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.     

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.     

An interaction energy integral method for computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions

A new interaction energy integral method for extracting mixed-mode stress intensity factors along the fronts of non-planar, three-dimensional cracks is described. In the method, interaction energy contour integrals are defined and expressed in domain form. The mixed-mode stress intensity factors are obtained by evaluating the domain integrals as a post-processing step in the finite element method. To assess the accuracy of the method, two benchmark problems are considered. The first problem is that of an arc crack in an infinitely extended solid subjected to equibiaxial tension. The second is that of a lens-shaped crack embedded in an infinite solid subjected to hydrostatic tension. Excellent agreement is obtained between the numerical and corresponding analytical results obtained from the literature.     

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.     

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.     

An interaction integral method for computing mixed mode stress intensity factors for curved bimaterial interface cracks in non-uniform temperature fields

In finite element analysis the interaction integral has been a useful tool for computing the stress intensity factors for fracture analysis. This work extends the interaction integral to account for non-uniform temperatures in the calculation of stress intensity factors for three dimensional curvilinear cracks either in a homogeneous body or on a bimaterial interface. First, the derivation of the computational algorithm, which includes the additional terms developed by the non-zero gradient of the temperature field, is presented in detail. The algorithm is then implemented in conjunction with commercial finite element software to calculate the stress intensity factors of a crack undergoing non-uniform temperatures on both a homogeneous and a bimaterial interface. The numerical results displayed path independence and showed excellent agreement with available analytical solutions.     

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.     

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.     

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     

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.     

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.     

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.     

Simple bounds for the stress intensity factors by the method of singular integral equations

The method of singular integral equations has become a classical method for solving plane and antiplane, static and dynamic crack problems in isotropic and anisotropic elasticity, particularly in cases where no closed-form solutions are available. In this paper, very simple methods are suggested for obtaining upper bounds for the stress intensity factors at the crack tips from the corresponding singular integral equation without solving this equation, even numerically, and with very few computations. Naturally, such a simplicity should lead to very conservative bounds and this is really the case. But, clearly, in a lot of practical cases such bounds are sufficient. Numerical results in simple crack problems show the efficiency of the proposed methods.     

Stress intensity factors for an interface crack along an elliptical inclusion

The problem of a crack along the interface of an elliptical elastic inclusion embedded in an infinite plate subjected to uniform stresses at infinity is analyzed by the body force method. The crack tip stress intensity factors are calculated for various inclusion geometries and material combinations. Based on numerical results, the effect of the inclusion geometry on the stress intensity factors is investigated. It is found that for small interface cracks the stress intensity factors are mainly determined by the stresses, occurring at the crack center point before the crack initiation, and interface curvature radius alone.     

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.     

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.