Abstract
The unstable growth of a crack in a large viscoelastic plate is considered within the framework of continuum mechanics. Starting from the local stress and deformation fields at the tip of the crack, a non-linear, first order differential equation is found to describe the time history of the crack size if the stress applied far from the crack is constant. The differential equation contains the creep compliance and the intrinsic surface energy of the material. The surface energy concept for viscoelastic materials is clarified. Inertial effects are not considered, but the influence of temperature is included for thermorheologically simple materials.
Initial crack velocities are given as a function of applied load in closed form, as well as a comparison of calculated crack growth history with experiments. Above a certain high stress, crack propagation ensues at high speeds controlled by material inertia while at a lower limit infinite time is required to produce crack growth. Thus an upper and lower limit criterion of the Griffith type exists. For rate insensitive (elastic) materials the two limits coalesce and only the brittle fracture criterion of Griffith exists. The implications of these results for creep fracture in metals and inorganic glasses are examined.
Résumé
L'accroissement instable d'une fissure dans une tôle viscoélastique de grande dimension est examiné sous Tangle de la mécanique des milieux continus.
En partant des distributions locales des contraintes et des déformations à l'extrémité d'une fissure, on a trouvé une équation différentielle non linéaire et du premier ordre, qui décrit l'évolution de la dimension d'une fissure, dans le cas où une contrainte constante est appliquée à une distance suffisante de cette dernière. Dans l'équation différentielle interviennent le fluage et l'énergie intrinsèque de surface du matériau. Le concept d'énergie de surface est éclairci dans le cas des matériaux viscoélastiques. Les effets d'inertie ne sont pas pris en considération, mais l'influence de la température est étudiée pour des matériaux à rhéologie thermique simple. On exprime les vitesses initiales de fissuration en fonction de la charge appliquée, et on établit une comparaison entre la progression de la propagation de la fissure, déduite de calcul, et les résultats fournis par l'expérience.
Au delà d'un certain seuil de contrainte, la propagation de la fissure se fait à une grande vitesse qui dépend de l'inertie du matériau; sous une certaine limite inférieure, l'accroissement de la fissure ne se produit qu'après un temps infmi. Il existe dès lors un critère, du type de celui de Griffith, à limites supérieure et inférieure.
Dans le cas de matériaux insensibles à l'effet de la vitesse (matériaux élastiques) ces deux limites convergent et seul demeure le critère de rupture fragile de Griffith.
On examine ce qu'impliquent ces résultats dans les ruptures par fluage des métaux et des verres inorganiques.
Zusammenfassung
Es wird das ungehinderte Wachstum eines Risses in einer grossen viskoelastischen Platte vom Gesichtspunkt der Kontinuumsmechanik betrachtet. Unter Gebrauch der lokalen Spannungen and Verformungen an den Spitzen des Risses wird eine Differentialgleichung erster Ordnung abgeleitet, welche die Rissgrösse in Abhängigkeit von der Zeit für eine vom Riss weft entfernte, konstante Spannung gibt. Die Differentialgleichung enthält die viskoelastische Kriechdehnungsfunktion and die dem Material eigene Oberflächenenergie. Die Auffassung der Oberflächenenergie für viskoelastische Materialien ist erläutert. Massenträgkeit ist nicht in Betracht genommen, aber der Einfluss der Temperatur ist für thermorheologisch einfache Materialien eingeschlossen.
Die Anfangsgeschwindigkeiten der Rissbildung werden als Funktion der angelegten Spannung ausgedrückt, wie auch Vergleiche zwischen Berechnung and Versuchsresultaten. Über einer gewissen Spannung breitet sich der Riss so schnell aus, dass die Geschwindigkeit von der Wellenmechanik kontrolliert wird, während an einer un t eren Spannungsgrenze unendliche Zeit für Rissausbreitung benötigt ist. So bestehen zwei Kriterien des Griffith Types mit einer oberen and einer unteren Grenze. Für Materialien die keine Dämpfung aufweisen (elastische Materialien) fallen die zwei Grenzen zusammen and ergeben die Griffith Formel für den Sprödbruch. Folgerungen für den zeitbedingten Bruch in Metallen and inorganischen Gläsern werden angestellt.
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This work was supported by the National Aeronautics and Space Administration Research Grant No. NGL-05-002-005 GALCIT 120.
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Knauss, W.G. Delayed failure — the Griffith problem for linearly viscoelastic materials. Int J Fract 6, 7–20 (1970). https://doi.org/10.1007/BF00183655
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DOI: https://doi.org/10.1007/BF00183655