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A general treatment of crack tip contour integrals

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Given a general balance statement we derive an expression for the associated crack tip flux integral. The conditions under which the integral is physically meaningful and yields a non-trivial result are outlined. To illustrate the approach a number of well known integrals in use in fracture mechanics are derived. It is demonstrated that complementary analogues to these integrals can be derived in a similar fashion and a result indicating the equality of dual integrals under quite general conditions is presented. We discuss the domain integral method as an alternative means of representing crack tip integrals and we show that the method may be interpreted as a particular form of Signorini's theorem of stress means. A discussion of some associated integral identities is presented.


Une expression pour l'intégrale de flux á l'extrémité d'une fissure est obtenue sur base d'un état général d'équilibre. On souligne les conditions pour lesquelles l'intégrale a un sens physique et ne mène pas á des résultats triviaux. L'approche adoptée est illustrée par 1'établissement d'une série d'intégrales bien connues, utilisées en mécanique de rupture. On démontre que l'on peut tirer d'une manière similaire des contreparties compl'ementaires à ces intégrales et on présente un résultat indiquant que les intégrales doubles ainsi établies sont égales dans des conditions très générales. On discute de la méthode d'intégration sur un domaine comme variante de représentation des intégrales à l'extrémité d'une fissure et on montre que cette méthode peut être interprêtée comme une forme particulière du théorème de Signorini des contraintes médianes. On présente enfin une discussion sur diverses formes d'intégrales associées.

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Moran, B., Shih, C.F. A general treatment of crack tip contour integrals. Int J Fract 35, 295–310 (1987). https://doi.org/10.1007/BF00276359

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