High speed fracture in brittle materials: Supersonic crack propagation

Microcircuit resistance grids were deposited on the surface of glass and glass-ceramic single edge notched beam (SENB) specimens. The individual strips were as small as ~ 10 μm in width. The specimens were broken in bending and the signal from the grids was captured using a waveform recorder. It was observed that the crack began to propagate with a nonzero initial velocity provided the initial notch was blunt. With continued crack propagation, the crack velocity decreased. In specimens with sharp notches, the crack began propagating with a near zero velocity and the velocity increased with increasing crack length. In some glass specimens with blunt notches, the initial crack velocity was found to be considerably greater than the sound velocity thereby showing that supersonic crack propagation can occur.     

Fracture toughness and crack morphology in indentation fracture of brittle materials

The relationship between the indentation fracture toughness, K _c, and the fractal dimension of the crack, D, has been examined on the indentation-fractured specimens of SiC and AIN ceramics, a soda-lime glass and a WC-8%Co hard metal. A theoretical analysis of the crack morphology based on a fractal geometry model was then made to correlate the fractal dimension of the crack, D, with the fracture toughness, K _IC, in brittle materials. The fractal dimension of the indentation crack, D, was found to be in the range 1.024–1.145 in brittle materials in this study. The indentation fracture toughness, K _c, increased with increasing fractal dimension, D, of the crack in these materials. According to the present analysis, the fracture toughness, K _IC, can be expressed as the following function of the fractal dimension of the crack, D, such that $$In K_{IC} = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\{ In[2\Gamma E/(1 - \nu ^2 )] - (D - 1)In r_L \}$$ Where Γ is the work done in creating a unit crack surface, E is Young's modulus, v is Poisson's ratio, and r _L is r _min/r _max, the ratio of the lower limit, r _min, to the upper limit, r _max, of the scale length, r, between which the crack exhibits a fractal nature (r _min ?r?r _max). The experimental data (except for WC-8%Co hard metal) obtained in this study and by other investigators have been fitted to the above equation. The factors which affect the prediction of the value of K _IC from the above equation have been discussed.     

Creep crack growth in brittle materials

During steady state crack growth by diffusive cavitation at grain boundaries, crack tip fields are relaxed due to the presence of a cavitation zone. In the present analysis, analytic solutions for the actual crack tip stress fields and the crack velocity in the presence of cavitation zone consisting of continuously distributed cavities ahead of the crack tip are derived using the smeared volume concept. Results indicate that the r ^–1/2 singularity is now attenuated to r ^{–1/2 +} (0<<1/2) singularity. The singularity attenuation parameter is a function of the crack velocity and material parameters. The crack growth rate is related to the mode I stress intensity factor K by K ² at relatively high load, K ⁿ at intermediate load, and approaches zero at small load near K _th. Meanwhile, the cavitation zone extends further into the material due to the stress relaxation at the crack tip and the subsequent stress redistribution. Such relaxation effects become very distinct at low crack velocity and low applied load. Key words: Creep crack growth, brittle material, diffusive cavity growth, sintering stress, crack tip stress field.     

Generalized fracture toughness of brittle materials

A generalization of the concept of fracture toughness of brittle materials, when subjected to multiaxial loadings, is presented. The theory characterizes the fracture strength of materials under any combination of the three basic modes of crack surfaces displacement.With reference to three-dimensional loading systems, the fracture toughness may be represented, in theK ₁ K ₂ K ₃ Cartesian orthogonal space, by a surface Fracture Envelope characteristic for a specified material, whose equation is determined by the (symmetric) fracture toughnessK _1c and Poisson's ratio .It is shown that the most general fracture process, resulting from the combination of the opening mode of the tangential stress component and the tearing mode of the antiplane shear, may be conveniently analyzed with the aid of the generalized fracture toughness concept. From the knowledge of the Fracture Envelope relative to a structural material, a simple fracture criterion permits forecasting crack propagation for any combination of loads and geometries.The theory is applied to mixed-mode problems to define the analytic threshold of fatigue crack growth.

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Résumé On présente une généralisation du concept de ténacité à la rupture de matériaux fragiles soumis à des contraintes multiaxiales. La théorie proposée caractérise la résistance à la rupture des matériaux sous toutes les combinaisons possibles des trois modes de base des déplacements des surfaces d'une fissure. Par rapport à un système de mise en charge à deux dimensions, la ténacité à la rupture peut être représentée dans un espace orthogonal cartésienK ₁ K ₂ K ₃ par une Enveloppe de Rupture caractéristique d'un matériau donné dont l'équation est déterminée par la ténacité à la rupture symétriqueK _c et le module de Poisson .On montre que le processus de rupture le plus général qui résulte de la combinaison d'une ouverture sous l'effet de la composante tangentielle de la contrainte et d'un arrachement sous l'effet du cisaillement antiplanaire peut être analysé d'une manière satisfaisanteà l'aide du concept de ténacité à la rupture généralisée. A partir de la connaissance de l'Enveloppe de Rupture relative à un matériau de construction déterminé, un critère simple de rupture permet de prévoir la propagation d'une fissure pour toutes combinaisons de contraintes et de géométries.La théorie est appliquée à des problèmes de fissure suivant des modes mixtes en vue de définir de manière analytique le seuil de propagation d'une fissure de fatigue.

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Operated for the U.S. Department of Energy, Contract No. DE-AC12-76N00052.     

Quasi-sudden brittle fracture at both edges of a finite crack

The diffraction of a plane horizontally polarized shear wave by a crack of finite length is analyzed and the extension of both crack edges prior to the arrival of the first diffracted waves, i.e. quasi-sudden fracture, is studied. In light of an energy rate balance criterion it is found that for an incident step-stress pulse, quasi-sudden fracture may occur but always at both crack edges, often initiating at the trailing edge first. For an incident wave whose stress vanishes at the wavefront, however, quasi-sudden fracture may occur only at the leading crack edge, or if at both edges, at the leading edge first. For both waveforms, the rate of crack extension is non-constant and increases rapidly so that crack branching may be expected. Finally instantaneous crack extension at a uniform rate is possible only if the incident wave stress possesses a square-root sinularity at the wavefront. This result agrees with earlier work by Achenbach.     

Mixed-mode fracture of brittle cellular materials

Dimensional argument analysis and near-tip singular in-plane shear stress of a continuum model have been employed to derive the expression for mode II fracture toughness of brittle cellular materials. It was found that both mode I and II fracture toughnesses have the same dependence on cell size, relative density and modulus of rupture of solid cell walls, except a microstructure coefficient included in their expressions. In addition, the linear superposition principle was applied to calculate the bending moment exerted at the first unbroken cell wall for brittle cellular materials under a combined loading of uniform tensile and in-plane shear stresses. The resulting mixed-mode fracture criterion was compared to existing experimental data in PVC foams; agreement was found to be good.     

The fractal effect of irregularity of crack branching on the fracture toughness of brittle materials

Microstructural observations of brittle materials indicated that a variety of microdefect events can be responsible not only for inelastic behaviour, but also for macroscopic crack front irregularity. This irregularity produces an increase in the fracture toughness of the material. In this paper, this irregularity is analysed by fractal geometry in a very simple manner; a fractal model of crack branching is established. Both microscopic and macroscopic analytical results show that the toughness can be raised appreciably as a fractal geometric effect of the irregularity.

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Résumé Des observations microscopiques sur des matériaux fragiles ont montré qu'une variété d'évènements à l'échelle du microdéfaut peuvent être responsables non seulement du comportement inélastique, mais aussi de l'irrégularité du front d'une fissure macroscopique. Cette irrégularité provoque un accroissement de la ténacité à la rupture du matériau. Dans cette étude, on analyse de manière très simple cette irrégularité par fractogéométrie (Mandelbrot) et on établit un modèle fractal relatif à une fissure qui se ramifie. Les résultats de l'analyse microscopique et macroscopique montrent qu'un effet fractogéométrique de l'irrégularité du front de fissure est d'accroitre de manière appréciable la ténacité.

     

Theory of dynamic crack branching in brittle materials

The problem of dynamic symmetric branching of a tensile crack propagating in a brittle material is studied within Linear Elastic Fracture Mechanics theory. The Griffith energy criterion and the principle of local symmetry provide necessary conditions for the onset of dynamic branching instability and for the subsequent paths of the branches. The theory predicts a critical velocity for branching and a well defined shape described by a branching angle and a curvature of the side branches. The model rests on a scenario of crack branching based on reasonable assumptions and on exact dynamic results for the anti-plane branching problem. Our results reproduce within a simplified 2D continuum mechanics approach the main experimental features of the branching instability of fast cracks in brittle materials.     

Analysis of smooth crack growth in brittle materials

When a fault in the crust extends, earthquake faults appear as a set of displacement discontinuities on the ground surface and cause strong motion and large deformation. For the prediction of such earthquake faults, one needs to analyze the propagation of smoothly growing cracks. This paper develops an analysis method based on a new formulation of growing crack problems. In this method, the change in the stress intensity factors due to a small extension is explicitly related to the curvature and length of the extension, and these geometrical parameters can be determined from assumed fracture criteria, without taking any trial-and-error routes of the extension geometry. The validity of the proposed method is numerically examined; in particular, the predicted stress intensity factor changes coincide with numerically computed ones. The proposed method is applied to reproduce two experimental observations. It is shown that in both cases, the configuration of the simulated crack is in good agreement with the observed one.     

On brittle fracture paths

Two classes of fracture are defined: I — fracture path completely predictable, and II — fracture path predictable only after initial random propagation. Class I fractures occur when there is a line of principal stress passing through the tip of the initiating notch or slit across which the stress is a maximum away from the tip. All Class II fractures eventually become Class I.Although the probability for incremental growth in any one direction is low, it is shown that the line of local symmetry is a highly probable path for macroscopic growth. This condition is equally true for slow or fast fractures and is the explanation of branching at high velocity since with fast fractures two lines of local symmetry pass through the crack tip.

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Zusammenfassung Die Brueche werden in zwei Klassen eingeteilt: I — Bruchweg voellig voraussagbar, and II — Bruchweg nur nach anfaenglich zufaelliger Bruchfortpflanzung voraussagbar. Brueche der Klasse I erfolgen wenn eine Hauptspannungslinie durch die Spitze der den Bruch verursachenden Kerbe oder des Schlitzes laeuft ueber die die Spannung ein von der Kerb- oder Schlitzspitze entferntes Maximum hat. Alle zur Klasse II gehoerigen Brueche werden nach gewisser Zeit zu Bruechen der Klasse I.Obgleich die Wahrscheinlichkeit microskopischer Bruchfortpflanzung in jeder anderen Richtung klein ist, wind gezeigt, dass die Linie oertlicher Symmetrie eine makroskopische Fortpflanzungsrichtung hoher Wahrscheinlichkeit darstellt. Diese Bedingung ist in gleicher Weise fuer langsame and Schnelle Brueche zutreffend und ist die Erklaerung fuer die bei hohen Bruchgeschwindigkeiten auftretenden Verzweigungen da bei schnellen Bruechen zwei Linien oertlicher Symmetrie durch die Bruchspitze laufen.

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Résumé Deux classes de rupture sont définies: I — trajectoire de la fracture entièrement prédite, et II, trajectoire de la fracture prédite uniquement après une propagation initiale probable. Les ruptures de classe I arrivent quand il y a une ligne de contrainte principale passant par l' extrémité d' une entaille ou fente naissante, au travers de laquelle la contrainte est maximum loin du bout. Toutes les ruptures de classe II deviennent éventuellement classe I. Bien que la probabilité pour avoir un accroissement différentiel, dans n'importe quelle direction soit faible, on a montré que la ligne de symmétrie locale est une trajectoire hautement probable pour un accroissement macroscopique. Cette condition est également vraie pour des fractures lentes ou rapides,et est l'explication de l'embranchement a grande vitesse, puisque avec des fractures rapides, deux lignes de symmétrie locale passent par l'extrémité de la rupture.