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Energy of a non-linear viscoelastic model compatible with fractional relaxation

Andrea Giusti, Andrea Mentrelli, Tommaso Ruggeri

Abstract

Recently, a non-linear model of viscoelasticity based on Rational Extended Thermodynamics was proposed in [arXiv:2312.05116]. This theory extends the evolution of the viscous stress beyond the linear framework of the Maxwell model to the non-linear realm, provided that the viscous energy function is given. This work aims at establishing a possible constitutive law for the viscous energy such that the relaxation modulus of the fractional Maxwell model of order $α\in (1/2, 1]$ is contained within the solutions of the (non-linear) relaxation experiment. Necessary and sufficient conditions for the existence of this coincident solution are discussed, together with a numerical evaluation of the viscous energy associated with the non-linear model.

Energy of a non-linear viscoelastic model compatible with fractional relaxation

Abstract

Recently, a non-linear model of viscoelasticity based on Rational Extended Thermodynamics was proposed in [arXiv:2312.05116]. This theory extends the evolution of the viscous stress beyond the linear framework of the Maxwell model to the non-linear realm, provided that the viscous energy function is given. This work aims at establishing a possible constitutive law for the viscous energy such that the relaxation modulus of the fractional Maxwell model of order is contained within the solutions of the (non-linear) relaxation experiment. Necessary and sufficient conditions for the existence of this coincident solution are discussed, together with a numerical evaluation of the viscous energy associated with the non-linear model.
Paper Structure (3 sections, 2 theorems, 35 equations, 2 figures)

This paper contains 3 sections, 2 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Relaxation modulus and viscous energy
  3. Discussion

Key Result

Theorem 1

Consider the family of constitutive equations $e^{(V)}(\sigma)$ depending on $\alpha$ and expressed in the following parametric form: with $s\geq 0$, $e_0$ being a real inessential constant, and $k_0$ a structural constant of the material. Then, there exists a solution $\sigma(t) = k_0 \, E_\alpha [ - (t/\tau_0)^\alpha ]$ of both Eqs. nonlineare and frazione if we choose as initial condition $\s

Figures (2)

  • Figure 1: Non-dimensional non-linear relaxation time $\bar{\tau}(\sigma) = \tau/\tau_0$ (left panel) and non-dimensional viscous energy $\bar{e}^{(V)}\left(\sigma\right )= \left[\rho^\ast \mu \left(\varepsilon_0\right)/\left(\tau_0 k_0^2\right)\right] \, e^{(V)}(\sigma)$ (right panel).
  • Figure 2: $\bar{\sigma} = \sigma/k_0$ and the dimensionles upper bound $\bar{\sigma}^{ub} = \left(k_0/\sigma_0\right) \sigma_F/k_0 = \sigma_F/\sigma_0$ as a function of $\bar{t} = t / \tau_0$, for $\alpha = 0.6$, $k_0 = 1$, $\sigma_0 = 1/2$.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof