Investigation For Microbeams’ Analytical Algorithm And Isogeometric Numerical Approach Based The Reformulated Strain Gradient Elasticity Theory

Micro-nano device/system,a device/system with a size of micron/nanometer that depends on the deformation,motion and dynamic behavior of beams,plates and other structures to achieve specific functions,is considered to be played an important role in many fields such as national production economy and military.However,micro-nano-scale experiments confirmed that when the feature size of the structure shrinks to the order of micrometers or even nanometers,the material will appear forceful inherent scale effects.These physical effects that occur at the microscopic size cause the material to exhibit properties different from those at the macroscopic size.The constitutive relationship of classical continuous mechanics theory does not contain any scale-dependent characteristic parameters,and it cannot explain the scale effect.Therefore,classic continuous mechanics theory cannot be used to design and optimize micro devices/systems.In recent years,the development of non-classical continuous theories that can explain the phenomena of scale effects has attracted the attention of scholars all over the world.Among the many non-classical continuous theories,the reformulated strain gradient elasticity theory only contains only two additional material scale parameters,which reduces the computational difficulty and thevelocity gradient is also introducted,so that the reformulated strain gradient elasticity theory can more effectively reflect the size effects and vibration response.In this paper,bernoulli-Euler and timoshenko beam model applied to static bending、free vibration response、buckling and nolinear postbuckling response of microstructure are formulated based on the reformulated strain gradient elasticity theory derived from Form I of Mindlin’s strain gradient elasticity theory.The bernoulli-Euler and timoshenko beam model includes strain gradient effect、couple stress effect and veiocity gradient effect simultaneously,and both of the beam model can be futher degenerated to the modified couple stess or classical continuous mechanics theory beam model by restrained strain gradient effect or couple stress effect.Based on this basis,the paper has mainly completed the following work content.(1)Based on reformulated strain gradient elasticity theory,the bernoulli-Euler beam model applied to static bendng、free vibration response and buckling response of microstructure is formulated.The governing equation、boundary conditions and corresponding analytic solutions of static bending 、 free vibration and buckling analysis for simple support beam model is obtained by utilized the variation method、Hamilton’s principle and the principle of minimum energy,and the numerical solution of static bending、free vibration and buckling analysis is acquired by utilized the isogeometric numerical method.Through the analysis of typical beam examples,the performance of developed method and the accuracy of isogeometric analysis are getting certification,and the effects of beam’s length、beam’s thickness、material scale parameters and boundary conditions on the bending deflection、natural frequency and critical buckling load valus predicted by beam models are discussed in detail.(2)Based on reformulated strain gradient elasticity theory,the Timoshenko beam model applied to static bendng、free vibration response and buckling response of microstructure is formulated.The governing equation、boundary conditions and corresponding analytic solutions of static bending 、 free vibration and buckling analysis for simple support beam model is obtained by utilized the variation method、Hamilton’s principle and the principle of minimum energy,and the numerical solution of static bendng、free vibration and buckling analysis is acquired by utilized the isogeometric numerical method.Through the analysis of typical beam examples,the performance of developed method and the applicability for varied boundary contidions of isogeometric analysis are getting certification,and the effects of beam’s length、beam’s thickness、material scale parameters and boundary conditions on the critical buckling load valus predicted by beam models are discussed.(3)Based on reformulated strain gradient elasticity theory,the Timoshenko beam and bernoulli-Euler beam model applied to nolinear postbuckling response of microstructure are formulated.The governing equation、boundary conditions and corresponding analytic solutions of nolinear postbucklng response for simple support beam model is obtained by utilized the variation method and principle of minimum energy.Through the analysis of typical beam examples,the performance of developed method is getting certification,and the effects of beam’s length、beam’s thickness and material scale parameters on the postbuckling load valus and postbuckling amplitude predicted by beam models are discussed.