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Nonlocal-to-local limit in linearized viscoelasticity

Manuel Friedrich, Manuel Seitz, Ulisse Stefanelli

Abstract

We study the quasistatic evolution of a linear peridynamic Kelvin-Voigt viscoelastic material. More specifically, we consider the gradient flow of a nonlocal elastic energy with respect to a nonlocal viscous dissipation. Following an evolutionary $Γ$-convergence approach, we prove that the solutions of the nonlocal problem converge to the solution of the local problem, when the peridynamic horizon tends to $0$, that is, in the nonlocal-to-local limit.

Nonlocal-to-local limit in linearized viscoelasticity

Abstract

We study the quasistatic evolution of a linear peridynamic Kelvin-Voigt viscoelastic material. More specifically, we consider the gradient flow of a nonlocal elastic energy with respect to a nonlocal viscous dissipation. Following an evolutionary -convergence approach, we prove that the solutions of the nonlocal problem converge to the solution of the local problem, when the peridynamic horizon tends to , that is, in the nonlocal-to-local limit.
Paper Structure (20 sections, 29 theorems, 132 equations)

This paper contains 20 sections, 29 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. The model and main result
  3. Notation
  4. The model
  5. Main result
  6. Differentials of the potentials
  7. Nonlocal potentials
  8. Local potentials
  9. Nonlocal well-posedness and energy-dissipation-equality
  10. Preliminary results
  11. Solutions to the nonlocal viscoelastic problem
  12. Energy dissipation equality and a priori estimates
  13. Compactness
  14. Nonlocal-to-local convergence of derivatives
  15. General nonlocal-to-local convergence results
  16. ...and 5 more sections

Key Result

Theorem 2.1

Under Assumptions assumptions, let a sequence of initial conditions $u_n^0 \in \mathcal{S}_{n}({\widetilde{\Omega}};\mathbb{R}^d)$ and $u^0 \in H^1_0(\Omega;\mathbb{R}^d)$ be given such that $u_{n}^0 \to u^0$ strongly in $L^2({\widetilde{\Omega}};\mathbb{R}^d)$ and $E_n(u_n^0) \to E (u^0)$ as $n \to

Theorems & Definitions (44)

  • Theorem 2.1: Evolutionary $\Gamma$-convergence
  • Remark 2.2: Regularity
  • Lemma 3.1: Gâteaux derivatives
  • proof
  • Corollary 3.2
  • proof
  • Lemma 4.1: mengeshaNONLOCALKORNTYPECHARACTERIZATION2012
  • Proposition 4.2: Nonlocal Poincaré-Korn inequality, mengeshaVariationalLimitClass2015
  • Proposition 4.3: Compactness, mengeshaVariationalLimitClass2015
  • Lemma 4.4: Korn inequality
  • ...and 34 more