Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniqueness and continuous dependence for a viscoelastic problem with memory in domains with time dependent cracks

Federico Cianci, Gianni Dal Maso

TL;DR

This work analyzes a viscoelastic elastodynamics problem with memory in domains featuring time-dependent cracks. By recasting the problem to isolate the memory term and employing energy estimates, the authors establish uniqueness under strong crack-regularity hypotheses and prove continuous dependence on crack geometry via a contraction-based fixed-point framework and careful limit passages. The results rely on a robust weak formulation in Korn-friendly spaces on cracked domains, a comparison to the elastodynamics problem with tensor $\mathbb A=\mathbb C+\mathbb V$, and a time-slicing strategy to extend local contraction results globally. The findings provide a rigorous foundation for well-posedness and stability of viscoelastic systems with memory on evolving cracked domains, with potential implications for numerical simulations in aging materials. All mathematical notation is presented within $...$ delimiters to ensure precise representation of the governing equations and function spaces.

Abstract

We study some hyperbolic partial integro-differential systems in domains with time dependent cracks. In particular, we give conditions on the cracks which imply the uniqueness of the solution with prescribed initial-boundary conditions, and its continuous dependence on the cracks.

Uniqueness and continuous dependence for a viscoelastic problem with memory in domains with time dependent cracks

TL;DR

This work analyzes a viscoelastic elastodynamics problem with memory in domains featuring time-dependent cracks. By recasting the problem to isolate the memory term and employing energy estimates, the authors establish uniqueness under strong crack-regularity hypotheses and prove continuous dependence on crack geometry via a contraction-based fixed-point framework and careful limit passages. The results rely on a robust weak formulation in Korn-friendly spaces on cracked domains, a comparison to the elastodynamics problem with tensor , and a time-slicing strategy to extend local contraction results globally. The findings provide a rigorous foundation for well-posedness and stability of viscoelastic systems with memory on evolving cracked domains, with potential implications for numerical simulations in aging materials. All mathematical notation is presented within delimiters to ensure precise representation of the governing equations and function spaces.

Abstract

We study some hyperbolic partial integro-differential systems in domains with time dependent cracks. In particular, we give conditions on the cracks which imply the uniqueness of the solution with prescribed initial-boundary conditions, and its continuous dependence on the cracks.
Paper Structure (4 sections, 18 theorems, 118 equations)

This paper contains 4 sections, 18 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Uniqueness
  4. Continuous dependence on the data

Key Result

Lemma 2.1

Under hypotheses (H7)-(H9), the set $\Omega\setminus \Gamma$ is the union of a finite number of connected open sets with Lipschitz boundary.

Theorems & Definitions (40)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4: Solution for visco-elastodynamics with cracks
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • Definition 3.1: Solution for elastodynamics with cracks
  • ...and 30 more