Variational Approaches to Fracture
Fracture Mechanics is largely based on an assumption of A.A. Griffith, according to which the evolution of a crack in a fractured body is governed by the competition between the elastic strain energy and the energy spent in the crack growth. In the recent variational formulation of Francfort and Marigo, some analytical difficulties due to the presence of discontinuous displacements are avoided using the Г-convergence theory. In it, the problem is approximated by a sequence of more regular problems, formulated in Sobolev spaces, and therefore solvable with standard finite elements techniques. A further regularization introduced by G.I. Barenblatt allows to solve the problem of determining the fracture onset in an initially unfractured body.
In this communication I discuss the role plaid by local energy minimizers in the post-fracture evolution. I show that a dissipation inequality leads to the formulation of an incremental problem, which determines a quasi-static evolution along a path made of local energy minimizers. The introduction of a dissipation inequality provides different responses at loading and unloading, in accordance with experimental evidence. Finally, I briefly discuss the possibility of bulk regularization, based on the assumption that the fracture energy be diffused in a three-dimensional region, instead of being concentrated on a singular surface.
KeywordsFracture Energy Elastic Strain Energy Discontinuous Displacement Half Line Fracture Problem
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This work was supported by the Research Project Mathematical Models for Materials Science - PRIN 2005 of the Italian Ministry for University.
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