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Nonlinear relations of viscous stress and strain rate in nonlinear Viscoelasticity

Lennart Machill

TL;DR

The paper develops a frame‑indifferent, nonlinear Kelvin–Voigt viscoelastic model for large deformations of second‑grade materials and proves the existence of weak solutions by recasting the evolution as a metric‑gradient flow with a frame‑indifferent time‑discretization. Central to the analysis is a rigidity estimate that enables control of displacement gradients via the nonlinear elastic strains, allowing nonquadratic viscous densities. The authors establish convergence of time‑discrete minimizers to curves of maximal slope and connect these to weak solutions, then derive long‑time behavior for small data, showing exponential, polynomial, or finite‑time convergence depending on the viscous exponent $\tilde{p}$. The results provide a robust variational framework for nonlinear viscoelasticity at large strains, encompassing second‑gradient regularization and anisotropic dissipation, with explicit decay laws in the small‑data regime.

Abstract

We consider a Kelvin-Voigt model for viscoelastic second-grade materials, where the elastic and the viscous stress tensor both satisfy frame indifference. Using a rigidity estimate by [Ciarlet-Mardare '15], existence of weak solutions is shown by means of a frame-indifferent time-discretization scheme. Further, the result includes viscous stress tensors which can be calculated by nonquadratic polynomial densities. Afterwards, we investigate the long-time behavior of solutions in the case of small external loading and initial data. Our main tool is the abstract theory of metric gradient flows.

Nonlinear relations of viscous stress and strain rate in nonlinear Viscoelasticity

TL;DR

The paper develops a frame‑indifferent, nonlinear Kelvin–Voigt viscoelastic model for large deformations of second‑grade materials and proves the existence of weak solutions by recasting the evolution as a metric‑gradient flow with a frame‑indifferent time‑discretization. Central to the analysis is a rigidity estimate that enables control of displacement gradients via the nonlinear elastic strains, allowing nonquadratic viscous densities. The authors establish convergence of time‑discrete minimizers to curves of maximal slope and connect these to weak solutions, then derive long‑time behavior for small data, showing exponential, polynomial, or finite‑time convergence depending on the viscous exponent . The results provide a robust variational framework for nonlinear viscoelasticity at large strains, encompassing second‑gradient regularization and anisotropic dissipation, with explicit decay laws in the small‑data regime.

Abstract

We consider a Kelvin-Voigt model for viscoelastic second-grade materials, where the elastic and the viscous stress tensor both satisfy frame indifference. Using a rigidity estimate by [Ciarlet-Mardare '15], existence of weak solutions is shown by means of a frame-indifferent time-discretization scheme. Further, the result includes viscous stress tensors which can be calculated by nonquadratic polynomial densities. Afterwards, we investigate the long-time behavior of solutions in the case of small external loading and initial data. Our main tool is the abstract theory of metric gradient flows.
Paper Structure (10 sections, 10 theorems, 100 equations)

This paper contains 10 sections, 10 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. The model and main results
  3. The variational setting
  4. Equations of nonlinear viscoelasticity
  5. Main results
  6. Metric gradient flows
  7. Definitions
  8. Curves of maximal slope as limits of time-discrete solutions
  9. Properties of the model
  10. Long-time behavior for small strains

Key Result

Theorem 2.2

Let $M>0$, $\tilde{p} \in (1,+\infty)$, and $y_0 \in \mathscr{S}_{M}$.

Theorems & Definitions (25)

  • Definition 2.1: Weak solutions of nonlinear viscoelasticity
  • Theorem 2.2: Existence of solutions
  • Theorem 2.3: Long-time behavior for small strains
  • Definition 3.1: Upper gradients, slopes, curves of maximal slope
  • Theorem 3.2
  • proof
  • Lemma 4.1: Bounds in $\mathscr{S}_{M}$
  • proof
  • Lemma 4.2: Rigidity estimate
  • proof
  • ...and 15 more