A modified formulation of quasi-linear viscoelasticity for transversely isotropic materials under finite deformation

TL;DR

This paper develops a modified quasi-linear viscoelasticity model for transversely isotropic materials under finite deformation, incorporating distinct relaxation responses and demonstrating unique effects like the Poynting effect due to anisotropic relaxation.

Contribution

It introduces a novel integral formulation of nonlinear viscoelasticity for TI materials, incorporating physically distinct relaxation responses and analyzing their effects under various deformations.

Findings

The model is consistent with linear viscoelasticity at small strains.

The Poynting effect appears in transversely isotropic, neo-Hookean, modified QLV materials.

Distinct relaxation responses influence the material's behavior under shear.

Abstract

The theory of quasi-linear viscoelasticity (QLV) is modified and developed for transversely isotropic (TI) materials under finite deformation. For the first time, distinct relaxation responses are incorporated into an integral formulation of nonlinear viscoelasticity, according to the physical mode of deformation. The theory is consistent with linear viscoelasticity in the small strain limit and makes use of relaxation functions that can be determined from small-strain experiments, given the time/deformation separability assumption. After considering the general constitutive form applicable to compressible materials, attention is restricted to incompressible media. This enables a compact form for the constitutive relation to be derived, which is used to illustrate the behaviour of the model under three key deformations: uniaxial extension, transverse shear and longitudinal shear. Finally, it is demonstrated that the Poynting effect is present in transversely isotropic, neo-Hookean, modified QLV materials under transverse shear, in contrast to neo-Hookean elastic materials subjected to the same deformation. Its presence is explained by the anisotropic relaxation response of the medium.