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A Nonlinear approach to Viscoelasticity via Rational Extended Thermodynamics

Tommaso Ruggeri

Abstract

In the one-dimensional isothermal case, we introduce a simple model of nonlinear viscoelasticity within the Rational Extended Thermodynamics (RET) framework. The differential system is determined by the universal principles of RET, exhibiting symmetric hyperbolic form and ensuring the existence of smooth solutions for appropriately small initial data. In the linear case, the equation for viscous stress reduces to the well-known Maxwell model, thereby representing a plausible nonlinear extension of the Maxwell-type model. The total stress instead satisfies a non-linear Zener model.

A Nonlinear approach to Viscoelasticity via Rational Extended Thermodynamics

Abstract

In the one-dimensional isothermal case, we introduce a simple model of nonlinear viscoelasticity within the Rational Extended Thermodynamics (RET) framework. The differential system is determined by the universal principles of RET, exhibiting symmetric hyperbolic form and ensuring the existence of smooth solutions for appropriately small initial data. In the linear case, the equation for viscous stress reduces to the well-known Maxwell model, thereby representing a plausible nonlinear extension of the Maxwell-type model. The total stress instead satisfies a non-linear Zener model.
Paper Structure (7 sections, 3 theorems, 58 equations)

This paper contains 7 sections, 3 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. The model via Rational Extended Thermodynamics
  3. Main field, Convexity, and Symmetric form
  4. Linear case
  5. Physical interpretation
  6. Characteristic velocities and K-Condition
  7. Conclusions

Key Result

Theorem 2.1

A necessary and sufficient condition such that the system elastvisco satisfies the dissipative supplementary equation energia is that the constitutive equations satisfy the following expressions: where $\Psi$ is an arbitrary function of $w$ and $\alpha$ is an arbitrary positive function of $(F,\sigma)$.

Theorems & Definitions (3)

  • Theorem 2.1
  • Theorem 2.2: Ruggeri & Strumia RS
  • Theorem 2.3