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Energy-dissipation balance of a smooth moving crack

Maicol Caponi, Ilaria Lucardesi, Emanuele Tasso

TL;DR

This work addresses the energy-dissipation balance for a dynamically evolving Mode III crack along a prescribed smooth path in a 2D elastic medium. The authors introduce a rigorous displacement representation that isolates a leading singular component at the crack tip, obtained through a four-step change-of-variables procedure that yields a cylindrical, time-fixed framework where semigroup theory applies. They prove a local and then global representation for the solution, enabling a precise energy balance formula: $\mathcal{E}(t)-\mathcal{E}(0)+\frac{\pi}{4}\int_{0}^{t} k^2(\tau) a(\tau) \dot s(\tau)\,d\tau = \int_{0}^{t} \langle f(\tau), \dot u(\tau)\rangle_{L^2(\Omega)}\,d\tau$, and they show this balance holds if and only if the stress-intensity factor satisfies $k(t)=\frac{2}{\sqrt{\pi a(t)}}$ during crack opening. The factor $a(t)$ encodes the influence of $A$, the crack geometry $\Gamma$, and the growth $s$, and for $A=I$ reduces to $a(t)=1$. Extending prior straight-crack results, the paper handles curved crack paths and general tensors $A$ by leveraging geometric measure theory and semigroup methods, offering a constructive framework for analyzing fracture dynamics in time-dependent domains.

Abstract

In this paper we provide necessary and sufficient conditions in order to guarantee the energy-dissipation balance of a Mode III crack, growing on a prescribed smooth path. Moreover, we characterize the singularity of the displacement near the crack tip, generalizing the result in [S.Nicaise, A.M.Sandig - \textit{J. Math. Anal. Appl.} 2007] valid for straight fractures.

Energy-dissipation balance of a smooth moving crack

TL;DR

This work addresses the energy-dissipation balance for a dynamically evolving Mode III crack along a prescribed smooth path in a 2D elastic medium. The authors introduce a rigorous displacement representation that isolates a leading singular component at the crack tip, obtained through a four-step change-of-variables procedure that yields a cylindrical, time-fixed framework where semigroup theory applies. They prove a local and then global representation for the solution, enabling a precise energy balance formula: , and they show this balance holds if and only if the stress-intensity factor satisfies during crack opening. The factor encodes the influence of , the crack geometry , and the growth , and for reduces to . Extending prior straight-crack results, the paper handles curved crack paths and general tensors by leveraging geometric measure theory and semigroup methods, offering a constructive framework for analyzing fracture dynamics in time-dependent domains.

Abstract

In this paper we provide necessary and sufficient conditions in order to guarantee the energy-dissipation balance of a Mode III crack, growing on a prescribed smooth path. Moreover, we characterize the singularity of the displacement near the crack tip, generalizing the result in [S.Nicaise, A.M.Sandig - \textit{J. Math. Anal. Appl.} 2007] valid for straight fractures.

Paper Structure

This paper contains 11 sections, 12 theorems, 175 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Standing assumptions
  5. The change of variables approach
  6. Proof of the representation result
  7. Preliminaries on semigroup theory
  8. Local representation result in the cylindrical domain
  9. Local representation result in the time-dependent domain
  10. Global representation result in the time-dependent domain
  11. The energy-dissipation balance

Key Result

Lemma 2.1

There exists a constant $c_4>0$ such that, for every $t\in [t_0,t_1]$ and $x\in \Omega^{(4)}$, Moreover, for every $t\in [t_0,t_1]$, there holds Finally, there exists a vector field $W:\partial_N \Omega^{(4)} \cup \Gamma^{(4)}(t_0)\to \mathbb R^2$ such that, for every $t\in[t_0,t_1]$ and $x\in \partial_N \Omega^{(4)} \cup \Gamma^{(4)}(t_0)$, and $W(x)=n(x)=e_2$ in a neighborhood of the tip of $

Figures (3)

  • Figure 1: The endpoints of $\Gamma$ are $\gamma(0)$ and $\gamma(\ell)$ and belong to $\partial \Omega$. We study the evolution of the fracture along $\Gamma$ from $\gamma(s(0))$ to $\gamma(s(T))$.
  • Figure 2: In polar coordinates, the function $S$ reads $S(r,\theta)=r^{1/2} \sin (\theta/2)$, where $r$ is the distance from the origin and $\theta\in [-\pi,\pi]$ is the angle which has a discontinuity on the horizontal half line $\{x_1\leq 0\}$.
  • Figure 3: A possible choice of determination of $Im(\sqrt{y_1+iy_2})$, with $\Gamma(t)$ as discontinuity set.

Theorems & Definitions (34)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • Remark 3.5
  • ...and 24 more