Traveling waves in one-dimensional nonlinear models of strain-limiting viscoelasticity

TL;DR

This paper studies traveling wave solutions in one-dimensional nonlinear strain-limiting viscoelastic models, establishing conditions for their existence and providing explicit, implicit, or numerical solutions for various nonlinear relations.

Contribution

It introduces a framework for analyzing traveling waves in nonlinear strain-limiting viscoelasticity and derives conditions for their existence in these models.

Findings

Conditions for existence of traveling wave solutions are established.

Traveling wave solutions are explicitly, implicitly, or numerically obtained.

The study enhances understanding of wave behavior in nonlinear viscoelastic materials.

Abstract

In this article we investigate traveling wave solutions of a nonlinear differential equation describing the behaviour of one-dimensional viscoelastic medium with implicit constitutive relations. We focus on a subclass of such models known as the strain-limiting models introduced by Rajagopal. To describe the response of viscoelastic solids we assume a nonlinear relationship among the linearized strain, the strain rate and the Cauchy stress. We then concentrate on traveling wave solutions that correspond to the heteroclinic connections between the two constant states. We establish conditions for the existence of such solutions, and find those solutions, explicitly, implicitly or numerically, for various forms of the nonlinear constitutive relation.