In-Plane Vibrations of Elastic Lattice Plates and Their Continuous Approximations

This paper presents an analytical study on the in-plane vibrations of a rectangular elastic lattice plate. The plane lattice is modelled considering central and angular interactions. The lattice difference equations are shown to coincide with a spatial finite difference scheme of the corresponding continuous plate. The considered lattice converges to a 2D linear isotropic elastic continuum at the asymptotic limit for a sufficiently small lattice spacing. This continuum has a free Poisson's ratio, which must be lower than that foreseen by the rare-constant theory, to preserve the definite positiveness of the associated discrete energy. Exact solutions for the in-plane eigenfrequencies and modes are analytically derived for the discrete plate. The stiffness characterising the lattice interactions at the boundary is corrected to preserve the symmetry properties of the discrete displacement field. Two classes of constraints are considered, i.e., sliding supports at the nodes, one normal and the other parallel to the boundary. For both boundary conditions, a single equation for the eigenfrequency spectrum is derived, with two families of eigenmodes. Such behaviour of the lattice plate is like that of the continuous plate, the eigenfrequency spectrum of which has been given by Rayleigh. The convergence of the spectrum of the lattice plate towards the spectrum of the continuous plate from below is confirmed. Two continuous size-dependent plate models, considering the strain gradient elasticity and non-local elasticity, respectively, are built from the lattice difference equations and are shown to approximate the plane lattice accurately.