Research On The Mechanical Behavior And The Finite Element Method Of Micro/Nano Structure

With the development of technology and manufacturing process,MEMS?Micro Electro Mechanical System:MEMS?,with many advantages,has been widely used in the field of electronics,medical equipment,machinery and aerospace and so on.So the study of MEMS becomes very important.When MEMS is designed and calculated,some structures can be simplified to some of the micro planes or micro beams,which are typical structures in the MEMS devices.Therefore,the analysis of the mechanical behavior of these microstructures becomes very important.However,there are many differences between the microstructures and the macrostructures.so the traditional theory does not apply to analyze the mechanical properties of the microstructures.Many researchers do some related mechanics tests of micro/nano structure,these tests shows that al l of the material of posite,polycrystalline silicon,polymer,and metal have size effect,namely with reducing in the microstructure size,The mechanical property of material becomes stronger gradually,and this kind of phenomenon cann't be explained by the traditional mechanics theory.So it is necessary to develop a theory and model applying to micro/nano structure.Although many scholars have developed many theories to explain the phenomenon of size effect,such as the nonlocal theory,coupling stress theory,surface energy theory,gradient elastic theory and the strain gradient theory,but the strain gradient theory is the one of the most successful theories.Nonlocal theory is more suitable for softening structure research.Surface energy theory only considers the influence of surface effect and ignores the influence of the body,so the theory is more suitable for nanostructure,which the ratio of surface to volume is very high.Though the calculation of gradient elastic theory is simple,it precision is low.The couple stress theory only considers a strain gradient?gradient tensor rotation symmetry component?.The strain gradient theory considers three kinds of strain gradient,congregating some advantages of many theories,so the theory has been widely applied.A size-dependent finite element model for micro/nano-scale Timoshenko beam is developed based on the strain gradient elasticity theory.The newly developed element contains three material length scale parameters which can capture the size effect.This element is a new comprehensive Timoshenko beam element which can reduce to the modified couple stress Timoshenko beam element or the classical Timoshenko beam element if two?l0 and l1?or three?l0,l1 and l2?material length scale parameters are set to zero.The element satisfies the C₀ continuity and C₁ continuity and is a two-node element which has 4-DOF at each node by only considering bending deformation.The deflection and cross-sectional rotation of the element are interpolated independently.The finite element formulations,the stiffness and mass matrices are derived by using the corresponding weak form equations.In order to verify the reliability and accuracy of the proposed element,the problems of convergence and shear locking are studied.Using this newly developed element,the static bending and free vibration problems of the clamped and simply supported Timoshenko microbeam are investigated.The results of a simply supported Timoshenko microbeam predicted by the new element model agree well with the results in existing literature.Moreover,the results illustrate that the size effect on Timoshenko microbeam can be effectively predicted by using the proposed element.To precisely modelling the size dependencies in nanostructures,the surface effect and bulk effect are incorporated.From the physical point of view,size dependencies stem from not only the surface but also the bulk.The surface energy theory and strain gradient elasticity theory are introduced to characterize the surface effect and bulk effect respectively.The new models for Bernoulli-Euler and Timoshenko beams are developed.Governing equations,initial conditions and boundary conditions are derived simultaneously by using Hamilton's principle.The new models,incorporating the Poisson effect,contain three material length scale parameters and three surface elasticity constants to capture the size effect in the bulk and surface layer of the beam,respectively.The models recover the models which either the bulk effect or the surface effect is considered,and also can degenerate into the corresponding modified couple stress models or the classical models when some constants are ignored.In addition,the new Timoshenko beam model recovers the new Bernoulli-Euler beam when shear deformation is ignored.To illustrate the new models,the static bending and free vibration problems of the simply supported nano-scale Bernoulli-Euler and Timoshenko beams are solved respectively.Numerical results reveal that the differences in the deflection,rotation and natural frequency predicted by the present model and the other models are large when the beam thickness is small.These differences,however,are decreasing or even diminishing with the increase of the size of the beams.The models may guide the precise design of nanobeam-based devices for a wide range of potential applications.A size-dependent Kirchhoff micro-plate model resting on elastic medium is developed based on the strain gradient elasticity theory.Three material length scale parameters are introduced in the model,and those parameters may effectively capture the size effect.The model can degenerate into the modified couple stress plate model or the classical plate model by setting two?l0 and l1?or all?l0,l1 and l2?of the material length scale parameters to be zero.Analytical solutions for the static bending,buckling and free vibration problems of a rectangular micro-plate with all edges simply supported are obtained.The results predicted by the present model are compared with those predicted by the degraded models.Influences of the elastic medium on the static bending,buckling,and free vibration are discussed.The results show that the present model can predict prominent size-dependent normalized stiffness,buckling load,and natural frequency with the reduction of structural size,especially when the plate thickness is on the same order of the material length scale parameter.The study may be helpful to guide the design of microplate-based devices resting on elastic medium for a wide range of potential applications.