Dynamic quasi-continuum model for plate-type nano-materials and analysis of fundamental frequency
C. X. Xia, W. L. Xu and G. H. Nie
Applied Mathematics and Mechanics, 42, 85-94 (2021).
ABSTRACT
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.