Anisotropic extended Burgers model, its relaxation tensor and properties of the associated Boltzmann viscoelastic system

TL;DR

This paper introduces a novel method for constructing the anisotropic relaxation tensor in the extended Burgers model, enabling analysis of its properties and the exponential decay of solutions in the associated Boltzmann viscoelastic system without restrictive assumptions.

Contribution

A new approach for deriving the relaxation tensor in the extended Burgers model that avoids the commutativity condition and establishes key properties and decay behavior.

Findings

The relaxation tensor is positive and smooth over time.

The relaxation tensor exhibits exponential decay and causality.

Solutions to the Boltzmann viscoelastic system decay exponentially over time.

Abstract

We provide a new method for constructing the anisotropic relaxation tensor and proving its exponential decay property for the extended Burgers model (abbreviated by EBM). The EBM is an important viscoelasticity model in rheology, and used in Earth and planetary sciences. Upon having this tensor, the EBM can be converted to a Boltzmann-type viscoelastic system of equations (abbreviated by BVS). Historically, the relaxation tensor for the EBM is derived by solving the constitutive equation using the Laplace transform. (We refer to this approach by the L-method.) Since inverting the inverse Laplace transform needs a partial fractions expansion, the L-method needs to assume that the EBM elasticity tensors satisfy a commutivity condition. The new method not only avoids this condition but also enables obtaining several important properties of the relaxation tensor, including its positivity, smoothness with respect to the time variable, its exponential decay property together with its derivative, and its causality. Furthermore, we show that the BVS converted from the EBM has the exponential decay property. That is, any solution for its initial boundary value problem with homogeneous boundary data and source decays exponentially as time tends to infinity.