Tensor decomposition for modified quasi-linear viscoelastic models: towards a fully nonlinear theory

TL;DR

This paper develops a tensor decomposition approach for Modified Quasi-Linear Viscoelastic models, enabling a fully nonlinear theory that captures strain-dependent relaxation phenomena in anisotropic materials.

Contribution

It introduces a tensor basis decomposition method ensuring consistency with elastic limits and models non-linear strain-dependent relaxation in viscoelastic materials.

Findings

Tensor decomposition ensures symmetric and consistent constitutive equations.

Bases are deformation-dependent in transversely isotropic cases.

The framework captures non-linear strain-dependent relaxation phenomena.

Abstract

We discuss the decomposition of the tensorial relaxation function for isotropic and transversely isotropic Modified Quasi-Linear Viscoelastic models. We show how to formulate the constitutive equation by using a convenient decomposition of the relaxation tensor into scalar components and tensorial bases. We show that the bases must be symmetrically additive, i.e they must sum up to the symmetric fourth-order identity tensor. This is a fundamental property both for isotropic and anisotropic bases that ensures the constitutive equation is consistent with the elastic limit. We provide two robust methods to obtain such bases. Furthermore, we show that, in the transversely isotropic case, the bases are naturally deformation-dependent for deformation modes that induce rotation or stretching of the fibres. Therefore, the Modified Quasi-Linear Viscoelastic framework allows to capture the non-linear phenomenon of strain-dependent relaxation, which has always been a criticised limitation of the original Quasi-Linear Viscoelastic theory. We illustrate this intrinsic non-linear feature, unique to the Modified Quasi-Linear Viscoelastic model, with two examples (uni-axial extension and perpendicular shear).