Issue 12

M. Angelillo et alii, Frattura ed Integrità Strutturale, 12 (2010) 63-78; DOI: 10.3221/IGF-ESIS.12.07 64 propagation. Essentially the analysis is based on the formulation o f Francfort and Marigo [3] the main difference being the fact that we rely on local rather than on global minimization. Nucleation and propagation of fracture is obtained by minimizing in a step by step process a form of energy that is the sum of bulk and interface terms. Recent attempts of producing numerical codes for variational fracture (see for example the works of Bourdin [4] or Del Piero et al [5]) are based on the approximation of the energy, in the sense of Gamma-convergence, by means of elliptic functionals (e.g. Ambrosio-Tortorelli approximations [6]). Here instead we adopt discontinuous finite elements and search for local minima of the energy, close to equilibrium trajectories, through descent methods. In constructing such a numerical model there are two main questions that must be answered. 1. The variational model for fracture requires the ability to accurately approximate the location of cracks, as well as their length. Cracks may not be restricted to propagate along the skeleton of a fixed finite element mesh. To overcome this difficulty our mesh is made variable in the sense that node positions in the reference configuration of the body are considered as further variables. 2. On adopting a Griffith type interface energy the initiation of fracture in a previously virgin body is always "brutal", that is the system cannot proceed along neutral equilibrium paths or, in other words, descent directions toward local minima do not exist. To reach local minima the system must have the ability to surmount small energy barriers and then proceed through “non-equilibrium” descent paths. Here we use a sort of mesh dependent relaxation of the interface energy to get out of small energy wells. The relaxation consists in the adoption of a carefully tailored cohesive type interface energy, tending to the Griffith limit as the mesh size tends to zero. In the first part of the paper we give the main ingredients of the theory and of the model implementation. In the final part of the paper, some examples concerning fracture nucleation and propagation are presented. G RIFFITH ’ S VERSUS VARIATIONAL FRACTURE MECHANICS n the theory of brittle fracture due to Griffith, cracking is associated to the existence of surface energy and the propagation of cracks to the competition between surface energy and potential energy release during an infinitesimal increase of the crack length “c”. A number of authors, since the pioneering work of Ambrosio and Braides [7] in 1995, has proposed variational models of brittle fracture based on global energy minimization with free discontinuities (for early thoughts on minimization with free discontinuities see De Giorgi [8]) . The basic idea is to use the competition between bulk and interface energy that is implicit in Griffith’s theory, [9], as the driving criterion for optimal crack selection. Global energy minimization remedies to two main shortcomings of Griffith’s theory, namely  Initiation: Griffith’s is a model for crack growth from a pre-existing crack, not a model for crack nucleation. Indeed the cost of surface energy for the creation of an infinitesimally long crack cannot be compensated by the elastic energy release in an originally “flawless” body.  Crack path selection: Griffith’s model is restricted to the study of crack propagation along a given path. In the original formulation the energy balance is expressed in terms of “c” only. The variational formulation based on global minimization which looks at the crack set as one of the sought unknowns, leads naturally to the search of crack paths without any restriction on crack topology, thus allowing, at least in principle, the study of initiation and, possibly, branching. There are two main criticisms to variational fracture: 1. The minimum energy formulation implies reversibility but fracture is an inherently irreversible phenomenon. 2. Global minimality does not seem physically correct. The “infinitesimal” local minimality of the Griffith’s theory is presumably not correct either but at least does not lead to paradoxes such as the impossibility of equilibrium under purely tensile loading. In 1998 Francfort & Marigo formulated a consistent variational model of fracture respecting the irreversibility of the phenomenon, but again based on global minimality. The problem is formulated as a quasi-static evolution problem in which the geometry and size of the cracks are limited by their precedessors. A more realistic approach that would investigate local minimizers still lacks of the necessary mathematical apparatus (though the recents attempts of Dal Maso & Toader [10] and Larsen [11] must be aknowledged). Our work, as described in the Introduction, is devoted to assess, through numerical experiments, the ability of a variational model of fracture based on local minimization to model fracture initiation and propagation. From the numerical point of view the main issue with local minimization is how the material moves from an equilibrium I