Nonlinear relations of viscous stress and strain rate in nonlinear Viscoelasticity

TL;DR

This paper studies a Kelvin-Voigt model for nonlinear viscoelastic materials, establishing the existence of weak solutions with frame indifference and analyzing their long-term behavior using gradient flow theory.

Contribution

It introduces a novel existence proof for weak solutions of nonlinear viscoelastic models with nonquadratic viscous stress tensors, employing a frame-indifferent discretization scheme.

Findings

Existence of weak solutions for the model.

Inclusion of nonquadratic polynomial densities for viscous stress.

Analysis of long-time behavior under small external forces.

Abstract

We consider a Kelvin-Voigt model for viscoelastic second-grade materials, where the elastic and the viscous stress tensor both satisfy frame indifference. Using a rigidity estimate by [Ciarlet-Mardare '15], existence of weak solutions is shown by means of a frame-indifferent time-discretization scheme. Further, the result includes viscous stress tensors which can be calculated by nonquadratic polynomial densities. Afterwards, we investigate the long-time behavior of solutions in the case of small external loading and initial data. Our main tool is the abstract theory of metric gradient flows.