A class of linear viscoelastic models based on Bessel functions

TL;DR

This paper introduces a new class of linear viscoelastic models using Bessel functions, characterized by infinite spectra of retardation and relaxation times, bridging fractional and standard Maxwell behaviors over different time scales.

Contribution

It proposes a novel class of viscoelastic models based on Bessel functions, providing explicit time-domain expressions and spectral properties.

Findings

Models exhibit fractional Maxwell behavior at short times.

Models transition to standard Maxwell behavior at long times.

Creep and relaxation functions are expressed as Dirichlet series.

Abstract

In this paper we investigate a general class of linear viscoelastic models whose creep and relaxation memory functions are expressed in Laplace domain by suitable ratios of modified Bessel functions of contiguous order. In time domain these functions are shown to be expressed by Dirichlet series (that is infinite Prony series). It follows that the corresponding creep compliance and relaxation modulus turn out to be characterized by infinite discrete spectra of retardation and relaxation time respectively. As a matter of fact, we get a class of viscoelastic models depending on a real parameter $\nu > -1$. Such models exhibit rheological properties akin to those of a fractional Maxwell model (of order $1/2$) for short times and of a standard Maxwell model for long times.