Highly accurate closed-form solutions for free vibration and eigenbuckling of rectangular nanoplates

This work presents the highly accurate closed-form solutions for free vibration and eigenbuckling of isotropic rectangular nanoplates with arbitrary homogeneous boundary conditions based on Eringen's nonlocal theory and classical thin plate theory. The iterative separation-of-variable (iSOV) method based on the Rayleigh quotient, which is the most accurate closed-form solution method among all separation-of-variable methods, is used to obtain the highly accurate solutions, including the exact well-known Navier and Levy types of solutions. The highly accurate closed-form solutions for free vibration of rectangular nanoplates and for eigenbuckling of different scale rectangular plates with arbitrary homogeneous boundary conditions are achieved for the first time, and all solutions are presented in excellent explicit forms. The present solutions coincide well with analytical and numerical solutions in literature, verifying the accuracy of the present method. The influences of nonlocal parameters, boundary conditions and lengths of nanoplates on frequencies and critical buckling loads are studied, and nonlocal effects are explained in physical sense. The present solutions can be taken as the benchmarks for the validation of numerical methods, a guide in parametric design of structure, and the basis of constructing new numerical methods.