ABSTRACT
This chapter deals with local/nonlocal and gradient one-dimensional models of elastic media. It considers modeling of a lattice of elastically couple particles. The chapter illustrates the classical continuous approximations of a discrete problem. The chapter describes splashes, envelope continualization and intermediate continuous models. It then outlines the benefits of the use of the Pade approximations for the construction of continuous models. The chapter also outlines the correspondence between functions of discrete arguments and continuous systems as well as relations between the kernels of integro-differential equations of the discrete and continuous systems. It discusses the following topics: dispersive wave propagation, Green function, double-dispersive equations, Toda lattices, discrete kinks, continualization of the Fermi-Pasta-Ulam lattice, anticontinuum limit and two-dimensional lattices. The chapter then addresses the problems regarding molecular dynamics vs. continualization, continualization vs. discretization, as well as opened problems. Discrete lattice-type models are widely used to describe vibrations in crystals, in foams, in cellular structures and bone tissues.
