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Dynamic crack growth in viscoelastic materials with memory

Federico Cianci

TL;DR

The paper develops a rigorous framework for dynamic crack growth in viscoelastic materials with memory, where damping incorporates the deformation history. By formulating a weak viscoelastic problem with memory on evolving cracked domains and introducing an energy-dissipation balance together with a maximal-dissipation principle, it derives an existence result for crack evolution in two dimensions. The analysis hinges on precise crack-path regularity, a memory-augmented energy balance, and a constructive time-discretization approach coupled with compactness and continuous dependence results. This work extends dynamic Griffith-type theories to memory-bearing viscoelastic media and provides a mathematically robust mechanism to select crack paths under the maximal-dissipation criterion, with potential implications for modeling fracture in complex materials.

Abstract

In this paper we introduce a model of dynamic crack growth in viscoelastic material, where the damping term depends on the history of the deformation. The model is based on a dynamic energy dissipation balance and on a maximal dissipation condition. Our main result is an existence theorem in dimension two under some a priori regularity constraints on the cracks.

Dynamic crack growth in viscoelastic materials with memory

TL;DR

The paper develops a rigorous framework for dynamic crack growth in viscoelastic materials with memory, where damping incorporates the deformation history. By formulating a weak viscoelastic problem with memory on evolving cracked domains and introducing an energy-dissipation balance together with a maximal-dissipation principle, it derives an existence result for crack evolution in two dimensions. The analysis hinges on precise crack-path regularity, a memory-augmented energy balance, and a constructive time-discretization approach coupled with compactness and continuous dependence results. This work extends dynamic Griffith-type theories to memory-bearing viscoelastic media and provides a mathematically robust mechanism to select crack paths under the maximal-dissipation criterion, with potential implications for modeling fracture in complex materials.

Abstract

In this paper we introduce a model of dynamic crack growth in viscoelastic material, where the damping term depends on the history of the deformation. The model is based on a dynamic energy dissipation balance and on a maximal dissipation condition. Our main result is an existence theorem in dimension two under some a priori regularity constraints on the cracks.
Paper Structure (4 sections, 14 theorems, 91 equations)

This paper contains 4 sections, 14 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Energy balance
  4. Existence for the coupled problem

Key Result

Lemma 2.4

There exist two constants $\hat{r}$ and $\hat{L}$, with $0 < \hat{r} < r$ and $\hat{L}> L$, depending only on $r$ and $L$, such that for every $\gamma \colon [a_0, b_\gamma] \to \overline\Omega$ with $\gamma \in \mathcal{G}_{ r, L}$ there exists an extension $\hat{\gamma} \colon [a_0, b_\gamma + \h

Theorems & Definitions (39)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • Lemma 2.7
  • Definition 2.8
  • Lemma 2.9
  • Remark 2.10
  • ...and 29 more