A Modified First-Order Shear Deformation Theory With Size Dependent Effect And Its Application In Micro-Nano Structures

Micro-nano plates are one of the most basic structural forms of micro-nano structures,and keeping the micro-nano structures in a stable state is always the premise to realize their functions.A large number of microscopic experiments have shown that the mechanical properties of micro-nano plates are quite different from those predicted by theories for macroscopic plates,so it is of great significance to establish a theoretical model for micro-nano plates.Based on continuum mechanics,theory for plates and shells,mechanics of composite materials,and Koiter’s couple stress theory,this paper studies the buckling and free vibration for isotropic micro-nano plates and for composite laminated micro-nano plates.The main contents of this thesis are as follows:A modified first-order shear deformation theory with size dependent effect is established.On the base of Koiter’s couple stress theory,the curvature-rotation relations are introduced to characterize the relative rotation between the internal atoms of the shear deformable laminated micro-nano plates.Three length scale parameters corresponding to three principal directions are adopted to describe the couple stress-curvature relationship,and the symmetric stress and the asymmetric stress expressed by displacement are deduced by employing continuum mechanics and by considering the equilibrium condition of an element.According to mechanics of composite materials,the constitutive equations and the differential equations of equilibrium for laminated micro-nano plate are established in which a higher-order constitutive matrix and higher-order stiffness matrices are named to reflect the size-dependent effect.Based on the established modified first-order shear deformation theory,the buckling problem of micro-nano plate is analyzed.According to the first order deformation theory,the governing equations for buckling of the micro-nano plates are derived in terms of the deflection and the slope functions.Using the variables separation method,a theoretical formula to calculate the critical buckling load for a simply supported laminated micro-nano plate subjected to the in-plane compression is derived.MATLAB programs are compiled to discuss the influence of scale effect,geometric dimensions,material parameters,stacking sequence on the buckling load of laminated micro-nano plates.The theoretical solutions are compared with the finite element results to verify the correctness of the established theoretical analysis.The analytical model for aminated micro-nano plates is degraded as one for isotropic micro-nano plates and the characteristic equation of buckling for isotropic micro-nano plate is established.The buckling problems of micro-nano plates with SSSS boundary conditions(simply supported at four edges)and SCSC boundary conditions(simply supported at two opposite edges and clamped at the other two edges)are analyzed,and the influence of boundary conditions on the critical buckling load is discussed.The influence of scale effect on the free vibration response for micro-nano plates is studied.Considering the influence of inertial force on the vibration for micro-nano plate,the equations of motion for free vibration of laminated micro-nano plates are established and are expressed in terms of displacements.The characteristic equation to determine the natural frequency for laminated plate with simply supported boundary conditions is deduced.MATLAB programs are compiled to discuss the influence of scale effect,the geometric dimensions and the material parameters on the natural frequency for the laminated micro-nano plates.On this basis,the free vibration of isotropic micro-nano plates of two scenarios of boundary conditions –SSSS and SCSC--is analyzed,and influence of boundary conditions on the natural frequency is discussed.It is found that the scale effect of micro-nano plates is very obvious.When the dimension of the plate decreases to that of microns,the critical buckling load and the natural frequency of the plate are much larger than those predicted by the macro continuum theory,and the scale effect on the mechanical performance of the isotropic micro-nano plate is stronger than that of the laminated micro-nano plate.