Stress gradient effects on the nucleation and propagation of cohesive cracks

Tuan Hiep  Pham; Jérôme  Laverne; Jean-Jacques  Marigo

doi:10.3934/dcdss.2016012

Article Contents

2016, Volume 9, Issue 2: 557-584. Doi: 10.3934/dcdss.2016012

This issue Previous Article Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions Next Article Cellular instabilities analyzed by multi-scale Fourier series: A review

Stress gradient effects on the nucleation and propagation of cohesive cracks

1.
CNRS, Ecole Polytechnique, Laboratoire de Mécanique des Solides, (UMR 7649), F-91128 Palaiseau Cedex
2.
Institute of Mechanical Sciences and Industrial Applications, (UMR EDF-CNRS-CEA-ENSTA Paristech 9219), 92141 Clamart

Received: April 30, 2015

Revised: October 31, 2015

Published: March 2016

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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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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