A local existence result for system of viscoelasticity with physical viscosity

TL;DR

This paper establishes the local existence of classical solutions for a general system of isothermal viscoelasticity equations with viscous stress tensors satisfying physical and mathematical conditions.

Contribution

It introduces a general form of viscous stress tensor and proves local existence under Korn-type conditions, extending previous results to more general viscoelastic models.

Findings

Proved local in time existence of classical solutions.

Established conditions on viscous stress tensors compatible with physical principles.

Provided examples of linear and nonlinear viscous tensors satisfying the conditions.

Abstract

We prove the local in time existence of the classical solutions to the system of equations of isothermal viscoelasticity with clamped boundary conditions. We deal with a general form of viscous stress tensor $\mathcal{Z}(F,\dot F)$, assuming a Korn-type condition on its derivative $D_{\dot F}\mathcal{Z}(F, \dot F)$. This condition is compatible with the balance of angular momentum, frame invariance and the Claussius-Duhem inequality. We give examples of linear and nonlinear (in $\dot F$) tensors $\mathcal{Z}$ satisfying these required conditions.