A Size-dependent Functionally Graded Micro-beam Based On The Strain Gradient Elasticity Theory

With the development of high technology, Micro-electro-mechanical system(MEMS) with micro-scale has been developed, which combines the function of micromechanical and microelectronics into one product. MEMS is composed of a variety ofmicrostructure, such as micro beam, micro plate, micro shell, micro tubes, micro thinfilm and micro spring, and micro beam is one of the common simple microstructure.With the decrease of the feature size of microstructure, its mechanical property changeswith the variation of the feature size of microstructure, which is named the size effect.As the phenomenon of size effect can’t be explained by the classical continuummechanics theory, it is very important and urgent to study the change rules of themicrostructure’s mechanical property and develop the theoretical model of itsmechanical property to the decision, optimization and experimental research of theMEMS products.In this thesis, the size effect of the functional graded material micro-beam is studied,the main research contents and conclusions are as follows:1. Chapter one introduces the research background and present situation of the sizeeffect of the microstructure in recent years detailed and expounds the research progressesof size effect in experimental and theoretical respect.2. Chapter two elaborates the theoretical basis of size effect analysis of thefunctional gradient material micro-beams in detail. The theories include the sinusoidalshear deformation theory, strain gradient elasticity theory, Mori–Tanaka homogenizationtechnique, Hamilton variational principle and basis of modal analysis.3. In chapter three, a novel size-dependent beam model made of functionally gradedmaterials (FGMs) is developed based on the strain gradient elasticity theory andsinusoidal shear deformation theory. The material properties of the functionally graded(FG) microbeams are assumed to vary in the thickness direction and are estimatedthrough the Mori–Tanaka homogenization technique. Governing equations and boundaryconditions are derived simultaneously by using Hamilton’s principle. The new model contains three material length scale parameters and can consequently capture the sizeeffect.4. In chapter four and five, The Navier-type solution is developed forsimply-supported boundary conditions. Numerical results are presented to investigate theinfluences the material length scale parameter, different material compositions, and sheardeformation on the bending and free vibration and buckling behavior of FG microbeams.Some of the present results are compared with the previously published results toestablish the validity of the present formulation.5. It is established that the present FG microbeams exhibit significantsize-dependence when the thickness of the microbeam approaches to the material lengthscale parameter.