Comptes Rendus
Edge singularities and structure of the 3-D Williams expansion
Comptes Rendus. Mécanique, Volume 336 (2008) no. 8, pp. 629-635.

The elastic solution in a vicinity of a re-entrant wedge can be described by a Williams like expansion in terms of powers of the distance to a point on the edge. This expansion has a particular structure due to the invariance of the problem by translation parallel to the edge. We show here that some terms, so-called primary solutions, derive directly from solutions to the 2-D corner problem posed in the orthogonal cross section of the domain. The others, baptized shadow functions, derive of the primary solutions by integration along the axis parallel to the edge. This 3-D Williams expansion is shown to be equivalent to the edge expansion proposed by Costabel et al. [M. Costabel, M. Dauge, Z. Yosibash, A quasidual function method for extracting edge stress intensity functions, SIAM J. Math. Anal. 35 (5) (2004) 1177–1202].

Les solutions élastiques au voisinage d'un dièdre rentrant peuvent être décrites par un développement de type Williams composé de termes en puissance de la distance à un point de l'arête du dièdre. Ce développement a une structure particulière due à l'invariance du problème par translation parallèle à l'arête. Certains termes, appelés solutions particulières, viennent directement des solutions du problème bidimensionnel autour d'un coin entrant, posé sur la section droite du dièdre. Les autres, baptisés ombres, sont déduits des solutions particulières par intégration le long de l'axe parallèle à l'arête du dièdre. Nous montrons que le développement de Williams tridimensionnel est alors équivalent au développement le long de l'arête proposé par Costabel et al. [M. Costabel, M. Dauge, Z. Yosibash, A quasidual function method for extracting edge stress intensity functions, SIAM J. Math. Anal. 35 (5) (2004) 1177–1202)].

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2008.05.008
Keywords: Elasticity, Edge singularities, Generalized stress intensity factors
Mot clés : Elasticité, Singularités d'arête, Facteurs d'intensité des contraintes généralisés

Thomas Apel 1; Dominique Leguillon 2; Cornelia Pester 3; Zohar Yosibash 4

1 Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, 85577 Neubiberg, Germany
2 Institut Jean Le Rond d'Alembert, CNRS UMR 7190, Université Pierre et Marie Curie, case 162, 4, place Jussieu, 75252 Paris cedex 05, France
3 CST GmbH – Computer Simulation Technology, Bad Nauheimer Str. 19, 64289 Darmstadt, Germany
4 Perlstone Center for Aeronautical Engineering Studies, Dept. of Mechanical Engrg., Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel
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Thomas Apel; Dominique Leguillon; Cornelia Pester; Zohar Yosibash. Edge singularities and structure of the 3-D Williams expansion. Comptes Rendus. Mécanique, Volume 336 (2008) no. 8, pp. 629-635. doi : 10.1016/j.crme.2008.05.008. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2008.05.008/

[1] M. Costabel; M. Dauge; Z. Yosibash A quasidual function method for extracting edge stress intensity functions, SIAM J. Math. Anal., Volume 35 (2004) no. 5, pp. 1177-1202

[2] Z. Yosibash; N. Omer; M. Costabel; M. Dauge Edge stress intensity functions in polyhedral domains and their extraction by a quasidual function method, Int. J. Fracture, Volume 136 (2005), pp. 37-73

[3] D. Leguillon; E. Sanchez-Palencia Computation of Singular Solutions in Elliptic Problems and Elasticity, John Wiley & Sons, New York, 1987 (and Masson, Paris)

[4] Z. Yosibash; B.A. Szabo Numerical analysis of singularities in two dimensions, Part 1: Computation of eigenpairs, Int. J. Num. Meth. Engrg., Volume 38 (1995) no. 12, pp. 2055-2082

[5] M.L. Williams The stresses around a fault or crack in dissimilar media, Bull. Seism. Soc. Amer., Volume 49 (1956), pp. 199-204

[6] D. Leguillon Computation of 3D-singularities in elasticity (M. Costabel; M. Dauge; S. Nicaise, eds.), Boundary Value Problems and Integral Equations on Non-Smooth Domains, Lecture Notes in Pure and Applied Math., vol. 167, Marcel Dekker, New York, 1995, pp. 161-170

[7] D. Leguillon Computation of 3D singular elastic fields for the prediction of failure at corners (F.G. Buchholz; H.A. Richard; M.H. Aliabadi, eds.), Advances in Fracture and Damage Mechanics, Key Engineering Materials, vols. 251–252, 2003, pp. 147-152

[8] A. Dimitrov; H. Andrä; E. Schnack Singularities near three dimensional corners in composite laminates, Int. J. Fract., Volume 115 (2002), pp. 361-375

[9] C. Pester A posteriori error estimation for non-linear eigenvalue problems for differential operators of second order with focus on 3-D vertex singularities, Logos Verlag, Berlin, 2006 (Dissertation Dr. rer. nat., T.U. Chemnitz, Germany)

[10] C. Pester Hamiltonian eigenvalue symmetry for quadratic operator eigenvalue problems, J. Integral Equations Appl., Volume 17 (2005) no. 1, pp. 71-89

[11] V.A. Kozlov; V.G. Maz'ya; J. Rossman Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Problems, Amer. Math. Soc., Providence, RI, 2001

[12] P.E.W. Labossiere; M.L. Dunn Calculation of stress intensities at sharp notches in anisotropic media, Engrg. Fract. Mech., Volume 61 (1998), pp. 635-654

[13] T. Apel; V. Mehrmann; D. Watkins Structured eigenvalue methods for the computation of corner singularities in 3D anisotropic elastic structures, Comp. Methods Appl. Mech. Engrg., Volume 191 (2002), pp. 4459-4473

[14] E. Sommer Formation of fracture ‘lances’ in glass, Engrg. Fract. Mech., Volume 1 (1969), pp. 539-546

[15] F.G. Buchholz; A. Chergui; H.A. Richard Fracture analyses and experimental results of crack growth under general mixed mode loading conditions, Engrg. Fract. Mech., Volume 71 (2004), pp. 455-468

[16] D. Leguillon Calcul du taux de restitution de l'énergie au voisinage d'une singularité, C. R. Acad. Sci. Paris, Volume 309 (1989) no. II, pp. 945-950

[17] Z. Yosibash; N. Omer Numerical methods for extracting edge stress intensity functions in anisotropic three-dimensional domains, Comp. Meth. Appl. Mech. Engrg., Volume 196 (2007), pp. 3624-3649

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