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.     

Analysis of stress intensity factor of a crack approaching welding bead

An analysis of the stress intensity factor is made for a crack approaching the welding line from the normal direction in an infinite plate.The equations proposed by one of the authors and others were employed in the analysis as the distribution function of the residual stress caused by welding. The effects of crack location and the length on the stress intensity factor were closely examined.For the constant crack length, the maximum stress intensity factor does not occur when the crack center coincides with the welding line, but does when the crack front approximately reaches the welding line.For a better understanding of the behavior of a crack approaching the weldment, analysis was also carried out for a crack with various lengths but constant eccentricity.

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Résumé On a procédé à une analyse du facteur d'intensité de contrainte qui correspond à une fissure qui se rapproche d'un dépôt de soudure normalement à la ligne de fusion et dans une tôle infinie.On a utilisé les équations élaborées par un des auteurs ainsi que par d'autres chercheurs pour décrire la distribution des contraintes résiduelles provoquées par le soudage. On a également examiné de près les effets sur le facteur d'intensité de contrainte de l'emplacement de la fissure et de sa longueur.A longueur de fissure constante, le facteur d'intensité de contrainte ne passe pas par sa valeur maximum lorsque le centre de la fissure coïncide avec l'axe de la soudure, mais bien lorsque l'extrémité de la fissure est sur le point d'atteindre cet axe.Pour mieux comprendre le comportement d'une fissure qui se rapproche d'une soudure, on a également analysé le cas de fissures de diverses longueurs, mais à excentricité constante.

     

Stress intensity factors for an elliptical crack approaching the surface of a semi-infinite solid

Stress intensity factors for an embedded elliptical crack approaching the free surface of the semi-infinite solid that is subjected to uniform tension perpendicular to the plane of crack are presented in a nondimensional form for various crack aspect ratios and crack distances from the free surface. Stress intensity factors are determined numerically using an alternating technique with two solutions. The first solution involves an elliptical crack in a solid and subjected to normal loading expressible in a polynomial of x and y. The second solution involves stresses in the half space due to prescribed normal and shear stresses on the surface. Effect of the Poisson's ratio on these stress intensity factors is also investigated. Stress intensity factors for a semi-elliptical surface crack in a tinite thickness plate are then estimated in a nondimensional form for various crack aspect ratios and crack depth to plate thickness ratios.Specialist Engineer, Aerospace Group, The Boeing Company, Seattle, Washington.Professor, Department of Mechanical Engineering, University of Washington, Seattle, Washington, and also Aerospace Group, The Boeing Company, Seattle.     

Biaxial-load effects on isopachics for a crack at the interface of dissimilar media

Summary The plane problem of two dissimilar materials, bonded together and containing a crack along their common interface, which were subjected to a biaxial load at infinity, is examined by giving a closed-form expression for the first stress invariant of the normal stresses, which is equally valid everywhere, near to, and far from, the crack-tip region. This exact expression for the first-stress invariant is compared by constructing the respective isopachic-fringe patterns, to the approximate expression with non-singular terms, due to the biaxiality factor, for the same quantity. Significant differences between respective isopachic-patterns were found and their dependence on the elastic properties of both materials and the applied loads was demonstrated. The relative errors between the computedK _I - andK _II -components by using the approximate expression for the first stress-invariant and the accurate one, derived from closed-form solution along either isopachic-fringes or along circles and radii from the crack-tip have been given, indicating in some cases large discrepancies between exact and approximate solutions.With 7 Figures     

On the numerical evaluation of stress intensity factors for an interface crack of a general shape

A numerical algorithm is presented for the problem of a crack along the interface of an elastic inclusion embedded in an elastic plane subjected to uniform stress at infinity. The algorithm is based on a Fredholm integral equation of the second kind and allows for fast and accurate solutions to geometries of great complexity. In an example crack opening displacement and stress intensity factors are computed for a crack in the interface of an inclusion with nineteen protruding arms. Copyright © 1999 John Wiley & Sons, Ltd.     

Dynamic stress intensity factors analysis of interface crack using line-spring model

In this study, a new method for calculating the dynamic stress intensity factors of a bimaterial bending specimen with an interface crack is proposed by making use of a line-spring model. A pre-cracked bending specimen is modeled by one-dimensional beam finite elements and a line-spring representing the stiffness or compliance of a cracked part. The proposed method enables the one-dimensional analysis of a two-dimensional crack problem; thus the time variations of the dynamic stress intensity factors of a bimaterial bending specimen with an interface crack can be obtained by making use of a personal computer within a few minutes. The results obtained by the proposed method agree reasonably well with those obtained by the two-dimensional finite element method, although a slight difference in period can be found. The proposed method enables rapid evaluation of dynamic stress intensity factors. So a rapid evaluation system of the dynamic fracture toughness of a bimaterial with an interface crack can be achieved by combining an instrumented impact test apparatus with a computer program based on the proposed method which runs on a personal computer.     

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.     

Determination of thermal stress intensity factors for an interface crack in a graded orthotropic coating-substrate structure

The thermal fracture problem of an interface crack between a graded orthotropic coating and the homogeneous substrate is investigated by two different approaches. For the case that most of the material properties in the graded orthotropic coating are assumed to vary as an exponential function, the integral transform and singular integral equation technique is used to obtain some analytical results. In order to analyze the case with more complex material distribution, an interaction integral is presented to evaluate the thermal stress intensity factors of cracked functionally graded materials (FGMs), and then the element-free Galerkin method (EFGM) is developed to obtain the final numerical results. The good agreement is obtained between the numerical results and the analytical ones. In addition, the influence of material gradient parameters and material distribution on the thermal fracture behavior is also presented.     

Calculation of mixed-mode stress intensity factors for a crack normal to a bimaterial interface using contour integrals

A pair of contour integrals J_kε are proposed in this paper. The integrals are shown to be path-independent in a modified sense and so they can be accurately evaluated without using any particular singular finite elements. Also, the relationship between J_kε and the generalized stress intensity factors (SIFs) is analytically derived and expressed as functions of the bimaterial mechanical constants. Once the J_kε-integrals are accurately computed, the generalized SIFs and, consequently, the asymptotic mixed-mode stress field can then be properly determined. Numerical results in this study show that the contribution from mode II stress component appears to be more dominant when the uncracked material is relatively stiffer, and vice versa.     

Crack tip stress intensity factors for a crack emanating from a sharp notch

Accurate calibrations are provided for the crack tip stress intensity factor for a crack of finite length emanating from the symmetric tip of a sharp notch, of arbitrary angle, in terms of the generalised stress intensity quantifying remote loading of the notch. The solution is applied to example problems and shown to be accurate for cases where the crack is much shorter then the notch depth.     

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.     

Determination of thermal stress intensity factors for an interface crack under vertical uniform heat flow

In the case where an interface crack exists in an infinite two-dimensional elastic bimaterial, the crack surface is insulated under traction-free conditions and the uniform heat flow vertical to the crack from an infinite boundary is given, temperature and stress potentials are obtained by using the complex variable approach to solve Hubert problems, and the results are used to obtain thermal stress intensity factors. The mode II thermal stress intensity factor only occurs if both the shear moduli, as well as the Poisson's ratios in the upper and lower material, are the same. Otherwise, mode I and II thermal stress intensity factors exist but the value of the mode I thermal stress intensity factor is much smaller than that of mode II.     

The effect of material combinations and relative crack size to the stress intensity factors at the crack tip of a bi-material bonded strip

Several types of singular stress fields may appear at the corner where an interface between two bonded materials intersects a traction-free edge depending on the material combinations. Since the failure of the multi-layer systems often originates at the free-edge corner, the analysis of the edge interface crack is the most fundamental to simulate crack extension. In this study, the stress intensity factors for an edge interfacial crack in a bi-material bonded strip subjected to longitudinal tensile stress are evaluated for various combinations of materials using the finite element method. Then, the stress intensity factors are calculated systematically with varying the relative crack sizes from shallow to very deep cracks. Finally, the variations of stress intensity factors of a bi-material bonded strip are discussed with varying the relative crack size and material combinations. This investigation may contribute to a better understanding of the initiation and propagation of the interfacial cracks.     

An experiment on stress intensity factors for an interface crack between epoxy and aluminum plates

A teflon tape (0.07 mm thickness) is placed at the center of an edge of an epoxy plate. The plate is used to fabricate a mold, and epoxy resin is cast in the mold so as to produce a cracked epoxy plate. A tensile test is conducted so as to determine the fracture toughness value of the epoxy plate. Next, a mold is fabricated from an aluminum plate having a teflon tape placed along its edge, and epoxy resin is cast in the mold so as to produce an epoxy-aluminum composite weakened by an interface crack. Tensile testing reveals that the crack always propagates into the epoxy plate at an angle measured from the interface. The stress intensity factor for an interface crack is defined in a manner similar to that for a crack in a homogeneous material, and is obtained for several values of a/h, 2a being the crack length and 2h being the width of the epoxy-aluminum composite.     

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.     

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.     

Stress intensity factors of a surface crack near an interface end

Due to the singular behavior of the stress field near the interface edge of bonded dissimilar materials, fracture generally initiates near the interface edge, or just from the interface edge point. In this paper, an edge crack near the interface, which can be considered as being induced by the edge singularity and satisfying two conditions, is analyzed theoretically, based on the singular stress field near the interface edge and the superposition principle. It is found that the stress intensity factor can be expressed by the stress intensity coefficient of the edge singular stress field, the crack length, the distance between the interface and the crack, as well as the material combination. Boundary element method analysis is also carried out. It is found that the theoretical result coincides well with the numerical result when the crack length is small. Therefore, the theoretical representation obtained by this study can be used to simply evaluate the stress intensity factor of an edge singularity induced crack for this case. However, when the crack length becomes larger than a certain value, a significant difference appears, especially for the case with large edge singularity.     

Evaluation of stress intensity factors for bimaterial bodies using numerical crack flank displacement data

Methods for obtaining stress intensity factors for bimaterial bodies using numerical crack flank displacement data are presented and compared. The stress analysis of a cracked bimaterial is reviewed. The analysis results in a data reduction scheme for the stress intensity factors via the crack flank displacement data. The data reduction scheme produces adequate resolution of the magnitude of the complex stress intensity factor but is incapable of resolving the angle when the moduli of the adjoining materials are close. An extended form of the J-integral is shown to provide increased accuracy for the magnitude of the complex stress intensity factor without yielding the angle.

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Résumé On présente et on compare des méthodes destinées à obtenir les facteurs d'intensité de contrainte pour des corps constitués de 2 matériaux en utilisant des données numériques de déplacements des flans d'une fissure. L'analyse des contraintes d'un corps constitué par 2 matériaux fissurés est passée en revue. L'analyse conduit à un schéma de réduction des données pour les facteurs d'intensité de contrainte utilisant les données de déplacement des flans de la fissure. Le schéma de réduction des données permet d'obtenir une résolution adéquate de l'amplitude du facteur d'intensité des contraintes complexes, mais n'est pas à même de résoudre l'angle caractéristique lorsque les modules des matériaux en présence sont proches. On montre qu'une forme dérivée de l'intégrale J est à même de fournir une précision accrue pour l'amplitude du facteur d'intensité de contrainte complexe, sans avoir à prendre en considération l'angle caractéristique.