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

Arnaud Münch; Pablo Pedregal

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 3, page 719-743
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Münch, Arnaud, and Pedregal, Pablo. "Relaxation of an optimal design problem in fracture mechanic: the anti-plane case." ESAIM: Control, Optimisation and Calculus of Variations 16.3 (2010): 719-743. <http://eudml.org/doc/250724>.

@article{Münch2010,
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. },
author = {Münch, Arnaud, Pedregal, Pablo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Fracture mechanics; optimal design problem; relaxation; numerical experiments; fracture mechanics; optimal design problem},
language = {eng},
month = {7},
number = {3},
pages = {719-743},
publisher = {EDP Sciences},
title = {Relaxation of an optimal design problem in fracture mechanic: the anti-plane case},
url = {http://eudml.org/doc/250724},
volume = {16},
year = {2010},
}

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
DA - 2010/7//
PB - EDP Sciences
VL - 16
IS - 3
SP - 719
EP - 743
AB - 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.
LA - eng
KW - Fracture mechanics; optimal design problem; relaxation; numerical experiments; fracture mechanics; optimal design problem
UR - http://eudml.org/doc/250724
ER -

References

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