Traveling waves in one-dimensional non-linear models of strain-limiting viscoelasticity

• Traveling waves in one-dimensional strain-limiting non-linear viscoelasticity are investigated.
• Six different non-linear models suggested in the literature are considered.
• Traveling wave solutions are obtained explicitly, implicitly or numerically.
• The traveling wave profiles for the stress become smoother as the viscosity increases.
• The traveling wave profiles for the strain are strongly distorted due to the non-linearity.

In this paper we investigate traveling wave solutions of a non-linear 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 non-linear 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 non-linear constitutive relation.