Fatigue notch factor and short crack propagation

This paper addresses the problem of high cycle fatigue at notches and the role of short crack propagation in the fatigue notch factor k_f. Ahead of a V-notched feature, the stress field is characterized by two parameters, i.e. the stress concentration factor k_t and the normalized notch stress intensity factor k_n. Whether fatigue strength at a given life is controlled by crack initiation (k_f = k_t) or by short crack propagation (k_f < k_t) depends on k_t, k_n and the material resistances to crack initiation and to short crack propagation. The analysis accounts for the effects of notch acuity, notch size, material and fatigue life on the fatigue notch factor k_f. It opens the door to a new method for predicting fatigue life using two S-N curves for a given material; one being measured from a smooth specimen, the other from a severe V-notch.     

Fatigue life predictions for notch geometries in particle reinforced metal matrix composites

Abstract

The potential use of particle reinforced metal matrix composites (MMCs) for demanding structural applications highlights the need for an effective method to predict the fatigue performance of notched components. The present paper evaluates the ‘critical strain’ technique and its suitability with respect to particle reinforced MMC alloys. Strain controlled fatigue data generated on the aluminium alloy 2124 reinforced by two different fractions of 2–3 μm SiC particles are used to predict the lives of a ‘pseudocomponent’ representative of engineering situations. It is demonstrated that both alloys are essentially cyclically stable compared with monotonic work hardening curves and, where available, data from tension and torsion modes superimpose on a Von Mises effective stress-strain criterion. The implications of the fatigue life analysis are discussed.     

Path-independent integrals and crack extension force for functionally graded materials

Path-independent integrals for functionally graded materials are constructed and calculated around a tip of crack. It is found that the stress intensity factor is not equivalent to the crack extention force in estimating crack growth. The effect of functional gradient on the crack extension force is presented.     

A review of the J and I integrals and their implications for crack growth resistance and toughness in ductile fracture

The application of the J and the I-integrals to ductile fracture are discussed. It is shown that, because of the finite size of the fracture process zone (FPZ), the initiation value of the J-integral is specimen dependent even if the plastic constraint conditions are constant. The paradox that the I-integral for steady state elasto-plastic crack growth is apparently zero is examined. It is shown that, if the FPZ at the crack tip is modelled, the I-integral is equal to the work performed in its fracture. Thus it is essential to model the fracture process zone in ductile fracture. The I-integral is then used to demonstrate that the breakdown in applicability of the J-integral to crack growth in ductile fracture is as much due to the inclusion in the J-integral of progressively more work performed in the plastic zone as it is to non-proportional deformation during unloading behind the crack tip. Thus J _R -curves combine the essential work of fracture performed in the FPZ with the plastic work performed outside of the FPZ. These two work terms scale differently and produce size and geometry dependence. It is suggested that the future direction of modelling in ductile fracture should be to include the FPZ. Strides have already been made in this direction.     

Time-dependent crack growth characterization using rate integrals

This paper investigates the utility of the rate forms of Kishimoto et al.'s integral () and Blackburn's integral (J^*) as parameters for correlating time-dependent crack growth rates. These rate integrals are computed from the results of finite element analyses of crack growth in button-head single edge notch specimens of Alloy 718. The specimens are tested under constant strain and constant load conditions at 593°C and 649°C. The crack tip deformation includes large-scale plasticity, primary and secondary creep. The measured crack growth rates are correlated with the computed rate integrals. The results show that both integrals can consolidate the crack growth data very well.     

Various limit processes for evaluating stresses at crack tip and notch crown tip

The stress at a crack tip point can be obtained in several ways. In the first way, after letting an elliptic notch become a crack, a moving point approaches the crack tip. In the second way, after letting a moving point approach the crown point of notch, the elliptic notch becomes a crack. It is found that for some particular stresses in question the finally obtained results by using two ways are different. The mentioned result is caused by change of the order of limitation for evaluating the stress. Assumed that there are two processes, one is by letting the moving point approach the crown point of elliptic notch, and other is by letting the ellipse become a crack. In the third way, the two processes are completed simultaneously. It is found that, for example, in case that a remote shear is applied to a elliptic notch of infinite plate, the shear stress at the crown point can reach an arbitrary value, provided an appropriate limit process is assumed in the third way. Similar result is found in the normal remote loading of an elliptic notch. In addition to the detail analysis, the obtained results are summarized in the form of theorem.     

On crack closure and the notch size effect in fatigue

A method for determining the fatigue notch-size-effect is presented based upon the development of closure in the wake of a newly formed crack growing from a notch. A comparison with experimental results is in accord with predictions. The effect of crack closure on the formation of non-propagating cracks at notches and on notch sensitivity is also discussed.     

The evaluation of domain integrals in complex multiply-connected three-dimensional geometries for boundary element methods

The treatment of domain integrals has been a topic of interest almost since the inception of the boundary element method (BEM). Proponents of meshless methods such as the dual reciprocity method (DRM) and the multiple reciprocity method (MRM) have typically pointed out that these meshless methods obviate the need for an interior discretization. Hence, the DRM and MRM maintain one of the biggest advantages of the BEM, namely, the boundary-only discretization. On the other hand, other researchers maintain that classical domain integration with an interior discretization is more robust. However, the discretization of the domain in complex multiply-connected geometries remains problematic. In this research, three methods for evaluating the domain integrals associated with the boundary element analysis of the three-dimensional Poisson and nonhomogeneous Helmholtz equations in complex multiply-connected geometries are compared. The methods include the DRM, classical cell-based domain integration, and a novel auxiliary domain method. The auxiliary domain method allows the evaluation of the domain integral by constructing an approximately C ¹ extension of the domain integrand into the complement of the multiply-connected domain. This approach combines the robustness and accuracy of direct domain integral evaluation while, at the same time, allowing for a relatively simple interior discretization. Comparisons are made between these three methods of domain integral evaluation in terms of speed and accuracy. This work was partially supported by the United States Department of Energy (DOE) grants DE-FG03-97ER14778 and DE-FG03-97ER25332. This financial support does not constitute an endorsement by the DOE of the views expressed in this paper.     

Numerical analysis of the notch effect and the behaviour of notch crack in adhesively bonded composite laminates

In this study the finite element method is used to analyse the notch effect and the behaviour of notch cracks in adhesively composite laminate under tension by computing respectively the stress concentration factor at the notch tip which characterize the notch strength and the stress intensity factor at the crack tip which characterize the resistance to the crack propagation. The effects of the adhesive properties and fiber orientation on the variation of both stress concentration and stress intensity factors are highlighted. The obtained results show that the notch strength is reduced in the layer of the laminate of which the fiber orientation is in the applied load direction; the resistance to the crack propagation is also reduced in this type of layer. The stress intensity factor at the tip of notch crack exhibits an asymptotic behaviour as the crack length increases.     

Computational studies on path independent integrals for non-linear dynamic crack problems

Recently the authors have derived various new types of path independent integrals in which the theoretical limitations of the so-calledJ integral are overcome. First, for elastodynamic crack problems, a path independent integralJ which has the physical meaning of energy release rate was derived. Later, more general forms of path independent integralsT ^* andT were derived, which are valid for any constitutive relation under quasi-static as well as dynamic conditions.This paper presents the theoretical and computational aspects of these integrals, of relevance in non-linear dynamic fracture mechanics. An efficient solution technique is also presented for non-linear dynamic finite element method in which a factorization of the assembled stiffness matrix is done only once throughout the computation for a given mesh pattern. Finite element analyses were carried out for an example problem of a center-cracked plate subject to a uniaxial impact loading. The material behavior was modeled by three different constitutive relations such as linear-elastic, elastic-plastic, elastic-viscoplastic cases. The applicability of theT ^* integral to non-linear dynamic fracture mechanics was shown with the numerical results.This study was supported by the Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, and in part by the Science and Technology Grant from Toray Science Foundation     

Two-dimensional principal value hypersingular integrals for crack problems in three-dimensional elasticity

Summary A principal value definition of the basic hypersingular integral in the fundamental integral equation for two-dimensional cracks in three-dimensional isotropic elasticity is proposed. As is the case with the corresponding definitions of Cauchy-type one-dimensional and two-dimensional principal value singular integrals, as well as Mangler-type one-dimensional principal value hypersingular integrals, the present definition is based on the special consideration of an appropriate region around the singular point. The cases of circular, square and equilateral triangular regions are considered in some detail.     

Finite element analysis of bi-material interface notch crack behaviour

The finite element method is extended to the analysis of the behaviour of an interface crack in bi-material specimen with a central hole. First, only the notch effect is considered, the field of stress and variation of the stress concentration factor as a function of the Young’s modulus ratio are determined. Secondly, the notch interface crack behaviour is investigated, the variations of the stress intensity factor versus the Young’s modulus ratio and crack length are shown as well as the distribution of stresses in the plate and along the interface.     

Application of conservation integrals to interfacial crack problems

Properties of the J, L and M integrals, proposed by Knowles and Sternberg for homogeneous materials are explored here for bi-material interfaces. The integrals are shown to satisfy the conservation law under certain conditions on the interfaces. Relations between the stress intensity factors and the conservation integrals for interfacial cracks in isotropic, linear-elastic materials are derived. The conservation integrals for some interfacial cracks are applied to get the stress intensity factors in a very simple way without solving the complicated boundary value problems. For interfacial cracks in finite-sized medium some numerical computations are carried out to verify the usefulness of the conservation integrals.     

A general treatment of crack tip contour integrals

Given a general balance statement we derive an expression for the associated crack tip flux integral. The conditions under which the integral is physically meaningful and yields a non-trivial result are outlined. To illustrate the approach a number of well known integrals in use in fracture mechanics are derived. It is demonstrated that complementary analogues to these integrals can be derived in a similar fashion and a result indicating the equality of dual integrals under quite general conditions is presented. We discuss the domain integral method as an alternative means of representing crack tip integrals and we show that the method may be interpreted as a particular form of Signorini's theorem of stress means. A discussion of some associated integral identities is presented.

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Résumé Une expression pour l'intégrale de flux á l'extrémité d'une fissure est obtenue sur base d'un état général d'équilibre. On souligne les conditions pour lesquelles l'intégrale a un sens physique et ne mène pas á des résultats triviaux. L'approche adoptée est illustrée par 1'établissement d'une série d'intégrales bien connues, utilisées en mécanique de rupture. On démontre que l'on peut tirer d'une manière similaire des contreparties compl'ementaires à ces intégrales et on présente un résultat indiquant que les intégrales doubles ainsi établies sont égales dans des conditions très générales. On discute de la méthode d'intégration sur un domaine comme variante de représentation des intégrales à l'extrémité d'une fissure et on montre que cette méthode peut être interprêtée comme une forme particulière du théorème de Signorini des contraintes médianes. On présente enfin une discussion sur diverses formes d'intégrales associées.

     

Evaluation of fatigue crack initiation life from a notch

Local strain at the notch-root and its effect on fatigue crack initiation was investigated in four metals by the real-time, fine-grid method. Special attention was focused on local notch-root strain behaviour until crack initiation. From the application of strain hysteresis at the notch root, the maximum strain under loading conditions during each cycle was investigated in detail. One of the main results was that the maximum strain value at the first cycle of the fatigue test coincided with that at crack initiation. Maximum strain defined from the cyclic strain changes at the notch root was proposed as one possible parameter for estimating fatigue crack initiation life. Based on the curvilinear relationship between maximum strain and number of cycles to crack initiation, a new life evaluation method for fatigue crack initiation is proposed. This approach differs fundamentally from the usual fracture mechanics method based on the stress intensity factor.     

Prediction of fatigue crack initiation lives at elongated notch roots using short crack concepts

Re-initiation lives of fatigue cracks departing from stop-holes roots, previously introduced at the tip of deep cracks on modified SE(T) specimens, have been satisfactorily predicted using their properly calculated notch sensitivity factor q, considering the notch tip stress gradient influence on the fatigue behavior of mechanically short cracks. This is an indispensable detail, since traditional q estimates are only applicable to semi-circular notches, whereas elongated slits can have q values which also depend on their shape, not only on their tip radius. Based on this experimental evidence, a criterion for acceptance of short cracks is proposed.