An analytical solution for the free vibration of FG nanoplates

In the present work, an analytical solution for the free vibration of nanoplates made of functionally graded materials (FGMs) under various boundary conditions is provided. In this context, a new refined plate theory with four variables based on the theory of non-local elasticity including the small-scale influences is adopted. Using the rule of mixture, the material properties of nanoplates are supposed to vary continuously across the thickness direction. Based on Eringen's non-local elasticity theory, the equations of motion of functionally graded (FG) nanoplate are derived using Hamilton's principle, and the obtained equations are solved analytically. Here, the number of unknowns and governing equations of the present model are reduced to four separating the vertical displacement into shear and bending components, and so the number of unknowns has become less than the other alternative theories. The influences of the different parameters such as vibration mode, the aspect ratio, boundary conditions, power-law index, and non-local parameter on the natural frequencies of the FG nanoplate are discussed, in detail. Finally, it is decided that the considered parameters have major influence on the natural frequencies of the FG nanoplates. Furthermore, the proposed solution method not only satisfactorily handled the present problem and yielded successful results but also it supplied ease in the examination of non-local free vibration of FG nanoplates.

Automated summary

In this study, an analytical solution for the free vibration of nanoplates made of functionally graded materials (FGMs) was provided using a new refined plate theory and Eringen's non-local elasticity theory. Various parameters affecting the natural frequencies of nanoplates were discussed, with significant influences observed. The proposed solution method successfully handled the problem and provided convenience in examining the non-local free vibration of functionally graded nanoplates.