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

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

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

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topMünch, 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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