Computation of three-dimensional weight functions for circular and elliptical cracks

Three-dimensional mode I fundamental fields for circular and elliptical cracks in isotropic, finite bodies with prescribed displacement and traction boundaries are analyzed by a previously introduced finite element method [1]. For the circular crack, we present a procedure for determining the Fourier coefficients of the stress intensity factor by using the ordinary fundamental fields.

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Résumé En recourant à une méthode par éléments finis présentée précédemment (1), on analyse les champs fondamentaux à trois dimensions de Mode I correspondant à des fissures circulaires et elliptiques dans des corps finis isotropes, qui sont soumis à des limites définies de déplacements et de traction.Dans le cas d'une fissure circulaire, on présente une procédure pour déterminer les coefficients de Fourier du facteur d'intensité de contrainte, en utilisant les champs fondamentaux ordinaires.

     

Computation of weight functions for three-dimensional cracks by the boundary element method

A boundary element procedure is presented for calculating weight functions for three-dimensional cracks with a smooth front. The weight function for the particular point at the crack front is represented as a sum of regular and singular parts. The known weight function for a circular crack in an infinite body is used as the singular part. The boundary integral equation is formulated for the regular part of the weight function in the vicinity of considered crack front point and for the whole weight function for the rest of the body. A discretized form of the boundary integral equation is given. Some examples are provided to test the accuracy of the proposed procedure.     

Engineerng method of constructing a weight function for plane normal-separation cracks in three-dimensional bodies

On the basis of an analysis of the known solutions for a circular and for a semiinfinite normal-separation crack with straight front in an infinite body we constructed, in analogy to the method of Burns and Oore, a weight function for elliptical cracks. Its use for finding the stress intensity factors K_I at the point of minimal curvature under conditions of uniform loading leads to a maximal error of 10%; at the point of minimal curvature of the crack front the error increases with decreasing ratio of the semiaxes of the ellipse. With the aid of this solution the weight function is found in bounded bodies with a quarter-elliptical, semielliptical, and elliptical crack. A comparison of the data obtained by this method with the values of K_I calculated by the finite-element method under nonuniform loading showed that the suggested method is very accurate.Translated from Problemy Prochnosti, No. 10, pp. 14–22, October, 1992.     

Calculation of the energy J-integral for bodies with notches and cracks

The approximate solutions for calculation of the energy J-integral of a body both with a notch and with a crack under elastic-plastic loading have been obtained. The crack is considered as the limit case of a sharp notch. The method is based on stress concentration analysis near a notch/crack tip and the modified Neuber's approach. The HRR-model and the method based on an equation of equilibrium were also employed to calculate the J-integral. The influence of the strain hardening exponent on the J-integral is discussed. New aspects of the two-parameter J ^* _c-fracture criterion for a body with a short crack are studied. A theoretical investigation of the effect of the applied critical stress (or the crack length) on the strain fields ahead of the crack tip has been carried out.     

A direct determination of weight functions for arbitrary three-dimensional cracks – a combined analytical and numerical approach

A combined analytical and numerical method is proposed for the determination of the weight functions of stress intensity factors of cracks in an arbitrary three-dimensional elastic body. Having defined the weight functions for a given geometry of a structure, the stress intensity factors for arbitrary loading conditions can be obtained by a simple inner product of the weight function and a traction vector. Traditionally weight functions are defined in the two ways; the one is defined by the hyper-singular term of the eigen-function expansion of the displacement field of a cracked body, and the other is defined by the variation of displacement field with respect to a virtual extension of a crack. In the present paper, the weight functions for stress intensity factors are defined by applying the Maxwell-Betti's reciprocal theorem to an original problem and the auxiliary problems subjected to three kinds of force-couples acting on the crack surfaces near the limiting periphery of an arbitrary three-dimensional crack. In the present formulation, weight functions can be calculated by using a general-purpose finite element code combined with analytical expressions near the condensation point, where hyper-singularities exist. The validity of the method is confirmed by two- and three-dimensional illustrative examples.     

Determination of approximative weight functions for straight through cracks

The method of determining approximative weight functions, well known for straight through surface cracks (edge cracks), is extended to internal straight through cracks. The procedure is outlined in detail for central cracks in endless strips of infinite and finite widths. For the finite width strip the coefficients of the weight function are given in a closed form.

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Résumé On étend aux fissure internes traversantes et directes la méthode de détermination des fonctions pondérales approximatives, dont l'application aux fissures droites débouchant en surface (fissures de bord) est bien connue.On décrit en détail la procédure utiisée dans le cas de fissures centrales dans des bandes sans fin de largeur infinie be finie.Pour les bandes de largeur finie, les coefficients de la fonction pondérale sont fournis sous une forme fermée.

     

Application of weight functions in the analysis of three-dimensional problems of fracture mechanics. Communication 2. Calculation of weight functions

The article deals with asymptotic and variation methods of calculating weight functions. For multiparameter configurations a system of equations was obtained which makes it possible to calculate stress intensity factors for any load from solutions obtained for one or several reference loads, without solution of a boundary-value problem. It is shown that the use of the method of linear springs is effective in the calculation of weight functions.Translated from Problemy Prochnosti, No. 4, pp. 65–68, April, 1991.     

Universal features of weight functions for cracks in mode I

An analysis of known analytical and numerical weight functions for cracks in mode I has revealed that they all have a similar singular term and that it is possible to approximate them with one universal expression with three unknown parameters. The unknown parameters can be determined directly from reference stress intensity factor expressions without using the crack opening displacement function. The universal weight function expression, with suitable reference stress intensity factors, was used to derive the weight functions for internal and external radial cracks in a thick cylinder. These weight functions were then further used to calculate the stress intensity factors for radial cracks in a cylinder subjected to several nonlinear stress fields and were compared to available numerical data.     

Analytical wide-range weight functions for various finite cracked bodies

Closed-form wide-range weight functions have been presented for various finite plane cracked bodies. A unified analytical procedure was used in the derivation. First, accurate crack face displacement expressions for center and edge cracks were determined for the polynomial type reference load case. These displacements were then used to derive analytical weight functions, whose accuracy was critically assessed using the related Green's functions. Stress intensity factors formulae for a number of basic load cases including concentrated forces, polynomial as well as a band of linearly varying stress, have been obtained. These basic solutions combined with superposition method enable stress intensity factors to be rapidly determined for complex loadings, as demonstrated by example engineering crack problems. Discussions were made on the reference load case dependence of the weight functions, and the significance of the number of terms contained in the crack face displacement representation on the solution accuracy at extended crack lengths. The analytical wide-range weight functions have been proved versatile, very cost-saving, easy-to-use, and accurate.     

Computation of shear mode weight functions for circular and elliptical cracks

Three-dimensional shear mode fundamental fields in finite bodies with mixed boundary conditions are analyzed by a special finite element method for circular and elliptical cracks. A procedure for determining the Fourier coefficients of the stress intensity factor for circular cracks is presented. A special series is proposed to represent the computed crack face weight functions for elliptical cracks.     

Determination of approximate point load weight functions for embedded elliptical cracks

This paper presents the application of weight function method for the calculation of stress intensity factors in embedded elliptical cracks under complex two-dimensional loading conditions. A new general mathematical form of point load weight function is proposed based on the properties of weight functions and the available weight functions for two-dimensional cracks. The existence of this general weight function form has simplified the determination of point load weight functions significantly. For an embedded elliptical crack of any aspect ratio, the unknown parameters in the general form can be determined from one reference stress intensity factor solution. This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body. The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of planer embedded cracks subjected to fluctuating non-linear stress fields resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

The weight functions of the configuration of collinear cracks

Collinear cracks have been the subject of many investigations and publications. This paper provides (once and for all) the weight functions for the general configuration. The functions are equally useful for modes I and II.     

A weight function method for the assessment of partially closed three-dimensional plane cracks

The interference and physical contact between mating fracture surfaces can lead, even under the action of traction loads, to the closure of a crack. A low-cost numerical tool for the assessment of three-dimensional partially closed mode-I cracks is presented in this paper. The devised tool is based on the weight function methodology and it allows computing the geometry of the open part of the crack and the stress intensity factor along the complete crack front. The accuracy and versatility of the proposed procedure is assessed by solving a number of examples and comparing the obtained results with those available in the literature.     

Slicing procedure for approximate three-dimensional Green's functions for cracks in plates of finite thickness

Exact results for the stress intensity factor are presented for an external circular crack with oppositely directed concentrated loads applied to the crack surfaces. This result is specialized to the case of a semi-infinite crack in an infinite body with concentrated loads on the crack. A procedure is then suggested by which one can obtain from the corresponding plane result the approximate three-dimensional Green's function (concentrated load result) for any straight crack in an infinite elastic body. This procedure is used to determine the Green's functions for a finite-length crack in an infinite body, and is then used in conjunction with a suggested slicing procedure to obtain approximate three-dimensional Green's function for plates of finite thickness and infinite extent, containing finite length cracks. Previously existing solutions for crack problems are compared with results obtained by application to plate tension and bending problems of the three-dimensional Green's functions. The results indicate that the procedure yields satisfactory results when stress gradients through the plate thickness are not excessive. However, an accurate assessment of the validity of the slicing procedure awaits further progress in three-dimensional crack analysis.

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Zusammenfasung Es werden exakte Werte für den Spannungsintensitätsfaktor im Falle eines kreisförmigen Oberflächenrisses, wobei die Rißoberflächen konzentrierten und entgegengesetzten Belastungen direkt unterworfen sind. Dieses Ergebnis ist anwendbar auf den Fall eines halbunendlichen Risses in einem Körper unendlicher Abmessungen, wobei der Riß konzentrierten Beanspruchungen unterworfen ist.Anschließend wird ein Verfahren vorgeschlagen, welches es ermöglicht die dreidimensionale angenäherte Funktion von Green aus dem entsprechenden planen Ergebnis zu ermitteln und dies für den Fall eines beliebigen geraden Risses in einem elastischen Körper unendlicher Größe. Nach diesem Verfahren werden die Green'schen Funktionen für einen Riß endlicher Größe in einem unendlichen Körper bestimmt. Dieser wird anschließend dazu benutzt um mit Hilfe eines Unterteilungsverfahrens die angenäherten dreidimensionalen Green'schen Funktionen für Feinbleche unendlicher Oberfläche mit Rissen endlicher Abmessungen zu ermitteln.Die unter Anwendung der dreidimensionalen Green'schen Funktionen auf die Probleme von Zug- und Biegebe-anspruchung von Platten erzielten Ergebnisse, werden mit den schon früher vorgeschlagenen Lösungen verglichen. Dieser Vergleich zeigt, daß der vorgeschlagene Weg befriedigende Lösungen ergibt, sofern die Spannungsgradienten über die Dicke des Bleches nicht übermässig groß sind.Um jedoch die Gültigkeit des angewandten Unterteilungsverfahren exakt zu prüfen, sind bedeutende Fortschritte auf dem Gebiet der dreidimensionalen Analyse von Rissen noch erfordert.

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Résumé On présente les valeurs exactes du facteur d'intensité de contrainte dans le cas d'une fissure circulaire périphérique où des charges concentrées opposées sont directement appliquées sur ses surfaces. Ce résultat s'applique au cas de la fissure semi-infinie, dans un corps infini, des charges concentrées étant appliquées à la fissure.On suggère ensuite une procédure permettant d'obtenir la fonction tridimensionnelle approchée de Green à partir du résultat plan correspondant, et ce pour tout cas de fissure droite dans un corps infini et élastique. On détermine selon cette procédure les fonctions de Green pour une fissure de longueur finie dans un corps infini, et on l'utilise ensuite, à l'aide d'un processus de découpage, à l'obtention des fonctions tridimensionnelles approchées de Green pour des tôles d'épaisseur fine et de surface infinie, comportant des fissures de dimensions finies.Les résultats obtenus par l'application des fonctions tridimensionnelles de Green aux problèmes de traction et de flexion des plaques sont comparés aux solutions proposées antérieurement. Il résulte de cette comparaison que la procédure suivie fournit des résultats satisfaisants pour autant que les gradients de contrainte suivant l'épaisseur ne soient pas excessifs.Toutefois, pour vérifier d'une manière exacte la validité du processus de découpage qui a été adopté, il est nécessaire d'attendre que des progrès plus substantiels aient été accomplis en matière d'analyse tridimensionnelle des fissures.

     

Acoustic emission during jumps in subcritical growth of cracks in three-dimensional bodies

A model for jumps in subcritical growth of plane cracks (of arbitrary shape) is proposed for three-dimensional bodies and the accompanying acoustic emission (AE). An approximation method for determining the displacement field resulting from one jump of the crack, based on the fact that the required solution is found in the comparison of displacement fields obtained from currently known problems in crack theory, is proposed. A series of new analytical relationships between parameters of plane cracks in three-dimensional bodies and parameters of AE is given.     

Domain-independent values of the J-integral for cracks in three-dimensional residual stress bearing bodies

This paper describes an approach for computing domain-independent values of the J-integral in the finite element context for three-dimensional bodies containing residual stress. In the analysis of cracked bodies containing residual stress, the usual domain integral formulation results in domain-dependent values of J, and this paper discusses modifications that yield domain independence. Two correction terms are defined. The first of these relates to the spatial gradients of non-mechanical strains in the crack-driving direction, and the second accounts for plastic dissipation included in the material state, but unrelated to fracture. The paper further presents results for two examples recently discussed in the literature. Application of the corrections in these two cases demonstrates the ability of the approach to obtain path-independent domain integral results in residual stress bearing bodies.     

Description of loadings and screenings of cracks with the aid of universal weight functions

The stress intensity vector K _i is defined as the limiting behaviour of the stress near the tip of a crack, the stress components being proportional to r ^?1/2 for any external loading. Internal stresses caused by dislocations show the same power dependence at the crack tip; the stress intensity associated with a loading can thus be screened (or amplified) by a plastic zone. Since for any particular specimen and crack geometry the stress intensity vector must be a functional of the loading and screening which are of vectorial character (lines of forces f _i or dislocations with a Burgers vector b _i) one can define two tensorial weight functions, one for screenings, D _si(x, a), and one for forces, F _si(x, a), so that the stress intensity K _s can be found by integration over the product of weight functions and dislocation or force density. In order to find the weight functions the displacement field and the Airy stress vector must be known for some, completely arbitrary, loading or screening.