Averaged strain energy density and J-integral for U- and blunt V-shaped notches under torsion

Analytical solutions for the J-integral and the strain energy density (SED) averaged over a control volume are presented for U- and blunt V-shaped notches under torsion. The influence of the notch tip radius and the notch opening angle are fully included in the new proposed relationships. These expressions take advantage of some recent solutions for the stress fields ahead of blunt notches under Mode III loading and can be seen as a synthesis of the efforts carried out during the last years by the present authors on this topic. Afterwards, the expressions are applied to a recent set of experimental data from cracked, notched and plain specimens tested under torsion at low temperature \((-60\,^{\circ }\hbox {C})\) . The large scatter shown by notched specimens in terms of maximum elastic stresses strongly reduces when the J-integral or the local SED are used in combination with the specific control volumes.     

Calculation of weight functions for three-dimensional bodies with cracks

A new algorithm is proposed for the numerical calculation of weight functions used in the determination of stress intensity factors by the finite elements method. The algorithm is based on the method of equivalent volume integration. It is shown that weight functions can be obtained for cracks in three-dimensional bodies with the stress intensity factor averaged over a small section of the crack front. A numerical example demonstrating the usefulness of the algorithm is presented.Moscow. Moscow Physico-Engineering Institute. Translated from Problemy Prochnosti, No. 11, pp. 32–36, November, 1989.     

Boundary elements for cracks and notches in three dimensions

Plane and curved cracks are modelled by boundary elements, of geometry defined by conforming quadratic and hybrid quadratic‐Hermitian cubic shape functions. Displacement and traction are interpolated by the quadratic functions, supplemented by singular functions by which are multiplied stress intensity factors corresponding to each of the three modes of crack opening displacement, for the first three eigenvalues of the Williams eigenfunction expansion and its equivalent for antiplane strain. Singular and hypersingular boundary integral equations are taken at nodes of elements and auxiliary collocation points. Singular and hypersingular components of integrals are evaluated by consideration of trial displacement fields (simple solutions) for subdomains lying to either side of the crack. Examples are shown of buried and surface cracks, and computed results compared with those obtained by other methods. For surface cracks, the computed results reveal the cause of significant discrepancies between values given by well established empirical and other formulae. The modelling of notches is demonstrated by an analysis of stress in rock near a tunnel intersection. Computational efficiency is discussed, and improvements and extensions of the analysis are proposed. Copyright © 2005 John Wiley & Sons, Ltd.     

Propagation of fatigue cracks from notches

Abstract

The traditional design approaches to fatigue at notches, based on stress level–endurance relationships, are briefly reviewed. It is shown, by considering crack propagation from notches and invoking a change in control mode from notch plasticity to crack-tip plasticity, that a critical stress condition can be obtained which must be exceeded if the crack is to propagate to failure. The traditional techniques are then reinterpreted and explained by this propagation method. An example is given of crack growth from a sharp defect at a weld toe. It is shown that the integration of an elastic fracture mechanics growth law can reproduce stress range–cycles to failure data for this situation. There are, however, complexities of stress analysis and crack shape. A simple treatment of residual stresses affecting the threshold and slow–growth regimes, shows some promise as a technique for accounting for residual stresses.

MST/70     

J-integral and CMOD for external inclined cracks on autofrettaged cylinders

Autofrettage is a well-known method to increase the load carrying capacity of the pressure vessels. The autofrettage outcomes can be contrary when the imperfections or material discontinuities are on the outer surface. In this study, the influences of the external inclined cracks on the fracture parameters in autofrettaged cylinders were studied. The inclination angles of cracks are different but they have the same depth. Crack mouth opening displacements and J-Integral values along the cracks front were investigated. The effects of autofrettage ratio, inclination angle, crack depth and length, the ratio of inner to outer radius and applied pressure were studied. The J-Integral values were calculated by using a modified equation taken from the recent literature, which includes the residual stress effects.     

J integral applications for short fatigue cracks at notches

Elastic and elastic-plastic fracture mechanics solutions are modified to predict the behaviour of short cracks. An effective crack length, l ₀ is introduced into the solutions for both the linear elastic stress intensity factor and the J integral. Crack growth results for short cracks, in both elastic and plastic strain fields of unnotched specimens, when interpreted in terms of the modified solutions, show excellent agreement with elastic long crack data. The modified J integral solutions are extended to plastically strained notches, and the solutions obtained are tested in the correlation of data for growth of sort cracks near notches of varying severity with data for long crack under elastic loading. Although constant stress amplitude tests of these notches gave crack growth rate versus crack length curves which varied from monotonically increasing for blunt notches, to an initial decrease followed by an increase of sharp notches, all the data fell within the long crack data when correlated by the J integral solutions. Conversely, these solutions can be used to predict elastic and inelastic short crack growth curves for notches of various severities.

---

Résumé On a modifié les solutions de mécanique de rupture élastique et élastoplastique afin de prédire le comportement de fissures courtes. On a introduit une longueur effective de fissure l ₀ dans les solutions donuant le facteur d'intensité de contrainte linéaire élastique et l'intégrale J. Les résultats de croissance de fissure dans le cas de fissures courtes dans des éprouvettes non entaillées soumises à des champs de déformation élastique ou plastique, font état d'un excellent accord avec les données afférant à des fissures longues en condition élastique, lorsqu'ils sont interprétés sous forme de solutions modifiées. Les solutions des intégrales J sont extrapolées aux cas des entailles sollicitées dans le domaine plastique, et les solutions obtenues sont éprouvées dans une corrélation des données de croissance de fissures courtes au voisinage d'une entaille de sévérités diverses, avec les données de croissance de fissures longues sous mise en charge élastique.Les essais à amplitude de contrainte constante sur ces entailles conduisent à une vitesse de croissance qui, en fonction de la longuer de fissure, varie d'un accroissement régulier dans le cas d'entailles arrondies, à une diminution suivie d'un accroissement, dans le cas d'entailles aiguës. Ce nonobstant, toutes les données se sont révélées similaires aux données pour de longues fissures, lorsque l'on établit la corrélation des solutions des intégrales J.Complémentairement, ces solutions peuvent être utilisées pour prédire les courbes de croissance des fissures courtes élastique et inélastique, dans le cas d'entailles de sévérités différentes.

     

Estimations of the T-stress for small cracks at notches

The T-stress is increasingly being recognized as an important additional stress field characterizing parameter in the analyses of cracked bodies. Using T-stress as the constraint parameter, the framework of failure assessments including the constraint effect has been established; and the effect of T-stress on fatigue crack propagation rate has been investigated by several researchers. In this paper, a simple method for determining the T-stress for small notch-emanating cracks is presented. First, the background on the T-stress calculation using the superposition principle and the similarities between the elastic notch-tip stress fields described by two parameters: the stress concentration factor (K_t) and the notch-tip radius (ρ), are summarized. Then, the method of estimating T-stress for both short and long cracks at the notches is presented. The method is used to predict T-stress solutions for cracks emanating from an internal hole in a wide plate, and cracks emanating from an U-shaped edge notch in a finite thickness plate. The results are compared to the T-stress results in the literature, and the T-stresses solutions obtained from finite element analysis. Excellent agreements have been achieved for small cracks. The method presented here can be used for a variety of notch crack geometries and loading conditions.     

Analytical evaluation of J-integral for elliptical and parabolic notches under mode I and mode II loading

In the present work the J-integral (indicated here as J_Vρ because two parallel flanks are not present) was calculated by using, along the free border, the exact analytical stress distribution for the ellipse and the asymptotic one for parabolic notches. The material was assumed as homogeneous isotropic and linear elastic. First, for an ellipse under remote tensile loading, the expression of J_Vρ has been analytically calculated on the basis of Inglis’ equations. The equations have been used to prove that, in terms of J-integral, the crack is the limit case of an equivalent elliptic notch. Furthermore, by distinguishing the symmetric and skew-symmetric terms, the well-known Stress Intensity Factors (SIF) of mode I and II for a crack in a wide plate under tension are obtained by adding a limiting condition. Second, by means of Creager–Paris’ equations, J_Vρ has been analytically calculated for a parabolic notch of assigned tip notch radius ρ. The asymptotic value of J_Vρ and the relationship between the peak stress and the relative SIF are the same as the ellipse. Finally, as an engineering application, we provide an accurate formula for the evaluation of the Notch Stress Intensity Factors of a crack, mainly subjected to tensile stress, from the peak stress of the equivalent ellipse under the same loading.     

SIF and threshold for small cracks at small notches under torsion

The basic approach to the problem of torsional fatigue strength of pieces containing defects is based on the stress concentration factor concept. However, experiments have shown that the torsional fatigue limit of specimens containing small holes is controlled by the threshold condition for small cracks emanating from small notches. Therefore, the ratio of torsional to bending fatigue limit ( τ _w / σ _w ) for specimens containing small defects must be studied from the viewpoint of fracture mechanics.
The scope of this paper is to address the calculation of the stress intensity factor for a small crack emanating from a three-dimensional hole under a biaxial state of stress by using the weight function method and to apply it to the fatigue limit prediction. The results obtained are in good agreement with experimental results on specimens with defects.     

Nucleation and growth of corrosion-fatigue cracks from semicircular notches

The initial stages of corrosion fatigue on the surfaces of semicircular notches are studied for the system of low-strength carbon steel and a 3% NaCl solution under different loading and electrochemical conditions on the notch surface. It is shown that the process of electrochemical dissolution of the metal plays the predominant role in the first stages of growth of short corrosion-fatigue crack. We suggest an expression for the prediction of the characteristic density of corrosion-fatigue cracks as a function of the synergistic action of cyclic stresses and parameters of the process of electrochemical dissolution on the notch surface. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Published in Fizyko-Khimichna Mekhanika Materialiv. Vol. 34, No. 5, pp. 53–60. September–October, 1998.