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Variational methods in brittle fracture

  • (journal) - A priori inequalities between energy release rate and energy concentration for 3D quasistatic brittle fracture propagation, Mathematics and Mechanics of Solids, accepted paper, arxiv (preliminary version)
    • We study the properties of absolute minimal and equilibrium states of generalized Mumford-Shah functionals, with applications to models of quasistatic brittle fracture propagation. The main results concern a priori inequalities between energy release rate and energy concentration for 3D cracks with complex shapes, seen as outer measures living on the crack edge.
  • (arxiv) - Microfractured media with a scale and Mumford-Shah energies (2007)
    • We want to understand the concentration of damage in microfractured elastic media. Due to the different scallings of the volume and area (or area and length in two dimensions) the traditional method of homogenization using periodic arrays of cells seems to fail when applied to the Mumford-Shah functional and to periodically fractured domains. In the present paper we are departing from traditional homogenization. The main result implies the use of Mumford-Shah energies and leads to an explanation of the observed concentration of damage in microfractured elastic bodies.
  • (arxiv) - Energy minimizing brittle crack propagation II
    • This is an alternative version of the published article "Energy Minimizing Brittle Crack Propagation" (see below), containing an improved Mumford-Shah model based on the functional K2, a generalization of the integral J of Rice.
  • (arxiv) - Perturbed area functionals and brittle damage mechanics. Preprint IMAR no. 27/1996.
    • Some Mumford-Shah functionals are revisited as perturbed area functionals in connection with brittle damage mechanics. We find minimizers "on paper" for the classical Mumford-Shah functional for some particular two dimensional domains and boundary conditions. These solutions raise the possibility of validating experimentally the energetic model of crack appearance. Two models of brittle damage and fracture are proposed after; in the one of these models the crack belongs to the set of integral varifolds. We have felt the necessity to start the paper with a preliminary section concerning classical results in equilibrium of a cracked elastic body reviewed in the context of Sobolev spaces with respect to a measure.
  • (arxiv) Energy concentration and brittle crack propagation (1997). This is a paper which has been submitted for publication in a well-known journal in 1997. The referee did not want to accept the paper unless substantial modifications are made. In my opinion the suggested modifications were against the philosophy of the paper, which is: brittle fracture propagation is a geometrical evolution problem, therefore geometrical treatment is highly significant, both from mechanical and mathematical point of view. Eventually unpublished, this paper circulated in manuscript.
    • The purpose of this paper is to fill the gap between the classical treatment of brittle fracture mechanics and the new idea of considering the crack evolution as a free discontinuity problem. Griffith and Irwin criterions of crack propagation are studied and transformed in order to be no longer dependent on any prescription of the geometry of the crack during its evolution. The inequality contained in theorem 6.1. represents the link between generalized Irwin and Griffith criterions of brittle crack propagation. The physical meaning of this inequality is explained in the last section.
  • (journal) (or try this pdf )- Energy Minimizing Brittle Crack Propagation, J. of Elasticity, 52, 3 (1999) (submitted in 1997), pp 201-238
    • I propose a minimizing movement model for quasi-static brittle crack appearance and/or evolution. The model is based on Mumford-Shah type functionals. By the discretization of the time variable we obtain a sequence of free discontinuity problems.
    • Exact solutions and estimations predict a non-physical crack appearance (the constant of Griffith G and the critical stress which causes the fracture in an uni-dimensional traction experiment cannot be both constants of material).
    • The model is of applicative interest for crack propagation. A partial existence result for the model is obtained under the assumption of uniformly bounded (in time) power communicated to the body by the rest of the universe. A numerical approach and examples, using an Ambrosio-Tortorelli variational approximation of the energy functional, are given in the last section.
  • (journal) - Brittle crack propagation based on an optimal energy balance, Rev. Roum. des Math. Pures et Appl., 45, 2 (2001), pp 201-209
  • (journal) - Geometric evolution problem and action-measures, in: Proceedings of PAMM Conference PC 122, Constanta 1998, Tech. Univ. Budapest, (1998)
    • The potential of a free-discontinuity model for crack propagation is not exploited yet. I propose a model of crack propagation in which the crack is driven by a diffeomorphisms flow in the reference configuration of the body. The idea is not new at all and somehow is present from the beginnings of the field (explicitly, for example, in Stumpf, Le). What I have proved is that on the edge of the crack live two concentrated measures. The first, named CF, contains informations about the energy release rate due only to the crack propagation. The second, named CM measures the elastic energy concentration at the edge of the crack. A Griffith type criterion for fracture propagation is naturally formulated with the help of CF, while a Irwin type criterion should be formulated in terms of CM. I prove that as measures, CM is absolutely continuous with respect to CF, moreover, for any borelian B CM(B) is smaller or equal than CF(B). This means that in the class of all fracture propagation criteria the Irwin criterion is the most restrictive and the Griffith criterion is the most permissive.