Simple bounds for the stress intensity factors by the method of singular integral equations

The method of singular integral equations has become a classical method for solving plane and antiplane, static and dynamic crack problems in isotropic and anisotropic elasticity, particularly in cases where no closed-form solutions are available. In this paper, very simple methods are suggested for obtaining upper bounds for the stress intensity factors at the crack tips from the corresponding singular integral equation without solving this equation, even numerically, and with very few computations. Naturally, such a simplicity should lead to very conservative bounds and this is really the case. But, clearly, in a lot of practical cases such bounds are sufficient. Numerical results in simple crack problems show the efficiency of the proposed methods.     

Effect of crack curvature on stress intensity factors for ASTM standard compact tension specimens

The stress intensity factors (SIF) are calculated using the method of lines for the compact tension specimen in tensile and shear loading for curved crack fronts. For the purely elastic case, it was found that as the crack front curvature increases, the SIF value at the center of the specimen decreases while increasing at the surface. For higher values of crack front curvatures, the maximum value of the SIF occurs at an interior point located adjacent to the surface. A thickness average SIF was computed for parabolically applied shear loading. It is assumed that it reflects the average stress environment near the crack edge. These results were used to assess the requirements of ASTM standards E399-71 and E399-81 on the shape of crack fronts.     

Simple methods of determining stress intensity factors

A prerequisite for any fracture mechanics analysis of a cracked structure, is a knowledge of the stress intensity factor at the tip of the crack. Many methods are available for evaluating stress intensity factors, but if the structural configuration is complex, they are usually costly in time and money. This paper describes some simpler approximate methods which are both quick and cheap. Their use is illustrated by examples typical of aerospace applications, e.g. cracks at holes and cracks in stiffened sheets. The errors introduced into calculations of residual static strength and fatique lifetimes by the use of such approximations are acceptable for many practical cases: They are usually no greater and often smaller than those due to uncertainties in other parameters such as service loads, material toughness, etc.     

Parametric equations for T-butt weld toe stress intensity factors

This paper describes the generation of parametric equations for weld toe stress intensity factors. The methodology employed used a two-dimensional finite element analysis to evaluate the ‘crack opening’ stress distribution in the uncracked plane of T-butt geometries. This was then used as input into a dedicated weight function solution for the determination of stress intensity factors. The final parametric equations describe the stress intensity factor distributions for tension and bending as a function of plate thickness, weld attachment width, weld angle, weld root radius, crack length and crack shape. The equations are compared and validated against a wide spectrum of published values and appear by comparison accurate and wide ranging. The validation exercise uncovered situations where present design guidance is unconservative.     

The exact calculation of stress intensity factors in transformation toughened ceramics by means of integral equations

The stress induced transformation toughening is described by means of a quantitative model. A Griffith crack in an infinite plate under uniform tension interacts with a transformed circular ZrO₂ particle. The crack is stabilized by the volume expansion of the ZrO₂ inclusion which accompanies the phase transition: An overlying inclusion compresses the flanks of the crack, whereas a particle in front of the crack opens them so that the crack will be attracted and finally absorbed! The stress intensity factors corresponding to these situations are calculated numerically using the technique of singular integral equations.

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Résumé On décrit le durcissement par transformation sous contrainte au moyen d'un modèle quantitatif. Une fissure de Griffith dans une plaque infinie sous tension uniforme est en interaction avec une particule circulaire de ZrO₂ transformée. La fissure est stabilisée par l'extension du volume de l'inclusion de ZrO₂ qui accompagne le changement de phase. Une inclusion décalée comprime les flancs de las fissure tandis qu'une particule en front de fissure les ouvre de telle manière que la fissure sera attirée et finalement absorbée. On calcule, par voie numérique, les facteurs d'intensité de contrainte correspondant à ces situations, en utilisant la technique des équations intégrales singulières.

     

On the calculation of derivatives of stress intensity factors for multiple cracks

In this paper, the work of Lin and Abel [Lin SC, Abel JF. Variational approach for a new direct-integration form of the virtual crack extension method. Int J Fract 1988;38:217-35] is further extended to the general case of multiple crack systems under mixed-mode loading. Analytical expressions are presented for stress intensity factors and their derivatives for a multiply cracked body using the mode decomposition technique. The salient feature of this method is that the stress intensity factors and their derivatives for the multiple crack system are computed in a single analysis. It is shown through two-dimensional numerical examples that the proposed method gives very accurate results for the stress intensity factors and their derivatives. It is also shown that the variation of mode I and II displacements at one crack-tip influence the mode I and II stress intensity factors at any other crack. The computed errors were about 0.4-3% for stress intensity factors, and 2-4% for their first order derivatives for the mesh density used in the examples.     

Experimental calculation of mixed-mode notch stress intensity factors for anisotropic materials

This study developed a least-squares method to find the notch stress intensity factors (SIFs) of anisotropic materials using image-correlation experiments without the requirement of smooth procedures. Complex displacement functions are deduced into a least-squares form, and then displacements from the image-correlation experiments are substituted into the least-squares equation to evaluate the notch SIFs for anisotropic materials. Experimental results compared with the H-integral and finite element analyses show that the SIFs evaluated from the current method are acceptably accurate if more than five displacement terms are included. Moreover, the least-squares SIFs are not sensitive to the maximum or minimum radius of the area from which data is included, so experimental data very near the notch tip is not necessary.     

A modified mapping-collocation technique for accurate calculation of stress intensity factors

A technique combining the advantages of conformal mapping and boundary collocation arguments for calculating stress intensity factors for cracks in plane problems is described. The difficulty of finding the mapping function on a rigidly prescribed parameter region is avoided at the expense of using boundary collocation methods on part of the boundary. Conventional collocation arguments are modified by prescribing stress, force, and moment conditions in a least-square collocation sense. These pseudo-redundant conditions provide a reasonable basis for estimation of the effects of inaccuracy of the boundary conditions. The technique is applied to the problem of a circular disk with an internal crack under a loading of external hydrostatic tension.

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Zusammenfassung Es wird ein Verfahren zur Berechnung von Spannungsintensitä tsfaktoren für Riße in einem ebenen Zustand beschrieben, welches sowohl die Vorteile der Methode der konformen Darstellung als auch die der Verfahren zur Bestimmung der Grenzbedingungen miteinander verbindet.Die Anwendung der Verfahren der Festlegung von Grenzbedingungen für einen Teil der Außenlinie ermöglicht es die Schwierigkeiten zu umgehen, welche sich dann ergeben wenn man versucht die darstellende Funktion für einen Bereich mit streng auferlegten Parameter zu bestimmen. Die konventionelle Behandlung dieser Verfahren wird dadurch abgeändert, daß die Bedingungen für Spannungen, Kräfte and Momente im Singe der kleinsten Quadratzahlen auferlegt werden.Diese pseudo-überflüssige Bedingungen ergeben eine Basis zur Beurteilung der Auswirkung einer Ungenauigkeit in der Definition des Umrisses.Diese Methode wird auf das Problem einer runden Scheibe mit inneren Rissen, welche der Wirkung von äußeren Spannungen unterworfen ist, angewendet.

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Résumé On décrit une technique de calcul des facteurs d'intensité de contraintes pour des fissures en état plan, qui combine les avantages de la méthode de la representation conforme et des méthodes de fixation des conditions aux limites.La difficulté que l'on rencontre à trouver la fonction de représentation qui correspond à une région à paramètres imposés est levée par l'emploi des méthodes de fixation des conditions aux limites sur une partie d'un contour. Le traitement conventionnel de ces méthodes est modifié en imposant les conditions de contraintes, de forces et de moments en un ajustement par moindres carrés.Les conditions pseudo-redondantes ainsi réunies procurent une base d'appréciation des effets d'une inexactitude dans la définition du contour.La technique est appliquée au problème du disque circulaire comportant une fissure interne et soumis a faction d'une tension extérieure uniforme.

     

Evaluation of the errors in equations for stress intensity factors of elliptical type cracks

A method of evaluation of the integral error in the approximate equations for stress intensity factors based on the energy balance equation is described. A review of the equations for calculation of stress intensity factors for cracks of elliptical, semielliptical, and quarter-elliptical form is given. The boundaries of applicability of the equations are given, the errors in approximation of the corresponding numerical solutions are noted, and the integral errors in the equations for stress intensity factors are also calculated.Institute of Problems of Strength, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Problemy Prochnosti, No. 10, pp. 53–58, October, 1989.     

The calculation of stress intensity factors for combined tensile and shear loading

Stress intensity calculations are presented for cases of combined tensile and shear loading for a linear elastic material. Using functions of a complex variable, a theory is developed to determine the direction of maximum energy release rate. A finite element method using virtual crack extensions is also used to determine the energy release rate for crack extensions in various directions and in particular that which gives the maximum energy release rate.Except when shear is more significant than tension, these results give good agreement with available experimental evidence. When shear is most significant, plasticity effects are probably becoming important, thereby invalidating the results of any linear theory. However, the results may still be used to determine K _I and K _II numerically from virtual crack extension calculations of J ₁ and J ₂ for general two-dimensional geometries.

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Résumé On présente des calculs de l'intensité des contraintes dans les cas de mises en charge combinée par traction et cisaillement d'un matériau redevable de la mécanique linéaire et élastique. En utilisant des fonctions d'une variable complexe, on développe une théorie pour la détermination de la direction du taux maximum de relaxation de l'énergie. Une méthode par éléments finis utilisant des extensions d'une fissure virtuelle est également employée pour déterminer le taux de relaxation de l'énergie correspondant à des extensions de la fissure dans des directions diverses et, en particulier, dans celle qui donne un taux de relaxation maximum de l'énergie.A l'exception du cas où le cisaillement est significativement plus important que la traction, les résultats sont en bon accord avec les observations expérimentales. Lorsque le cisaillement est proportionnellement le plus significatif, les effets de la plasticité deviennent probablement importants, et rendent invalides les résultats de toute théorie élastique linéaire. Toutefois, la méthode peut être encore utilisée pour la détermination numérique de K _I et K _II au départ de calculs de J ₁ et J ₂ correspondant à une extension d'une fissure virtuelle dans des géométries bidimensionnelles.

     

Dynamic stress intensity factors studied by boundary integro-differential equations

The boundary integro-differential equation method is illustrated by two numerical examples concerning the study of the dynamic stress intensity factor around a penny-shaped crack in an infinite elastic body. Harmonic and impact load on the crack surface has been considered. Applying the Laplace transform with respect to time to the governing equations of motion the problem is solved in the transformed domain by the boundary integro-differential equations. The Laplace transformed general transient problem can be used to solve the steady-state problem as a special case where no numerical inversion is involved.     

Boundary element crack closure calculation of three-dimensional stress intensity factors

A general method for boundary element-crack closure integral calculation of three-dimensional stress intensity factors is presented. An equation for the strain energy release rate in terms of products of nodal values of tractions and displacements is obtained. Embedded and surface cracks of modes I, II, and III are analyzed using the proposed method. The multidomain boundary element technique is introduced so that the crack surface geometry is correctly modeled and the unsymmetrical boundary conditions for mode's II and III crack analysis are handled conveniently. Conventional quadrilateral elements are sufficient for this method and the selection of the size of the crack front elements is independent of the crack mode and geometry. For all of the examples demonstrated in this paper, 54 boundary elements are used, and the most suitable ratio of the width of the crack front elements to the crack depth is 1/10 and the calculation error is kept within ±1.5 percent. Compared to existing analytical and finite element solutions the boundary element-crack closure integral method is very efficient and accurate and it can be easily applied to general three-dimensional crack problems.     

The calculation of dynamic stress intensity factors for a cracked thick walled cylinder

The finite element method for calculating dynamic stress intensity factors of a thick walled cylinder is studied in this paper, reasonable computational schemes are provided, and the regularity of stress intensity factors for a thick walled cylinder subjected to dynamic inner pressure is obtained. Furthermore, the superposition integral method is presented to compute dynamic stress intensity factors for the thick walled cylinder, which is convenient for calculating dynamic stress intensity factors of different kinds of dynamic inner pressures     

A numerical model for calculation of stress intensity factors in particle-reinforced metal–matrix composites

A two-dimensional finite element model was developed to study effects of particle diameter, mechanical properties of the fiber and matrix materials and loading conditions (Mode 1 and Mixed-Mode). A theoretical model was proposed to calculate the stress intensity factor for an interface crack in Particle-Reinforced Metal–Matrix Composites. The displacement Correlation Method was used to calculate the stress intensity factors K ₁ and K ₂. In the present model the fiber and matrix materials were modeled in linear elastic conditions. The interface crack was considered between the fiber and matrix, without the presence of the interphase. Obtained results show that the key role on the stress intensity factors played by the relative elastic properties of the fiber and matrix. The results also show that K ₁ and absolute K ₂ values increase for both Mode 1 and mixed-mode loading condition once Young’s modulus of the fiber material increases. The values of K ₁ and K ₂ stress intensity factors decrease when the fiber volume fraction increases for Mode 1 loading.