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.     

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.     

Analysis of stress intensity factors for three-dimensional interface crack problems in electronic packages using the virtual crack closure technique

In this study the fracture mechanics parameters, including the strain energy release rate, the stress intensity factors and phase angles, along the curvilinear front of a three-dimensional bimaterial interface crack in electronic packages are considered by using finite element method with the virtual crack closure technique (VCCT). In the numerical procedure normalized complex stress intensity factors and the corresponding phase angles (Rice, J Appl Mech 55:98–103, 1988) are calculated from the crack closure integrals for an opening interface crack tip. Alternative procedures are also described for the cases of crack under inner pressure and crack faces under large-scale contact. Validation for the procedure is performed by comparing numerical results to analytical solutions for the problems of interface crack subjected to either remote tension or mixed loading. The numerical approach is then applied to study interface crack problems in electronic packages. Solutions for semi-circular surface crack and quarter-circular corner crack on the interface of epoxy molding compound and silicon die under uniform temperature excursion are presented. In addition, embedded corner delaminations on the interface of silicon die and underfill in flip-chip package under thermomechanical load are investigated. Based on the distribution of the fracture mechanics parameters along the interface crack front, qualitative predictions on the propensity of interface crack propagation under thermomechanical loads are given.     

Boundary element method based computation of stress intensity factor by modified crack closure integral

The local smoothing scheme in conjunction with the modified crack closure integral technique has been adopted in the boundary element method to improve the accuracy of computed stress intensity factors. Simple relations have been derived for the case of linear, quadratic and quarter point elements around the crack tip. Case studies are presented to demonstrate improvement in the accuracy. While the displacement method gives a difference with the standard handbook solution up to 26%, the suggested method helps to reduce it to within 2%. Communicated by S. N. Atluri, 14 August 1996     

Calculation of stress intensity factors for an interfacial crack between dissimilar anisotropic media,using a hybrid element method and the mutual integral

Using Beom and Atluri's complete eigen-function solutions for stresses and displacements near the tip of an interfacial crack between dissimilar anisotropic media, a hybrid crack tip finite-element is developed. This element, as well as a mutual integral method are used to determine the stress intensity factors for an interfacial crack between dissimilar anisotropic media. The hybrid element has, for its Galerkin basis functions, the eigen-function solutions for stresses and displacements embedded within it. The mutual integral approach is based on the application of the path-independent J integral to a linear combination of two solutions: one, the problem to be solved, and the second, an auxiliary solution with a known singular solution. A comparison with exact solutions is made to determine the accuracy and efficiency of both the methods in various mixed mode interfacial crack problems. The size of the hybrid element was found to have very little effect on the accuracy of the solution: an acceptable numerical solution can be obtained with a very coarse mesh by using a larger hybrid element. An equivalent domain integral method is used in the application of the mutual integral instead of the line integral method. It is shown that the calculated mutual integral is domain independent. Therefore, the mutual integral can be evaluated far away from the crack-tip where the finite element solution is more accurate. In addition, numerical examples are given to determine the stress intensity factors for a delamination crack in composite lap joints and at plate-stiffener interfaces.This work was supported by a grant from the NASA Langley Research Center.     

Contour integral method for stress intensity factors of interface crack

A general Betti's reciprocal work theorem with interface cracks of a bimaterial is established in this paper, and a path independent contour integral method for the stress intensity factor (SIF) of the interface crack was obtained. When the stress and displacement fields in a specimen are calculated by the finite element method, the SIF K _I and K _II of interface cracks can be obtained immediately by a contour integral. Some solutions of interesting examples, such as two collinear interface cracks, are also given.Presented at the Far East Fracture Group (FEFG) International Symposium on Fracture and Strength of Solids, 4–7 July 1994 in Xi'an China.     

Finite element analysis of postbuckling and delamination of composite laminates using virtual crack closure technique

The two-dimensional and three-dimensional parametric finite element analysis (FEA) of composite flat laminates with two through-the-width delamination types: 0₄/(±θ)₆//0₄ and 0₄//(±θ)₆//0₄ (θ = 0°, 45°, and “//” denotes the delaminated interface) under compressive load are performed to explore the effects of multiple delaminations on the postbuckling properties. The virtual crack closure technique which is employed to calculate the energy release rate (ERR) for crack propagation is used to deal with the delamination growth. Three typical failure criteria: B-K law, Reeder law and Power law are comparatively studied for predicting the crack propagation. Effects of different mesh sizes and pre-existing crack length on the delamination growth and postbuckling properties of composite laminates are discussed. Interaction between the delamination growth mechanisms for multiple cracks for 0₄//(±θ)₆//0₄ composite laminates is also investigated. Numerical results using FEA are also compared with those by existing models and experiments.     

A comparison of stress intensity factors obtained through crack closure integral and other approaches using eXtended element-free Galerkin method

In this paper, a new approach for extracting stress intensity factors (SIFs) by the extended element-free Galerkin method, through a crack closure integral (CCI) scheme, is proposed. The CCI calculation is used in conjunction with a local smoothing technique to improve the accuracy of the computed SIFs in a number of case studies of linear elastic fracture mechanics. The cases involve problems of mixed-mode, curved crack and thermo-mechanical loading. The SIFs by CCI, displacement and stress methods are compared with those based on the M-integral technique reported in the literature. The proposed CCI method involves very simple relations, and still gives good accuracy. The convergence of the results is also examined.     

Finite elements for determination of crack tip elastic stress intensity factors

A new type of finite element is introduced which embodies the inverse square root singularity present near a crack in an elastic medium. Using this element near the tip in two typical cracked configurations, stress intensity factors within 5 per cent of accepted values were obtained with meshes having as few as 250° of freedom.     

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.

     

Fracture criteria and stress intensity factors including the effect of crack closure

Two- and three-dimensional thermo-mechanical failure criteria, including the effects of crack/cavity closure, are developed in terms of thermal and mechanical loading by extending the work of McClintock and Walsh. General 2- and 3-D fracture criteria in terms of soley stress intensity factors are developed and it is shown that they are expressed in the single relation, $\text{(k}{}_2\text{k}{}_{\mathrm{2c}}\text{)}{}^2\text{+}\text{k}{}_1\text{k}{}_{\mathrm{1c}}\text{= 1}$, on the basis of Griffith theory and fracture mechanics. General expressions of stress intensity factors in 3-D crack problems under arbitrary thermo-mechanical loading with the effect of crack closure are also deduced.     

Finite element visualization of fatigue crack closure in plane stress and plane strain

Elastic-plastic finite element simulations of growing fatigue cracks in both plane stress and plane strain are used as an aid to visualization and analysis of the crack closure phenomenon. Residual stress and strain fields near the crack tip are depicted by both color fringe plots and x-y graphs. Development of the residual plastic stretch in the wake of a growing plane stress fatigue crack is shown to be associated with the transfer of material from the thickness direction to the axial direction. Finite element analyses indicate that crack closure does occur under pure plane strain conditions. The development of the residual plastic stretch in plane strain is shown to be associated with the transfer of material from the in-plane transverse direction to the axial direction. This in-plane contraction also leads to the generation of complex residual stress fields. The total length of closed crack at minimum load in plane strain is shown to be a small fraction of the total crack length, especially for positive stress ratios. This suggests that experimental measurement of plane strain closure would be extremely difficult, and may explain why some investigators have concluded that closure does not occur in plane strain.     

Calculation of stress intensity factors based on mode I crack opening integral parameters

We present stress intensity factor assessment using nodal displacements of the crack surfaces determined by the finite element method for cracked bodies. The equation is solved by expanding the crack opening displacement in the Chebyshev function, where crack front asymptotic behavior corresponds to the regulations of the linear elastic fracture mechanics. Results of the stress intensity factor calculations are obtained for test problems with analytical solution. Crack opening displacements are defined with the help of the 3D SPACE software package designed to model mixed variational formulation of the finite element method for displacements and strains of the thermoelastic boundary value problems. Translated from Problemy Prochnosti, No. 6, pp. 122–127, November–December, 2008.     

Finite element analysis of stress intensity factors for cracks at a bi-material interface

The stress singularity at the tip of a crack, either lying along or perpendicular to the interface of the two materials, is first investigated by the complex variable method. The order of the singularity is shown to be dependent on both the crack geometry and two parameters , which are related to the four elastic constants of the two materials. A hybrid crack element is constructed to properly account for the crack tip singularity. The stress intensity factors and energy release rate for cracks in different bi-material continua are then calculated using the finite element method. The results show that the present finite element analysis makes possible a highly accurate and efficient numerical solution of fracture mechanics problems.

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Résumé On étudie la singularité de la contrainte à l'extrémité d'une fissure située le long de ou perpendiculairement à l'interface de deux matériaux, en recourant d'abord à la méthode des variables complexes. On montre que l'ordre de la singularité dépend à la fois de la géométrie de la fissure et de deux paramètres et , en relation avec les quatre constantes élastiques des deux matériaux. On construit un élément de fissure hybride propre à tenir compte de la singularité d'extrémité de fissure, et on calcule par éléments finis les facteurs d'intensité des contraintes et le taux de relaxation de l'énergie, pour des fissures dans différents continuum à deux matériaux. Les résultats montrent que les techniques actuelles d'analyse aux éléments finis permettent de trouver une solution numérique efficace et de haute précision aux problèmes de mécanique de la rupture.