Investigation of mixed mode brittle fracture in rounded-tip V-notched components

A criterion is proposed for brittle fracture analysis in rounded-tip V-notched components. This criterion, called RV-MTS, is developed based on the maximum tangential stress (MTS) criterion proposed earlier for investigating mixed mode brittle fracture in sharp cracks. Using the RV-MTS criterion, a set of fracture curves is presented based on the notch stress intensity factors (NSIFs) for predicting mixed mode and also pure mode II fracture toughness of rounded-tip V-notches. The criterion is also able to predict fracture initiation angles under mixed mode loading. The validity of the criterion is evaluated by several fracture tests performed on the rounded-tip V-notched Brazilian disc (RV-BD) specimens made of PMMA. A good agreement is shown to exist between the theoretical predictions and the experimental results for various notch opening angles and different notch radii.     

Effect of second non-singular term of mode I near the tip of a V-notched crack

The effect of a second non-singular term of mode I in the vicinity of the tip of a V-notched crack is discussed. The undetermined eigenvector coefficients of singular and non-singular terms of eigenfunction expansion were calculated by the numerical analysis method using the reciprocal work contour integral method (RWCIM) based on Betti's reciprocal work theorem and finite-element analysis. This study demonstrates that the second non-singular term of mode I can have a significant effect on the size and shape of plastic zones as an additional fracture mechanics parameter along with the conventional parameter K .     

A criterion study for non-singular stress concentrations in brittle or quasi-brittle materials

In engineering applications, it is difficult to avoid the non-singular stress concentrations that often play an important role in structure designs. The simplest engineering strength criteria are in general not appropriate due to their incapacity in dealing with important size effect induced by stress gradients. In this paper, we present first a simple experimentation, which consists of plates with a central hole under uniaxial tensile loading, showing important size effect. Second, numerous criteria, including commonly used engineering criteria, crack initiation criteria based on the finite fracture mechanics, or cohesive criteria, were adapted to fit the experimental results. We found that most of these criteria, including criteria with a single material parameter and those with two material parameters, are not suitable for fracture prediction of materials under non-singular stress concentrations. It seems that three material parameters would be the minimum to establish an adequate fracture criterion for arbitrary stress concentrations. By analysing the energy dissipation of micro-crack bands under different stress concentrations, we proposed a new fracture criterion with three material parameters based on the finite fracture mechanics. It is shown that this criterion can provide accurate critical remote loads comparing with experimental data. We believe that the three parameter concept is physically reasonable and can be used in establishing fracture criteria in both the cases of singular and non-singular stress concentrations.     

Temperature dependence of fracture stress in the brittle fracture region

The temperature dependence of the fracture stress of bcc metals in the brittle fracture region is examined. A model taking into account the effect of local plastic deformation of the crack tip on transition to brittle fracture is proposed. The model is then used to derive analytical expression for the temperature dependence of the fracture stress in the material with an initial crack or notch.Translated from Problemy Prochnosti, No. 11, pp. 57–62, November, 1991.     

On the role of stress fluctuations in brittle fracture

Cracks in random stress fields are assumed to be originated in regions with high local tension. As a legacy of this special location, additional local tractions opening the crack in its centre are developed even in self-equilibrating stress fields. As the crack becomes a mesocrack it will deviate its path to meet the regions with higher possible local tension. The necessary statistical properties of the microcrack-generated random stress field can be calculated using the dipole asymptotics to approximate the stresses generated by each microcrack. The microcracks are assumed to be noninteracting and surrounded by nonintersecting excluded volumes. For the case of spherical excluded volumes the correlation radius is found to be less than the microcrack radius, which suggests that the stresses acting on each microcrack can be assumed to be statistically independent. In brittle fracture under uniaxial tension the effect of the stress fluctuations is shown to be able to significantly reduce the macroscopic strength. In fracture of brittle materials under uniaxial compression wing cracks are developed which, in real 3-D situations, cannot grow extensively and therefore cannot themselves cause failure. Instead, they induce stress fluctuations which generate mesocracks growing towards compression in such a way as to avoid the wing cracks. Hence, only stresses outside excluded volumes around the wing cracks will affect the mesocrack growth. These stresses have positive mean even if the full stress field is self-equilibrating. This results in a background tension acting perpendicular to the compression axis, amplifying the mesocrack growth and eventually causing failure. The growth and opening of mesocracks results in a specific dependence between dilatancy, i.e. inelastic increase of the sample volume, and the applied compressive stress. This dependence has a universal nature independent of the particular model of wing cracks. It corresponds well to the data of uniaxial compressive tests on 4 samples of Oshima granite (Sano et al. 1981) despite markedly different loading rates and resulted strengths. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

The thermodynamics of brittle fracture initiation under triaxial stress conditions

A calculation has been carried out of the changes in Gibbs free energy associated with the initiation of crack propagation in an isotropic, brittle solid of the Griffith type when the solid is subjected to a homogeneous, uniform but otherwise arbitrary triaxial system of stresses at infinity. The cracks have been taken to be flattened ellipsoidal cavities which are free from surface tractions. In accordance with the two-dimensional treatment of the problem and with experimental observations it is shown that minimisation of the Gibbs free energy, though necessary, is not a sufficient condition for brittle fracture initiation. There is in general more than enough energy available in the system to enable fracture to be initiated; the excess energy being particularly large in bodies fractured in compression.

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Zusammenfassung Die Änderung der freien Gibbs'schen Energie bei Beginn einer Rißfortpflanzung in einem spröden isotropen Festkörper des Griffith Typs wurde berechnet für den Fall wo these Körper einem homogenen und gleichmäßigen, ansonsten jedoch arbiträre System triaxialer Spannungen im Unendlichen ausgesetzt ist.Es wurden Risse gewählt, welche die Form flacher ellipsoïdaler Aushöhlungen hatten und welche frei von irgendwelchen Oberflächenspannungen waren. In Übereinstimmung mit der zweidimensionalen Betrachtung des Problems und mit den experimentellen Beobachtungen wird gezeigt, daß die Minimisierung der Gibbs'schen freien Energie eine zwar notwendige jedoch keinesfalls genügende Bedingung für das Entstehen eines Sprödbruchs ist. Im Allgemeinen ist mehr als genügend Energie im System vorhanden, um einen Bruch zu ermöglichen; die Überschußenergie ist besonders groß für Körper die unter Druck gebrochen werden.

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Résumé On a calculé les changements de l'énergie libre de Gibbs lors du début de la propagation d'une fissure dans un solide isotrope et fragile, du type Griffith, lorsque ce solide est soumis à un systeme arbitraire de contraintes triaxiales à l'ínfini, de caractère homogène et uniforme.On a considéré des fissures ayant la forme de cavités ellipsoïdales aplaties, libres de toutes tensions de surface. En vertu d'un traitement bidimensionnel du problème, et en relation avec les observations expérimentales, on montre qu'il ne suffit pas, pour qu'une fissure fragile s'amorce, que l'énergie libre de Gibbs passe par un minimum, encore que cette condition soit nécessaire. En général, il existe plus que suffisamment d'énergie potentielle dans le système pour qu'une fissure puisse s'amorcer. L'énergie en excès est particulièrement importante dans le cas des corps rompus en compression.

     

Singular finite elements for the fracture analysis of V-notched plate

Special accurate and efficient hybrid elements were developed to account for notch-tip singularities of type r^γ, where γ can be complex. Proper normalization of the notch-tip element size was used to minimize the oscillation of the assumed function around the boundaries of the element and to improve the accuracy of the solution. Examples of various notch angles and sizes are presented.     

Quantitative analysis of delayed fracture observed in stress rate tests on brittle materials

Delayed fracture in brittle materials may be demonstrated by observing the change in bend strength over a range of constant stress rates to failure. A simple technique has been developed to analyse this behaviour making efficient use of the experimental data. Based on a model incorporating theories of stress corrosion and brittle fracture, with Weibull statistics, the technique provides estimates of relevant parameters using the method of maximum likelihood. Confidence intervals of estimates, the significance of any observed rate effect, and the validity of the model are also assessed. The technique is demonstrated by applying it to data from bend tests on soda-lime glass and WC-Co materials.     

Effects of atoms on brittle fracture

This article aims to answer two related sets of questions. First: in principle, how large an effect can structure at the atomic scale have upon the fracture of two macroscopically identical samples? The answer to this question is that the effects can be very large. Perfectly sharp cracks can be pinned and stationary under loading conditions that put them far beyond the Griffith point. Crack paths need not obey the rule K _II=0. Crack speeds can vary from zero to the Rayleigh wave speed under identical loading conditions but depending upon microscopic rules. These conclusions are obtained from simple solvable models, and from techniques that make it possible to extrapolate reliably from small numerical calculations to the macroscopic limit. These techniques are described in some detail. Second: in practice, should any of these effects be visible in real laboratory samples? The answer to this second question is less clear. The qualitative phenomena exhibited by simple models are observed routinely in the fracture of brittle crystals. However, the correspondence between computations in perfect two-dimensional numerical samples at zero temperature and imperfect three-dimensional laboratory specimens at nonzero temperature is not simple. This paper reports on computations involving nonzero temperature, and irregular crack motion that indicate both strengths and weaknesses of two-dimensional microscopic modeling.     

Thermomechanics of brittle fracture

On the basis of the thermodynamics of thermoelastic deformation we propose an energy condition of the Griffith type for brittle fracture of solids under single loading. An analysis is given of the proposed condition under the plane stressed (strained) state for the two known models of an isolated defect. In the first model, stresses on the external (distant) surface of a solid with a defect are prescribed the same as in a similar solid without a defect. In the second model, displacements are prescribed on the external surface of a solid with a defect, which correspond to the load applied to the solid, but before the defect or crack was formed. Stresses on the surface of the defect in both models are equal to zero. It is shown that the first model in the isothermal case of deformation leads to the Griffith condition. The second model meets the energy condition of the Griffith type from which, under additional assumptions concerning the shape of the defect, and from conditions of isotropy and convexity, we obtain a curve of fracture (macroscopic criterion of fracture) in an ellipse form in the space of principal stresses. The orientation of the crack was determined. Coefficients of the curve of fracture obviously depend upon elastic constants, temperature, linear coefficient of thermal expansion and crack dimensions.     

Thermomechanics of brittle fracture

On the basis of the thermodynamics of thermoelastic deformation we propose an energy condition of the Griffith type for brittle fracture of solids under single loading. An analysis is given of the proposed condition under the plane stressed (strained) state for the two known models of an isolated defect. In the first model, stresses on the external (distant) surface of a solid with a defect are prescribed the same as in a similar solid without a defect. In the second model, displacements are prescribed on the external surface of a solid with a defect, which correspond to the load applied to the solid, but before the defect or crack was formed. Stresses on the surface of the defect in both models are equal to zero. It is shown that the first model in the isothermal case of deformation leads to the Griffith condition. The second model meets the energy condition of the Griffith type from which, under additional assumptions concerning the shape of the defect, and from conditions of isotropy and convexity, we obtain a curve of fracture (macroscopic criterion of fracture) in an ellipse form in the space of principal stresses. The orientation of the crack was determined. Coefficients of the curve of fracture obviously depend upon elastic constants, temperature, linear coefficient of thermal expansion and crack dimensions.     

Dynamics of brittle fracture

The equation of motion and the stability conditions of surface cracks which are subjected to stress waves are derived from an energy balance and the law of angular momentum conservation. From the resulting differential equation the terminal crack velocity and dynamic threshold conditions can be derived. It will be shown that not only a critical stress (amplitude of the wave) but also a critical time (duration of the wave) are necessary to move the crack. These two critical quantities can be combined to a critical action. The specific action of the wave must exceed a certain minimal value for crack propagation to occur. Quantum considerations allow us to generalize this criterion further. Some simple applications of the least action law are mentioned. Supercritical stress pulses are also treated briefly. This leads to the concept of the cross section of a stress wave. Crack stability can also be treated as an eigenvalue problem.     

The effect of welding residual stress on brittle fracture of plates with surface cracks

Two types of butt-welded wide plates (steel OX 522D) containing half-elliptical surface cracks (30 mm thick plates with the crack perpendicular to the weld and 100 mm thick plates with the crack parallel to the weld) were tested in the as-welded and thermally stress relieved conditions. The effects of welding residual stress and stress relief on brittle fracture were investigated. It was found that the influence of welding residual stress on surface crack fracture may be substantial, depending on the stress field and the crack orientation and geometry. The influence was found to be predictable with linear elastic fracture mechanics. Post weld heat treatment was found to remove most of the residual stresses and therefore reduce the fracture risk, provided the material toughness is not impaired.