| In recent years,piezoelectric materials have been widely used in intelligent devices in aerospace,medical devices,vehicle engineering and other fields,practical devices are developing towards miniaturization and intelligence.At the nanoscale,the strain gradient between piezoelectric material and structure is very large.At this time,the mechanicalelectrical coupling phenomenon caused by the strain gradient,namely the flexoelectric effect,is very obvious,which will lead to significant changes in the mechanical behavior of the structure.The flexoelectric effect has a huge influence on piezoelectric nanodevices,such as nanorobots,nanosensors in automobiles,energy harvesters,etc.In micro/nano electromechanical system or equipment,the flexoelectric rectangular plate structure is one of the most widely used components,and it is of great significance to study its mechanical properties.In this paper,based on the theories of three different shear deformation plates,the free vibration characteristics of axially moving flexoelectric nanoplates and the influence of each parameter on vibration frequency are studied based on the application of the flexoelectric nanoplates in engineering structures such as nanorobots,nanosensors in automobiles and energy harvesters.At present,most of the researches on the flexoelectric effect are focused on the nanobeams structure,while the researches on the two-dimensional nanoplates structure are less,and the nonlocal effect is generally not considered.In fact,at the nanoscale,the internal and external characteristic scales of the structure are of the same order of magnitude,and nonlocal effects must be considered.In addition,in some structures,large shear deformation also needs to be considered to improve the accuracy of the calculation results.Therefore,based on the nonlocal theory and combined with Kirchhoff plate theory,Mindlin plate theory and Reddy plate theory,this paper studies the free vibration of axially moving flexural electric nanoplates.Firstly,according to the piezoelectric linear theory,the coupling effect of strain gradient and electric field is added to obtain the constitutive relation of the nanoplates considering the flexoelectric effect.Then,the motion differential equation and three mechanical boundary conditions were obtained by using Hamilton’s principle.Then,the differential quadrantation method is used to discrete and solve the higher order differential equations of motion and the boundary conditions on arbitrary at other two sides and simply supported at opposite sides,and the first four order dimensionless vibration frequencies of the free vibration of the flexoelectric nanoplates are obtained.Finally,the following results are obtained through the analysis of numerical calculation results:(1)Under the three shear deformation plate theories,the frequencies of axially moving flexoelectric nanoplates are related to nonlocal parameters,plate size,flexoelectric coefficient and axial motion speed.Nonlocal effect has softening effect on the stiffness of nanoplates,while flexoelectric effect has hardening effect on the stiffness of nanoplates.(2)The axial motion will reduce the natural frequencies of the nanoplates,and its frequencies decrease with the increase of the axial velocity,and the more rapidly it decreases.When the velocity reaches a certain value(critical velocity),the natural frequencies disappear,and the nanoplates become unstable,or called critical velocity instability,The flexoelectric effect increases the stiffness and critical velocity of the nanoplates.(3)The flexoelectric effect increases the dimensionless natural frequencies of the nanoplates,and the more constraints imposed on the boundary of the nanoplates,the greater the impact.Compared with the nonlocal effect,the dimensionless natural frequencies of the nanoplates are more sensitive to the flexural effect.(4)The flexoelectric effect has obvious scale effect.The thinner the nanoplates are,the greater the flexoelectric effect is.When the thickness of the nanoplates reaches several hundred nanometers,the flexoelectric effect can be almost ignored.The relationship between the frequencies and the flexoelectric coefficient is approximately linear.(5)Kirchhoff plate theory only applies to thin plates;Mindlin plate theory is recommended for medium thick plate.For thick plate,Reddy plate theoretical calculation results are more accurate.The flexoelectric effect has a certain influence on the calculation results of different shear deformation plate theories.When the flexoelectric effect is considered,the higher order theory should be used as far as possible to get more accurate results.In addition,when the flexoelectric effect is considered,the distinction of the results by the three theories under different slenderness ratio has obvious boundary effect.When the boundary conditions are different,the application scope of the three shear deformation plate theories is also different. |
