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Acceleration Waves and the K-Condition in Viscoelastic Solids and Non-Newtonian Fluids

Tommaso Ruggeri

TL;DR

This work analyzes the weaker K-condition for global existence of smooth solutions in dissipative hyperbolic systems by studying acceleration waves in a one-dimensional Rational Extended Thermodynamics model that unifies viscoelastic solids and non-Newtonian fluids. The acceleration-wave evolution is governed by a Bernoulli-type equation with coefficients $a$ and $b$ determined by material properties and the production term, yielding a critical amplitude $G_{ m cr}=b/|a|$ that separates global decay from finite-time blow-up. For viscoelastic solids the weaker K-condition is always satisfied and acceleration waves remain bounded, while for non-Newtonian fluids the outcome depends on the power-law index $m$: Newtonian $(m=1)$ admits a finite blow-up threshold, shear-thinning $(m<1)$ violates the weaker K-condition and blows up for any positive amplitude, and shear-thickening $(m>1)$ can be globally damped via a regularization that enforces instantaneous regularization in the singular limit. These results emphasize the constitutive structure and energy-dissipation mechanism as central to predicting rapid-loading responses and to the mathematical theory of global well-posedness in hyperbolic-relaxation models.

Abstract

The K-condition introduced by Shizuta and Kawashima provides a sufficient criterion for the global existence of smooth solutions to dissipative hyperbolic systems. For genuinely nonlinear characteristic fields, a weaker K-condition becomes necessary, although not sufficient. In this paper, we analyze this weaker K-condition through the study of acceleration waves propagating in an equilibrium state. We investigate two classes of hyperbolic models: one describing viscoelasticity with linear dissipation, and the other non-Newtonian fluids asymptotically converging to a power-law behavior. For viscoelastic models, the weaker K-condition is always satisfied and acceleration waves remain bounded. For non-Newtonian fluids, the validity of the condition depends on the power-law index $m$: it holds for Newtonian fluids ($m=1$), is violated for shear-thinning fluids ($m<1$), and leads to an instantaneous regularization of acceleration waves for shear-thickening fluids ($m>1$).

Acceleration Waves and the K-Condition in Viscoelastic Solids and Non-Newtonian Fluids

TL;DR

This work analyzes the weaker K-condition for global existence of smooth solutions in dissipative hyperbolic systems by studying acceleration waves in a one-dimensional Rational Extended Thermodynamics model that unifies viscoelastic solids and non-Newtonian fluids. The acceleration-wave evolution is governed by a Bernoulli-type equation with coefficients and determined by material properties and the production term, yielding a critical amplitude that separates global decay from finite-time blow-up. For viscoelastic solids the weaker K-condition is always satisfied and acceleration waves remain bounded, while for non-Newtonian fluids the outcome depends on the power-law index : Newtonian admits a finite blow-up threshold, shear-thinning violates the weaker K-condition and blows up for any positive amplitude, and shear-thickening can be globally damped via a regularization that enforces instantaneous regularization in the singular limit. These results emphasize the constitutive structure and energy-dissipation mechanism as central to predicting rapid-loading responses and to the mathematical theory of global well-posedness in hyperbolic-relaxation models.

Abstract

The K-condition introduced by Shizuta and Kawashima provides a sufficient criterion for the global existence of smooth solutions to dissipative hyperbolic systems. For genuinely nonlinear characteristic fields, a weaker K-condition becomes necessary, although not sufficient. In this paper, we analyze this weaker K-condition through the study of acceleration waves propagating in an equilibrium state. We investigate two classes of hyperbolic models: one describing viscoelasticity with linear dissipation, and the other non-Newtonian fluids asymptotically converging to a power-law behavior. For viscoelastic models, the weaker K-condition is always satisfied and acceleration waves remain bounded. For non-Newtonian fluids, the validity of the condition depends on the power-law index : it holds for Newtonian fluids (), is violated for shear-thinning fluids (), and leads to an instantaneous regularization of acceleration waves for shear-thickening fluids ().
Paper Structure (9 sections, 2 theorems, 85 equations)

This paper contains 9 sections, 2 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Balance laws, K-condition and global existence
  3. Weak-discontinuity waves
  4. Nonlinear model for uniaxial isothermal viscoelasticity and non-Newtonian fluids
  5. Acceleration waves for the present model
  6. Acceleration waves in viscoelastic solids under uniaxial deformation
  7. Evaluation of $R$ with the Mooney--Rivlin potential
  8. Acceleration waves in a non-Newtonian fluid
  9. Concluding Remarks

Key Result

theorem 1

Assume that system conservation is strictly dissipative and that the K-condition holds. Then there exists $\delta>0$ such that, if there exists a unique global smooth solution satisfying

Theorems & Definitions (5)

  • definition thmcounterdefinition: K-condition
  • theorem 1: Global existence
  • theorem 2: Stability of constant state
  • remark thmcounterremark
  • remark thmcounterremark: Singular limit for $b$