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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.
@article{COCV_2010__16_3_719_0,
author = {M\"unch, Arnaud and Pedregal, Pablo},
title = {Relaxation of an optimal design problem in fracture mechanic : the anti-plane case},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {719--743},
publisher = {EDP-Sciences},
volume = {16},
number = {3},
year = {2010},
doi = {10.1051/cocv/2009019},
mrnumber = {2674634},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2009019/}
}
TY - JOUR AU - Münch, Arnaud AU - Pedregal, Pablo TI - Relaxation of an optimal design problem in fracture mechanic : the anti-plane case JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 719 EP - 743 VL - 16 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2009019/ DO - 10.1051/cocv/2009019 LA - en ID - COCV_2010__16_3_719_0 ER -
%0 Journal Article %A Münch, Arnaud %A Pedregal, Pablo %T Relaxation of an optimal design problem in fracture mechanic : the anti-plane case %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 719-743 %V 16 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2009019/ %R 10.1051/cocv/2009019 %G en %F COCV_2010__16_3_719_0
Münch, Arnaud; Pedregal, Pablo. Relaxation of an optimal design problem in fracture mechanic : the anti-plane case. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 3, pp. 719-743. doi: 10.1051/cocv/2009019
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