On aspects of continualization of 2D lattices antiplane dynamical problem

Abstract

This paper is devoted to the continualization of a 2D lattice. PDEs describing the standard long-wavelength continuous approximation, π-vibrational modes in the x- and y-directions, and π – π-vibrational mode are derived. These limiting cases are used when constructing an asymptotically equivalent continuous approximation described by the model with modified inertia. The principle of asymptotic equivalence is reduced to the requirement that the solutions of the dispersion equations of the original lattice model and the improved continuous model coincide when the limiting cases are considered. It is worth noting that the second-order PDE with respect to the spatial variables is obtained. In this regard, its use in a finite region does not require the formulation of additional boundary conditions. The improved continuous approximation provides a sufficiently accurate description of the frequency spectrum throughout the entire first Brillouin zone. The kinetic and elastic potential energy densities of this model are positive definite. The positive definiteness of the elastic potential energy density allows for the application of this model in static problems. To assess the accuracy of the obtained continuous approximation and the range of its applicability, 3D visualizations comparing various models, as well as 2D sectional views of these 3D plots are presented. Moreover, additional criteria for assessing the accuracy of the continuous approximation (the coefficient of determination, residual variance, relative mean squared and absolute errors) are provided.