Vibration And Wave Propagation Analysis Of Axially Moving Functionally Graded Nanobeams

In recent years,nanomaterials and nanotechnologies have been widely used in biological monitoring and microelectronics,such as micro-drug transporters,nanoblood detection robots.As we know,nanostructures have special small-scale effect when compared with the traditional macro-scale mechanical structures.The nonlocal elasticity theory,Proposed by Eringen and his assistants,can well capture and explain the nonlocal scale effect of nanostructures.Mechanical behavior of axially moving homogeneous nanostructures are widely researched using the nonlocal elasticity theory,while the researches on the mechanical behavior of axially moving inhoruogeneous nanostructures are lacking.Therefore,it is very important to investigate the mechanical behavior of axially moving inhomogeneous nanostructures.In this paper,the vibration and wave propagation behaviors of axially moving functionally graded nanobeams are investigated using the nonlocal elasticity theory and three different deformation theories.It is assumed that the material properties of functionally graded nanobeams vary along the thickness direction according to power index law.By complex mode method and differential quadrature method,effects of gradient index,nonlocal parameter and axial velocity on vibration and wave propagation are investigated in detail.Firstly,the free vibration and wave propagation of functionally graded Euler nanobeams with axial motion are investigated.The neutral plane of functionally graded materials is introduced to express the axial displacement of geometry surface.Subsequently,the governing equation of this model are derived using Hamilton's principle.The complex mode method is utilized to solve the governing equation and the exact solutions of frequency and phase velocity for wave propagation are obtained accordingly.In order to solve the derived transcendental equations of vibration,the Newton iteration method is applied.Effects of gradient index,axial velocity and nonlocal scale parameter on natural frequencies are analyzed.Also,the relationship between wave propagation frequency,phase velocity and wave number is revealed.Secondly,on the basis of Euler theory,considering the shear deformation of the cross-section,the governing equations of the functionally graded Timoshenko nanobeams with axial motion are derived.The analytical solutions of frequency and phase velocity for wave propagation are obtained by complex mode method.The governing equations and two boundary conditions are discretized by differential quadrature method,and the first three order vibration frequencies are solved.The accLracy of the results and differential quadrature method are verified by numerical examples,and the effects of material parameters on the first three order vibration frequencies,wave frequencies and phase velocities are discussed.Thirdly,for the thick nanobeams model,considering the axial displacement of the geometrically middle plane,the governing equations of axially moving functionally graded Reddy nanobeams are derived based on the Reddy shear deformation theory.The analytical solutions of the wave frequency and phase velocity are obtained,and the first three-order vibration frequencies of the model are solved using the differential quadrature method.Moreover,the influence of axial velocity,nonlocal parameter and gradient index on the vibration characteristics and wave propagation characteristics of the model are discussed.Finally,the influence of different deformation theories on vibration characteristics is further analyzed.In detail,the relationship between vibration frequencies and slendermess ratio of nanobeams is discussed to quantitatively analyze the application scope of deformation theory.