C. X. Xia, W. L. Xu and G. H. Nie
Applied Mathematics and Mechanics, 42, 85–94 (2021).
A dynamic quasi-continuum model is presented to analyze free vibration of plate-type cubic crystal nano-materials. According to the Hamilton principle, fundamental governing equations in terms of displacement components and angles of rotations are given. As an application of the model, the cylindrical bending deformation of the structure fixed at two ends is analyzed, and a theoretical formula evaluating the fundamental frequency is obtained by using Galerkin's method. Meanwhile, the solution for the classical continuous plate model is also derived, and the size-dependent elastic modulus and Poisson's ratio are taken in computation. The frequencies corresponding to different atomic layers are numerically presented for the plate-type NaCl nano-materials. Furthermore, a molecular dynamics (MD) simulation is conducted with the code LAMMPS. The comparison shows that the present quasi-continuum model is valid, and it may be used as an alternative model, which reflects scale effects in analyzing dynamic behaviors of such plate-type nano-materials.