Computing stress intensity factors for curvilinear cracks

The use of the interaction integral to compute stress intensity factors around a crack tip requires selecting an auxiliary field and a material variation field. We formulate a family of these fields accounting for the curvilinear nature of cracks that, in conjunction with a discrete formulation of the interaction integral, yield optimally convergent stress intensity factors. In particular, we formulate three pairs of auxiliary and material variation fields chosen to yield a simple expression of the interaction integral for different classes of problems. The formulation accounts for crack face tractions and body forces. Distinct features of the fields are their ease of construction and implementation. The resulting stress intensity factors are observed converging at a rate that doubles that of the stress field. We provide a sketch of the theoretical justification for the observed convergence rates and discuss issues such as quadratures and domain approximations needed to attain such convergent behavior. Through two representative examples, a circular arc crack and a loaded power function crack, we illustrate the convergence rates of the computed stress intensity factors. The numerical results also show the independence of the method from the size of the domain of integration. Copyright © 2015 John Wiley & Sons, Ltd.     

The stress intensity factors and crack profiles for centre and edge cracks in plates subject to arbitrary stresses

Expressions are given for the stress intensity factors for centre and edge cracks in plates subject to an arbitrary stress. The stress intensity factors for cracks subject to a stress expressed in the form of a polynomial of order 8 are given and their application to cracks in service components are indicated. The results are in good agreement with known solutions. Expressions for the crack profiles of centre and edge cracks subject to arbitrary stress are constructed. They predict profiles in good agreement with the results of a finite element stress analysis.

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Résumé On présente des expressions pour des facteurs d'intensité de contraintes relatives à des fissures centrales et des fissures latérales dans des tôles sujettes à des contraintes arbitraires. Les facteurs d'intensité des contraintes correspondant à des contraintes exprimées par une loi polynômique de l'ordre de 8 sont établies, et l'on indique leur possibilité d'application à des fissures dans des composants en service.Les résultats obtenus sont en bon accord avec les solutions connues. On établit des expressions permettant de décrire les profils de fissuration dans le cas de fissures centrales et latérales sujettes à des mises en charge arbitraires. Les prédictions de ces expressions sont en bon accord avec les résultats d'une analyse des contraintes par éléments finis.

     

Plane strain stress intensity factors for branched cracks

A method of calculating stress intensity factors for branched and bent cracks embedded in an infinite body has been developed. The branches are always assumed to be sharp cracks and are modelled by dislocation distributions. The original crack may be either sharp or of elliptical cross-section with finite root radius. Hence, the method which has a precision ±2%, is also applicable to the study of crack branches emanating from elliptical holes and, approximately, also from notches. The following detailed calculations have been made assuming mode I loading: branched sharp crack with branches of equal and different length, bent sharp crack, and one and two crack branches emanating from the crack with a finite root radius. Bending of a sharp crack under mixed mode loading has also been studied. The criteria of maximum tensile stress and maximum energy release rate used in the study of direction of crack propagation are discussed.     

Averaged stress intensity factors for embedded elliptical cracks

As an extension of well-known two-degree-of-freedom models describing virtual crack extensions in weight function applications for surface crack configurations different possibilities are suggested of four-degree-of-freedom models used for embedded elliptical cracks. As a result, the geometric function for an elliptical crack in a strip of finite width is presented for tension and bending. FEM-calculations have been carried out to check the quality of the solution.     

Stress intensity factors for cracks of arbitrary shape near an interfacial boundary

Modes I, II and III stress intensity factors for a crack of arbitrary planar shape near a bimaterial interface are calculated. The solution utilizes the body-force method and requires Green's functions for perfectly bonded elastic half-spaces. The formulation leads to a system of two-dimensional singular integral equations whose solutions represent the three modes of crack opening displacement. Numerical examples of a semicircular or semielliptical crack terminating at the interface and circular or elliptical cracks contained in one material are given for both internal pressure and farfield tension.     

Static finite element stress intensity factors for annular cracks

Results of finite element static stress intensity factor calculations for an annular crack around a spherical inclusion (void) are presented and compared with those from approximate analytical methods.     

Analysis of stress intensity factors for three-dimensional cruciform cracks

The stress intensity factors for three-dimensional cruciform surface cracks in a semi-infinite body are numerically calculated by the body force method. Mindlin's point force solution is used for the derivation of basic equations to express the influence coefficient of triangular elements, into which the crack is divided. The interactions between crossed crack planes as well as contact between crack surfaces are considered in the iterative manner. Stress intensity factors for a cruciform median crack and a cruciform semicircular crack under a point force on the surface of a semi-infinite solid are analyzed. The possibility of growth of a median crack toward the free surface of the semi-infinite solid is discussed. A cruciform semicircular surface crack under remote uniaxial tension, or under combined tension and compression is also analyzed. The effect of contact of crack surfaces on stress intensity factors is discussed.     

Compounded stress intensity factors for cracks at fastener holes

The compounding technique, a method for obtaining stress intensity factors for complex geometrical configurations from those for simple configurations, is applied to cracks at the edges of the holes in a row of fastener holes. The holes are assumed to be loaded on their perimeters; the original technique requires modification in order to incorporate these loads into the “equivalent crack” concept. The accuracy of the method is tested by comparing the solution obtained by compounding with that obtained by a collocation technique for cracks at the edges of a row of pressurized holes. Finally stress intensity factors are obtained for cracks at a row of fastener holes near the edge of a sheet.     

Opening mode stress intensity factors for cracks in pin-loads joints

Opening mode stress intensity factors are determined using point matching for two diametrically opposed radial cracks at a hole in a rectangular sheet where the cracks lie along the line of the minimum section. The sheet is subjected to either a biaxial stress on its edge or a pressure distribution on the hole boundary. The pressure on the hole is uniform or has a cosine distribution, symmetrical about the line of the cracks and having zero value on the crack-line. By applying the principle of superposition it is shown that the results can be applied to pin-loaded joints to determine stress intensity factors for different combinations of pin-load and interference fit and the effects of load transfer ratio between the pin or rivet load and the remote loading. Detailed results are given for stress intensity factors of cracks in a wide range of typical pin-joint configurations and it is shown that the stress intensity factors, for small cracks, are strongly dependent on the type of pressure distribution assumed to represent the pin-loading. Reducing the distance from the pin to the stress-free end of the joint is shown to increase the stress intensity factor more in the case of pin-loading than in the case of a uniaxial stress. The stress intensity factors, determined for simple lug-joints, are in agreement with available results from existing theoretical and experimental work and, for cracks at fastener holes, comparisons are made with other more approximate solutions.

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Résumé Les facteurs d'intensité de contrainte sous des conditions d'ouverture sont déterminés en utilisant le point de rencontre de deux fissures radiales diamétralement opposées partant d'un trou dans un feuillard rectangulaire et telles que les fissures se situent sur la ligne de la section minimale. Le feuillard est soumis soit à des contraintes biaxiales sur ses bords ou à une distribution de pression sur les bords du trou. La pression sur le trou est uniforme ou a une distribution en cosinus symétrique par rapport à la ligne des fissures, en passant par zéro sur la ligne des fissures. En appliquant le principe de superposition, on montre que les résultats peuvent être appliqués à des liaisons mécaniques concentrées, en vue de déterminer les facteurs d'intensité de contrainte pour différentes combinaisons de charge appliquées sur ces liaisons, l'effet d'interférence, et les effets de transfert de charge entre la charge appliquée sur le rivet et la charge éloignée. Des résultats détaillés sont fournis pour les facteurs d'intensité de contrainte relatifs à des fissures correspondant à une large gamme de configurations typiques de joints; on montre que les facteurs d'intensité de contrainte dans le cas de petites fissures dépendent largement du type de distribution de pression supposée représenter la mise en charge locale. On montre qu'en réduisant la distance entre le rivet et l'extrémité libre de contrainte de l'assemblage, on augmente le facteur d'intensité de contrainte d'avantage dans le cas de la liaison mécanique locale que dans le cas d'une contrainte uniaxiale. Les facteurs d'intensité de contrainte déterminés dans le cas d'assemblages simples sont en accord avec les résultats disponibles à partir des travaux existants sur le plan théorique et expérimental et, en ce qui concerne les fissures à des trous de rivet, des comparisons peuvent être faites avec d'autres solutions plus approximatives.

     

Calculation of stress intensity factors for curved cracks in anisotropic FGMs

Abstract

A combined analytical and numerical method is proposed for computation of mixed-mode stress intensity factors (SIFs) for arbitrary curved cracks in anisotropic functionally graded materials (FGMs). By developing a pair of closed-form expressions that relate the SIFs and the J_k-integrals, it is anticipated that the SIFs can be properly extracted should the J_k-integrals be accurately evaluated. To this end, a novel method for calculating the J_k-integrals is presented and has proved reasonably accurate in numerical computations. Since neither a priori information nor extra auxiliary solutions corresponding to the singular behavior is required, this proposed scheme appears to be applicable to problems containing arbitrary shapes of curvature in generally anisotropic FGMs.     

Boundary element analysis of stress intensity factors for Z-shaped cracks

The stress intensity factors for Z-shaped cracks are computed by the boundary element method employing the multiregion technique and the double-point concept. To demonstrate the validity of the current method, the stress intensity factors of other well-known simple models such as a slanted edge crack and an arcular crack are determined, in advance, which are proved to be in good agreement within 5% with the preexisting solutions. Z-shaped cracks are analyzed with various branch crack lengths and branching angles.     

Thermal stress intensity factors of interface cracks in bimaterials

In this communication numerical results for thermal stress intensity factors (TSIFs) of interface cracks are presented for arbitrary material combinations which are characterized by Dundurs' parameters and . It is shown that TSIFs are linear in crack length ratio a/w and quadratic in and depend also on . The local phase angle at the interface crack tip is a linear function of . The striking feature of residual thermal stresses is their strong mode II character at the tip of an interface crack. In the framework of linear elasticity these TSIFs can be linearly superimposed on the stress intensity factors (SIFs) from applied loads for interface cracks in composites.     

Stress intensity factors for through-thickness cracks in a wide plate: Derivation and application to arbitrary weld residual stress fields

Exact closed-form stress intensity factor (SIF) solutions have been developed for mode-I, II and III through-thickness cracks in an infinite plate. Centre-crack problems have been analysed comprehensively in the literature, but the focus has been on the effect of simple loading about the crack centre. In the current work, the formula of Sih-Paris-Erdogan was extended to consider the difference in SIF on the left and right crack tips under an asymmetric stress field. Mathematical manipulations were performed to derive exact stress magnification factors for SIF computations and simultaneously circumvent the problem of crack-tip stress singularity. The solutions so obtained are applied to derive the residual SIFs that would act on a crack growing under the influence of the residual stress fields associated with VPPA (variable polarity plasma arc) and friction stir welds, using measured residual stress profiles.     

Analysis of stress intensity factors of modes I,II and III for inclined surface cracks of arbitrary shape

The analysis of stress intensity factors K_I, K_II and K_III by the body force method is developed for an arbitrarily shaped surface crack. The stress intensity factors for basic problems as semielliptical cracks, rectangular cracks and triangular cracks inclined to tensile axis at the surface of a semiinfinite body are numerically calculated.Defining the polar coordinate system (r, θ) on the plane which is perpendicular to both the plane of crack and the line of crack front, we can determine the stress intensity factor K_gq which prescribes the stress field of the tangential stress σ_gq. The maximum value K_θmax of K_gq, along the crack front can be expressed by the approximate formula: K_{θ max} ≅ 0.650 σ₀√π√area_p; Poisson's ratio v = 0.3, where area_pis the crack area projected in the direction of the maximum principal stress σ₀. The limitation and application rule of the approximate formula is also described.     

The stress intensity factors of slightly undulating interface cracks of bimaterials

A modified interface crack with slightly undulating profile, which has a good agreement with reality and retains the simplicity of a mathematical model, is presented in this paper. This model is utilized to reveal some of the properties of uneven cracks, especially the stress intensity factors. As we know, many failures occurring in the interface are induced by crucial lateral stresses which are parallel to the interface. Hence, when the lateral stresses are much stronger than others, the corresponding solution is also derived for understanding how the lateral stresses affect the stress intensity factors as the crack is uneven. In the present paper, the Hilbert's problem enables different perturbed-interface cracks to be solved in an unified manner. Muskhelishvili's potential formulation is used to derive, by means of a perturbation analysis technique, an homogeneous and general Hilbert's problem.