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

top
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

top

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

top
  1. G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences146. Springer-Verlag, New York (2002).  Zbl0990.35001
  2. G. Allaire, F. Jouve and N. Van Goethem, A level set method for the numerical simulation of damage evolution. Internal report 629, CMAP, École polytechnique, France (2007).  Zbl05587328
  3. H.D. Bui, Mécanique de la rupture fragile. Masson, Paris (1983).  
  4. M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction. Interface and Free Boundaries5 (2003) 301–329.  Zbl1081.35134
  5. B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences78. Springer-Verlag, Berlin (1989).  Zbl0703.49001
  6. F. De Gournay, G. Allaire and F. Jouve, Shape and topology optimization of the robust compliance via the level set method. ESAIM: COCV14 (2008) 43–70.  Zbl1245.49054
  7. P. Destuynder, Calculation of forward thrust of a crack, taking into account the unilateral contact between the lips of the crack. C. R. Acad. Sci. Paris, Sér. II296 (1983) 745–748.  Zbl0572.73106
  8. P. Destuynder, An approach to crack propagation control in structural dynamics. C. R. Acad. Sci. Paris, Sér. II306 (1988) 953–956.  Zbl0633.73099
  9. P. Destuynder, Remarks on a crack propagation control for stationary loaded structures. C. R. Acad. Sci. Paris, Sér. IIb308 (1989) 697–701.  Zbl0665.73075
  10. P. Destuynder, Computation of an active control in fracture mechanics using finite elements. Eur. J. Mech. A/Solids9 (1990) 133–141.  Zbl0706.73060
  11. P. Destuynder, M. Djaoua and S. Lescure, Quelques remarques sur la mécanique de la rupture élastique. J. Mec. Theor. Appl.2 (1983) 113–135.  Zbl0529.73081
  12. M. Djaoua, Analyse mathématique et numérique de quelques problèmes en mécanique de la rupture. Thèse d'état, Université Paris VI, France (1983).  
  13. G.A. Francfort and J.J. Marigo, Revisiting brittle fracture as an energy minimisation problem. J. Mech. Phys. Solids46 (1998) 1319–1342.  Zbl0966.74060
  14. A.A. Griffith, The phenomena of rupture and flow in solids. Phil. Trans. Roy. Soc. London46 (1920) 163–198.  
  15. P. Grisvard, Singularities in boundary value problems, Research in Applied Mathematics. Springer-Verlag, Berlin (1992).  Zbl0766.35001
  16. P. Hild, A. Münch and Y. Ousset, On the control of crack growth in elastic media. C. R. Acad. Sci. Paris Sér. Méc.336 (2008) 422–427.  Zbl1143.74367
  17. P. Hild, A. Münch and Y. Ousset, On the active control of crack growth in elastic media. Comput. Methods Appl. Mech. Engrg.198 (2008) 407–419.  Zbl1228.74052
  18. J.-B. Leblond, Mécanique de la rupture fragile et ductile. Hermes Sciences Publications (2003) 1–197.  
  19. K.L. Lurie, An introduction to the mathematical theory of dynamic materials, Advances in Mechanics and Mathematics15. Springer (2007).  Zbl1125.74001
  20. F. Maestre, A. Münch and P. Pedregal, A spatio-temporal design problem for a damped wave equation. SIAM J. Appl Math.68 (2007) 109–132.  Zbl1147.35052
  21. A. Münch, Optimal design of the support of the control for the 2-D wave equation: numerical investigations. Int. J. Numer. Anal. Model.5 (2008) 331–351.  Zbl1242.49091
  22. A. Münch and Y. Ousset, Energy release rate for a curvilinear beam. C. R. Acad. Sci. Paris, Sér. IIb328 (2000) 471–476.  Zbl0994.74058
  23. A. Münch and Y. Ousset, Numerical simulation of delamination growth in curved interfaces. Comput. Methods Appl. Mech. Engrg.191 (2002) 2045–2067.  Zbl1131.74338
  24. A. Münch, P. Pedregal and F. Periago, Optimal design of the damping set for the stabilization of the wave equation. J. Diff. Eq.231 (2006) 331–358.  Zbl1105.49005
  25. A. Münch, P. Pedregal and F. Periago, Relaxation of an optimal design problem for the heat equation. J. Math. Pures Appl.89 (2008) 225–247.  Zbl1147.35015
  26. A. Münch, P. Pedregal and F. Periago, Optimal internal stabilization of the linear system of elasticity. Arch. Rational Mech. Analysis193 (2009) 171–193.  Zbl1169.74008
  27. F. Murat and J. Simon, Études de problèmes d'optimal design. Lect. Notes Comput. Sci.41 (1976) 54–62.  Zbl0334.49013
  28. M.T. Niane, G. Bayili, A. Sène and M. Sy, Is it possible to cancel singularities in a domain with corners and cracks? C. R. Acad. Sci. Paris, Sér. I343 (2006) 115–118.  Zbl1143.35021
  29. O. Pantz and K. Trabelsi, A post-treatment of the homogenization for shape optimization. SIAM J. Control. Optim.47 (2008) 1380–1398.  Zbl1161.49042
  30. P. Pedregal, Parametrized measures and variational principles. Birkhäuser (1997).  
  31. P. Pedregal, Vector variational problems and applications to optimal design. ESAIM: COCV11 (2005) 357–381.  Zbl1089.49022
  32. P. Pedregal, Optimal design in two-dimensional conductivity for a general cost depending on the field. Arch. Rational Mech. Anal.182 (2006) 367–385.  Zbl1104.74052
  33. P. Pedregal, Div-Curl Young measures and optimal design in any dimension. Rev. Mat. Comp.20 (2007) 239–255.  Zbl1140.49010
  34. L. Tartar, An introduction to the Homogenization method in optimal design, in Lecture Notes in Mathematics1740, A. Cellina and A. Ornelas Eds., Springer, Berlin/Heidelberg (2000) 47–156.  Zbl1040.49022

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.