Stress gradient effects on the nucleation and propagation of cohesive cracks

Jean-Jacques Marigo,Jérôme Laverne,Tuan Hiep Pham

doi:10.3934/dcdss.2016012

Abstract

The aim of the present work is to study the nucleation and propagation of cohesive cracks in two-dimensional elastic structures. The crack evolution is governed by Dugdale's cohesive force model. Specifically, we investigate the stabilizing effect of the stress field non-uniformity by introducing a length $l$ which characterizes the stress gradient in a neighborhood of the point where the crack nucleates. We distinguish two stages in the crack evolution: the first one where the entire crack is submitted to cohesive forces, followed by a second one where a non-cohesive part appears. Assuming that the material characteristic length $d_c$ associated with Dugdale's model is small in comparison with the dimension $L$ of the body, we develop a two-scale approach and, using the methods of complex analysis, obtain the entire crack evolution with the loading in closed form. In particular, we show that the propagation is stable during the first stage, but becomes unstable with a brutal crack length jump as soon as the non-cohesive crack part appears. We also discuss the influence of the problem parameters and study the sensitivity to imperfections.

Highlights

IntroductionGriffith’s theory of fracture [Griffith, 1920] is based on the concept of critical energy release rate Gc which comes from the fundamental but somewhat too restrictive assumption that the surface energy associated with a crack is proportional to the area of the crack (at least in a homogeneous and isotropic body), or, equivalently, that there is no interaction between the lips of a crack
Since Dugdale’s law contains a critical stress σc, one can account for the nucleation of a crack in a sound body at a finite loading te, in contrast with Griffith’s law
Assuming that is small by comparison to the size of the body, situation the most frequent in practice, all the solutions can be obtained in a closed form which renders easy the study of the size effects

Summary

Introduction

Griffith’s theory of fracture [Griffith, 1920] is based on the concept of critical energy release rate Gc which comes from the fundamental but somewhat too restrictive assumption that the surface energy associated with a crack is proportional to the area of the crack (at least in a homogeneous and isotropic body), or, equivalently, that there is no interaction between the lips of a crack. It remains the most used approach in fracture mechanics thanks to its simplicity in terms of material behavior. Following the ideas of [Dugdale, 1960] and [Barenblatt, 1962], many such models have been proposed and tested, see for instance[Tvergaard, 1990; Needleman, 1992; Keller et al, 1999; Roe and Siegmund, 2002; Talon and Curnier, 2003; Del Piero and Raous, 2010]

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Discussion

Conclusion