Nonlinear Anisotropic Viscoelasticity

TL;DR

This paper revisits the mathematical foundations of nonlinear viscoelasticity, analyzing deformation geometry, and deriving governing equations for various material symmetries, with semi-analytical studies of creep and relaxation.

Contribution

It clarifies the geometric and physical assumptions behind the multiplicative decomposition and derives comprehensive governing equations for different anisotropic viscoelastic materials.

Findings

Corrected the physical interpretation of the viscous deformation tensor

Derived governing equations using a two-potential approach

Semi-analytical solutions for creep and relaxation in specific deformations

Abstract

In this paper we revisit the mathematical foundations of nonlinear viscoelasticity. We study the underlying geometry of viscoelastic deformations, and in particular, the intermediate configuration. Starting from the multiplicative decomposition of deformation gradient into elastic and viscous parts $\mathbf{F}=\Fe\Fv\,$, we point out that $\Fv$ can be either a material tensor ($\Fe$ is a two-point tensor) or a two-point tensor ($\Fe$ is a spatial tensor). We show that based on physical grounds the second choice is unacceptable. It is assumed that the free energy density is the sum of an equilibrium and a non-equilibrium part. The symmetry transformations and their action on the total, elastic, and viscous deformation gradients are carefully discussed. Following a two-potential approach the governing equations of nonlinear viscoelasticity are derived using the Lagrange-d'Alembert principle. We discuss the constitutive and kinetic equations for compressible and incompressible isotropic, transversely isotropic, orthotropic, and monoclinic viscoelastic solids. We finally semi-analytically study creep and relaxation in three examples of universal deformations.