Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mellin-Space Prony Representability of Linear Viscoelastic Models

Dimiter Prodanov

TL;DR

This work develops a universal Mellin-space criterion for when linear viscoelastic moduli admit a finite Prony representation. By expressing the constitutive relation in Mellin space, the authors analyze the pole-lattice geometry of Gamma factors and derive two necessary-and-sufficient conditions—Lattice-Alignment and Residue-Compatibility—for existence of a finite Prony spectrum $\tilde{H}(s)\in\mathcal{P}$. The theory classifies classical models (e.g., Maxwell, Standard Linear Solid) as finitely representable, while fractional models (power-law, Cole–Cole, Havriliak–Negami, Gaussian) are transcendental and require an infinite Prony ladder, with Cole–Davidson representing a borderline case failing residue compatibility. A practical test is provided to decide finite representability from a given modulus, and the framework yields a complete pole-lattice taxonomy linking complex-analytic structure to viscoelastic modeling. The results offer a rigorous basis for distinguishing discrete-network representations from continuous-spectrum models and guide exact infinite-ladder constructions where finite representations fail.

Abstract

Linear viscoelasticity is universally described by relaxation functions, which are continuous in relaxation times. Yet experimentally, measurements are necessarily discrete and band-limited, creating a fundamental gap between the continuous mathematical description and empirical observations. While Laplace-domain rational moduli admit finite Prony representations, using Mellin space analysis requires a deeper structural criterion. This work proves that finite Prony series representation holds if and only if the arithmetic pole lattices of Gamma factors in the Mellin transform of the complex modulus align exactly with those of a trial kernel, with residues satisfying decoupled first-order recurrences along aligned sublattices. This maximal criterion classifies classical models (Maxwell, standard linear solid) as finitely representable and fractional models (power-law, Cole--Cole, Zener, Gaussian) as transcendentally representable via infinite Prony ladders, yielding a complete analytical pole-lattice taxonomy of viscoelastic material models.

Mellin-Space Prony Representability of Linear Viscoelastic Models

TL;DR

This work develops a universal Mellin-space criterion for when linear viscoelastic moduli admit a finite Prony representation. By expressing the constitutive relation in Mellin space, the authors analyze the pole-lattice geometry of Gamma factors and derive two necessary-and-sufficient conditions—Lattice-Alignment and Residue-Compatibility—for existence of a finite Prony spectrum . The theory classifies classical models (e.g., Maxwell, Standard Linear Solid) as finitely representable, while fractional models (power-law, Cole–Cole, Havriliak–Negami, Gaussian) are transcendental and require an infinite Prony ladder, with Cole–Davidson representing a borderline case failing residue compatibility. A practical test is provided to decide finite representability from a given modulus, and the framework yields a complete pole-lattice taxonomy linking complex-analytic structure to viscoelastic modeling. The results offer a rigorous basis for distinguishing discrete-network representations from continuous-spectrum models and guide exact infinite-ladder constructions where finite representations fail.

Abstract

Linear viscoelasticity is universally described by relaxation functions, which are continuous in relaxation times. Yet experimentally, measurements are necessarily discrete and band-limited, creating a fundamental gap between the continuous mathematical description and empirical observations. While Laplace-domain rational moduli admit finite Prony representations, using Mellin space analysis requires a deeper structural criterion. This work proves that finite Prony series representation holds if and only if the arithmetic pole lattices of Gamma factors in the Mellin transform of the complex modulus align exactly with those of a trial kernel, with residues satisfying decoupled first-order recurrences along aligned sublattices. This maximal criterion classifies classical models (Maxwell, standard linear solid) as finitely representable and fractional models (power-law, Cole--Cole, Zener, Gaussian) as transcendentally representable via infinite Prony ladders, yielding a complete analytical pole-lattice taxonomy of viscoelastic material models.
Paper Structure (34 sections, 18 theorems, 87 equations, 1 figure, 2 tables)

This paper contains 34 sections, 18 theorems, 87 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and Conventions
  3. Mellin transforms
  4. The Finite Prony Class $\mathcal{P}$
  5. Mellin‑Space Formulation of Linear Viscoelasticity
  6. Constitutive relation
  7. Mellin transform of the spectral kernel
  8. Constitutive equation in Mellin space
  9. Criteria for Finite Prony Representation
  10. Degenerate case
  11. Generic case
  12. Finite Prony exactness as a relaxation-time question
  13. Step 1: Notation and preliminary setup
  14. Step 2: Partial fraction expansion of $K(s)\tilde{H}(-s)$
  15. Step 3: Structure of the left‑hand side $\tilde{G}(s)$
  16. ...and 19 more sections

Key Result

Proposition 1

Let $H(\tau)$ be a positive measure on $(0,\infty)$ with Mellin transform $\tilde{H}(s)$ converging in some vertical strip. Then $H(\tau)$ is a finite Prony series, i.e., if and only if $\tilde{H}(s)$ is a finite exponential sum of the form where $c_k = g_k > 0$ and $\lambda_k = \ln\tau_k \in \mathbb{R}$.

Figures (1)

  • Figure 1: Schematic pole-lattice (“ladder”) structure in the Mellin constitutive identity $\tilde{G}(s)=\tilde{K}(s)\,\tilde{H}(-s)$. Crosses mark the simple poles of the kernel factor $\tilde{K}(s)$, located at the integer lattice $s\in\mathbb Z$. Filled dots mark a representative arithmetic pole ladder of $\tilde{H}(-s)$ lying on the real axis. Red circles highlight an aligned sublattice where poles of $\tilde{H}(-s)$ coincide with kernel poles; along such an aligned ladder the residue matching yields the first-order recurrence of Theorem \ref{['th:main']}.

Theorems & Definitions (40)

  • Definition 1: Finite Prony class $\mathcal{P}$
  • Remark 1
  • Proposition 1: Characterization of finite Prony spectra
  • proof
  • Definition 2: Trial state
  • Theorem 1: Finite-Prony Representation Criterion
  • proof
  • Remark 2
  • Theorem 2: Infinite Prony ladder representation
  • proof
  • ...and 30 more