A non-linear crack model

Elliott's crack model which accounts for non-linearity of material behaviour in the vicinity of a crack tip is difficult to investigate analytically in view of the mathematical complexities associated with the governing singular integral equation. However, with a particular force law, the governing equation can be transformed into one that arises in the theory of the lift coefficient of a thin jet-flapped wing, and for which a solution has been obtained. The displacement variation ahead of the crack tip is therefore known, and the results show that the width of the crack front decreases as the force law becomes sharper, a state of affairs which is analogous to that for the dislocation width in the Peierls-Nabarro model. The fact that a solution of the governing equation has been obtained without the use of inverse methods, should act as a spur to the search for solutions for other force laws.     

A fatigue crack propagation model

A model for fatigue crack propagation has been developed which incorporates mechanical, cyclic and fatigue properties as well as a length parameter. The latter can be associated with the microstructure of the material. The fatigue failure criterion is based on a measure of the dissipated plastic strain energy. This model predicts crack propagation at low and intermediate ΔK values, i.e. stage I crack growth rate as well as that of the stage II. A number of crack growth rate models proposed earlier, are shown to be particular cases of the one developed herein. Predictions of the model are in good agreement with the experimental data. The required data for predicting the crack growth rate, can be found in standard material handbooks where fatigue properties are listed.     

A model for fatigue crack closure

The phenomenon of fatigue crack closure has attracted continued interest over recent years. This paper concerns itself with one aspect of the phenomenon namely the effects of a single asperity on the crack face close to the crack tip and under dominantly plane strain Mode 1 loading conditions. The model as developed is two dimensional in character but the nature of the closure processes allows it to be applied to real materials. The model illustrates the influence of asperity height, the distance from the crack tip of the asperity and rigidty of the asperity on the crack closure stress intensity. Application of the model to the case of microstructural asperities in nickel alloys and to oxide asperities in steels has met with reasonable success.     

A continuous-discontinuous model for crack branching

A new continuous-discontinuous model for fracture that accounts for crack branching in a natural manner is presented. It combines a gradient-enhanced damage model based on nonlocal displacements to describe diffuse cracks and the extended finite element method (X-FEM) for sharp cracks. Its most distinct feature is a global crack tracking strategy based on the geometrical notion of medial axis: the sharp crack propagates following the direction dictated by the medial axis of a damage isoline. This means that, if the damage field branches, the medial axis automatically detects this bifurcation, and a branching sharp crack is thus easily obtained. In contrast to other existing models, no special crack-tip criteria are required to trigger branching. Complex crack patterns may also be described with this approach, since the X-FEM enrichment of the displacement field can be recursively applied by adding one extra term at each branching event. The proposed approach is also equipped with a crack-fluid pressure, a relevant feature in applications such as hydraulic fracturing or leakage-related events. The capabilities of the model to handle propagation and branching of cracks are illustrated by means of different two-dimensional numerical examples.     

A model for fatigue crack propagation

A model for fatigue crack propagation is presented which incorporates low cycle fatigue, mechanical properties and a microstructurally-associated process zone. Comparison of the model to published date for 4340 (hard and soft), a series of TRIP steels, Ti-6A1-4V, 2024-T6 and 300 grade maraging steel shows good agreement.     

A cumulative model of fatigue crack growth

A model of fatigue crack growth based on an analysis of elastic/plastic stress and strain at the crack tip is presented. It is shown that the fatigue crack growth rate can be calculated by means of the local stress/strain at the crack tip. The local stress and strain calculations are based on the general solutions given by Hutchinson, Rice and Rosengren. It is assumed that a small highly strained area existing at the crack tip is responsible for the fatigue crack growth. It is also assumed that the fatigue crack growth rate depends mainly on the width, x¹, of the highly strained zone and on the strain range, $\text{Δ?}{}^1$, within the zone. A relationship between stress intensity factor K and the local strain and stress has been developed. It is possible to calculate the local strain for a variety of crack problems. Then, the number of cycles N¹ required for material failure inside the highly strained zone is calculated. The fatigue crack growth rate is calculated as the ratio $\text{x}{}^1\text{N}{}^1$.The calculated fatigue crack growth rates were compared to the experimental ones. Two alloys steels and two aluminium alloys were analyzed. Good agreement between experimental and theoretical results is obtained.     

A mixed mode fatigue crack closure model

Fatigue crack closure conditions leading to mixed mode crack-tip stress states are discussed. A closure model based on contact between impinging fracture surface asperities is introduced. Asperity angle, magnitude of mode II interference, friction between contacting surfaces and distance of the contact surfaces from the crack-tip are features included in the model. The magnitudes of mode I and mode II stress intensity factors due to closure contact are found to vary substantially with asperity angle. Also, because of a reversal of friction force direction during cycle loading, the stress intensity factors exhibit a complex behavior. During the closure phase of the loading cycle, the mode II contribution is found to be quite significant. For loading through an entire cycle, which includes both the closure and crack open phases, nonproportional stress states are developed. Also, it is concluded that the mixed mode states developed could provide the conditions required for crack branching.     

A multiple asperity fatigue crack closure model

The plane strain, fatigue crack problem in which fracture surface asperities prevent closure is considered. A model in which closure contact can develop on a distribution of asperities is introduced. A comparison of closure behavior for one and for two asperities is made, and the effects of variations in the model parameters are examined. The relationships of these parameters to the value of the closure stress intensity factor and crack opening displacement are discussed. The possible use of the model in examining conditions in which asperity welding occur is considered. The resulting closure behavior is briefly described.     

A damage model for creep crack growth

A model is developed for damage produced by the growth of isolated grain boundary cavities under power law creep. This damage model is combined with the small scale yielding stress and strain fields to predict the damage ahead of a stationary and a steadily propagating crack tip in an elastic-power law creeping material. A failure criterion, based upon the damage ahead of the crack tip attaining a critical value, is invoked. This criterion leads to predictions for the incubation time prior to initiation of crack growth and for the relationship between the remote stress intensity factor and the steady state crack speed. Results are presented for both elastic-primary and -secondary creep crack growth. In either case, there exists a minimum stress intensity factor below which steady state crack growth is not possible. Comparisons of the predictions of this model with others for steady state crack propagation in elastic-secondary creeping materials are also made.

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Résumé On développe un modèle décrivant le dommage produit par la croissance de cavités isolées aux frontières des grains sous l'effet d'un fluage selon une loi parabolique. Ce modèle d'endommagement est combiné aux champs de contrainte et de déformation conduisant à une plastification à petite échelle, en vue de prédire le dommage qui se produit en avant de l'extrémité d'une fissure stationaire et en propagation lente, dans un matériau soumis à fluage selon une loi élastique. On invoque un critère de rupture basé sur la valeur critique atteinte par le dommage en avant de l'extrémité de la fisure. Ce critère conduit à prédire le temps d'incubation avant amorçage de la croissance d'une fissure, ainsi que la relation entre le facteur d'intensité des contraintes appliquées à une certaine distance et la vitesse de croissance stable de la fissure.On présente les résultats dans les cas de croissance d'une fissure de fluage au stade secondaire. Dans les deux cas, il existe un facteur d'intensité de contrainte minimum en dessous duquel une croissance stable de la fissure n'est pas possible. On procède également à des comparaisons des prédictions données par ce modèle avec d'autres, pour la propagation en état stable d'une fissure dans des matériaux soumis à fluage secondaire élastique.

     

A new stochastic model for crack propagation

In the following paper we present a new model for crack propagation. The new model is able to involve sudden growth of crack length (jumps) and is a generalization of the well-known Paris–Erdogen law. We obtain the lifetime distribution and the residual lifetime distributions.     

A cohesive model of fatigue crack growth

We investigate the use of cohesive theories of fracture, in conjunction with the explicit resolution of the near-tip plastic fields and the enforcement of closure as a contact constraint, for the purpose of fatigue-life prediction. An important characteristic of the cohesive laws considered here is that they exhibit unloading-reloading hysteresis. This feature has the important consequence of preventing shakedown and allowing for steady crack growth. Our calculations demonstrate that the theory is capable of a unified treatment of long cracks under constant-amplitude loading, short cracks and the effect of overloads, without ad hoc corrections or tuning.     

A cumulative model of fatigue crack growth and the crack closure effect

A model of fatigue crack growth based on an analysis of elastic/plastic stress and strain at the crack tip is presented. It is shown that the fatigue crack growth rate can be calculated using the local stress/strain at the crack tip by assuming that a small highly strained area $\text{x}{}^1$, existing at the crack tip, is responsible for the fatigue crack growth, and that the fatigue crack growth may be regarded as the cumulation of successive crack re-initiations over a distance $\text{x}{}^1$. It is shown that crack closure can be modelled using the effective contact zone g behind the crack tip. The model allows the fatigue crack growth rate over the near threshold and linear ranges of the general da/dN versus ΔK curve to be calculated. The fatigue crack growth retardation due to overload and fatigue crack arrest can also be analysed in terms of g and $\text{x}{}^1$.Calculated fatigue crack growth rates are compared with experimental ones for low and high strength steel.     

A third-order state-space model for fatigue crack growth

A second‐order state‐space model of fatigue crack growth in ductile alloys was presented by Patankar et al., 1 - 4 where the crack length and the crack opening stress were treated as two state variables. Simulation results showed that this model gave good predictions when compared with experimental data for aluminium alloy 7075‐T6 and 2024‐T3 at constant‐amplitude load as well as with overloads. These model predictions were, however, poor for cases with over/under load or under/over load sequences where load excursion effects were underestimated. A third‐order state‐space model is presented is this paper that is believed to be more accurate for predictions of fatigue crack growth for ductile alloys under various loadings. The constraint factor calculated from an algebraic equation in the second‐order state‐space model is treated as the third state variable in this model. Through simulations, it is shown that the third‐order state‐space model gives better predictions than the second‐order state‐space model and FASTRAN II, especially when the effects of over/under load and under/over load are necessary considerations.     

A finite element model for single-sided crack patchings

A finite element model is established for analyzing the behavior of cracked plates which are repaired with a single-sided patch. The formulation is based on the Reissner-Mindlin plate theory with an assumed variation of the transverse shear and normal stresses through the thickness of the cracked -plate and patch. The generalized stress-strain relations relating the transverse shear stress resultants and the adhesive stresses to the displacements of the plate and patch are established by using a variational principle. By means of the finite element model presented herein, single-sided crack patching problems can be solved with a reasonable estimate of the adhesive stresses and the stress intensity factor. Numerical examples are provided to illustrate the effects of the patch size on the stress intensity factor in the cracked plate and the stress distribution in the adhesive layer, and compared with results from the previous analysis.     

A statistical model of crack formation in welds

It is shown that an exponential distribution of defect depth follows from the results of a Monte Carlo simulation and from combinatorical arguments, if the formation of cracks in welds is considered as a random process. Simple relations between the model parameters, the parameter of the exponential distribution function and the expected number of defects per unit length are deduced. The obtained distribution of the depth to length ratio of the defects cannot be described by a simple distribution function.