- (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.
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