A note on the Lambert W function: Bernstein and Stieltjes properties for a creep model in Linear Viscoelasticity

TL;DR

This paper explores the application of the Lambert W function in linear viscoelasticity, specifically in modeling creep behavior using spectral functions and leveraging its Bernstein and Stieltjes properties.

Contribution

It introduces a novel use of the Lambert W function to characterize creep models in linear viscoelasticity through spectral functions and symmetry properties.

Findings

Identified the role of the Lambert W function in creep modeling.

Computed spectral functions and relaxation functions.

Demonstrated the importance of conjugate symmetry in analysis.

Abstract

The purpose of this note is to propose an application of the Lambert W function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular we recognize the role of the main branch W_0(t) in a peculiar model of creep with two spectral functions in frequency that completely characterize the creep model. In order to calculate these spectral functions, it turns out that the conjugate symmetry property of the Lambert W function along its branch cut on the negative real axis is essential. We supplement our analysis by computing the corresponding relaxation function and providing the plots of all computed functions.