Archive for Rational Mechanics and Analysis

, Volume 180, Issue 3, pp 399–447

A Γ-Convergence Approach to Stability of Unilateral Minimality Properties in Fracture Mechanics and Applications


DOI: 10.1007/s00205-005-0392-3

Cite this article as:
Giacomini, A. & Ponsiglione, M. Arch. Rational Mech. Anal. (2006) 180: 399. doi:10.1007/s00205-005-0392-3


We prove a stability result for a large class of unilateral minimality properties which arise naturally in the theory of crack propagation proposed by Francfort & Marigo in [14]. Then we give an application to the quasistatic evolution of cracks in composite materials. The main tool in the analysis is a Γ-convergence result for energies of the form Open image in new window where S(u) is the jump set of u and Open image in new window is a sequence of rectifiable sets with Open image in new window We prove that no interaction occurs in the Γ-limit process between the bulk and the surface part of the energy. Relying on this result, we introduce a new notion of convergence for (N−1)-rectifiable sets called σ-convergence, which is useful in the study of the stability of unilateral minimality properties.

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di BresciaBresciaItaly
  2. 2.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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