# On the passage from nonlinear to linearized viscoelasticity

## TL;DR

This paper develops a framework connecting nonlinear viscoelastic models to their linearized counterparts, demonstrating convergence of solutions and approximations using gradient flow and Gamma-convergence techniques.

## Contribution

It introduces a nonlinear quasistatic viscoelastic model at finite strains and proves the rigorous passage to linearized viscoelasticity, including convergence of solutions and discretizations.

## Key findings

- Nonlinear model formulated for nonsimple viscoelastic materials.
- Solutions of nonlinear model converge to linearized solutions.
- Time-discrete approximations are consistent with the linear limit.

## Abstract

We formulate a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin's-Voigt's rheology where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. We identify weak solutions in the nonlinear framework as limits of time-incremental problems for vanishing time increment. Moreover, we show that linearization around the identity leads to the standard system for linearized viscoelasticity and that solutions of the nonlinear system converge in a suitable sense to solutions of the linear one. The same property holds for time-discrete approximations and we provide a corresponding commutativity result. Our main tools are the theory of gradient flows in metric spaces and Gamma-convergence.