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 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.     

Determining stress intensity factors of composites using crack opening displacement

The main purpose of this paper is to find the mixed-mode stress intensity factors of composite materials using the crack opening displacement (COD). First, a series solution of the composite material with a crack was used to evaluate COD values. Then, the least-squares method was used to calculate mixed-mode stress intensity factors. This algorithm can be applied to any method that generates or measures COD values. The major advantage of this method is that COD values very near the crack tip are not necessary. Both finite element simulations and laboratory experiments were applied to validate this least-squares method with acceptable accuracy if the even terms of the series solution are removed.     

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.     

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.     

Measuring stress intensity factors during fatigue crack growth using thermoelasticity

Thermoelastic stress analysis has been developed in recent years as a direct method of investigating the crack tip stresses in a structure under cyclic loading. This is a consequence of the fact that stress intensity factors obtained from thermoelastic experiments are determined from the cyclic stress field ahead of a fatigue crack, rather than inferred from measurement of the crack length and load range. In the present paper the results of fatigue crack growth tests performed on welded ferritic steel plates are reported. From the results it can be observed that the technique is sensitive to the effects of crack closure and the presence of tensile and compressive residual stresses due to welding.     

Finite element calculation of stress intensity factors for interfacial crack using virtual crack closure integral

This paper presents a successful implementation of the virtual crack closure integral method to calculate the stress intensity factors of an interfacial crack. The present method would compute the mixed-mode stress intensity factors from the mixed-mode energy release rates of the interfacial crack, which are easily obtained from the crack opening displacements and the nodal forces at and ahead of the crack tip, in a finite element model. The simple formulae which relate the stress intensity factors to the energy release rates are given in three separate categories: an isotropic bimaterial continuum, an orthotropic bimaterial continuum, and an anisotropic bimaterial continuum. In the example of a central crack in a bimaterial block under the plane strain condition, comparisons are made with the exact solution to determine the accuracy and efficiency of the numerical method. It was found that the virtual crack closure integral method does lead to very accurate results with a relatively coarse finite element mesh. It has also been shown that for an anisotropic interfacial crack under the generalized plane strain condition, the computed stress intensity factors using the virtual crack closure method compared favorably with the results using the J integral method applied to two interacting crack tip solutions. In order for the stress intensity factors to be used as physical variables, the characteristic length for the stress intensity factors must be properly defined. A study was carried out to determine the effects of the characteristic length on the fracture criterion based the mixed-mode stress intensity factors. It was found that the fracture criterion based on the quadratic mixture of the normalized stress intensity factors is less sensitive to the changes in characteristic length than the fracture criterion based on the total energy release rate along with the phase angle.This work has been supported by ONR, with Dr. Y. Rajapakse as the program official.     

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.     

Stress intensity factor analysis of an interface crack between dissimilar anisotropic materials under thermal stress using the finite element analysis

New numerical methods were presented for stress intensity factor analyses of two-dimensional interfacial crack between dissimilar anisotropic materials subjected to thermal stress. The virtual crack extension method and the thermal M-integral method for a crack along the interface between two different materials were applied to the thermoelastic interfacial crack in anisotropic bimaterials. The moving least-squares approximation was used to calculate the value of the thermal M-integral. The thermal M-integral in conjunction with the moving least-squares approximation can calculate the stress intensity factors from only nodal displacements obtained by the finite element analysis. The stress intensity factors analyses of double edge cracks in jointed dissimilar isotropic semi-infinite plates subjected to thermal load were demonstrated. Excellent agreement was achieved between the numerical results obtained by the present methods and the exact solution. In addition, the stress intensity factors of double edge cracks in jointed dissimilar anisotropic semi-infinite plates subjected to thermal loads were analyzed. Their results appear reasonable.     

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.     

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.     

Elastic-plastic analysis of surface flaws using a simplified line-spring model

The line-spring model has proven to be an effective tool for evaluating fracture parameters in surface-cracked plates and shells. However, application of the model requires detailed numerical computations, necessitating the availability of a specialized computer code. For approximate engineering calculations a version of the model which is more convenient to implement computationally, would be useful.In this paper a simplified line-spring model is presented along with detailed illustration of its application. The simplification is accomplished by replacing the crack front with a crack of constant depth and treating the ligament “spring” as elastic perfectly plastic. Despite its simplicity the model gives reasonably accurate predictions of fracture parameters, such as the J-integral or crack opening displacement (COD) at the root of surface cracks. This will be demonstrated by comparing analytical results for J and COD with previously published experimental data for surface-cracked steel plates.     

Evaluation of the stress intensity factors for a crack approaching the bonded half-plates interface from isopachics

Summary The optical evaluation of the stress intensity factors from isopachic fringes is presented for a straight crack approaching the free boundary of a half-plate or the interface of two bonded plates. It is based on appropriate numerical approximation of the exact stress fields obtained by the method of singular integral equations. The proposed evaluation of the stress intensity factors is either by a numerical procedure or through the use of concise nomograms. Also, isopachic fringe patterns have been analytically constructed for a crack perpendicular to the interface and at various distances from it, to show the significant influence from the free boundary or the interface.     

Determination of stress intensity factors for surface cracks using fatigue crack growth data

The fatigue growth of semi-elliptical surface cracks in a structural steel under constant amplitude tensile cycling loading is investigated. The AC potential technique is used for sizing and monitoring the profile development of the cracks. The data are used to determine the stress intensity factors along the entire crack front at different stages of crack growth where the effects of front face, finite thickness and finite width are continuously changing. Various numerical solutions are compared with the experimental results     

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.