Dynamics Characteristics Research Of Nb₄AlC₃ Nano-plate Based On The Nonlocal Theory

With the widely application of nano-electromechanical system,the dynamics of nano-scale components,especially nano-plate,has become an important research topic.However,in the nano-meter scale,the classical continuum mechanics theory is no longer available because the influence of small scale effects should be considered.In this paper,the free vibration of nano-plate was studied based on a new continuum mechanics theory,non-local theory.For the nano-scale components,the choice of material is particularly important,therefore,the anisotropic Nb4AlC3 nano-plate which has excellent mechanical properties was selected as the research object in this paper.In this paper,according to the basic theory of dynamics,the differential equation of motion about the isotropic classical plate was established.And then the differential equation of motion about the isotropic classical plate was translated into the differential equation of motion about the anisotropic classical plate through the physical equations of anisotropic plate.Finally the differential equation of motion about the anisotropic classical plate was translated into the differential equation of motion about the anisotropic nano-plate according to the mathematical expressions of nonlocal theory.Secondly,based on the density functional theory,the crystal structure of Nb₄AlC₃ was built by using Materials Studio software.The crystal structure of Nb4AlC3 was optimized,and the density of states and the elastic constants were also calculated.The results of density of states showed that the Nb4AlC3 is a metal and has covalent bonds.And the results of elastic constants revealed the Nb4AlC3 crystal is mechanically stable.Finally,the differential equation of free vibration about Nb₄AlC₃ nano-plate was derived by substituting the elastic constants of Nb4AlC3 into the differential equation of motion about the anisotropic nano-plate.By using the Navier type solution and the Levy type method,the differential equation of motion about simply supported nano-plate and two opposite edge simply supported nano-plate were solved,respectively.The nature frequency of free vibration about nano-plate was obtained,and the relationships between the nature frequency ratio and nonlocal parameter,mode number,aspect ratio of nano-plate and the length of nano-plate were discussed.The results showed that the nature frequency ratio increases with the increase of nonlocal parameter,mode number,and aspect ratio of nano-plate,the nature frequency ratio increases with the decrease of the length of nano-plate.It revealed that the influence of small scale effect increases with the increase of nonlocal parameter,mode number,and aspect ratio of nano-plate,the influence of small scale effect increases with the decrease of the length of nano-plate.