Relaxation of an optimal design problem in fracture mechanic: the anti-plane case

Arnaud Münch¹ and Pablo Pedregal²

¹ UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France. arnaud.munch@univ-fcomte.fr (On leave to Laboratoire de Mathématiques de Clermont-Ferrand)
² ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain. Pablo.Pedregal@uclm.es

Received: 13 February 2008
Revised: 15 November 2008

Abstract

In the framework of the linear fracture theory, a commonly-used tool to describe the smooth evolution of a crack embedded in a bounded domain Ω is the so-called energy release rate defined as the variation of the mechanical energy with respect to the crack dimension. Precisely, the well-known Griffith's criterion postulates the evolution of the crack if this rate reaches a critical value. In this work, in the anti-plane scalar case, we consider the shape design problem which consists in optimizing the distribution of two materials with different conductivities in Ω in order to reduce this rate. Since this kind of problem is usually ill-posed, we first derive a relaxation by using the classical non-convex variational method. The computation of the quasi-convex envelope of the cost is performed by using div-curl Young measures, leads to an explicit relaxed formulation of the original problem, and exhibits fine microstructure in the form of first order laminates. Finally, numerical simulations suggest that the optimal distribution permits to reduce significantly the value of the energy release rate.