Nonstandard continualization of 1D lattice with next-nearest interactions. Low order ODEs and enhanced prediction of the dispersive behavior

Abstract

In this article, different standard and nonstandard continualization techniques are applied to a one-dimensional solid consisting in a chain of masses interacting with nearest and next-nearest neighbors through linear springs. The study focuses on the reliability of the different continua in capturing the dispersive behavior of the discrete, on the order of the continuous governing equation because of its effect on the need for including nonclassical boundary conditions, as well as on the physical inconsistencies that appear for short wavelengths. The Regularization method, used by Bacigalupo and Gambarotta for a lattice with nearest interactions, presents advantages over the others.