Tilting cracks: the evolution of fracture surface topology in brittle solids

For a growing crack the conventional definitions of tilting and twisting are inadequate. New definitions are proposed, based on differential geometry, and it is shown that, in homogeneous, isotropic, brittle solids, non-planar crack growth must occur entirely by tilting movements. Examples are given of the growth of cracks on curved surfaces which illustrate that the no-twist condition produces significant constraints on the path of fracture. The development of fracture surfaces when cracks are subject to mixed-mode loading conditions are described with particular reference to the influence of mode III, twisting conditions. It is shown that the requirement that only tilting can occur leads to many characteristic fractographic features.     

A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids

This paper presents a single-domain boundary element method (BEM) analysis of fracture mechanics in 2D anisotropic piezoelectric solids. In this analysis, the extended displacement (elastic displacement and electrical potential) and extended traction (elastic traction and electrical displacement) integral equations are collocated on the outside boundary (no-crack boundary) of the problem and on one side of the crack surface, respectively. The Green's functions for the anisotropic piezoelectric solids in an infinite plane, a half plane, and two joined dissimilar half-planes are also derived using the complex variable function method. The extrapolation of the extended relative crack displacement is employed to calculate the extended `stress intensity factors' (SIFs), i.e., K_I, K_II, K_III and K_IV. For a finite crack in an infinite anisotropic piezoelectric solid, the extended SIFs obtained with the current numerical formulation were found to be very close to the exact solutions. For a central and inclined crack in a finite and anisotropic piezoelectric solid, we found that both the coupled and uncoupled (i.e., the piezoelectric coefficient e_ijk=0) cases predict very similar stress intensity factors K_I and K_II when a uniform tension σ_yy is applied, and very similar electric displacement intensity factor K_IV when a uniform electrical displacement D_y is applied. However, the relative crack displacement and electrical potential along the crack surface are quite different for the coupled and uncoupled cases. Furthermore, for a inclined crack within a finite domain, we found that while a uniform σ_yy (=1 N m⁻²) induces only a very small electrical displacement intensity factor (in the unit of Cm^−3/2), a uniform D_y (=1 C m⁻²) can produce very large stress intensity factors (in the unit of Nm^−3/2).     

The prediction of fracture in brittle solids subjected to very short duration tensile stresses

The failure mechanism of brittle solids under very short duration tensile loading differs basically from that under prolonged tensile loads. Under static loading, or long stress pulses, the initiation and propagation of a single inherent flaw may lead to complete separation of the part. By contrast, if the loading is of very short duration, many cracks have to initiate and propagate so that they can link up and create a fracture surface during the loading period. A simple theory is proposed which relates the strength values observed under short duration stress pulses to the distribution of inherent flaws in the material. The prediction is shown to compare favorably with experimental results obtained by subjecting several types of rock to electron beams which produce stress pulses lasting from fractions of a microsecond to several microseconds.

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Résumé Le mécanisme de rupture d'un solide fragile soumis à une charge de traction de courte durée diffère fondamentalement de celui obtenu par une mise en charge prolongée. Pour une charge statique ou dans le cas des impulsions à contrainte prolongée, l'initiation de la propagation d'un simple défaut inhérent peut conduire à une séparation complète des parties. Par contraste, si la mise en charge est de courte durée, plusieurs microfissures s'initient et se propagent de telle façon qu'elles peuvent s'enchainer et créer une surface de rupture pendant la période de charge. Une théorie simple est proposée reliant les valeurs de la résistance observée pour des impulsions de contraintes de courte durée à la distribution des défauts inhérents dans le matériau. Une corrélation favorable est démontrée avec les résultats expérimentaux obtenus en soumettant divers types de roches à l'action de faisceaux d'électrons durant des fractions de microsecondes et jusqu'à plusieurs microsecondes.

     

Numerical prediction of slant fracture with continuum damage mechanics

Ductile specimens always exhibit an inclined fracture surface with an angle relative to the loading axis. This paper reports a numerical study on the cup-cone fracture mode in round bar tensile tests and the slant fracture in plane-strain specimens based on continuum damage mechanics. A combined implicit-explicit numerical scheme is first developed within ABAQUS through user defined material subroutines, in which the implicit solver: Standard, and the explicit solver: Explicit, are sequentially used to predict one single damage/fracture process. It is demonstrated that this numerical approach is able to significantly reduce computational cost for the simulation of fracture tests under quasi-static or low-rate loading. Comparison with various tensile tests on 2024-T351 aluminum alloy is made showing good correlations in terms of the load-displacement response and the fracture patterns. However, some differences exist in the prediction of the critical displacement to fracture.     

Phonon theory of brittle fracture of solids

In this article we outline the basic principles of a new approach to the solution of the problem of brittle fracture of solids. The kinetic theory of long-time strength formulated by S. N. Zhurkov et al. is analyzed and generalized.     

Fractoemission during indentation fracture of brittle solids

Journal of Materials Science Letters -     

On the location of fracture in brittle solids—II

The fracture produced in a slender rod, when a compressive stress pulse reflects from a free end as tension, has often been used to measure the tensile strength of brittle solids. In the present paper it is shown that the distribution of fracture location, when many such tests are made with a rectangular pulse of fixed stress level, may be used to deduce the distribution in strength of the material. The procedure is illustrated using tests on glass rods.

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Zusammenfassung Der in einem schlanken Stab unter Einwirkung einer sich an einem freien Ende reflektierenden Druckwelle hervorgerufene Bruch, wurde oftmals als Maß für die Zugfestigkeit spröder Werkstoffe herangezogen. Dieser Bericht zeigt, daß die Bestimmung der örtlichen Verteilung der Bruchstellen, fur eine große Anzahl solcher Versuche mit einem rechteckigen Impuls konstanter Stärke, es ermöliglicht die statistische Verteilung der Zugfestigkeit eines Materials zu ermitteln. Das Verfahren wird an Hand von Versuchen an Glasstäben erlaütert.

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Résumé La rupture qui se produit dans un barreau élancé sous l'effet d'une onde de compression qui, se réfléchissant sur une extrémité libre, se transforme en onde de traction, a été souvent utilisée dans l'analyse de la résistance à la traction des matériaux fragiles.Dans ce mémoire, on montre que l'examen de la distribution de l'emplacement de la rupture pent permettre une analyse statistique de la résistance du matériau, lorsque l'on procède à de nombreux essais de ce genre avec une impulsion de compression rectangulaire d'amplitude fixe. La procédure d'essais est illustrée dans le cas de barreaux de verre.

     

Integral estimates to the fast brittle fracture mechanics

In the terms of a general approach the problem of the fast crack propagation in elastic bodies is considered. The time dependence of, so called domain of possible fractures due to the fast crack propagation, is obtained. The dynamic domain of possible fractures is the upper estimate of the zone of real fractures and thereby it takes into account such important features of the fast brittle fracture as the instability of the fast crack propagation, branching, waving, the influence of boundary conditions and so on. The problem of dynamic crack initiation is considered in some details. Within the linear fracture mechanics the dynamic criterion of crack initiation is offered.     

Use of chevron notches for fracture toughness determination in brittle solids

Abstract

The chevron notch testpiece geometry has been used in three and four point bending to evaluate the toughness of silicon carbide and silicon carbide reinforced with particulate titanium diboride. The toughness values obtained do not appear to depend either on initial chevron notch depth m the range 0·2≤α_o≤0·4 or on chevron notch included angle in the range 40≤θ≤120°. This testpiece geometry allows relatively subtle effects of reinforcement in increasing the toughness of the monolithic material to be assessed with confidence. It also provides a convenient method of measuring toughness over a wide range of test temperatures.

MST/1595     

Grain-size dependence of fracture stress in anisotropic brittle solids

The stress concentrations that occur at grain boundaries due to thermal expansion anisotropy and elastic stress concentration are discussed, and the stress intensity factor that results from these stresses is estimated. The procedure for the stress intensity factor calculation is based on the model in which a spherical crystal (grain) is forced into a cavity of equal size possessing annular or radial cracks emanating from the boundary. The stress intensity factor equation thus obtained is extended to include the effect of elastic stress concentration due to the presence of a cavity, and is subsequently used to predict the grain-size dependence of strength in anisotropic brittle ceramics. In assessing the degradation of strength with increasing grain size in non-cubic ceramics, it is shown that, in addition to grain size, the effect of pre-existing crack size must also be considered. Cubic ceramics, on the other hand, are known to exhibit no thermal expansion anisotropy and, based on the present model, their strength is predicted to be governed by the pre-existing flaw size, rather than the grain size.     

Atomically sharp cracks in brittle solids: an electron microscopy study

The issue of bond rupture versus microplasticity as an essential mechanism of crack propagation in brittle solids is addressed. A detailed survey of existing theoretical and experimental evidence relating to this issue highlights the need for direct observations of events within the crack-tip process zone, at a level approaching 10 nm. Transmission electron microscopy is accordingly used to study arrested cracks about sharp-contact (Vickers indentation and particle impact) sites in Si, Ge, SiC and Al₂O₃. The nature of the deformation which accommodates the irreversible contact impression is first investigated, in the light of Marsh's proposal of an equivalence between indentation and crack-tip zone processes. Interfacial and tip regions of the surrounding cracks are then examined for any trace of a plasticity-controlled fracture process. Dislocation-like images are indeed evident at the crack planes, but these are shown to be totally inconsistent with any conventional slip mechanism. The close connection between the dislocation patterns and moiré fringe systems along the cracks points to lattice mismatch contrast in association with a partial closure and healing operation at the interface. Analysis of all other details in the crack patterns, e.g. the presence of a crack-front contrast band indicative of a residual strain field and the disposition of interfacial fracture steps relative to the dislocation/moiré system, reinforces this interpretation. It is concluded that the concept of an atomically sharp crack provides a sound basis for the theory of fracture of brittle solids.     

Brittle fracture of solids with arbitrary cracks

One of the problems of fracture mechanics is the prediction of the propagation of cracks in solids. The present paper deals mainly with linear fracture mechanics which owes its origin to the works of A. A. Griffith [1, 2] and studies the development of cracks under sufficiently low loads when the behaviour of the material within a region sufficiently remote from the edges of cracks may be regarded as linearly elastic, At present, linear fracture mechanics [3] is restricted mainly to special kinds of loading geometry, with the crack extending rectilinearly (in a plane case) or in its plane (in a three-dimensional case). The main problem here is to establish a relationship between the dimensions of cracks and the loads applied. Within the framework of linear fracture mechanics the fracture itself and other non-linear phenomena that precede it are assumed to take place only within local regions which are small compared to the dimensions of cracks. The possibility that such a situation exists is associated with the fact that when the crack dimensions are sufficiently large the characteristic dimension of the end region is fully determined by a certain intrinsic dimension of the material structure. Therefore, if the material does not exhibit time dependency, the state of the end region at the moment of rupture becomes fully independent of the loads applied and the geometry of the solid, i.e. autonomous. The notion of autonomy [4] leads to the formulation of this theory as one of limit equilibrium.

If the conditions of rectilinear extension of the crack (or those of the crack extension in its plane) are disturbed, there arises a problem of determining not only the dimensions of the crack, but also the path of the crack extension under such conditions of loading that a slow, quasi-static crack development is possible. This problem can be actually subdivided into two: (1) Criteria for the determination of the dimensions and paths of the crack extension, and (2) Expressions for the characteristics of the stress-strain state which are constituents of these criteria through the geometry of solid with cracks and the loads applied.

As regards (1), there have been many assertions, and the connections between them are not quite clear at present. The first of the suggested criteria, namely that of local symmetry for the plane problem formulated by Barenblatt and Cherepanov [5, 6] and by Erdogan and Sih [7] can be within certain limits substantiated and generalized for the three-dimensional case. The guiding principle here is the treatment of the theory of cracks from the standpoint of the method of inner and outer expansions or that of singular perturbations [8]. The concept of the stress intensity factor which is basic in linear fracture mechanics is decisive in matching inner and outer expansions to find the main term of the asymptotic solution of the complete problem. Actually the construction of the theory of equilibrium cracks [4] implicitly employs this technique for a certain specific model. More explicit indications are given in Ref. 9. In the treatment of the problem of plastic zones in the vicinity of notches, the idea of the boundary layer is employed in Ref. 10. The problem of fracture of a solid is analysed from this standpoint in Ref. 11.

As regards (2), progress has been hampered by the lack of efficient techniques for fording the stress-strain state of a solid having non-rectilinear cuts. A number of investigations have been carried out for cuts of a particular kind an arc of a circumference [12, 13], an arc of a parabola [l4], and a three-link broken line which is close to a straight line to such an extent that the boundary conditions are assumed referable to the direction of the middle portion [15, 16]. The problem of a semi-infinite curvilinear cut slightly deviating from a rectilinear one by expanding complex elastic potentials in the magnitude of deviation of the cut from the rectilinear axis tangent to the line of cut at its end is considered in Ref. 17. An exact solution of the problem of a semi-infinite cut having the form of a two-link broken line is given in Ref. 18.

The present paper is devoted to the investigation of the development of cracks under arbitrary loading conditions.

In Section 1 the criterion of local symmetry is substantiated and generalized for the three-dimensional case. In Section 2 an effective procedure of finding stress intensity factors for the plane case is given, in terms of which the criterion is formulated. Closed first approximation formulas for these magnitudes are presented in the case of a slightly curved crack, numerical calculations showing the applicability of the latter with an error not exceeding 10 to 15 with the angles of deviation of the crack from the straight line coming to 20°. In Section 3 equations of extension of curvilinear cracks are derived on the basis of the first approximation formulas and criterion of local symmetry. In Section 4 some examples are considered.