Analysis of stress intensity factors in crack problems of periodic two-layered elastic composites

Summary The stress intensity factors in the problem of an interface crack in a periodic two-layered elastic composite subjected to the action of isolated body forces are discussed. The analysis is performed within the framework of the homogenized model with microlocal parameters. The effects of the periodic layered structure of the body on the stress intensity factors are presented.     

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 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.     

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.     

Convenient closed form stress intensity factors for common crack configurations

Linear elastic crack-tip solutions for twelve different shapes of cracked body of interest, are given. The purpose is to provide efficient closed formulations of data previously presented in a tabular or graphical manner. The formulae assist the user of fracture mechanics in that they carry out the interpolative step accurately and therefore may be usefully incorporated in other crack computational procedures, such as fatigue crack growth prediction, crack-tip plasticity corrections, etc. The method used to generate the formulae can be applied to other cracked body geometries.

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Résumé On fournit les solutions linéaires élastiques correspondant aux extrémités de fissures dans le cas de 12 différentes formes de corps fissurés présentant un certain intérêt. Le but de cette étude est de procurer une formulation fermée efficace pour les données présentées antérieurement d'une manière tabulaire ou graphique. Les formules aident l'usager de la mécanique de rupture parce qu'elles effectuent les étapes d'interpolation d'une manière précise et, dès lors, elles peuvent être avantageusement incorporées dans d'autres procédures de calcul de fissure tel que la prédiction de la croissance de fissure de fatigue, les corrections de plasticité aux extrémités d'une fissure, etc... La méthode utilisée pour créer les formules peut être appliquée à d'autres géométries de corps fissuré.

     

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.     

Prediction of threshold stress intensity for fatigue crack growth using a dislocation model

It has been shown that the ratio of threshold stress intensity for fatigue crack growth to the shear modulus is nearly a constant for many materials. This implies that fatigue crack growth is related to some fundamental phenomenon occurring at the crack tip. In the following a dislocation model has been developed to predict the threshold stress intensity. It is shown that the stress intensity can be related to the stress necessary to nucleate a dislocation at the crack tip. The most important outcome of the present analysis is that the threshold stress intensity depends more on the elastic modulus rather than on any other material property in agreement with many experimental results.

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Résumé On a démontré que le rapport de l'intensité de seuil de la contrainte provoquant une fissuration par un accroissement de la fissuration par fatigue au module de cisaillement est sensiblement une constante pour de nombreux matériaux. Ceci implique que la croissance d'une fissure de fatigue est reliée à certains phénoménes fondamentaux qui se produisent à l'extrémité d'une fissure. Dans le mémoire, on développe un modèle de dislocation qui permet de prédire l'intensité critique de la contrainte. On montre que l'intensité de la contrainte peut être mise en relation avec la contrainte nécessaire pour créer une dislocation à l'extrémité d'une fissure. La conséquence la plus importante de cette analyse est que l'intensité critique de seuil dépend davantage du module d'élasticité que de toutes autres propriétés du matériau et ce en accord avec de nombreux résultats expérimentaux.

     

Stress intensity factors for interacting cracks and complex crack configurations in linear elastic media

In this paper, an effective numerical method for analyzing interacting multiple cracks and complex crack configurations in infinite linear elastic media is presented. By extending Bueckner’s principle suited for a crack to a general system containing multiple interacting cracks, the original problem is divided into a homogeneous problem (the one without cracks) subjected to remote loads and a multiple-crack problem in an unloaded body with applied tractions on the crack surfaces. Thus, the results in terms of the stress intensity factors (SIFs) can be obtained by taking into account the latter problem, which is analyzed easily by means of the displacement discontinuity method with crack-tip elements proposed recently by the author. Test examples are included to illustrate that the method is very simple and effective for analyzing interacting multiple cracks and complex crack configurations in an infinite linear elastic media under remote uniform stresses. Specifically, analysis of perpendicular cracks under general in-plane loading is performed using the numerical approach and many numerical results are given in the form of tables.     

Computing a-posteriori bounds for stress intensity factors in elastic fracture mechanics

This paper mainly focuses on computing the lower and upper bounds on stress intensity factors in elastic fracture mechanics with an efficient finite element output bound procedure on quantities of interest in engineering. The bounds procedure is obtained by minimizing the quadratic energy functional of output with constraints of equilibrium conditions of mechanics and continuity conditions of finite element space. The computation is based on solving the elemental Neumann residual problems for the bounds on energy norm of error in finite element solutions. The lower and upper bounds on the intensity factors of an open mode and a shear mode elastic fracture problems are computed in this paper.     

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.

     

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.     

Fractal effects of crack propagation on dynamic stress intensity factors and crack velocities

Crack extension paths are often irregular, producing rough fracture surfaces which have a fractal geometry. In this paper, crack tip motion along a fractal crack trace is analysed. A fractal kinking model of the crack extension path is established to describe irregular crack growth. A formula is derived to describe the effects of fractal crack propagation on the dynamic stress intensity factor and on crack velocity. The ratio of the dynamic stress intensity factor to the applied stress intensity factor K(L(D, t), V)/K(L(t), 0), is a function of apparent crack velocity V_o, microstructure parameter d/a (grain size/crack increment step length), fractal dimension D, and fractal kinking angle of crack extension path . For fractal crack propagation, the apparent (or measured) crack velocity V_o, cannot approach the Rayleigh wave speed Cr. Why V_o is significantly lower than Cr in dynamic fracture experiments can be explained by the effects of fractal crack propagation. The dynamic stress intensity factor and apparent crack velocity are strongly affected by the microstructure parameter (d/a), fractal dimension D, and fractal kinking angle of crack extension path . This is in good agreement with experimental findings.     

Evaluation of the stress intensity factors and the T-stress in periodic crack problem

This paper investigates the T-stress at crack tips in the periodic crack problem. Remote tension in the y-direction is applied to cracks with an arbitrary inclined angle. The original stress field can be considered a superposition of a uniform stress field and a perturbation stress field. The problem of evaluating the stresses in the perturbation field can be considered a superposition of many single crack problems. A Fredholm integral equation is suggested for the solution of the perturbation stress field. In the equation, the loading on the crack face is chosen as unknown quantity. Once the integral equation is solved, the stress intensity factors and the T-stress at the crack tip can be evaluated immediately. For solving the integral equation and evaluating stresses in the perturbation field, the remainder estimation technique is suggested for evaluating the influences on the central crack from infinite cracks. The technique can considerably improve convergence in computation. Many results for the stress intensity factors and the T-stresses in periodic cracks are presented. It is shown that the interaction is significant for the closer cracks.     

A superposition procedure for computing stress intensity factors in cracked elastic composites

A simple, yet accurate, procedure is developed for computing stress intensity factors via finite element analyses of cracked, elastic composites. Attention is focused on bodies with piece-wise constant, isotropic, elastic properties which are deforming in plane strain. The crack-tip is taken to be wholly enclosed in a single homogeneous region; thus, the well-known square-root singular field prevails. The superposition procedure, in effect, regularizes the original singular problem by deducing the error incurred in numerically computing the non-regular (singular) part. The method is illustrated by considering a few test problems.

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Résumé On a développé une procédure simple, encore que précise, pour le calcul par élémente finis des facteurs d'intensité de contraintes dans des composites élastiques fissurés. On a concentré son attention sur des corps présentant, à l'instar de pièces réelles, des propriétés isotropes, élastiques et constantes, et déformés en état plan de déformation. On suppose que l'extrémité de la fissure est complètement enrobée d'une région homogène simple, ce qui permet de faire prévaloir la relation singulière bien connue de la racine carrée du champ. En fait, une procédure de superposition régularise le problème singulier initial, en déduisant l'erreur résultant d'un calcul numérique d'une portion non régulière. On illustre la méthode en prenant en considération quelques problèmes d'essais.

     

Analytic stress intensity factors for finite elastic disk using symplectic expansion

An analytic method to determine the stress intensity factors of finite elastic disk in polar coordinates is introduced with extension to various geometric domains using least-square method. It first finds the symplectic eigenfunctions after expressing the governing equilibrium equations in a Hamiltonian form for variable separation. The displacements and stresses are expanded by the symplectic eigenfunctions with coefficients determined from the boundary conditions. The stress intensity factors are actually the first two coefficients of the series and no post-processing is required. The higher coefficient gives the T-stress. Examples for discontinuous boundary conditions are included and new results are given.