Abstract
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 
where S(u) is the jump set of u and 
is a sequence of rectifiable sets with 
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.
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Communicated by L. Ambrosio
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Giacomini, A., Ponsiglione, M. A Γ-Convergence Approach to Stability of Unilateral Minimality Properties in Fracture Mechanics and Applications. Arch. Rational Mech. Anal. 180, 399–447 (2006). https://doi.org/10.1007/s00205-005-0392-3
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DOI: https://doi.org/10.1007/s00205-005-0392-3