Critical examination of cohesive-zone models in the theory of dynamic fracture

TL;DR

This paper critically examines cohesive-zone models in dynamic fracture, revealing mathematical issues in existing models, extending them with dissipation, and finding limited conditions for stability transitions, ultimately questioning their suitability for dynamic fracture analysis.

Contribution

The study reformulates existing cohesive-zone models, introduces dissipation to improve mathematical well-posedness, and analyzes stability, highlighting inherent limitations of these models in dynamic fracture.

Findings

Existing models are mathematically ill-posed.

Inclusion of dissipation can yield well-posed models.

Stability transition may occur via Hopf bifurcation at finite wavelength.

Abstract

We have examined a class of cohesive-zone models of dynamic mode-I fracture, looking both at steady-state crack propagation and its stability against out-of-plane perturbations. Our work is an extension of that of Ching, Langer, and Nakanishi (CLN) (Phys. Rev. E, vol. 53, no. 3, p. 2864 (1996)), who studied a non-dissipative version of this model and reported strong instability at all non-zero crack speeds. We have reformulated the CLN theory and have discovered, surprisingly, that their model is mathematically ill-posed. In an attempt to correct this difficulty and to construct models that might exhibit realistic behavior, we have extended the CLN analysis to include dissipative mechanisms within the cohesive zone. We have succeeded to some extent in finding mathematically well posed systems; and we even have found a class of models for which a transition from stability to instability may occur at a nonzero crack speed via a Hopf bifurcation at a finite wavelength of the applied perturbation. However, our general conclusion is that these cohesive-zone models are inherently unsatisfactory for use in dynamical studies. They are extremely difficult mathematically, and they seem to be highly sensitive to details that ought to be physically unimportant.