[Sur le contrôle de la propagation de fissure en milieu élastique]
Dans le cadre de la mécanique linéaire de la rupture, le critère de Griffith postule la croissance d'une fissure si le taux de restitution de l'énergie associé excède une valeur critique. On considère dans cette Note le problème d'optimisation de position qui consiste à minimiser ce taux en appliquant à la structure un chargement de frontière additionnel de support disjoint du chargement initial. On donne une condition suffisante d'existence de solution, on introduit une relaxation du problème dans le cas général, puis on présente une simulation numérique suggérant que ce problème non linéaire est en fait bien posé.
In the framework of linear fracture theory, the Griffith criterion postulates the growth of any crack if the corresponding so-called energy release rate, defined as the variation of the mechanical energy, reaches a critical value. We consider in this Note the optimal location problem which consists in minimizing this rate by applying to the structure an additional boundary load having a support which is disjoint from the support of the initial load possibly responsible of the growth. We give a sufficient well-posedness condition, introduce a relaxed problem in the general case, and then present a numerical experiment which suggests that the original nonlinear problem is actually well-posed.
Accepté le :
Publié le :
Mots-clés : Solides et structures, Mécanique linéaire de la rupture, Contrôle
Patrick Hild 1 ; Arnaud Münch 1 ; Yves Ousset 2
@article{CRMECA_2008__336_5_422_0, author = {Patrick Hild and Arnaud M\"unch and Yves Ousset}, title = {On the control of crack growth in elastic media}, journal = {Comptes Rendus. M\'ecanique}, pages = {422--427}, publisher = {Elsevier}, volume = {336}, number = {5}, year = {2008}, doi = {10.1016/j.crme.2008.02.005}, language = {en}, }
Patrick Hild; Arnaud Münch; Yves Ousset. On the control of crack growth in elastic media. Comptes Rendus. Mécanique, Volume 336 (2008) no. 5, pp. 422-427. doi : 10.1016/j.crme.2008.02.005. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2008.02.005/
[1] Shapes and Geometries – Analysis, Differential Calculus and Optimization, Advances in Design and Control, SIAM, 2001
[2] An approach of the crack propagation control in structural dynamics, C. R. Acad. Sci. Paris, Sér. II, Volume 306 (1988), pp. 953-956
[3] Remarks on a crack propagation control for stationary loaded structures, C. R. Acad. Sci. Paris, Sér. IIb, Volume 308 (1989)
[4] Computation of an active control in fracture mechanics, Eur. J. Mech. A Solids, Volume 9 (1990)
[5] Quelques remarques sur la mécanique de la rupture élastique, J. Mec. Theor. Appl., Volume 2 (1983) no. 1, pp. 113-135
[6] The phenomena of rupture and flow in solids, Philos. Trans. Roy. Soc. London, Volume 46 (1920) no. 8, pp. 163-198
[7] P. Hild, A. Münch, Y. Ousset, On the active control of crack growth in elastic media, Preprint 38-2007, Univ. Franche-Comté
[8] Modelling and control in solids mechanics, Inter. Series Numer., Volume 122 (1997), pp. 1-366
[9] Mécanique de la rupture fragile et ductile, Hermes Sciences, 2003 (pp. 1–197)
[10] Numerical simulation of delamination growth in curved interfaces, Comput. Methods Appl. Mech. Engrg., Volume 191 (2002), pp. 2045-2067
[11] Optimal design of the damping set for the stabilization of the wave equation, J. Differential Equations, Volume 231 (2006), pp. 331-358
[12] et al. Is it possible to cancel singularities in a domain with cracks?, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006), pp. 115-188
- Necessary conditions of first-order for an optimal boundary control problem for viscous damage processes in 2D, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 24 (2018) no. 2, pp. 579-603 | DOI:10.1051/cocv/2017041 | Zbl:1406.35392
- Rigidity parameter identification for thin inclusions located inside elastic bodies, Journal of Optimization Theory and Applications, Volume 172 (2017) no. 1, pp. 281-297 | DOI:10.1007/s10957-016-1025-8 | Zbl:1357.74050
- On crack propagations in elastic bodies with thin inclusions, Sibirskie Èlektronnye Matematicheskie Izvestiya, Volume 14 (2017), pp. 586-599 | DOI:10.17377/semi.2017.14.050 | Zbl:1386.35394
- Optimal control of rigidity parameters of thin inclusions in composite materials, ZAMP. Zeitschrift für angewandte Mathematik und Physik, Volume 68 (2017) no. 2, p. 12 (Id/No 47) | DOI:10.1007/s00033-017-0792-x | Zbl:1371.35125
- Rupture of intermetallic networks and strain localization in cast AlSi12Ni alloy: 2D and 3D characterization, Acta Materialia, Volume 112 (2016), p. 162 | DOI:10.1016/j.actamat.2016.04.020
- A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media, M
AS. Mathematical Models Methods in Applied Sciences, Volume 25 (2015) no. 14, pp. 2749-2793 | DOI:10.1142/s0218202515500608 | Zbl:1325.35212 - Shape-Topological Differentiability of Energy Functionals for Unilateral Problems in Domains with Cracks and Applications, Optimization with PDE Constraints, Volume 101 (2014), p. 243 | DOI:10.1007/978-3-319-08025-3_8
- Shape control of thin rigid inclusions and cracks in elastic bodies, Archive of Applied Mechanics, Volume 83 (2013) no. 10, pp. 1493-1509 | DOI:10.1007/s00419-013-0759-0 | Zbl:1293.74136
- Optimal control of inclusion and crack shapes in elastic bodies, Journal of Optimization Theory and Applications, Volume 155 (2012) no. 1, pp. 54-78 | DOI:10.1007/s10957-012-0053-2 | Zbl:1287.49046
- Relaxation of an optimal design problem in fracture mechanic: the anti-plane case, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 16 (2010) no. 3, pp. 719-743 | DOI:10.1051/cocv/2009019 | Zbl:1221.49074
Cité par 10 documents. Sources : Crossref, zbMATH
Commentaires - Politique