Analytical and numerical evaluation of crack-tip plasticity of an axisymmetrically loaded penny-shaped crack
Comptes Rendus. Mécanique, Volume 336 (2008) no. 1-2, pp. 54-68.

Analytical and numerical approaches are used to solve an axisymmetric crack problem with a refined Barenblatt–Dugdale approach. The analytical method utilizes potential theory in classical linear elasticity, where a suitable potential is selected for the treatment of the mixed boundary problem. The closed-form solution for the problem with constant pressure applied near the tip of a penny-shaped crack is studied to illustrate the methodology of the analysis and also to provide a fundamental solution for the numerical approach. Taking advantage of the superposition principle, an exact solution is derived to predict the extent of the plastic zone where a Tresca yield condition is imposed, which also provides a useful benchmark for the numerical study presented in the second part. For an axisymmetric crack, the numerical discretization is required only in the radial direction, which renders the programming work efficient. Through an iterative scheme, the numerical method is able to determine the size of the crack tip plasticity, which is governed by the nonlinear von Mises criterion. The relationships between the applied load and the length of the plastic zone are compared for three different yielding conditions.

On utilise des approches analytiques et numériques pour résoudre un problème de fissure axisymétrique avec un modèle de Barenblatt–Dugdale raffiné. La méthode analytique utilise la théorie du potentiel en élasticité linéaire classique, un potentiel approprié étant choisi pour le traitement du problème aux limites mixte. La solution complète du problème comportant une pression uniforme appliquée au voisinage du front d'une fissure en forme de pièce de monnaie est étudiée afin d'illustrer la méthodologie de l'analyse et également fournir une solution de référence pour l'approche numérique. Grâce au principe de superposition, une solution exacte est obtenue afin de prédire l'étendue de la zone plastique où une condition de Tresca est imposée, ce qui fournit aussi un test utile pour qualifier l'étude numérique présentée dans la seconde partie. Pour une fissure axisymétrique, seule une discrétisation dans la direction radiale est requise, ce qui permet une programmation efficace. Grâce à une procédure itérative, la méthode numérique est capable d'évaluer la taille de la zone plastique, qui est déterminée par le critère non-linéaire de von Mises. Les relations entre le chargement appliqué et la taille de la zone plastique sont comparées pour trois conditions d'écoulement différentes.

Published online:
DOI: 10.1016/j.crme.2007.10.015
Keywords: Penny-shaped crack, Crack tip plasticity, Dugdale approach, Tresca criterion, Von Mises criterion
Mot clés : Fissure en forme de pièce de monnaie, Plasticité en pointe de fissure, Approche de Dugdale, Critère de Tresca, Critère de von Mises

Sumitra Chaiyat 1; Xiaoqing Jin 2; Leon M. Keer 2; Kraiwood Kiattikomol 1

1 Department of Civil Engineering, King Mongkut's University of Technology, Thonburi, Bangkok 10140, Thailand
2 Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA
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Sumitra Chaiyat; Xiaoqing Jin; Leon M. Keer; Kraiwood Kiattikomol. Analytical and numerical evaluation of crack-tip plasticity of an axisymmetrically loaded penny-shaped crack. Comptes Rendus. Mécanique, Volume 336 (2008) no. 1-2, pp. 54-68. doi : 10.1016/j.crme.2007.10.015. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.10.015/

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