong,and the further extension of the method is expected to obtain analytic solutions of plate and shell problems under more complex theories and types.