A comparative analysis on variational models for quasi-static brittle crack propagation

Abstract

We consider a brittle material under displacement control with a mode III crack running in quasi-static regime along a straight line. Several definitions of propagation, based on variational criteria, have been proposed in the last decade. Our aim is a detailed study of their properties, in particular as far as energy, energy release rate and jump discontinuities is concerned. In order to make similarities and differences appear more clearly, the analysis of the evolutions follows a common structure and theoretical results are illustrated by a prototype numerical example.

A variational characterization of rate‐independent evolution

“…More recently, local minimization has been considered by Negri and Ortner [34], Knees, Mielke, and Zanini [16], Toader and Zanini [39], and Larsen [19]. See also the comparative analysis in Negri [33]. A second example in the frame of associative (linearized) elasto-plasticity with softening is due to Dal Maso, Desimone, Mora, and Morini [7]: by requiring global stability, no softening actually occurs.…”

“…As for the abstract situation, we shall refer to the finite dimensional results by Efendiev and Mielke [10] and the infinite-dimensional analysis in Mielke, Rossi, and Savaré [24]. Before closing this subsection let us mention that alternative (in some sense intermediate) ideas in order to localize stability are that of Mielke [21] and Larsen (see [33]). Both these approaches are intended to restrict minimization to neighboring states with respect to dissipation (the former) or smaller in energy up to a positive tolerance (the latter, in the frame of brittle fractures).…”

“…For instance, they arise in elasto-plasticity [6,12,14,15,38], phase-transformation in shape memory alloys [2,25,29,31,35], fracture propagation [3,9,13,16,33], delamination and damage phenomena [17,26], ferro-electricity [32], as well as in many other contexts. The common feature of these systems is that they show no (or very short) intrinsic time-scale and react immediately to changes of external actions [22,40], see below.…”

Asymptotic analysis of small defects near a singular point in antiplane elasticity, with an application to the nucleation of a crack at a notch

“…The nucleation or the onset of cracking is governed by either the so-called G-law or the so-called FM-law and one goal of this section is to compare those laws. The interested reader can also refer to [Bourdin et al 2008;Francfort and Marigo 1998;Negri 2010;Negri and Ortner 2008; where other comparisons between the G-law and the FM-law are proposed.…”

Energy release rate along a kinked path

“…In these works the above-mentioned results of Irwin and Rice are settled in a rigorous way in the framework of Sobolev spaces, considering the setting of regular cracks. For plane strain configurations, the technical tools provided in [4] are enough to study the quasi-static evolution in the case of a predefined crack path, at least of class C 2 (the reader interested in evolution problems may find in [6] an overview of the most recent results). Although smooth cracks occur frequently, especially in material testing, it seems more realistic to study the propagation without any geometrical restriction on the fracture set.…”