Comptes Rendus
L’approche variationnelle de la rupture : un exemple de collaboration fructueuse entre mécaniciens et mathématiciens
Comptes Rendus. Mécanique, Online first (2023), pp. 1-23.

Cet article revient sur quelques étapes importantes dans l’élaboration et l’application de l’approche variationnelle de la rupture. Il met l’accent sur le rôle essentiel qu’a joué le dialogue permanent entre mécaniciens et mathématiciens appliqués pour faire émerger ce nouveau paradigme et en assurer un incontestable succès.

This article reviews some important steps in the development and application of the variational approach to fracture. It emphasizes the essential role played by the permanent dialogue between mechanicians and applied mathematicians in bringing about this new paradigm and ensuring its undeniable success.

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DOI : 10.5802/crmeca.170
Mot clés : Endommagement, Rupture, Calcul des Variations, Stabilité, Discontinuités libres
Keywords: Damage, Fracture, Calculus of Variations, Stability, Free discontinuities
Jean-Jacques Marigo 1

1 Institut Jean le Rond d’Alembert (UMR 7190),Sorbonne Université & CNRS, 75005 Paris, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Jean-Jacques Marigo. L’approche variationnelle de la rupture : un exemple de collaboration fructueuse entre mécaniciens et mathématiciens. Comptes Rendus. Mécanique, Online first (2023), pp. 1-23. doi : 10.5802/crmeca.170.

[1] D. C. Drucker A definition of stable inelastic material, ASME J. Appl. Mech., Volume 26 (1959), pp. 101-106 | DOI | MR | Zbl

[2] A. A. Ilyushin On the postulate of stability, J. Appl. Math. Mech., Volume 25 (1961), pp. 746-752 | DOI

[3] G. A. Francfort; J.-J. Marigo Stable damage evolution in a brittle continuous medium, Eur. J. Mech. A/Solids, Volume 12 (1993) no. 2, pp. 149-189 | MR | Zbl

[4] G. Allaire Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002 | DOI

[5] A. Griffith The phenomena of rupture and flow in solids, Philos. Trans. R. Soc. Lond. A, Volume 221 (1921), pp. 163-198

[6] B. Bourdin; G. A. Francfort; J.-J. Marigo The variational approach to fracture, J. Elast., Volume 91 (2008) no. 1-3, pp. 5-148 | DOI | MR | Zbl

[7] G. A. Francfort; J. J. Marigo Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, Volume 46 (1998) no. 8, pp. 1319-1342 | DOI | MR | Zbl

[8] A. Mielke Evolution of rate-independent systems, Evolutionary Equations (Handbook of Differential Equations), Volume II, Elsevier/North-Holland, Amsterdam, 2005, pp. 461-559 | Zbl

[9] A. Chambolle; V. Crismale Existence of strong solutions to the Dirichlet problem for the Griffith energy, Calc. Var. Partial Differ. Equ., Volume 58 (2019) no. 4, 136 | DOI | MR | Zbl

[10] A. Giacomini; M. Ponsiglione A Γ-convergence approach to stability of unilateral minimality properties in fracture mechanics and applications, Arch. Ration. Mech. Anal., Volume 180 (2006) no. 3, pp. 399-447 | DOI | MR | Zbl

[11] L. Ambrosio; V. Tortorelli Approximation of functionals depending on jumps by elliptic functional via gamma-convergence, Commun. Pure Appl. Math., Volume 43 (1990), pp. 999-1036 | DOI

[12] L. Ambrosio; V. Tortorelli On the approximation of free discontinuity problems, Bollettino della unione matematica italiana, Volume 6-B (1992) no. 1, pp. 105-123 | MR | Zbl

[13] A. Braides Γ-convergence for Beginners, Lecture Series in Mathematics and its Applications, 22, Oxford University Press, New York, 2002 | DOI

[14] K. Pham; J.-J. Marigo Approche variationnelle de l’endommagement : I. Les concepts fondamentaux, C. R. Méc., Volume 338 (2010) no. 4, pp. 191-198 | DOI | Zbl

[15] K. Pham; J.-J. Marigo Approche variationnelle de l’endommagement : II. Les modèles à gradient, C. R. Méc., Volume 338 (2010) no. 4, pp. 199-206 | DOI | Zbl

[16] K. Pham; H. Amor; J.-J. Marigo; C. Maurini Gradient damage models and their use to approximate brittle fracture, Int. J. Damage Mech., Volume 20 (2011) no. 4, pp. 618-652 (Isiweb) | DOI

[17] B. Bourdin; G. A. Francfort; J.-J. Marigo Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids, Volume 48 (2000) no. 4, pp. 797-826 | DOI | MR | Zbl

[18] B. Bourdin Numerical implementation of the variational formulation for quasi-static brittle fracture, Interfaces Free Boundaries, Volume 9 (2007), pp. 411-430 | DOI | MR

[19] P. Sicsic; J.-J. Marigo From gradient damage laws to Griffith’s theory of crack propagation, J. Elast., Volume 113 (2013) no. 1, pp. 55-74 | DOI | MR | Zbl

[20] B. Bourdin; J.-J. Marigo; C. Maurini; P. Sicsic Morphogenesis and propagation of complex cracks induced by thermal shocks, Phys. Rev. Lett., Volume 112 (2014) no. 1, 014301 | DOI

[21] P. Sicsic; J.-J. Marigo; C. Maurini Initiation of a periodic array of cracks in the thermal shock problem: a gradient damage modeling, J. Mech. Phys. Solids, Volume 63 (2014), pp. 256-284 | DOI | MR | Zbl

[22] E. Tanné; T. Li; B. Bourdin; J.-J. Marigo; C. Maurini Crack nucleation in variational phase-field models of brittle fracture, J. Mech. Phys. Solids, Volume 110 (2018), pp. 80-99 | DOI | MR

[23] A. Takei; B. Roman; J. Bico Forbidden directions for the fracture of thin anisotropic sheets: an analogy with the Wulff plot, Phys. Rev. Lett., Volume 110 (2013), 144301 | DOI

[24] A. Ibarra; J. F. Fuentealba; B. Roman; F. Mela Predicting tearing paths in thin sheets, Phys. Rev. E, Volume 100 (2019), 023002 | DOI | MR

[25] G. Allaire Conception Optimale de Structures, Mathématiques & Applications, Springer-Verlag, Berlin Heidelberg, 2007

[26] J.-J. Marigo Modelling of brittle and fatigue damage for elastic material by growth of microvoids, Eng. Fract. Mech., Volume 21 (1985) no. 4, pp. 861-874 | DOI

[27] S. Andrieux; Y. Bamberger; J.-J. Marigo A model of micro-craked material for concretes and rocks, J. de Méc. Théor. Appl., Volume 5 (1986) no. 3, pp. 471-513

[28] J.-J. Marigo Constitutive relations in plasticity, damage and fracture mechanics based on a work property, Nucl. Eng. Des., Volume 114 (1989), pp. 249-272 | DOI

[29] G. A. Francfort; A. Garroni A variational view of partial brittle damage evolution, Arch. Ration. Mech. Anal., Volume 182 (2006), pp. 125-152 | DOI | MR | Zbl

[30] J.-F. Babadjian; F. Iurlano; F. Rindler Concentration versus oscillation effects in brittle damage, Commun. Pure Appl. Math., Volume 74 (2021) no. 9, pp. 1803-1854 | DOI | MR | Zbl

[31] G. Barenblatt The mathematical theory of equilibrium cracks in brittle fracture., Adv. Appl. Mech., Volume 7 (1962), pp. 55-129 | DOI

[32] G. D. Maso; F. Iurlano Fracture models as Γ-limits of damage models, Commun. Pure Appl. Anal., Volume 12 (2013), pp. 1657-1686 | DOI | MR

[33] Jean-Jacques Marigo La mécanique de l’endommagement au secours de la mécanique de la rupture : l’évolution de cette idée en un demi-siècle, Comptes Rendus Mécanique, Volume 351 (2023) no. S1 (à paraitre)

[34] Dov Bahat Tectonofractography, Springer-Verlag, Berlin Heidelberg, 1991 | DOI | Zbl

[35] A. Yuse; M. Sano Transition between crack patterns in quenched glass plates, Letters to Nature, Volume 362 (1993), pp. 329-331 | DOI

[36] Y. Shao; Y. Zhang; X. Xu; Z. Zhou; W. Li; B. Liu Effect of crack pattern on the residual strength of ceramics after quenching, J. Am. Ceram. Soc., Volume 94 (2011) no. 9, pp. 2804-2807 | DOI

[37] A. Chambolle; G. A. Francfort; J.-J. Marigo When and how do cracks propagate ?, J. Mech. Phys. Solids, Volume 57 (2009) no. 9, pp. 1614-1622 | DOI | MR | Zbl

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