The Research On Some Nonlinear Problems Of Functionally Graded Nanostructure Material Using Perturbation Methods

Functional gradient materials(FGMs)are a new type of composite material that is composed of two or more completely different materials.Due to its excellent mechanical properties,such as reducing stress concentration and optimizing stress distribution,various components made of functionally graded materials have been one of the most frequently used and important components in many industries.Functionally gradient multilayer graphene reinforced composites is a kind of graphene and its derivatives as a body to be added to the polymer matrix of the new type composite material,with a multi-layer structure of functionally graded GPLRC effectively combined with the merits of both functional gradient materials and the graphene platelet,which significantly improves the material performance and attracts the huge attention of the researchers.Therefore,it is of great significance to study the various mechanical behaviors of components made of these materials subjected to complex loads(electromagnetic loads,mechanical loads,thermal loads).On the other hand,according to experimental researches and molecular dynamics simulation results,the mechanical behaviors of nanostructures have small scale effects,surface effects and quantum effects.In the range of several nanometers to several hundred nanometers,the small-scale effect and the surface effect play a dominant role,which suggests that researchers must consider the small-scale effect and the surface effect when the components or systems are studied in the above range.In view of the characteristics described above,this article uses the non-classical continuum mechanics theory,the theory of elastic superposition principle,a two step perturbation-mixed Galerkin method and so on,to establish the corresponding mathematical models,and then utilized them to investigate those nonlinear mechanical behaviors of straight beams,circular tubes,curved beams and rectangular plates,including snap-through buckling,nonlinear vibration,nonlinear bending,buckling and post-buckling.The main research works can be listed as follows:In this paper,the nonlinear vibration of beams with different functional gradient distributions is studied in the theoretical framework of nonlocal strain gradient theory.They are beams with bottom-up functionally graded distribution,beams with inside-out functionally graded distribution and beams with mirror symmetrical functionally graded distribution.It should be pointed out that the latter two distributions make corresponding boundary conditions of functionally graded structure exactly match that of homogeneous isotropic materials under thermal load and mechanical load in a steady temperature field.According to the proposed hypothesis and approximate model,the effective material parameters of FGM beams are determined,and then the nonlinear vibration governing equations of functionally gradient nanobeams under thermal load is derived by using different displacement functions satisfying the stress boundary conditions and generalized variational principle.Subsequently,the equations are solved by a two step perturbation method as well as galerkin method.Based on the obtained asymptotic analytical solutions,the effects of various parameters on nonlinear vibration problems are studied in detail,including temperature,nonlocal parameters,strain gradient parameters,scale parameter ratio,slenderness ratio,volume index and various types of functional gradient distribution.The results show two new approaches to change the linear and nonlinear frequencies of beams.Next,this paper employs a hollow cylinder displacement function without considering a shear correction factor.With the aid of it and a generalized variational principle combined with the nonlocal strain gradient theory,we establish the corresponding mathematical model of functionally graded nanotubes with different boundary conditions under thermal loading.After using the perturbation method as well as the variational method to resolve the problem,the analytical solution of the equations are given.Based on this analytical solution,the effects of transverse shear deformation,temperature dependence of materials,functionally gradient index and radius of circular tubes on linear and nonlinear frequencies of functionally gradient circular tubes are discussed.Further,the author investigated the axial buckling and post-buckling of geometrically imperfect single-layer graphene sheets(GSs)under in-plane loading.Based on the generalized variational principle and the nonlocal strain gradient theory,a four-sided simply supported nanoplate model is set up.Subsequently,with the aid of a two-step perturbation method,a new analytical solution that includes the maximum dimensionless deflection to the sixth power is the first to derive historically,and then a detailed parametric study is performed based on the obtained analytical solution.The results show that both the size-dependent effect and a geometrical imperfection can’t be ignored in analyzing GSs.Finally,based on the theory of nonlocal strain gradient and surface elasticity,a general size-dependent model for buckling analysis of curved nanobeams is established.The model can capture the nonlocal effect,the strain gradient effect and the surface energy effect simultaneously.In addition,the model can be simplified into the Euler beam model,the Timoshenko beam model aligned with high-order shear Reddy beam model by selecting appropriate shape function.Then,the model is used to study the snap-through buckling of functionally graded multilayer graphene reinforced composite curved nanobeams with geometric imperfection.The results show that the snap-through buckling is more likely to occur when initial geometric dimensions become larger and larger.Moreover,curved beams subjected to the X-GPLRC distribution have the largest structural carrying capacity compared with curved beams subjected to the other distribution.Finally,the effective stiffness and the snap-through buckling of nanostructures are mainly influenced by the surface effect,and depends on the coupling effects of non-local effects and microstructural effects.