Bending,Buckling And Vibration Of Functionally Graded Based On The Nonlocal Strain Gradient Theory

In recent decades,nanotechnology has developed rapidly and is known as one of the three cutting-edge technologies in the 21 st century.Nanomaterials,as one of the main objects of nanotechnology research,have attracted lots of research from domestic and foreign scholars.Among them,the mechanical properties of nanomaterials occupy an important position in their applications.With the research around the mechanical properties of nanomaterials,micromechanics theory has also been developed and perfected,such as the nonlocal strain gradient theory.With the advancement of science and technology,the performance of a single nanomaterial can no longer meet the application requirements of some complex or extreme environments.To solve this problem,the concept of functional gradient was introduced,and functionally graded nanomaterials were successfully designed and manufactured.Due to its excellent physical,chemical,mechanical,and electrical properties,functionally graded nanomaterials have been widely used in many fields,such as railway turnout wing rails,nanorobots,and spacecraft structures.Functionally graded nanoplate structures are one of the common components in nanodevice systems.There are many basic mechanical problems in design,optimization,and applications that require systematic research.Based on the application of functionally graded nanoplates in engineering structures such as turnout wing rails,nanorobots,etc.,based on the theory of the nonlocal strain gradients,bending,buckling,vibration and axial motion stability of functionally graded nanoplates are analyzed.The effects of small-scale parameters and gradient indices on their mechanical behavior are described.The main research work and conclusions are as follows:(1)Based on the nonlocal strain gradient theory,the bending problem of functionally graded nanoplates is studied,and the explicit expression of bending deflection under the condition of simply supported boundary is obtained.The results show that the maximum deflection under different non-classical continuum mechanics theories increases with the increase of the gradient index,and the larger the square plate and the thicker the plate,the smaller the maximum deflection.The maximum deflection increases with the increase of the nonlocal parameter and decreases with the increase of the material characeristic scale parameter,which reflects the existence of stiffness softening and hardening phenomena in high-order bending.(2)Based on the bending problem,buckling of functionally graded nanoplates under opposite pressure is studied,and the analytical solution of the critical buckling load is obtained by the Navier method.The results show that the critical buckling load decreases with increasing gradient index,increases with increasing plate thickness,and increases with increasing aspect ratio.The critical buckling load decreases with the increase of the nonlocal parameter,and increases with the increase of the material characeristic scale parameter.It can be seen that there is also a phenomenon of stiffness softening and hardening in the critical buckling load,and there is a coupling effect between the two scale parameters.(3)Based on the above static analysis,the dynamic mechanical properties of functionally gradient nanoplates are studied.In particular,combining the vibration of the deformed body with the overall motion,the free vibration and stability of the functionally gradient nanoplates with axial motion were discussed based on the nonlocal strain gradient theory.Firstly,the governing equations of vibration are deduced,and then they are numerically solved using the complex modal method and Galerkin method.The results show that the natural frequency of the functionally graded nanoplates decreases with increasing axial velocity in the subcritical region,and the nanoplates in the supercritical region will experience divergent instability or flutter instability.The natural frequency and critical speed increase with the increase of the axial tension,increase with the increase of the aspect ratio,and decrease with the increase of the gradient index.The natural frequency and critical velocity decrease with the increase of the nonlocal parameter,and increase with the increase of the material characeristic scale parameter,reflecting the nonlocal softening mechanism and the strain gradient hardening mechanism...