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)].
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Mots-clés : Elasticité, Singularités d'arête, Facteurs d'intensité des contraintes généralisés
Thomas Apel 1; Dominique Leguillon 2; Cornelia Pester 3; Zohar Yosibash 4
@article{CRMECA_2008__336_8_629_0,
author = {Thomas Apel and Dominique Leguillon and Cornelia Pester and Zohar Yosibash},
title = {Edge singularities and structure of the {3-D} {Williams} expansion},
journal = {Comptes Rendus. M\'ecanique},
pages = {629--635},
publisher = {Elsevier},
volume = {336},
number = {8},
year = {2008},
doi = {10.1016/j.crme.2008.05.008},
language = {en},
}
TY - JOUR AU - Thomas Apel AU - Dominique Leguillon AU - Cornelia Pester AU - Zohar Yosibash TI - Edge singularities and structure of the 3-D Williams expansion JO - Comptes Rendus. Mécanique PY - 2008 SP - 629 EP - 635 VL - 336 IS - 8 PB - Elsevier DO - 10.1016/j.crme.2008.05.008 LA - en ID - CRMECA_2008__336_8_629_0 ER -
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/
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