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Pulse waves in the viscoelastic Kelvin-Voigt model: a revisited approach

Juan Luis Gonzalez-Santander, Francesco Mainardi, Andrea Mentrelli

TL;DR

The paper revisits transient uniaxial wave propagation in a Kelvin-Voigt viscoelastic medium occupying a semi-infinite domain, focusing on boundary pulse excitations. It develops integral-form solutions for step- and delta-pulse inputs by introducing a convolution kernel in dimensionless coordinates that removes the need for direct inverse Laplace transforms. The authors derive explicit asymptotic formulas for short and long times and distances, including $erfc$-type front behavior and hypergeometric-type corrections, enabling fast, accurate evaluation of the response. Numerical validation shows strong agreement with traditional inverse-Laplace solutions and demonstrates substantial computational efficiency improvements over previous integral representations.

Abstract

We calculate the mechanical response $r(x,t$) of an initially quiescent semi-infinite homogeneous medium to a pulse applied at the origin, and this is achieved within the framework of the Kelvin-Voigt model. Although this problem has been extensively studied in the literature because of its wide range of applications -- particularly in seismology -- here, we present a solution in a novel integral form. This integral solution avoids the numerical computation of the solution in terms of the inverse Laplace transform; that is, numerical integration in the complex plane. In particular, we derive integral form expressions for both delta-pulse and step-pulse excitations which are simpler and more computationally efficient than those previously reported in the literature. Furthermore, the obtained expressions allow us to obtain simple asymptotic formulas for $r(x,t$ as $x,t \to 0,\infty$ for both step- and delta-type pulses.

Pulse waves in the viscoelastic Kelvin-Voigt model: a revisited approach

TL;DR

The paper revisits transient uniaxial wave propagation in a Kelvin-Voigt viscoelastic medium occupying a semi-infinite domain, focusing on boundary pulse excitations. It develops integral-form solutions for step- and delta-pulse inputs by introducing a convolution kernel in dimensionless coordinates that removes the need for direct inverse Laplace transforms. The authors derive explicit asymptotic formulas for short and long times and distances, including -type front behavior and hypergeometric-type corrections, enabling fast, accurate evaluation of the response. Numerical validation shows strong agreement with traditional inverse-Laplace solutions and demonstrates substantial computational efficiency improvements over previous integral representations.

Abstract

We calculate the mechanical response ) of an initially quiescent semi-infinite homogeneous medium to a pulse applied at the origin, and this is achieved within the framework of the Kelvin-Voigt model. Although this problem has been extensively studied in the literature because of its wide range of applications -- particularly in seismology -- here, we present a solution in a novel integral form. This integral solution avoids the numerical computation of the solution in terms of the inverse Laplace transform; that is, numerical integration in the complex plane. In particular, we derive integral form expressions for both delta-pulse and step-pulse excitations which are simpler and more computationally efficient than those previously reported in the literature. Furthermore, the obtained expressions allow us to obtain simple asymptotic formulas for as for both step- and delta-type pulses.
Paper Structure (9 sections, 106 equations, 4 figures)

This paper contains 9 sections, 106 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Kelvin--Voigt Model
  3. General Solution
  4. Step Pulse Solution
  5. Asymptotic Solution for $\tau \rightarrow 0$ or $\xi \rightarrow \infty$
  6. Asymptotic Solution for $\tau \rightarrow \infty$
  7. Asymptotic Solution for $\xi \rightarrow 0$
  8. Delta Pulse Solution
  9. Conclusions

Figures (4)

  • Figure 1: Graph of $r\left( \xi _{0},\tau \right)$ with $\xi _{0}=0.5$ for the step pulse solutions and the asymptotic approximations in the Kelvin--Voigt model.
  • Figure 2: Graph of $r\left( \xi ,\tau _{0}\right)$ with $\tau _{0}=0.5$ for the step pulse solutions and the asymptotic approximations in the Kelvin--Voigt model.
  • Figure 3: Graph of $r\left( \xi _{0},\tau \right)$ with $\xi _{0}=0.5$ for the delta pulse solutions and the asymptotic approximations in the Kelvin--Voigt model.
  • Figure 4: Graph of $r\left( \xi ,\tau _{0}\right)$ with $\tau _{0}=0.5$ for the delta pulse solutions and the asymptotic approximations in the Kelvin--Voigt model.