A Representation Theorem for Fractional Viscoelastic Networks
Abstract
We prove a representation theorem for finite series-parallel viscoelastic networks composed of springs and spring-pots of a common order 0<α<1. The associated complex shear modulus, expressed in an appropriately chosen auxiliary variable, is rational and admits a finite partial-fraction representation with simple poles on the negative real axis and negative residues. Its reciprocal admits an analogous representation with positive residues and a possible additional pole at the origin. Inverting the corresponding Laplace transforms gives finite fractional Prony representations of the relaxation modulus and creep compliance with non-negative amplitudes. These results provide a rigorous proof of a corrected version of the Mainardi-Spada conjecture for this network class. A six-element network illustrates the construction.
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BibTeX
@misc{giusti2026representation,
title = {{A Representation Theorem for Fractional Viscoelastic Networks}},
author = {Andrea Giusti and Ivano Colombaro and Andrea Mentrelli and Francesco Mainardi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00750},
eprint = {2611.00750},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00750},
}
RIS
TY - UNPB
TI - A Representation Theorem for Fractional Viscoelastic Networks
AU - Andrea Giusti
AU - Ivano Colombaro
AU - Andrea Mentrelli
AU - Francesco Mainardi
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00750
UR - https://arxiv.org/abs/2611.00750
AB - We prove a representation theorem for finite series-parallel viscoelastic networks composed of springs and spring-pots of a common order 0<α<1. The associated complex shear modulus, expressed in an appropriately chosen auxiliary variable, is rational and admits a finite partial-fraction representation with simple poles on the negative real axis and negative residues. Its reciprocal admits an analogous representation with positive residues and a possible additional pole at the origin. Inverting the corresponding Laplace transforms gives finite fractional Prony representations of the relaxation modulus and creep compliance with non-negative amplitudes. These results provide a rigorous proof of a corrected version of the Mainardi-Spada conjecture for this network class. A six-element network illustrates the construction.
ER -