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

Published online by Cambridge University Press:  02 July 2009

Arnaud Münch  and
Pablo Pedregal
Arnaud Münch
Affiliation:
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)
Pablo Pedregal
Affiliation:
ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain. Pablo.Pedregal@uclm.es
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Abstract

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

Keywords

Fracture mechanicsoptimal design problemrelaxationnumerical experiments
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 3 , July 2010 , pp. 719 - 743
Copyright
© EDP Sciences, SMAI, 2009

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References

G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences 146. Springer-Verlag, New York (2002).
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).
H.D. Bui, Mécanique de la rupture fragile. Masson, Paris (1983).
Burger, M., A framework for the construction of level set methods for shape optimization and reconstruction. Interface and Free Boundaries 5 (2003) 301–329. CrossRef
B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences 78. Springer-Verlag, Berlin (1989).
De Gournay, F., Allaire, G. and Jouve, F., Shape and topology optimization of the robust compliance via the level set method. ESAIM: COCV 14 (2008) 43–70. CrossRef
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. II 296 (1983) 745–748.
P. Destuynder, An approach to crack propagation control in structural dynamics. C. R. Acad. Sci. Paris, Sér. II 306 (1988) 953–956.
P. Destuynder, Remarks on a crack propagation control for stationary loaded structures. C. R. Acad. Sci. Paris, Sér. IIb 308 (1989) 697–701.
Destuynder, P., Computation of an active control in fracture mechanics using finite elements. Eur. J. Mech. A/Solids 9 (1990) 133–141.
Destuynder, P., Djaoua, M. and Lescure, S., Quelques remarques sur la mécanique de la rupture élastique. J. Mec. Theor. Appl. 2 (1983) 113–135.
M. Djaoua, Analyse mathématique et numérique de quelques problèmes en mécanique de la rupture. Thèse d'état, Université Paris VI, France (1983).
Francfort, G.A. and Marigo, J.J., Revisiting brittle fracture as an energy minimisation problem. J. Mech. Phys. Solids 46 (1998) 1319–1342. CrossRef
Griffith, A.A., The phenomena of rupture and flow in solids. Phil. Trans. Roy. Soc. London 46 (1920) 163–198.
P. Grisvard, Singularities in boundary value problems, Research in Applied Mathematics. Springer-Verlag, Berlin (1992).
Hild, P., Münch, A. and Ousset, Y., On the control of crack growth in elastic media. C. R. Acad. Sci. Paris Sér. Méc. 336 (2008) 422–427.
Hild, P., Münch, A. and Ousset, Y., On the active control of crack growth in elastic media. Comput. Methods Appl. Mech. Engrg. 198 (2008) 407–419. CrossRef
J.-B. Leblond, Mécanique de la rupture fragile et ductile. Hermes Sciences Publications (2003) 1–197.
K.L. Lurie, An introduction to the mathematical theory of dynamic materials, Advances in Mechanics and Mathematics 15. Springer (2007).
Maestre, F., Münch, A. and Pedregal, P., A spatio-temporal design problem for a damped wave equation. SIAM J. Appl Math. 68 (2007) 109–132. CrossRef
Münch, A., 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.
A. Münch and Y. Ousset, Energy release rate for a curvilinear beam. C. R. Acad. Sci. Paris, Sér. IIb 328 (2000) 471–476.
Münch, A. and Ousset, Y., Numerical simulation of delamination growth in curved interfaces. Comput. Methods Appl. Mech. Engrg. 191 (2002) 2045–2067. CrossRef
Münch, A., Pedregal, P. and Periago, F., Optimal design of the damping set for the stabilization of the wave equation. J. Diff. Eq. 231 (2006) 331–358. CrossRef
Münch, A., Pedregal, P. and Periago, F., Relaxation of an optimal design problem for the heat equation. J. Math. Pures Appl. 89 (2008) 225–247. CrossRef
Münch, A., Pedregal, P. and Periago, F., Optimal internal stabilization of the linear system of elasticity. Arch. Rational Mech. Analysis 193 (2009) 171–193. CrossRef
Murat, F. and Simon, J., Études de problèmes d'optimal design. Lect. Notes Comput. Sci. 41 (1976) 54–62. CrossRef
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. I 343 (2006) 115–118.
Pantz, O. and Trabelsi, K., A post-treatment of the homogenization for shape optimization. SIAM J. Control. Optim. 47 (2008) 1380–1398. CrossRef
P. Pedregal, Parametrized measures and variational principles. Birkhäuser (1997).
Pedregal, P., Vector variational problems and applications to optimal design. ESAIM: COCV 11 (2005) 357–381. CrossRef
Pedregal, P., Optimal design in two-dimensional conductivity for a general cost depending on the field. Arch. Rational Mech. Anal. 182 (2006) 367–385. CrossRef
Pedregal, P., Div-Curl Young measures and optimal design in any dimension. Rev. Mat. Comp. 20 (2007) 239–255.
L. Tartar, An introduction to the Homogenization method in optimal design, in Lecture Notes in Mathematics 1740, A. Cellina and A. Ornelas Eds., Springer, Berlin/Heidelberg (2000) 47–156.