Fracture statistics of brittle materials with surface cracks

Several different statistical fracture theories are developed for materials with cracks confined to the surface. All assume that crack planes are normal to the surface, but are otherwise randomly oriented. The simplest theory assumes that only the component of stress normal to the crack plane contributes to fracture. This theory is in fair agreement with biaxial fracture data on Pyrex glass obtained by Oh. When the contribution of shear is included in the analysis, the crack shape has to be considered. Several shapes are examined and the corresponding fracture statistics are derived. Two failure criteria are employed. In one the fracture occurs when the maximum tensile stress on some part of the crack surface reaches the intrinsic strength of the material. The other is based on a critical strain energy release rate. The assumption of shear-sensitive cracks leads to improved agreement with experiment, but really good agreement appears to require the assumption that the cracks have a preferred orientation.     

On the relationship between the length of Palmqvist cracks and residual surface stresses in WC-Co

WC-Co cylinders were subjected to cyclic compressive stresses for varying numbers of cycles and maximum compressive stresses. The fractional changes in the length of Palmqvist cracks and in the residual surface stresses were measured after precompression. It is found that the relationship between Palmqvist crack length and residual surface stresses depends on the conditions of precompression, and thus Palmqvist crack lengths cannot provide quantitative information on residual surface stresses.     

A new constraint based fracture criterion for surface cracks

A new methodology for predicting the location of maximum crack extension along a surface crack front in ductile materials is presented. Three-dimensional elastic-plastic finite element analyses were used to determine the variations of a constraint parameter (α_h) based on the average opening stress in the crack tip plastic zone and the J-integral distributions along the crack front for many surface crack configurations. Monotonic tension and bending loads are considered. The crack front constraint parameter is combined with the J-integral to characterize fracture, the critical fracture location being the location for which the product Jα_h is a maximum. The criterion is verified with test results from surface cracked specimens.     

The relation between theory and application in statistics

Summary General comments on the relation between theory and application in statistics are made and emphasis placed on issues and principles of model formulation. Three examples are described in outline. Criteria for the choice of models are discussed.     

The application of fracture mechanics to surface cracks in shafts

A theoretical method was developed for calculating the stress intensity factor (K) for semielliptical surface cracks in shafts subjected to a constant moment load. A number of tests using 2-and 3-dimensional photoelasticity showed good agreement with the theoretical model. The calculated K was then used in computer predictions of crack propagation characteristics which were subsequently tested using high strength steel shafts. The theoretical model is useful for predicting critical crack depths and lifetimes for the particular geometry, crack shape and load configuration investigated.     

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.     

Correlation between fracture and damage for quasi-brittle bi-material interface cracks

Fracture at a bi-material interface is essentially mixed-mode, even when the geometry is symmetric with respect to the crack and loading is of pure Mode I, due to the differences in the elastic properties across an interface which disrupts the symmetry. The linear elastic solutions of the crack tip stress and displacement fields show an oscillatory type of singularity. This poses numerical difficulties while modeling discrete interface cracks. Alternatively, the discrete cracks may be modeled using a distributed band of micro-cracks or damage such that energy equivalence is maintained between the two systems. In this work, an approach is developed to correlate fracture and damage mechanics through energy equivalence concepts and to predict the damage scenario in quasi-brittle bi-material interface beams. The study is aimed at large size structures made of quasi-brittle materials failing at concrete-concrete interfaces. The objective is to smoothly move from fracture mechanics theory to damage mechanics theory or vice versa in order to characterize damage. It is concluded, that through the energy approach a discrete crack may be modeled as an equivalent damage zone, wherein both correspond to the same energy loss. Finally, it is shown that by knowing the critical damage zone dimension, the critical fracture property such as the fracture energy can be obtained.     

The relationship between cracks,holes and surface dislocations

International Journal of Fracture - It has been shown that surface cracks as well as holes can be represented in terms of surface dislocations. These surface dislocations exist in order to insure...     

Cyclic fracture toughness of structural alloys with surface cracks at low temperatures

In the present paper, we study specific features of crack propagation from surface flaws in full-scale sheets used in manufacturing pressure vessels for cryogenic applications. The process of crack propagation consists of several stages and terminates in stages of surface through or central through cracks under conditions of low-frequency repeated tension. The effect of a decrease in temperature from 292 to 77°K on crack growth behavior was studied for sheets with a thickness of 2, 8, and 12mm. We describe a procedure for testing for crack-growth resistance at cryogenic temperatures and construct fatigue crack growth diagrams. It is shown that zones of influence of the front and rear faces of the specimen on the stress and strain fields near the crack front arise in the plane of a semielliptic crack. The shape of the interface of these zones can be approximated by a second-order curve. Variations in the thickness of the specimen and the test temperature affect the slope of the curve, i.e., the interface of the zones of influence. Specific features of the fracture process in the material of the plate with surface cracks manifest themselves most adequately at points of the crack front located on the indicated interface. We suggest a procedure for estimating the cyclic crack growth resistance of highly ductile stainless steels that is based on the use of the cyclic J-integral. We propose to regard the lengthl of the interface of the zones of influence of the front and rear faces of the specimen as a geometric parameter of the crack. It is used to construct kinetic fatigue crack growth diagrams for specimens with semielliptic surface cracks.Published in Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 31, No. 1, pp. 9–19, January – February, 1995.     

Using criteria of the nonlinear fracture mechanics in examining the growth of surface fatigue cracks

The authors propose a method of evaluating the cyclic cracking resistance of high-ductility corrosion-resisting steels using the method of the cyclic J-integrals. In constructing kinetic diagrams of fatigue failure of specimens with surface semielliptical cracks of 07Kh13N4AG20 and 12Kh18N10T steels the geometrical parameter of the crack was represented by the quantity l proposed by the authors. Comparison of the diagrams constructed for the parameters a (depth of the surface crack), C (the length of the surface crack), and l showed that the evaluation of the cracking resistance by the dependence dl/dN-J_l; is accurate and that the information content of da/Dn-J_a, dC/dN-gDJ_c diagrams is low. Variation of the thickness of the examined material (t = 4–12 mm) and of test temperature (T = 293-77 K) has no marked effect on the form of the dl/dN-gDJ_l dependence.Translated from Problemy Prochnosti, No. 6, pp. 3–12, June, 1993.     

On fracture initiation toughness and crack sharpness for Mode II cracks

Fracture toughness of brittle materials is calibrated in experiments where a sample with a preexisting crack is loaded up to a critical point of the onset of static instability. Experiments with ceramics, for example, exhibit a pronounced dependence of the toughness on the sharpness of the crack/notch: the sharper is the crack the lower is the toughness. These experimental results are not entirely compatible with the original Griffith theory of brittle fracture and Linear Elastic Fracture Mechanics which both ignore the crack sharpness.To explain the experimental observations qualitatively we earlier considered Mode I cracks [Volokh KY, Trapper P. Fracture toughness from the standpoint of softening hyperelasticity. J Mech Phys Solids 2008;56:2459-72.] and in the present work we extend our considerations to Mode II cracks. We simulate pure shear of a thin plate with a small crack of a finite and varying sharpness. In simulations we introduce the failure energy as a limiter for the stored energy of the Hookean solid. The energy limiter induces softening, indicating material failure. Thus, elasticity with softening allows capturing the critical point of the onset of static instability of the cracked plate, which corresponds to the onset of the failure propagation at the tip of the crack. In numerical simulations we find that the magnitude of the fracture toughness can not be determined uniquely because it depends on the sharpness of the crack: the sharper is the crack the lower is the toughness.Based on the obtained results we argue that a stable magnitude of the toughness of brittle materials can only be reached when a characteristic size of the crack tip is comparable with a characteristic length of the material microstructure, e.g. grain size, atomic distance etc. In other words, the toughness can be calibrated only under conditions where the hypothesis of length-independent continuum fails.     

Elastic-plastic fracture mechanics analyses of circumferential through-wall cracks between elbows and pipes

The present work proposes a method for elastic-plastic fracture mechanics analysis of the circumferential through-wall crack in weldment joining elbows and attached straight pipes, subject to in-plane bending. Heterogeneous nature of weldment is not explicitly considered and thus, the proposed method assumes cracks in homogeneous materials. Based on small strain finite element limit analyses using elastic-perfectly plastic materials, closed-form limit loads for circumferential through-wall cracks between elbows and straight pipes under bending are given. Then applicability of the reference stress-based method to approximately estimate J and crack opening displacement (COD) is evaluated. It was found that the limit moments for circumferential cracks between elbows and attached straight pipes can be much lower than those for cracks in straight pipes, particularly for a crack length of less than 30% of the circumference; this result is of great interest in practical cases. This result implies that, if one assumes that the crack locates in the straight pipe, limit moments could be overestimated significantly, and accordingly, reference stress-based J and COD could be significantly overestimated. For the leak-before-break analysis, accurate J and COD estimation equations based on the reference stress approach are proposed.     

Relationships between fracture parameters and fracture surface roughness of brittle polymers

Fracture parameters such as crack velocity à, stress intensity factor K _d and a specific crack extension resistance R ^* were measured for Homalite-100, PMMA and epoxy in the course of fast crack propagation using a Cranz-Schardin type high speed camera. Fracture surface roughness was evaluated as a function of crack length a so that it could be correlated with the fracture parameters above. The results showed that none of those parameters could be uniquely related to . Instead, there was a good correlation between and a product R ^* à.