Quasi-three-dimensional Functionally Graded Microbeam Theory And The Related Differential Quadrature Finite Element Method

With the development of ultra-fine processing and material preparation technology,micro-electro-mechanical-systems(MEMS)have rapidly emerged and are widely used in aerospace,biomedicine,intelligent driving,electronic communications and other fields.The characteristic size of the main components of MEMS is often on the order of micrometers or sub-micrometers,and microbeams are the most common microstructure among them.Although the microstructure is the geometric miniaturization of the macrostructure in the scale,as the characteristic size of the component decreases,the performance of the material will change significantly.This phenomenon that cannot be explained by the classical continuum mechanics is called the size effect.To quantitatively predict the size effect,scholars have proposed many non-classical continuum mechanics theories,including modified couple stress theory and strain gradient theory.On the other hand,since MEMS often work in harsh environments of high temperature,high pressure and strong corrosiveness,it has become an urgent need for the materials of micro-components to have good mechanical properties.Functionally graded materials(FGMs)have been widely used in frontier fields such as aerospace and MEMS due to their excellent mechanical properties and good designability.FGMs are usually composited by ceramic and metal materials.Its notable feature is the volume fraction of the two component materials changes with continuous changes in spatial position on a macro scale.Furthermore,the stress concentration and interlayer shedding phenomenon existing in traditional composite materials can be overcome by changing the volume fraction of each component material.Therefore,the establishment of a reasonable theoretical model of functionally graded microbeams is of great significance for the design and performance optimization of load-bearing components in MEMS.This thesis takes FGMs as the research object,based on the Reddy-type quasi-three-dimensional high-order shear-normal expansion theory and modified couple stress theory.A novel theory of functionally graded microbeams is proposed,and the corresponding differential quadrature finite element method is constructed.The influence of scale effect and high-order shear-normal expansion and contraction effects on the mechanical properties of microbeams is discussed in detail.The present work mainly includes:(1)Based on the modified couple stress theory and the Reddy type quasi-three-dimensional igh-order shear-normal expansion and deformation theory,a scale-dependent unctionally graded microbeam mechanical model was established.The second type of agrange equation is used to derive the differential equations of motion and boundary conditions of the microbeam.The analytical solutions of the static bending and free vibration of simply supported microbeams are derived by using the Navier series solution.(2)Aiming at the high-order boundary value problem of quasi-three-dimensional functionally graded microbeam,a two-node sixteen-degree of freedom differential quadrature finite element is proposed.Combining the Gauss-Lobatto numerical integration criterion and the differential quadrature criterion to discretize the potential energy functional of microbeams,a fourth-order differential quadrature geometric mapping strategy is constructed to realize the conversion between the Gauss-Lobatto integration points and the displacement parameters of the element nodes.The C~1 continuity requirements of the deflection field and the corner field are met.With the help of MAPLE software,a symbol tools,the corresponding elements of the element stiffness matrix,element mass matrix,element geometric stiffness matrix and element load vector are obtained.(3)Based on MATLAB software,a differential finite element program was developed.Based on this,the static bending,free vibration,and linear buckling of the quasi-three-dimensional functionally graded microbeam were analyzed,the deflection,normal stress, shear stress,natural frequency,natural frequency,mode shape and critical buckling loads of the microbeam were obtained,and the problem of neutral axis correction is discussed. Through comparative research,the correctness of the theoretical model and solution method is verified.The effects of functional gradient index,intrinsic characteristic length,geometric parameters and boundary conditions on the static response,vibration characteristics and stability of microbeams are explored.The results show that the beam model and the beam element in this thesis are suitable for analyzing the static/dynamic problems of functionally graded microbeams with various slenderness ratios.The consideration of size effects will significantly change the mechanical properties of microbeams.The static/dynamic response results of the microbeam obtained by considering the normal expansion effect are more accurate.