Relaxation of an optimal design problem in fracture mechanic : the anti-plane case
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 3, pp. 719-743.

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.

DOI: 10.1051/cocv/2009019
Classification: 49J45,  65N30
Keywords: fracture mechanics, optimal design problem, relaxation, numerical experiments
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Münch, Arnaud; Pedregal, Pablo. Relaxation of an optimal design problem in fracture mechanic : the anti-plane case. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 3, pp. 719-743. doi : 10.1051/cocv/2009019. http://www.numdam.org/articles/10.1051/cocv/2009019/

[1] G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences 146. Springer-Verlag, New York (2002). | Zbl

[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).

[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 Boundaries 5 (2003) 301-329. | Zbl

[5] B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences 78. Springer-Verlag, Berlin (1989). | Zbl

[6] F. De Gournay, G. Allaire and F. Jouve, Shape and topology optimization of the robust compliance via the level set method. ESAIM: COCV 14 (2008) 43-70. | Numdam

[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. II 296 (1983) 745-748. | Zbl

[8] P. Destuynder, An approach to crack propagation control in structural dynamics. C. R. Acad. Sci. Paris, Sér. II 306 (1988) 953-956. | Zbl

[9] P. Destuynder, Remarks on a crack propagation control for stationary loaded structures. C. R. Acad. Sci. Paris, Sér. IIb 308 (1989) 697-701. | Zbl

[10] P. Destuynder, Computation of an active control in fracture mechanics using finite elements. Eur. J. Mech. A/Solids 9 (1990) 133-141. | Zbl

[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. | Zbl

[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. Solids 46 (1998) 1319-1342. | Zbl

[14] A.A. Griffith, The phenomena of rupture and flow in solids. Phil. Trans. Roy. Soc. London 46 (1920) 163-198.

[15] P. Grisvard, Singularities in boundary value problems, Research in Applied Mathematics. Springer-Verlag, Berlin (1992). | Zbl

[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. | Zbl

[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.

[18] J.-B. Leblond, Mécanique de la rupture fragile et ductile. Hermes Sciences Publications (2003) 1-197. | Zbl

[19] K.L. Lurie, An introduction to the mathematical theory of dynamic materials, Advances in Mechanics and Mathematics 15. Springer (2007). | Zbl

[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. | Zbl

[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.

[22] A. Münch and Y. Ousset, Energy release rate for a curvilinear beam. C. R. Acad. Sci. Paris, Sér. IIb 328 (2000) 471-476. | Zbl

[23] A. Münch and Y. Ousset, Numerical simulation of delamination growth in curved interfaces. Comput. Methods Appl. Mech. Engrg. 191 (2002) 2045-2067. | Zbl

[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. | Zbl

[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. | Zbl

[26] A. Münch, P. Pedregal and F. Periago, Optimal internal stabilization of the linear system of elasticity. Arch. Rational Mech. Analysis 193 (2009) 171-193. | Zbl

[27] F. Murat and J. Simon, Études de problèmes d'optimal design. Lect. Notes Comput. Sci. 41 (1976) 54-62. | Zbl

[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. I 343 (2006) 115-118. | Zbl

[29] O. Pantz and K. Trabelsi, A post-treatment of the homogenization for shape optimization. SIAM J. Control. Optim. 47 (2008) 1380-1398. | Zbl

[30] P. Pedregal, Parametrized measures and variational principles. Birkhäuser (1997). | Zbl

[31] P. Pedregal, Vector variational problems and applications to optimal design. ESAIM: COCV 11 (2005) 357-381. | Numdam | Zbl

[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. | Zbl

[33] P. Pedregal, Div-Curl Young measures and optimal design in any dimension. Rev. Mat. Comp. 20 (2007) 239-255. | Zbl

[34] 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. | Zbl

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