Static And Dynamic Responses Of Functionally Graded Micro-beams Subjected To Multi-field Coupling Effects

Micro-electro-mechanical-systems(MEMS)engineering is extensively utilized in many fields,such as aerospace,biomedical science,automotive electronics,information and communication,and environmental monitoring etc.MEMS is mainly composed of microcomponents,which are integrated on microcircuits with advanced composite materials by micromachining technology.These core microcomponents can generally be simplified as typical microstructural units,such as micro-beams,micro-plates and micro-shells etc.Both functionally graded material(FGM)and functionally graded piezomagnetic or piezoelectric material(FGPM)are two kinds of advanced composite and intelligent materials and FGM structure can be designed to satisfy particular working environment for its different parts and intelligent control.As we all known that beam structure is a simple and effective model and can be widely used in many engineering fields,such as micro-sensor,micro-actuators,micro-resonator etc.In recent years,with the development of testing technologies in micro or nano structure,many experimental micro-testing results for different material have proven the fact that micro-structures show size effect in its mechanical behaviors.But classical continuum theory is not applicable to interpret the size-dependent effect of those micro-structures.To overcome this drawback,different non-classical continuum theories have been developed to investigate the mechanical behaviors of micro-structures,such as nonlocal elasticity theory,couple stress theory and nonlocal strain gradient theory etc.So the analysis of static and dynamic mechanical behaviors of this kind of FGM micro-structures in multi-field is of great significance for MEMS in practical engineering for its safety and design,function optimization,intelligent control in the future when it face to variously extreme environments such as hygro-thermal environment and magnetic-electric-thermal environment.Furthermore,the mechanical behaviors of composite material micro-structures subjected to multi-field coupling effect is one of the cutting-edge research focuses in micromechanics and nanomechanics,as a results,researchers attach more and more importance to it.Focusing on the background of the size-dependent effect for static and dynamic mechanical behaviors of composite micro-beam in MEMS in this dissertation,based on two different types of displacement fields using a n-th order generalized beamtheory and within the framework of Eringen’s nonlocal elasticity theory and Hamilton system,several mechanical models are developed for static and dynamic responses analyses of functionally graded materials micro-beam subjected to particular multiple physical fields.Some effective and optimized numerical methods are implemented to solve coupling static and dynamic responses of FGM micro-beams under multi-field effect and multiple factors.In order to simplify decoupling,a modified generalized differential quadrature(MGDQ)method is proposed to solve the coupled vibration of FGM micro-beams under the action of multiple physical fields.For the first time,based on the binary coupling relations between static and dynamic behaviors of structures,both coupling vibration and buckling responses are obtained by using MGDQ method and writing the MATLAB computational procedure as a unity.This formulation can simplify decoupling process and avoid the re-decoupling for solving the buckling response of FGM micro-beams.When considering the damping effect,these of eigenfrequencies of viscoelastic FGM/FGPM micro-beams are complex.It is difficult to quickly and accurately identify the effective frequencies by applying the MGDQ method for this time.Therefore,an extended generalized Navier method is adopted to solve the free vibration problem of viscoelastic FGM/FGPM micro-beams under three different classical boundary conditions subjected to particular multiple physical fields.Specifically,the key research points of this dissertation are as follows:(1)Based on the n-th order GBT and Eringen nonlocal elasticity theory,the mechanical model for the analysis of static and dynamic responses of porous FGM micro-beams under hygro-thermal-mechanical loadings is developed.The governing equations and non-local boundary conditions for the micro-structure are derived from Hamilton principle as a unity,in which the unknown basic variables are axial displacement,bending and shear components of transverse displacement.The effect of micro-pore for material processing defects is considered and a two-parameter Winkler-Pasternak elastic foundation model is adopted.Various types of steady hygro-thermal distribution through the thickness of a micro-beam is assumed.The material properties are temperature-dependent and described via modified Voigt mixture power-law rule with porosity.The MGDQ method is used to study the coupling vibration and buckling characteristics of porous FGM micro-beams resting on an elastic foundation in hygro-thermal environment and subjected to initial axial mechanical forces.In addition to vibration and buckling problems,the coupling bending behaviors of porous FGM simply supported micro-beams subjected tohygro-thermal-mehcanical loadings and static transverse force is investigated by Navier method.(2)Consider the structural internal damping and external damping effect from medium,a model for porous viscoelastic FGM micro-beam model resting on three-parameters viscoelastic foundation is presented.Based on the n-th order GBT and Eringen nonlocal theory,the dynamic equations for these of micro-beams in hygro-thermal environment and subjected to initial axial mechanical forces is developed and those of non-local boundary conditions are also derived from Hamilton principle,in which the unknown basic variables are axial displacement,cross-section angle and transverse displacement.Finally,an extended generalized Navier method is utilized to analyze the damping free vibration characteristics of porous viscoelastic FGM micro-beams under three types of classical boundary conditions.(3)Consider a magneto-electro functionally graded materials,using the n-th order GBT,based on Eringen nonlocal elasticity theory in multi-field and Maxwell’s equation,the mechanical model for the analysis of static and dynamic responses of FGPM micro-beams under magnetic-electric-thermal-mechanical loading is developed.The governing equations and non-local boundary conditions for FGPM micro-beams are derived from Hamilton principle system as a unity,in which the unknown basic variables are axial displacement,bending and shear components of transverse displacement,electric potential and magnetic potential.The polarization of external electric field,magnetization of external magnetic field and temperature distribution along the thickness direction of FGPM micro-beams is assumed.A two-parameter elastic foundation model is applied.The MGDQ method is utilized to investigate the coupling vibration and buckling characteristics of FGPM micro-beams resting on an elastic foundation in magnetic-electric-thermal environment and subjected to initial axial mechanical forces.Considering the action of static transverse loads and initial axial mechanical forces,the coupling bending behaviors of FGPM for simply supported micro-beams in magnetic-electric-thermal environment is also studied by Navier method.(4)From the point of energy dissipation,a three-parameter viscoelastic foundation is adopted and the internal damping effect from material structure is considered.A magneto-electro-thermo-mechanical-viscoelastic model of FGPM micro-beams for dynamic analysis is proposed.Based on Eringen nonlocal theory in multi-field and the n-th order GBT,the dynamic governing equations of this model is formulated through Hamilton’s principle,in which the unknown basic displacementvariables are axial displacement,cross-section angle and transverse displacement.Finally,an extended generalized Navier method is used to analyze the damping free vibration behavior of viscoelastic FGPM micro-beams in magneto-electro-thermo environment and subjected to initial axial mechanical forces.The influence of multiple factors on the key dynamic parameters for micro-beams is analyzed in detail.(5)In terms of numerically solving coupling problems,the quantitative simulation and calculation for complex systems is carried out by using the optimized numerical method in this dissertation.By introducing the control parameters of various boundary conditions,the MGDQ method can be used to solve vibration response of FGM or FGPM micro-beam for three different classical boundary conditions by implementing MATLAB programming as a unity.Then,based on the mechanical behaviors on the duality between static buckling and dynamic vibration,the corresponding loop subroutine is written to obtain the static buckling response of FGM or FGPM micro-beam system.The results show that the presented method is effective and accurate,avoiding the quadratic decoupling for buckling response,and greatly improving the calculation efficiency,and optimizing the numerical method again.(6)The duality relationship between static buckling and dynamic vibration of micro-beams is shown by several numerical examples.The influence of size-dependent,multi-parameter and multi-factor on the static and dynamic responses of FGM/FGPM microbeams subjected to multi-field as well as those of mechanisms are analyzed in detail.The research is beneficial for composite material micro-beam structure unit in MEMS on its safety design,function optimization,intelligent control by providing the necessary theoretical basis and reference for application in future.The presented formulations are two feasible and effective methods for the analysis of mechanical behaviors of composite material such as FGM micro-structures subjected to multi-field effect.