Stress intensity factor analyses of interface cracks between dissimilar anisotropic materials using the finite element method

Delamination along an interface between dissimilar materials is the primary cause of failure in microstructures like electronic packages, micro-electro-mechanical systems (MEMS), and so on. Fracture mechanics is a powerful tool for the evaluation of delamination. However, many materials used in microstructures such as composite materials and single crystals are anisotropic materials. Stress intensity factors of an interface crack between dissimilar anisotropic materials, which were proposed by Hwu, are useful for evaluating the reliability of microstructures. However, numerical methods that can analyze the stress intensity factors of an interface crack between anisotropic materials have not been developed. We propose herein a new numerical method for the analysis of an interface crack between dissimilar anisotropic materials. The stress intensity factors of an interface crack are based on the generalized plane strain condition. The energy release rate is obtained by the virtual crack extension method in conjunction with the finite element method for the generalized plane strain condition. The energy release rate is separated into individual modes of the stress intensity factors K_I, K_II, and K_III, using the principal of superposition. The target problem to be solved is superposed on the asymptotic solution of displacement in the vicinity of an interface crack tip, which is described using the Stroh formalism. Analyses of the stress intensity factors of center interface cracks between semi-infinite dissimilar anisotropic media subjected to concentrated self-balanced loads on the center of crack surfaces and to uniform loads are demonstrated. The present method accurately provides mode-separated stress intensity factors using relatively coarse meshes for the finite element method.     

Analysis of three-dimensional interface cracks using enriched finite elements

Many important interface crack problems are inherently three-dimensional in nature, e.g., debonding of laminated structures at corners and holes. In an effort to accurately analyze three-dimensional interface fracture problems, an efficient computational technique was developed that utilizes enriched crack tip elements containing the correct interface crack tip asymptotic behavior. In the enriched element formulation, the stress intensity factors K _I, K _II, and K _III are treated as additional degrees of freedom and are obtained directly during the finite element solution phase. In this study, the results that should be of greatest interest are obtained for semi-circular surface and quarter-circular corner cracks. Solutions are generated for uniform remote tension and uniform thermal loading, over a wide range of bimaterial combinations. Of particular interest are the free surface effects, and the influence of Dundurs’ material parameters on the strain energy release rate magnitudes and corresponding phase angles.     

Stress intensity factor evaluation using singular finite elements

The classical theory of linear elastic fracture mechanics proposes that the stress and energy field near a crack tip can be accurately evaluated by determining the stress intensity factors. Several recent investigations, however, have demonstrated the previously unrecognized importance of the higher-order terms also present in the series eigenfunction representation of the near-tip crack environment. The finite element method has been shown to quite effectively yield these higher-order coefficients, with the method previously utilized only to determine the first term of the series expansion (the stress intensity factor). By numerically evaluating the higher-order coefficients for several finite geometries, the near-tip environment has been shown to be much more sensitive to variations in these terms, than previously believed. This is a phenomenon that no accurate crack propagation study, regardless of specific propagation theory, should disregard without careful consideration, particularly because of the inherent accumulated error in any incremental propagation study.     

Stress intensity factors for three‐dimensional surface cracks using enriched finite elements

The analysis of three‐dimensional crack problems using enriched crack tip elements is examined in this paper. It is demonstrated that the enriched finite element approach is a very effective technique for obtaining stress intensity factors for general three‐dimensional crack problems. The influence of compatibility, integration, element shape function order, and mesh refinement on solution convergence is investigated to ascertain the accuracy of the numerical results. It is shown that integration order has the greatest impact on solution accuracy. Sample results are presented for semi‐circular surface cracks and compared with previously obtained solutions available in the literature. Good agreement is obtained between the different numerical solutions, except in the small zone near the free surface where previously published results have often neglected the change in the stress singularity at the free surface. The enriched crack tip element appears to be particularly effective in this region, since boundary conditions can be easily imposed on the stress intensity factors to accurately represent the correct free surface condition. Copyright © 2002 John Wiley & Sons, Ltd.     

Stress intensity factors for three-dimensional cracks

The method of superposition of analytical and finite-element solutions is proposed for determining three-dimensional distributions of the stress intensity factor; the singular part of the solution is expressed by a linear combination of analytical solutions, and the rest by a finite-element solution. The method is applied to a round bar with a circumferential crack and plates with penetrating cracks. Detailed distributions of the stress intensity factor near the plate surfaces are investigated with the aid of Benthem's theory, which shows that less than 0.5% of the plate thickness is severely influenced by the plate surfaces in the case of a compact tension specimen. Computations for the present method can be performed with a general purpose program for finite element analysis without using special elements.

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Résumé La méthode de superposition des solutions analytiques et par éléments finis est proposée pour déterminer les distributions tri-dimensionnelles du facteur d'intensité des contraintes; la partie singulière de la solution est exprimée par une combinaison linéaire des solutions analytiques et le reste de la solution par une solution à éléments finis. La méthode est appliquée aux cas d'une barre ronde comportant une fissure circonférentielle, et de tôles comportant des fissures pénétrantes. Les distributions détaillées du facteur d'intensité de contrainte au voisinage des surfaces de la tôle sont analysées à l'aide de la théorie de Benthem, qui montre que moins de 0,5% de l'épaisseur de la tôle est sévèrement influencée par l'effet de surfaces dans le cas d'éprouvette de tension compacte. Les calculs de la méthode présentée peuvent être exécutés à l'aide d'un programme à objectifs généraux destiné à l'analyse par élément fini sans recourir à des éléments spéciaux.

     

Stress intensity factors for slanted through-wall cracks based on elastic finite element analyses

Based on detailed 3‐dimensional (3‐D) elastic finite element (FE) analyses, the present paper provides stress intensity factors (SIFs) for plates with slanted through‐wall crack (TWC) and cylinders with slanted circumferential TWC. Regarding loading conditions, axial tension was considered for the plates, whereas axial tension, global bending and internal pressure were considered for the cylinders. To cover a practical range, the geometric variables affecting the SIF were systematically varied. Based on FE results, SIFs along the crack front, including the inner and outer surface points, were provided. The present results can be used to evaluate the fatigue crack growth or stress corrosion cracking behaviour of a slanted TWC and furthermore to perform detailed Leak‐Before‐Break analysis considering a more realistic crack shape.     

Stress intensity factors for two-dimensional crack problems using constrained finite elements

Two-dimensional crack problems, in common with other elliptic problems containing a boundary singularity, may be solved efficiently with the aid of a constrained finite element. The singularity is surrounded by a superelement containing a refined mesh whose interior nodal values are constrained to agree with the first few terms of the known expansion for the solution. The superelement conforms with linear or bilinear elements, and may thus be included in standard finite element programs. The calculation yields the expansion coefficients directly, and the method has been applied to determine stress intensity factors for a variety of two-dimensional configurations, including mixed-mode. The results are in excellent agreement with those obtained by other methods.

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Résumé Les problèmes de fissures à deux dimensions ont en commun avec d'autres problèmes elliptiques comportant une singularisation de frontière, qu'ils peuvent être solutionnés de manière efficace à l'aide d'éléments finis sous sollicitations.Pour ce faire, on englobe la singularité dans un élément comportant lui-même un maillage fin dont les noeuds sont placés sous contrainte de manière à satisfaire aux premiers termes du développement en série de la solution. L'élément d'enrobage est compatible avec les éléments linéaires ou bilinéaires utilisés par ailleurs, et peut donc être intégré dans des programmes standard d'analyse par éléments finis.Les calculs permettent de déduire directement les coefficents du développement en série, et la méthode a été appliquée avec succès à la détermination des facteurs d'intensité de contrainte dans une large gamme de configurations bidimensionelles, y compris afférentes à des modes de rupture combinés. Les résultats sont en excellent accord avec ceux que fournissent d'autres méthodes de calcul.

     

Stress intensity factor analysis of a three-dimensional interface crack between dissimilar anisotropic materials

A new numerical method to calculate the stress intensity factors (SIFs) of a three-dimensional interface crack between dissimilar anisotropic materials was developed. In this study, the M-integral method was employed for mode separation of the SIFs. The moving least-square method was utilized to calculate the M-integral. Using the M-integral with the moving least-square method, SIFs can be automatically calculated with only the nodal displacements from the finite element method (FEM). Here, SIFs analyses of some typical three-dimensional problems are demonstrated. Excellent agreement was achieved between the numerical results obtained by the present method and the corresponding results proposed by other researchers. In addition, the SIFs of a single-edge crack, a through crack, and a semi-circular crack between two anisotropic solids in three-dimensional structures were analyzed.     

Stress intensity factors by enriched finite elements

The finite element method is extended to direct calculation of combined modes I and II stress intensity factors for axisymmetric and planar structures of arbitrary geometry and loading. High order conventional isoparametric elements are combined with a fracture mechanics enrichment of the same element so that a corner node is made to correspond to a crack-tip. The development is explained in detail and necessary modifications to a standard finite element computer program are identified. Stress intensity factors as well as a complete stress analysis are obtained directly from the computer printout. Few elements are generally required, minimizing engineering costs and eliminating the need for data generators. Example problems demonstrate the ease of using the method, the high accuracy to be expected from the method, and the fact that accuracy is relatively insensitive to variations of the element mesh in the vicinity of crack tips. Important applications include complex geometries for which other methods fail and situations wherein multiple crack tips may interfere with one another.     

Stress intensity factor error index for finite element analysis with singular elements

An error index for the stress intensity factor (SIF) obtained from the finite element analysis results using singular elements is proposed. The index was developed by considering the facts that the analytical function shape of the crack tip displacement is known and that the SIF can be evaluated from the displacements only. The advantage of the error index is that it has the dimension of the SIF and converges to zero when the actual error of the SIF by displacement correlation technique converges to zero. Numerical examples for some typical crack problems, including a mixed mode crack, whose analytical solutions are known, indicated the validity of the index. The degree of actual SIF error seems to be approximated by the value of the proposed index.     

Stress intensity factor error index for finite element analysis with singular elements

An error index for the stress intensity factor (SIF) obtained from the finite element analysis results using singular elements is proposed. The index was developed by considering the facts that the analytical function shape of the crack tip displacement is known and that the SIF can be evaluated from the displacements only. The advantage of the error index is that it has the dimension of the SIF and converges to zero when the actual error of the SIF by displacement correlation technique converges to zero. Numerical examples for some typical crack problems, including a mixed mode crack, whose analytical solutions are known, indicated the validity of the index. The degree of actual SIF error seems to be approximated by the value of the proposed index.     

Stress intensity factors by enriched mixed finite elements

An enriched finite element model for linear elastic fracture mechanics is developed for a mixed variational statement. The independent approximations for the displacement and stress components are enriched by adding the near-field analytic expressions for a cracked body to the polynomial approximations of a conventional element. This allows for an accurate representation of the stress and displacement fields near the crack tip and also results in the direct calculation of the appropriate stress intensity factors. The accuracy of this formulation is demonstrated through several numerical examples.     

Stress intensity factor for the cylindrical interface crack between nonhomogeneous coaxial finite elastic cylinders

The problem of determining the stress intensity factor for a cylindrical interface crack between two dissimilar nonhomogeneous coaxial finite elastic cylinders under axially symmetric longitudinal shear stress is considered. The mixed boundary conditions lead to a pair of dual series equations which are reduced to a Fredholm integral equation of the second kind and then finally to a system of algebraic equations. Numerical values of the stress intensity factor are presented graphically.     

Stress intensity factors of a central interface crack in a bonded finite plate and periodic interface cracks under arbitrary material combinations

Although a lot of interface crack problems were previously treated, few solutions are available under arbitrary material combinations. This paper deals with a central interface crack in a bonded finite plate and periodic interface cracks. Then, the effects of material combination and relative crack length on the stress intensity factors are discussed. A useful method to calculate the stress intensity factor of interface crack is presented with focusing on the stress at the crack tip calculated by the finite element method.     

Stress intensity factor analysis for part-elliptical cracks in structures

A method based on the generalized weight function theory is used for solving three-dimensional linear elastic fracture mechanics problems. A complete system of equations of the weight function method (WFM) has been obtained for the calculation of stress intensity factors (SIF) for part-elliptical cracks subjected to arbitrary normal loading.A procedure of the WFM is described to analyze structural components containing surface (semi-elliptical), corner (quarter-elliptical) and embedded (elliptical) flaws. The efficiency of the proposed method is illustrated by solving a number of methodical problems     

Stress intensity factor solutions for cracks in railway axles

The aim of this paper is a collection of stress intensity factor solutions for cracks in railway axle geometries which the authors of the present special issue developed and/or used for damage tolerance analyses. These solutions comprise closed form analytical as well as tabled geometry functions and they refer to solid as well as hollow axles and various crack sites such as the T- and V-notch and the axle body.     

Stress intensity factors of parallel cracks in a finite width sheet

Two and three parallel cracks in a finite sheet subjected to remote tensile loading have been studied. This paper presents empirical stress intensity factor formulae for these crack configurations. The stress intensity factors used to develop these formulae were obtained from finite element analysis. For central cracks and edge cracks, the formulae were within 1 and 3% of the finite element results, respectively.     

Stress intensity factors for interface cracks between orthotropic and monoclinic material

Different expressions are used in the literature for the stress intensity factors of interface cracks between anisotropic material. In particular, two of these approaches will be discussed and compared for orthotropic and monoclinic materials. Relations between the stress intensity factors will be found. Expressions for the interface energy release rate G_i{\mathcal{G}_i} are presented. Although the expressions appear different, they are shown to be the same by using the relations between the stress intensity factors. Phase angles are defined which may be used in a fracture criterion.