Structural Mechanics Analysis Of The Micro-Size Curved Beam Based On The Modified Couple Stress Theory

With the development of MEMS and NEMS,micro-electromechanical devices with various functions have been produced and used in important areas such as biomedicine,aerospace and so on.As the main components of micro-electromechanical devices,the mechanical properties of micro-and nano-plates,films and beams have a crucial impact on product design and manufacturing.A large number of experimental phenomena show that the mechanical properties of components at the micro-scale are very different from those at the macro-scale.Therefore,it is very important to propose an effective theoretical model to analyze the size effect of microstructures.In this paper,a functionally graded material curved beam model is developed on the basis of modified couple stress theory and surface elasticity theory.The analytic solutions to the bending,buckling and vibration of curved microbeam are obtained.The Poisson effect is also considered.The curved beam is described under polar coordinates.The contravariant and covariant basis in polar coordinates are introduced.Three beam theories are studied: Euler beam theory(classical deformed beam theory),Timoshenko beam theory(first-order deformed beam theory),and parabolic shear beam theory(third-order deformed beam theory).A variational formulation based on Hamilton's principle is employed,the governing equations and boundary conditions of the beam can be obtained simultaneously.Modified couple stress theory includes a material scale parameter to describe the size effect.Surface elasticity theory includes two material parameters of surface elastic modulus and residual surface tension.The Young-Laplace equation is used to describe the "jump phenomenon" of free surface stress above and below the beam.Euler beams can be solved analytically.The basic concept and detailed derivation of the differential quadrature method are introduced,and the solution of Timoshenko beam and parabolic shear beam is obtained by using DQM.The effects of surface tension,material scale parameter,beam size and functional gradient parameters on the mechanical properties of the beam are analyzed through the numerical results.The study found that curved beam can degenerate into straight beams when the radius is larger,and can degenerate into the classic beam theory with the increase of beam size.At the micro-scale,the couple stress has a significant impact on the mechanical properties of the beam,which shows that the beam's deflection decreases,the critical buckling load and the natural frequency of free vibration increase,and the couple stress theory is no longer applicable at the nanoscale,the surface elasticity theory is the main cause of the scale effect,and the positive residual surface tension will increase the beam stiffness,while the negative residual surface tension reduces the stiffness of the beam.