On présente dans cette Note une méthode dʼidentification du front dʼune fissure débouchant à la surface dʼun solide élastique, à partir de la donnée des composantes tangentielles du champ de déplacement sur une partie libre de charge de la surface extérieure. La méthode comporte deux étapes. Dans un premier temps, on résoud un problème de Cauchy pour prolonger le champ en surface jusquʼà une surface englobant la fissure inconnue. Dans une seconde étape, on utilise la méthode dʼécart la réciprocité pour identifier le saut de déplacement à la traversée de la fissure, ce qui conduit, par lʼidentification du support, à lʼidentification de la fissure elle-même. On prouve ainsi un résultat dʼidentifiabilité. La méthode est illustrée sur deux exemples synthétiques : une fissure traversante à front rectiligne et une fissure elliptique débouchante.
We present in this Note an identification method for the crack front of a crack emerging at the surface of an elastic solid, provided displacements field or its tangential components are given on a part free of loading of the external surface. The method is based on two steps. The first one is the solution of a Cauchy problem in order to expand the displacement field within the solid up to a surface enclosing the unknown crack. Then the reciprocity gap method is used in order to determine the displacement jump on the crack and then the crack itself. We prove then an identifiability result. The method is illustrated with two synthetic examples: a crossing crack with linear crack front and an elliptic emerging crack.
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Mots clés : Solides et structures, Problèmes inverses, Identification de fissure, Problème de Cauchy, Elasticité, Mesures de champs, Complétion de données
@article{CRMECA_2012__340_8_565_0,
author = {St\'ephane Andrieux and Thouraya Nouri Baranger},
title = {Emerging crack front identification from tangential surface displacements},
journal = {Comptes Rendus. M\'ecanique},
pages = {565--574},
publisher = {Elsevier},
volume = {340},
number = {8},
year = {2012},
doi = {10.1016/j.crme.2012.06.002},
language = {en},
}
TY - JOUR AU - Stéphane Andrieux AU - Thouraya Nouri Baranger TI - Emerging crack front identification from tangential surface displacements JO - Comptes Rendus. Mécanique PY - 2012 SP - 565 EP - 574 VL - 340 IS - 8 PB - Elsevier DO - 10.1016/j.crme.2012.06.002 LA - en ID - CRMECA_2012__340_8_565_0 ER -
Stéphane Andrieux; Thouraya Nouri Baranger. Emerging crack front identification from tangential surface displacements. Comptes Rendus. Mécanique, Volume 340 (2012) no. 8, pp. 565-574. doi : 10.1016/j.crme.2012.06.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2012.06.002/
[1] Application of an optimized digital correlation method to planar deformation analysis, Image Vis. Comput., Volume 4 (1986), pp. 143-150
[2] Spectral approach to displacement evaluation from image analysis, Eur. Phys. J., Appl. Phys., Volume 17 (2002), pp. 247-252
[3] S. Andrieux, T.N. Baranger, Three-dimensional recovery of stress intensity factors and energy release rates from surface full-field measurements, submitted for publication.
[4] Noise-robust stress intensity factor determination from kinematic field measurements, Eng. Fract. Mech., Volume 75 (2008), pp. 3763-3781
[5] Evaluating mixed-mode stress intensity factors from full-field displacement fields obtained by optical methods, Eng. Fract. Mech., Volume 74 (2006) no. 9, pp. 1399-1412
[6] Identification of internal cracks in a three-dimensional solid body via Steklov–Poincaré approaches, C. R. Mecanique, Volume 339 (2011) no. 10, pp. 674-681
[7] Constitutive law gap functionals for solving the Cauchy problem for linear elliptic PDE, Appl. Math. Comput., Volume 218 (2011) no. 5, pp. 1970-1989
[8] An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity, Comput. Methods Appl. Mech. Engrg., Volume 197 (2008) no. 9–12, pp. 902-920
[9] An optimization approach for the Cauchy problem in linear elasticity, J. Multidiscip. Optim., Volume 35 (2008) no. 2, pp. 141-152
[10] Reciprocity principle and crack identification, Inverse Problems, Volume 15 (1999) no. 1, pp. 59-65
[11] Optimal Control of Distributed Systems—Theory and Applications, Transl. Math. Monogr., vol. 187, American Mathematical Society, Providence, RI, 2000
[12] Numerical analysis of an energy-like minimization method to solve the Cauchy problem with noisy data, J. Comput. Appl. Math., Volume 235 (2011), pp. 3257-3269
[13] Analyse fonctionnelle : Théorie et applications, Dunod, Paris, 1999
[14] Comsol Multiphysics, Finite Element Analysis Software http://www.comsol.com
[15] MATLAB, The MathWorks Inc., 2012.
[16] Why is the Cauchy problem severely ill-posed?, Inverse Problems, Volume 23 (2007), pp. 823-836
[17] Image recovery via total variation minimization and related problems, Numer. Math., Volume 76 (1997), pp. 167-188
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