Structural Mechanical Analysis Of Micobeam Based On Two-Variable Deformation Theory

Modern science and technology continue to develop on a macro and large scale,and at the same time,they continue to progress on a micro scale.Nanotechnology and nanoscience help to introduce structures and devices with good accuracy at the nanoscale.Nanobeams are a core structure widely used in many systems,such as nanosensor and actuator applications for sensing and energy harvesting.At the macro scale,the effect of the scale effect on the structural performance is negligible,but at the micro scale,the scale effect can have a large impact on the structural performance,and the classical theory cannot accurately predict and solve micro-scale beams.It is therefore necessary to develop theoretical models that can describe and explain the performance of microscale structures.In this paper,a new uniform microbeam model is established.Based on the Hamilton principle,the theory of couple stress theory and surface energy effects are considered,and the governing equations and boundary conditions of the beam are given.The new model includes material length scale parameters that take into account the microstructure effects of the beam body and three surface elastic constants that describe the mechanical behavior of the beam surface layer.Then the differential quadrature element method(DQEM)was used to obtain the numerical solutions of bending,buckling and free vibration under different boundary conditions.Analytical solution of simply supported beams at both ends is solved by Naiver method.The two are compared to verify the accuracy of the differential quadrature method.Then use the differential quadrature element method to analyze the effect of the dual stress theory and the surface energy effect.When ignoring the effects of the dual stress theory and surface energy,the model can be degraded to a model that satisfies the beam on a macro scale.Then,based on the linear buckling analysis,the von Carmen nonlinearity assumption was introduced to simulate the post-buckling stage of the model,and a two-variable nonlinear post-buckling beam model was established.Due to the small effect of surface effects,the nonlinear post-buckling model only considers the effect of the dual stress theory.The governing equations and boundary conditions of the model are also obtained based on the Hamilton principle and the variational method.A combination of Newton's method and incremental deflection method Solve non-linear equations and compile corresponding MATLAB programs.By analyzing the load-displacement curve and load-deflection curve,the influence of the thickness and length of the beam on the post-buckling path is summarized.Finally,the importance of boundary conditions to the bearing capacity of the structure is demonstrated.Finally,the mechanical response of bivariate functionally graded beams is studied.The model takes into account bidirectional functional gradient changes and also considers the effects of couple stresses.The effect of the functional gradient index on bending,buckling and free vibration is studied in detail.The bending stiffness of the beam changes with the change of the gradient index.Due to the existence of the gradient index,the functional gradient beam also has a lower bending stress distribution compared with a uniform beam the same.These conclusions are very useful for the safety design of linear or nonlinear beams.The effects of beam thickness,length,scale effect,and boundary conditions on static bending,first-order buckling load,and first-order natural frequency are studied.When the functional gradient is not considered,the model can be transformed into a corresponding uniform beam model.