On the finite element method for calculating stress intensity factors with a modified elliptical model

A finite element method based on the modification of the elliptical displacement function model developed earlier by the authors is presented for the determination of stress intensity factors in cracked bodies. The modified elliptical model not only retains the simplicity of the original method in describing stress conditions at the crack tips but also extends the application of the method to cases where the elliptical shape of the crack surface is not entirely preserved. The present method avoids the need for successive computations of several strain energies in a cracked body as required by the strain energy approach and of high concentration of very fine elements at the crack tip by the conventional stress approach.

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Résumé On présente une méthode par éléments finis basée sur la modification du modèle de la fonction de déplacements elliptiques développée précédemment par les auteurs, pour déterminer les facteurs d'intensité des contraintes dans des solides fissurés. Le modèle modifié conserve la simplicité de la méthode originale pour décrire les conditions de contraintes aux extrémités d'une fissure. En outre, il permet d'étudier son application à des cas où la forme de la surface de la fissure n'est pas rigoureusement elliptique.La méthode proposée évite de devoir procéder à des calculs en série de plusieurs énergies de déformation dans le corps fissuré, ainsi que le requiert l'approche basée sur l'énergie de déformation. Elle permet aussi de ne pas devoir calculer des concentrations importantes d'éléments à très petites mailles à l'extrémité de la fissure, ainsi que l'exige l'approche concentionnelle basée sur les contraintes.

     

A semi-infinite-crack model for determining mode I stress intensity factors using crack surface displacements

A semi-infinite-crack model is used to supplement the conic section simulation method for determining stress intensity factors of finite cracked bodies under mode I loadings. The actual displaced crack surface profile is found by finite element analysis. For each crack surface segment between two neighbouring nodes, a set of model parameters is found by using the displacements of these two nodes. A stress intensity factor estimate is then calculated from the closed-form formula associated with the model. It is found that near-tip crack surface displacements produce model parameters that are sufficient for quantifying the stress intensity factor. The semi-infinite-crack model can be used either as a stand alone model or in conjunction with the ellipse simulation procedure to form a systematic approach. It is shown that this model can be applied to different geometries and loadings with excellent accuracy.     

A finite element calculation of stress intensity factors by a modified crack closure integral

An efficient technique for evaluating stress intensity factors is presented. The method, based on the crack closure integral, can be used with a constant strain finite element stress analysis and a coarse grid. The technique also permits evaluation of both Mode I and Mode II stress intensity factors from the results of a single analysis. Example computations are performed for a double cantilever beam test specimen, a finite width strip with a central crack, and a pin loaded circular hole with radial cracks. Close agreement between numerical results given by this approach and reference solutions were found in all cases.     

A stiffness derivative finite element technique for determination of crack tip stress intensity factors

A finite element technique for determination of elastic crack tip stress intensity factors is presented. The method, based on the energy release rate, requires no special crack tip elements. Further, the solution for only a single crack length is required, and the crack is advanced by moving nodal points rather than by removing nodal tractions at the crack tip and performing a second analysis. The promising straightforward extension of the method to general three-dimensional crack configurations is presented and contrasted with the practical impossibility of conventional energy methods.

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Zusammenfassung Das Verfahren der endlichen Elementen wird angewandt zur Bestimmung der elastischen Spannungsintensitäts-faktoren an einer Rißspitze. Begründet auf die Geschwindigkeit der Energiefreilassung braucht dieses Verfahren keine spezielle Rißspitzenangaben. Weiterhin braucht man nur die Lösung für eine einzige Rißlänge, und der Rill wird fortbewegt eher durch Versetzung von Knotenpunkten als durch Entziehung von Knotenzugspannung an der Rißspitze und durch Ausführung einer zweiten Analyse. Die vielversprechende direkte Ausdehnung der Methode auf allgemeine dreidimensionale Rißgestaltungen wird vorgestellt and der praktischen Unmöglichkeit der klassischen Energie-methoden entgegengestellt.

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Résumé Une technique d'analyse par éléments finis est présentée pour la détermination des facteurs d'intensité des contraintes élastiques à la pointe d'une fissure. Basée sur le taux de relaxation d'énergie la méthode ne nécessite pas d'éléments de forme particulière à la pointe de la fissure. En outre, seule est requise la solution pour une longueur déterminée de fissure simple: le processus d'extension de la fissure est obtenu en déplaçant les points nodaux du réseau plutôt qu'en ôtant les composantes de traction nodale à la pointe de la fissure et en procédant à une seconde analyse. On présente les possibilités prometteuses d'extension de la méthode à des configurations tridimensionnelles plus générales de fissures, en contraste avec les impossibilités auxquelles se heurtent les méthodes conventionnelles basées sur des considérations énergétiques.

     

A conic-section simulation analysis of two-dimensional fracture problems using the finite element method

A conic-section simulation analysis to determine the stress intensity factors for fracture mechanics problems of practical interest using the finite element method is presented. The method makes use of elliptic displacement functions which are satisfied by the introduction of an equivalent ellipse obtained through first simulating the actual crack surface displacements as a part of a parabola or a hyperbola. Unlike other finite element approaches that incorporate no special crack-tip treatment, the present approach requires neither extremely small finite elements in the vicinity of the crack tip nor the computation of several strain energies. The cases examined include not only problems of the opening mode (I) or the edge-sliding mode (II), but also the combined modes of crack deformation.

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Résumé On présente une analyse par simulation permettant de déterminer les facteurs d'intensité de contraintes pour des problèmes de mécanique de rupture d'intérêt pratique, en utilisant la méthode des éléments finis. On recourt à des fonctions de déplacement elliptique, qui sont satisfaites par l'introduction d'une ellipse équivalente, obtenue en assimilant les déplacements réels en surface de la rupture à une portion de parabole ou d'hyperbole. Au contraire des autres approches par éléments finis qui ne prévoient pas un traitement particulier de l'extrémité de la fissure, la méthode proposée ne requiert ni de mailles extrêmement fines au voisinage de cette extrémité, ni de nombreux calculs d'énergies de déformation. Les cas examinés ne se limitent pas aux problèmes d'ouverture de mode I ou de mode II, mais couvrent aussi des modes combinés déformation de la fissure.

     

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.     

The stress intensity factor for a deep surface crack in a finite plate

A stress intensity factor solution has been determined for the case of a surface crack in a finite width plate. This solution is for tension or bending and includes a finite area correction factor. It has been shown that using this new stress intensity factor solution it is possible to correlate fatigue crack growth data measured on surface cracked plate specimens with conventional through crack data.     

A relaxation technique for evaluating stress intensity factors by the finite element method

A simple two-step corrective technique is presented in this paper for evaluating stress intensity factors in crack problems. In the first step an approximate evaluation of the stress intensity factor was made by considering the cracked plate to be of infinite size. The stresses of the problem were relaxed by the stresses of the infinite body which corresponds to the approximate value of the stress intensity factor. The expected discrepancy in the value of SIF by the infinite plate approximation was corrected in the second step where the existing residual stresses are equilibrated at the cracked plate by using any of the conventional finite element techniques and the corrective value of the stress intensity factor is calculated by using an appropriate collocation formula. The method was applied to three typical plane problems of cracked plates with satisfactory results.     

The calculation of stress intensity factors using the finite element method with cracked elements

The calculation of stress intensity factors for complicated crack configurations in finite plates usually presents substantial difficulty. A version of the finite element method solves such problems approximately by means of special cracked elements. A general procedure for evaluating the stiffness matrix of a cracked element is developed, and numerical results obtained by the simplest elements are compared with those provided by other methods.

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Zusammenfassung Die Berechnung von Spannungsintensitätsfaktoren für komplizierte Rißgefüge in endlichen Platten bereitet gewöhnlich erhebliche Schwierigkeiten. Fine Variante finite element method löst annähernd solche Probleme mit Hilfe von spezieller gerissenen Elementarteilen.Es wird ein allgemeines Verfahren zur Ermittlung der Steifheits-Matrix eines gerissenes Elementarteilchens aufgestellt. Die numerischen Ergebnisse welche mit den einfachsten Elementarteilen bestimmt wurden, werden mit den nach anderen Verfahren erzielten Ergebnissen verglichen.

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Résumé Pour une plaque finie et une configuration de fissures compliquée, le calcul des coefficients d'intensité de contraintes s'avère normalement difficile, voire pratiquement impossible.Toutefois, une variante de la méthode des éléments finis permet de résoudre ce genre de problèmes de façon approximative moyennant l'adoption d'un élément fissure.Dans cet article l'auteur développe une méthode générale permettant d'évaluer la matrice de raideur d'un élément fissuré.Ensuite il procède pour des éléments simples à une comparaison des résultats numériques obtenus respectivement par d'autres méthodes et par la sienne.

     

The calculation of stress intensity factors ssing the finite element method with cracked elements

The calculation of stress intensity factors for complicated crack configurations in finite plates usually presents substantial difficulty. A version of the finite element method solves such problems approximately by means of special cracked elements. A general procedure for evaluating the stiffness matrix of a cracked element is developed, and numerical results obtained by the simplest elements are compared with those provided by other methods.

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Zusammenfassung Die Berechnung von Spannungsintensitätsfaktoren für komplizierte Rißgefüge in endlichen Platten bereitet gewöhnlich erhebliche Schwierigkeiten. Eine Variante finite element method löst annähernd solche Probleme mit Hilfe von spezieller gerissenen Elementarteilen.Es wird ein allgemeines Verfahren zur Ermittlung der Steifheits-Matrix eines gerissenes Elementarteilchens aufgestellt. Die numerischen Ergebnisse welche mit den einfachsten Elementarteilen bestimmt wurden, werden mit den nach anderen Verfahren erzielten Ergebnissen verglichen.

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Résumé Pour une plaque finie et une configuration de fissures compliquée, le calcul des coefficients d'intensité de contraintes s'avère normalement difficile, voire pratiquement impossible.Toutefois, une variante de la méthode des éléments finis permet de résoudre ce genre de problèmes de façon approximative moyennant l'adoption d'un élément fissuré.Dans cet article l'auteur développe une méthode générale permettant d'évaluer la matrice de raideur d'un élément fissuré.Ensuite il procède pour des éléments simples à une comparaison des résultats numériques obtenus respectivement par d'autres méthodes et par la sienne.

     

Calculating stress intensity factors for a semi-elliptical surface crack in a heterogeneous stress field

The author proposes an equation for calculating the stress intensity factor (SIF) for a semi-elliptical surface crack for uniform, linear, and quadratic laws of variation of the load applied to its edges. The derivation of the equation is based on the well-known Newman—Raju solution for a bent plate. The distribution of the values of SIF along the crack front, obtained using the empirical equations, coincides with the results of calculations carried out using the finite element method (FEM).Translated from Problemy Prochnosti, No. 7, pp. 38–41, July, 1990.     

A finite element modelling for directly determining intensity factors of piezoelectric materials with cracks

Recently, authors(Cao et al., Acta Aeronautica et Astronautica Sinica 25(5): 470–472, 2004) extended the singular crack element originally introduced by Wang et al. (Eng Fract Mech 37(6):1195–1201, 1990) for evaluating the stress intensity factors (SIFs). Extensive studies have proved the versatility and accuracy of the element. This study is to show the versatility of the element for piezoelectric materials. In this paper, electric potential and displacement fields near a crack tip of piezoelectric materials are first used to construct a finite element version for directly determining intensity factors of piezoelectric materials with cracks. A singular finite element is constituted and a new method to calculate intensity factors of piezoelectric materials with cracks is obtained without any post-processing procedures. Detailed derivations are given and the results obtained with present method are good agreement with those of theoretical results, the FEM data by ANSYS and singular electromechanical crack tip elements. The results to the different selections of the structural dimensions are carried out. Numerical examples demonstrate the accuracy and validity of the novel element of present method.     

A finite element method for calculating stress intensity factors and its application to composites

A concept which allows for the development of efficient finite element techniques in the analysis of plane elastic structures containing cracks is discussed. It consists of combining a specially defined finite element in the region surrounding each crack tip with conventional CST elements describing the remaining portion of the geometry considered. For the special element a pair of displacement functions is chosen which adequately represents the singular character of the elastic solution at the crack tip. The application of this concept is illustrated through a specific numerical method developed by W. K. Wilson for the calculation of mode I stress intensity factors.

Wilson's method was coded and used to analyze an infinitely long strip under tension with a line crack perpendicular to its axis of symmetry. Circular inclusions of different material properties were assumed to be present near the tips of the crack and their effect on the mode I stress intensity factor was investigated.

It was found that more flexible inclusions increase the intensity factor while more rigid inclusions decrease it. These results are quite similar to those obtained by analytical methods in an analogous problem involving an infinite sheet, but in the case of a strip, the influence of inclusions on the intensity factor was found to be more pronounced.     

Experimental and finite element analyses on stress intensity factors of an elliptical surface crack in a circular shaft under tension and bending

Experimental backtracking technique and finite element analysis have been employed to evaluate the stress intensities along the front of an elliptical surface crack in a cylindrical rod. The finite element solution covers a wide range of crack shapes loaded under end-free and end-constrained axial tension and pure bending. Convenient closed form stress intensity expressions along the whole crack front for each of the loading cases have been given in terms of the crack aspect ratio, crack depth ratio and place ratio.The closed form solutions have been compared against a number of representative solutions collected from the literature. It has been found that different finite element results for the interior points are generally in good mutual agreement, while solutions derived from other methods may sometimes indicate different trends. At the surface interception point agreement is less good because of a complication in the interpretation of stress intensity there.Experimental backtracking results on the end-constrained axial tension case corroborate well with the closed form solution presented. It suggests that the current closed form solution is adequate in describing the stress intensities along the whole crack front of real surface cracks in cylindrical rods.     

Determination of stress intensity factors in cracked plates by the finite element method

Stress fields near crack tips in an elastic body can be specified by the stress intensity factors which are closely related to the stress singularities arising from the crack tips. These singularities, however, cannot be represented exactly by conventional finite element models. A new method for the analysis of stresses around cracks is proposed in this paper on the basis of the superposition of analytical and finite element solutions. This method is applied to several two-dimensional problems whose solutions are obtained analytically, and it is shown that their numerical results are in excellent agreement with analytical ones. Sufficiently accurate results can be obtained by the conventional finite element analysis with rather coarse mesh subdivision. Computational efforts are then considerably reduced compared with other methods.     

Application of a hybrid finite element method to determine stress intensity factors in unidirectional composites

In this paper a hybrid finite element method is applied in evaluation of the stress intensity factors K _I and K _II of unidirectional fiber reinforced composites. In order to satisfy the stress singularity at the crack tip a singular super-element based on a modified complementary energy principle is developed. The stress and displacement fields in the super-element are expressed in terms of polynomials of two complex variables ₁ and ₂ in the transformed -plane. The stiffness matrix of the super-element was determined by using a line integral along the boundary of the super-element. The displacement vector was expressed in terms of the element nodal displacement vector {q} and a properly selected shape function defined along the element boundary.Numerical results for K _I and K _II of glass-epoxy and graphite-epoxy unidirectional composites with cracks along the diameter of a circular cut out as well as elliptical cut outs were evaluated

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Résumé On applique, dans la présente étude, une méthode d'éléments finis hybrides à l'évaluation des facteurs d'intensité de contrainte K_I et K_II pour des composites renforcés de fibres unidirectionnelles. Pour tenir compte de la singularité de la contrainte à l'extrémité de la fissure, on développe un super élément singulier en se basant sur un principe modifié d'énergie complémentaire. Les champs de contraintes et de déplacements dans le super-élément sont exprimés sous forme polynormale de deux variables complexes ₁, et ₂ dans le plan de la transformée. La matrice de rigidité du super élément est, quant à elle, définie en utilisant une intégrale linéaire le long du contour de l'élément. Le vecteur de déplacement est exprimé par un vecteur (9) de déplacement nodal de l'élément, et par une fonction de forme appropriée, définie le long du contour de l'élément.On évalue les résultats numériques pour K_I et K_II, correspondant à des composites à fibres unidirection-nelles de types verre-epoxy et graphite-epoxy, oú des fissures se situeraient sur le diamètre de découpes circulaires et elliptiques

     

A theoretical-experimental method of determining stress intensity factors

Translated from Fiziko-Khimicheskaya Mekhanika Materialov, No. 4, pp. 97–101, July–August, 1989.     

The stress intensity factors of a radial crack in a finite elastic disc

In this paper the authors make use of a Mellin transform technique to find formulae expressing the stress intensity factor and crack energy of a radial crack in a finite elastic disc directly in terms of the solution of a Fredholm integral equation. The constant loading case is considered in detail and numerical results are given.