Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity

TL;DR

This paper proves the local existence of solutions for a one-dimensional strain-limiting viscoelasticity model with nonlinear constitutive relations, using fixed point methods and heat equation theory.

Contribution

It establishes local existence results for a generalized nonlinear Kelvin-Voigt model under strain-limiting assumptions, expanding the mathematical understanding of such viscoelastic systems.

Findings

Proved local existence of strong solutions for the model.

Validated the assumptions for widely used constitutive functions.

Extended the theory to nonlinear strain-limiting viscoelasticity models.

Abstract

In this work we prove local existence of strong solutions to the initial-value problem arising in one-dimensional strain-limiting viscoelasticity, which is based on a nonlinear constitutive relation between the linearized strain, the rate of change of the linearized strain and the stress. The model is a generalization of the nonlinear Kelvin-Voigt viscoelastic solid under the assumption that the strain and the strain rate are small. We define an initial-value problem for the stress variable and then, under the assumption that the nonlinear constitutive function is strictly increasing, we convert the problem to a new form for the sum of the strain and the strain rate. Using the theory of variable coefficient heat equation together with a fixed point argument we prove local existence of solutions. Finally, for several constitutive functions widely used in the literature we show that the assumption on which the proof of existence is based is not violated.