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On the passage from nonlinear to linearized viscoelastodynamics

Barbora Benešová, Malte Kampschulte, Martin Kružík

Abstract

The equations of linearized viscoelastodynamics in Kelvin-Voigt rheology are rigorously derived from a nonlinear model that satisfies the time-dependent frame indifference in the sense of Antman. Besides showing the convergence of corresponding solutions of both systems, we also prove the convergence of time-discrete solutions on various scales and of continuous solutions of nonlinear problems to linearized ones.

On the passage from nonlinear to linearized viscoelastodynamics

Abstract

The equations of linearized viscoelastodynamics in Kelvin-Voigt rheology are rigorously derived from a nonlinear model that satisfies the time-dependent frame indifference in the sense of Antman. Besides showing the convergence of corresponding solutions of both systems, we also prove the convergence of time-discrete solutions on various scales and of continuous solutions of nonlinear problems to linearized ones.
Paper Structure (24 sections, 39 theorems, 229 equations)

This paper contains 24 sections, 39 theorems, 229 equations.

Table of Contents

  1. Introduction
  2. Modeling and background on linearization
  3. Viscoelastic nonsimple materials
  4. Assumptions on the underlying potentials
  5. Auxiliary results
  6. Properties of the elastic energy
  7. Properties of the dissipation functional
  8. Dual dissipation term
  9. Gamma-convergence of the underlying functionals
  10. Gamma-convergence of the energy
  11. Gamma-convergence of the dissipation functional
  12. Gamma-limit of the nonlinear discrete cost functional
  13. Discrete iterations and a-priori estimates
  14. Passing to the limit from (NL-disc) or (L-disc)
  15. The limit tau to zero from (NL-disc) and (L-disc)
  16. ...and 9 more sections

Key Result

Lemma 3.1

Fix a number $E_{\max}>0$. Then, there exists a constant $C>0$, such that for all $\delta>0$ small enough and all $y_{\delta} \in W^{2,p}(\Omega;\mathbb{R}^d)$ with $y_{\delta}|_{\partial\Omega} = \mathbf{id}$ and we have where $u_{\delta}:= \delta^{-1}(y_{\delta}- \mathbf{id})$.

Theorems & Definitions (88)

  • Lemma 3.1: Rigidity and uniform bounds
  • proof
  • Lemma 3.2: Quantified linerization of the energy
  • proof
  • Lemma 3.3: Healey-Krömer estimate
  • Lemma 3.4: Quantified linearization of the dissipation
  • proof
  • Lemma 3.5: Generalized Korn's inquality Neff02KFINPomp03KFIV
  • Corollary 3.6: Discrete Korn-type rigidity estimate
  • Corollary 3.7: Continuous Korn-type rigidity estimate
  • ...and 78 more