Vol. 2, No. 2, 2014

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ISSN: 2325-3444 (e-only)
ISSN: 2326-7186 (print)
Asymptotic analysis of small defects near a singular point in antiplane elasticity, with an application to the nucleation of a crack at a notch

Thi Bach Tuyet Dang, Laurence Halpern and Jean-Jacques Marigo

Vol. 2 (2014), No. 2, 141–179
Abstract

We use matching asymptotic expansions to treat the antiplane elastic problem associated with a small defect located at the tip of a notch. In a first part, we develop the asymptotic method for any type of defect and present the sequential procedure which allows us to calculate the different terms of the inner and outer expansions at any order. This requires in particular separating in each term its singular part from its regular part. In a second part, the asymptotic method is applied to the case of a crack of variable length located at the tip of a given notch. We show that the first two nontrivial terms of the expansion of the energy release rate are sufficient to well approximate the dependence of the energy release rate on the crack length in the range of values of the length which are sufficient to treat the problem of nucleation. This problem is considered in the last part where we compare the nucleation and the propagation of a crack predicted by two different models: the classical Griffith law and the Francfort–Marigo law based on an energy minimization principle. Several numerical results illustrate the interest of the method.

Keywords
brittle fracture, variational methods, asymptotic methods, singularities
Mathematical Subject Classification 2000
Primary: 35A15, 35B40, 35C20, 74A45, 74G70, 74R10
Milestones
Received: 18 December 2012
Revised: 30 April 2013
Accepted: 5 June 2013
Published: 9 June 2014

Communicated by Francesco dell'Isola
Authors
Thi Bach Tuyet Dang
Laboratoire de Mécanique des Solides
École Polytechnique
CNRS, UMR 7649
91128 Palaiseau cedex
France
Laurence Halpern
LAGA
Université Paris 13
Sorbonne Paris Cité, CNRS, UMR 7539
93430 Villetaneuse
France
Jean-Jacques Marigo
Laboratoire de Mécanique des Solides
École Polytechnique
CNRS, UMR 7649
91128 Palaiseau cedex
France