A Comparative View of Becker, Lomnitz, and Lambert Linear Viscoelastic Models

Juan Luis González-Santander () and Francesco Mainardi
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Juan Luis González-Santander: Department de Mathematics, University of Oviedo, C Leopoldo Calvo Sotelo 18, 33007 Oviedo, Spain
Francesco Mainardi: Department of Physics and Astronomy, University of Bologna and INFN, Via Irnerio 46, I-40126 Bologna, Italy

Mathematics, 2024, vol. 12, issue 21, 1-14

Abstract: We compare the classical viscoelastic models due to Becker and Lomnitz with respect to a recent viscoelastic model based on the Lambert W function. We take advantage of this comparison to derive new analytical expressions for the relaxation spectrum in the Becker and Lomnitz models, as well as novel integral representations for the retardation and relaxation spectra in the Lambert model. In order to derive these analytical expressions, we have used the analytical properties of the exponential integral and the Lambert W function, as well as the Titchmarsh’s inversion formula of the Stieltjes transform. In addition, we prove some interesting inequalities by comparing the different models considered, as well as the non-negativity of the retardation and relaxation spectral functions. This means that the complete monotonicity of the rate of creep and the relaxation functions is satisfied, as required by the classical theory of linear viscoelasticity.

Keywords: laplace transform; stieltjes transform; exponential integral; Lambert W function; linear viscoelasticity models (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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