Continuous and discrete models of bounded one-dimensional media in the theory of viscoelasticity

Abstract

The relation between the discrete and continuous models of a one-dimensional viscoelastic medium is discussed. Beginning with the discrete model, for the linear case it is shown how to construct a series of partial differential equations which might be thought of as intermediate between the differential-difference equation for a chain of discrete masses and the equation for a continuous medium. For these intermediate equations (one of which will be, in particular, the equation of a vibrating continuous medium) we give the conditions under which one is justified in replacing the initial continuous model by a discrete chain. For the non-linear case, the analogue of the Fermi-Pasta-Ulam (FPU) problem with strongly non-linear constraints is considered. This version of the FPU problem can be applied to a chain with non-linear viscoelasticity.