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A review of Finite Fracture Mechanics: crack initiation at singular and non-singular stress raisers

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Abstract

Crack initiation in brittle materials is not covered by classical fracture mechanics that deals only with the growth of pre-existing cracks. In order to overcome this deficiency, the Finite Fracture Mechanics concept assumes the instantaneous formation of cracks of finite size at initiation. Within this framework, a coupled criterion was proposed at the beginning of the 2000’s requiring two necessary conditions to be fulfilled simultaneously. The first one compares the tensile stress to the tensile strength, while the other uses an energy balance and the material toughness. The present analysis is restricted to the 2D case, and, through a wide list of references, it is shown that this criterion gives predictions in agreement with experiments in various cases of stress concentration, which can be classified in two categories: the singularities, i.e. indefinitely growing stresses at a point, and the non-singular stress raisers. It is applied to different materials and structures: notched specimens, laminates, adhesive joints or embedded inclusions. Of course, a lot of work remains to do in these domains but also in domains that are almost not explored such as fatigue loadings and dynamic loadings as well as a sound 3D extension. Some ideas in these directions are issued before concluding that FFM and the coupled criterion have filled a gap in fracture mechanics.

Résumé

L’approche classique de la mécanique de la rupture des matériaux fragiles n’aborde pas les problèmes d’initiation de nouvelles fissures, elle ne traite que la croissance de fissures préexistantes. Afin de surmonter cette déficience, l’approche connue sous la désignation anglo-saxonne de Finite Fracture Mechanics suppose la formation instantanée de fissures de taille finie à l’initiation. Développé dans ce cadre, le critère couplé requiert la vérification simultanée de deux conditions nécessaires : la première compare la contrainte de traction à la résistance en traction du matériau tandis que l’autre utilise une équation de conservation de l’énergie et fait appel à la ténacité. La présente analyse se limite au cas 2D et, à travers une longue liste de références, il est montré que le critère couplé donne, aux points de concentration de contraintes, des prédictions d’amorçage de fissures qui sont en accord avec les expériences. On peut distinguer deux catégories : les singularités, lorsque les contraintes croissent indéfiniment en s’approchant d’un point, et les simples concentrations de contraintes, lorsque celles-ci tout en étant élevées restent bornées. Le critère est appliqué à différents matériaux et structures: éprouvettes entaillées, composites stratifiés, joints adhésifs, inclusions. Bien sûr, beaucoup de travail reste à faire dans ces domaines, mais il existe aussi des sujets qui ne sont pratiquement pas explorés comme la fatigue, les chargements dynamiques ainsi qu’une généralisation aux situations tridimensionnelles. Quelques idées sont émises dans ces directions avant de conclure que la FFM et le critère couplé ont comblé une lacune en mécanique de la rupture.

Zusammenfassung

Die Initiierung von Rissen wird in der klassischen Bruchmechanik nicht umfasst, da sich diese auf die Beschreibung des Verhaltens vorhandener Risse beschränkt. In der Bruchmechanik finiter Risse wird diese Einschränkung durch die Betrachtung der instantanen Entstehung von Rissen endlicher Länge aufgehoben. Im Rahmen dieses Konzeptes wurde ein gekoppeltes Kriterium vorgeschlagen, das eine hinreichende Versagensbedingung in Form zweier gleichzeitig zu erfüllender notwendiger Bedingungen darstellt: eine Bedingung der Festigkeitsmechanik und eine Energiebedingung für den Bruchprozess. Die gegenwärtige Formulierung ist auf zweidimensionale Modelle beschränkt und ist, wie anhand zahlreicher Referenzen dargestellt, geeignet experimentelle Ergebnisse über Versagen an unterschiedlichsten Spannungskonzentratoren korrekt abzubilden. Diese lassen sich klassifizieren in singuläre Spannungskonzentratoren mit lokal unendlich hohen Spannungen und in nicht-singuläre Spannungskonzentratoren mit lokal stark erhöhten aber endlichen Spannungen. Das Kriterium wurde auf verschiedenste Struktursituationen und Materialen angewendet: gekerbte Bauteile, Laminate, Klebverbindungen oder auch Materialeinschlüsse. Jedoch verbleiben weitere unerschlossene Felder für weitere Untersuchungen wie etwa die Betrachtung von zyklischen und dynamischen Lasten oder eine gründliche Erweiterung auf dreidimensionale Risse. Mögliche Ansätze für solche Erweiterungen werden vorgestellt bevor der Schluss gezogen wird, dass die Bruchmechanik finiter Risse mit dem gekoppelten Kriterium eine Lücke in der Bruchmechanik geschlossen hat.

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Notes

  1. It should be noted that Li, Hong and Bažant have proposed a related criterion [19, 130] some years earlier. They developed a crack initiation criterion that also uses the concept of finite cracks to analyse the formation of crack patterns on smooth surfaces. A detailed discussion of the relationship and the fundamental differences to Leguillon’s coupled criterion is given in Sect. 5.1.4.

  2. Sometimes also denoted as Notch Stress Intensity Factor.

  3. First non-singular term of the asymptotic expansion corresponding to tensile normal stresses parallel to the crack.

  4. See Eq. (31) for \(\mathcal {G} = \lim (\varDelta a \rightarrow 0) \varDelta \varPi / \varDelta a \) in the 2D case.

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  1. Fachgebiet Strukturmechanik, Technische Universität Darmstadt, Franziska-Braun-Straße 7, 64287, Darmstadt, Germany

    Philipp Weißgraeber & Wilfried Becker

  2. Institut Jean-Le-Rond-d’Alembert, CNRS UMR 7190, Université Pierre et Marie Curie (Paris-6), Couloir 55-65, Case Courrier 162, 4 Place Jussieu, 75252, Paris Cedex 05, France

    Dominique Leguillon

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Weißgraeber, P., Leguillon, D. & Becker, W. A review of Finite Fracture Mechanics: crack initiation at singular and non-singular stress raisers. Arch Appl Mech 86, 375–401 (2016). https://doi.org/10.1007/s00419-015-1091-7

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