Abstract
A numerical method for the determination of weight functions relevant to cracked bodies with finite dimensions is presented. The discretized nodal weight function is determined by means of the finite element method, and the assumed form of the crack-face weight function is successfully demonstrated by using the least squares fitting procedure. The explicit weight functions of mode I are presented for plates of finite width and length with single edge, double edge and center cracks. With the use of the uncracked stress field in the crack faces and the crack-face weight function, the efficient calculation of the stress intensity factor is illustrated. The size of the Dugdale plastic zone ahead of the crack tip for a finite plate is estimated from the available weight functions. Some practical examples for various crack configurations and loading systems are given and a very satisfactory degree of accuracy is obtained from the results when compared with the findings of earlier studies.
Résumé
On présente une méthode numérique pour la détermination des fonctions pondérales relatives à des corps fissurés de dimensions finies. Au moyen de la méthode des éléments finis, on détermine la fonction correspondant à une discrétisation nodale, et la forme supposée de la fonction relative à la surface de la fissure se trouve confirmée grâce à une procédure de positionnement par moindres carrés. On présente les fonctions pondérales explicites de Mode 1 pour des tôles de largeur et longueur finies, comportant des fissures sur un seul bord, sur deux bords ou au centre. En appliquant aux faces de la fissure le champ de constraintes en condition non fissurée et la fonction pondérale relative à ces faces, on montre comment calculer efficacement le facteur d'intensité de contrainte. Les fonctions disponibles permettent d'estimer la taille de la zone plastique de Dugdale en avant de la fissure dans le cas d'une tôle de grandeur finie. On fournit quelques exemples pratiques correspondant à diverses configurations de fissures et de modes de sollicitation. Quand on compare les résultats obtenus à ceux résultant d'autres approches, on est frappé par le degré de précision auquel la méthode permet d'accéder.
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Abbreviations
- a :
-
crack length
- E :
-
Young's modulus
- v :
-
Poisson's ratio the prospective crack plane
- K (1) :
-
stress intensity factor for load system (1)
- K (2) :
-
stress intensity factor for load system (2)
- T (2) (s, a):
-
surface traction on the boundary
- s :
-
arc length of the boundary
- u(1) (s, a):
-
displacement field on the tractionS 1
- h (s, a) :
-
weight function on the traction boundaryS 1
- x :
-
distance from the crack mouth
- T (2) y (x, a):
-
stresses in the crack faces normal to the prospective crack plane
- u (1) y (x, a):
-
crack opening displacement
- h (x, a) :
-
weight function on the crack faces
- σ :
-
remote stress
- σ y :
-
yield stress
- c :
-
Dugdale plastic zone size
- W :
-
width of plate
- L :
-
length of plate boundary
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Tsai, CH., Ma, CC. Weight functions for cracks in finite rectangular plates. Int J Fract 40, 43–63 (1989). https://doi.org/10.1007/BF01150865
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DOI: https://doi.org/10.1007/BF01150865