Modified Crack Closure Research Articles

Fracture analysis of an adhesively bonded sandwich beam under Mode-I loading

In this paper, an analytical closed-form solution for Mode-I energy release rate of an adhesively bonded sandwich beam is derived from the modified elastic foundation model. For validation purpose, a model based on the finite element method (FEM) was also constructed using the modified crack closure integral technique to calculate the compliance and Mode-I energy release rate of the sandwich beam. Comparison between the closed-form solution and the FEM solution showed a good agreement. The results of calculations also showed a significant effect of the adhesive layer of the sandwich beam on fracture analysis. This means that the effect of the adhesive layer of the sandwich beam cannot be ignored.

Comparison of finite element techniques for 2D and 3D crack analysis under impact loading

For several technical applications the dynamic aspect in fracture mechanics cannot be neglected. When the reliability of components with macroscopic cracks has to be assessed, the consideration of dynamic effects may lead to much higher stress intensity factors than under static conditions. In this paper three different methods to calculate the dynamic stress intensity factor for the mode-I loading of stationary cracks are compared. Based on two- and three-dimensional finite element simulations, the dynamic stress intensity factor is computed with the dynamic J-integral, the modified crack closure integral and the displacement interpretation method. The theoretical fundamentals of all three methods are summarized in the paper and the numerical implementation is explained briefly. Results for different models are shown and compared to findings in the literature.

Mixed-mode fracture of orthotropic functionally graded materials using finite elements and the modified crack closure method

A finite element methodology is developed for fracture analysis of orthotropic functionally graded materials (FGMs) where cracks are arbitrarily oriented with respect to the principal axes of material orthotropy. The graded and orthotropic material properties are smooth functions of spatial coordinates, which are integrated into the element stiffness matrix using the isoparametric concept and special graded finite elements. Stress intensity factors (SIFs) for mode I and mixed-mode two-dimensional problems are evaluated and compared by means of the modified crack closure (MCC) and the displacement correlation technique (DCT) especially tailored for orthotropic FGMs. An accurate technique to evaluate SIFs by means of the MCC is presented using a simple two-step (predictor–corrector) process in which the SIFs are first predicted (e.g. by the DCT) and then corrected by Newton iterations. The effects of boundary conditions, crack tip mesh discretization and material properties on fracture behavior are investigated in detail. Many numerical examples are given to validate the proposed methodology. The accuracy of results is discussed by comparison with available (semi-) analytical or numerical solutions.

Three-dimensional thermal crack growth in self-stressed bimaterial joints: Analysis and experiment

A systematic examination of the initiation and growth of thermal cracks in three-dimensional self-stressed bimaterial joints has been performed by using special models consisting of two segments of different materials, for example of glass and aluminum, and by using a well-defined cooling procedure. Besides, the initiation and development of these thermal cracks originating at one corner of such a bimaterial joint have been measured by means of a special image-processing technique. The corresponding mixed boundary value problem has been formulated, and the fracture mechanical assessment of the three-dimensional thermal crack growth has been evaluated by applying the finite element method. Fracture mechanical parameters have been obtained by two different numerical approaches both based on the FE-method. The first approach is based on Irwin's modified crack closure integral by using the so-called 3DMVCCI-technique. As a second method, the well-known J-integral has been implemented in a FE-postprocessor on the basis of the equivalent domain integral (EDI) technique. An appropriate crack growth criterion has been established in accordance with the experimental results obtained.

Dual boundary element method for axisymmetric crack analysis

In this paper a dual boundary element formulation is developed and applied to the evaluation of stress intensity factors in, and propagation of, axisymmetric cracks. The displacement and stress boundary integral equations are reviewed and the asymptotic behaviour of their singular and hypersingular kernels is discussed. The modified crack closure integral method is employed to evaluate the stress intensity factors. The combination of the dual formulation with this method requires the adoption of an interpolating function for stresses after the crack tip. Different functions are tested under a conservative criterion for the evaluation of the stress intensity factors. A crack propagation procedure is implemented using the maximum principal stress direction rule. The robustness of the technique is assessed through several examples where results are compared either to analytical ones or to BEM and FEM formulations.

Variable thermal singularity boundary elements in the study of neighbouring singularities

AbstractTwo new boundary elements are presented for the simulation of variable order thermal singularities in two dimensions. The first can model the variable order temperature derivative and heat flux singularities at one end of the element. The second element can simulate the temperature derivative and heat flux singularities at both the ends of the element simultaneously. These elements are useful for studying the interaction of variable order thermal neighbouring singularities. They are employed here for the computation of stress intensity factors in the crack–crack interaction problems under thermal load. To improve the accuracy of such computations a modified crack closure integral based method is adapted. Examples of mode I and mode II centre crack, two collinear neighbouring cracks, kinked crack in a plate, and tee joint with a kinked edge crack under thermal or thermal and mechanical loading are studied to illustrate the usefulness of these elements in the study of neighbouring thermal singularities. Copyright © 2001 John Wiley & Sons, Ltd.

Numerical investigation of residual stress fields and crack behavior in TBC systems

Due to thermal expansion mismatch and bond coat (BC) oxidation, high residual stresses are induced in the thermal barrier coating (TBC), leading to failure by spalling and delamination. Using the finite element method (FEM), an analysis of the stress distributions in TBC systems, which is a prerequisite for the understanding of failure mechanisms, was performed. As cracking usually occurs at or near the interfaces between BC/thermally grown oxide (TGO) and TBC/TGO depending on the processing mode of the TBC, cracks in the interface region were considered in the FE models in order to determine the loading conditions for their propagation and, thus, the failure criteria of the TBCs. Due to the mode mixity of these cracks, suitable methods are required for the determination of the fracture mechanics parameters needed for their assessment, such as the strain energy release rate G. The modified crack closure integral method (MCCI) was found to be a very efficient tool which can be combined easily with an FE analysis and leads to highly accurate energy release rate values. Moreover, this method enables the determination of mode-dependent energy release rates. Using this tool and appropriate crack propagation criteria, TBC failure models could be developed and verified.

Further applications of variable singularity boundary elements in the study of neighbouring singularities

The application of two new variable singularity boundary elements (BEs) in the solution of interaction of neighbouring singularities is presented. The first element can model the variable order strain and traction singularities at one end of the element. The second element can simulate the variable strain and traction singularities at both ends of the element simultaneously. These elements are employed for the computation of stress intensity factors in crack–crack interaction problems. To improve the accuracy of such computations a modified crack closure integral (MCCI) method is adopted. To illustrate the usefulness of these elements in the study of neighbouring singularities, a number of examples are presented including, Z-shaped crack, multiple kinked surface crack, multiple edge cracks, interaction of a microcrack with a cross-shaped crack at a bimaterial strip.

Tearing mode interlaminar fracture toughness of composite materials

AbstractThe Simplified Split Cantilever Beam (SSCB) is proposed in this work and compared with the Split Cantilever Beam (SCB) to obtain the tearing mode interlaminar fracture toughness. The materials considered are single‐fiber system composites and interply hybrid composites. For interply hybrid composites, three different types of stacking sequence for SSCB specimens, which are [0/0//0],[0/0//0]. and [0/0//0], are tested to compare their suitability. Finite element analysis combined with a modified crack closure integral has been applied to separate the different components of the strain‐energy release rate. In addition, the method of compliance calibration was used to calculate Gc values. The effects of crack growth, initial crack length, specimen width, and number of glass fiber plies were also studied. The results show that SSCB testing has a more dominant Mode III component and more stable Gc values than SCB testing. For SSCB testing, the crack growth and the specimen width for the range considered have no clear effects on the interlaminar fracture toughness, but the initial crack length should be carefully selected to obtain corrected values. The tearing mode interlaminar fracture toughness of interply hybrid composites is higher than that of carbon/epoxy composites, and the three different types of stacking sequence considered are all suitable to approximate the Mode III interlaminar fracture toughness for interply hybrid composites.

ＦＲＰ複合材パッチで接着補修されたき裂を有する矩形構造パネルにおける補修効率の評価

Repair with bonded fiber-reinforced polymer composite patches has been recognized as an efficient and cost-effective method to extend the service life of aging aircraft. In this paper, a two-dimensional three-layered finite element model consisting of layers of composite patch, adhesive and cracked structural panel is made to analyze the stresses in cracked rectangular structural panels repaired with a bonded circular composite patch. The composite patch and structural panel are modeled as separate layers which are covered with Mindlin plate elements, whereas the adhesive is modeled as a thin elastic layer. The modified crack closure method is used to obtain stress intensity factors at the crack tip. First, by calculating the stress intensity factors, the effects of the diameter and thickness of the circular composite patch on the patching efficiency are examined. The stress intensity factors are also calculated at various hygrothermal conditions and the effect of the environment on the patching efficiency is clarified. Finally, the diagram of patching efficiency is presented to evaluate easily the patching efficiency at hygrothermal conditions.

Near-threshold fatigue crack growth behavior of 2195 aluminum-lithium-alloy—prediction of crack propagation direction and influence of stress ratio

Tensile properties and fatigue crack propagation behavior of a 2195-T8 Al-Li alloy were investigated at different stress ratios, with particular emphasis on their dependence on specimen orientation. Specimens with orientations of 0, 15, 30, 45, and 90 deg to the rolling direction were tested. The alloy contained a strong brass-type texture and a profuse distribution of platelike precipitates of T 1 (Al2CuLi) phase on {111} matrix planes. Both tensile strength and fatigue thresholds were found to be strongly dependent on the specimen orientation, with the lowest values observed along the direction at 45 deg to the rolling direction. The effect of stress ratio on fatigue threshold could generally be explained by a modified crack closure concept. The growth of fatigue crack in this alloy was found to exhibit a significant crystallographic cracking and especially macroscopic crack deflection. The specimens oriented in the L-T + 45 deg had the smallest deflection angle, while the specimens in the L-T and T-L orientations exhibited a large deflection angle. The dependence of the fatigue threshold on the specimen orientation could be rationalized by considering an equivalent fatigue threshold calculated from both mode I and mode II values due to the crack deflection. A four-step approach on the basis of Schmid’s law combined with specific crystallographic textures is proposed to predict the fatigue crack deflection angle. Good agreement between the theoretical prediction and experimental results was observed.

Variable singularity boundary element and its applications in computation of SIFs

Two boundary elements have been proposed for simulation of variable order singularities in two dimensions. The first can model the variable order strain singularity near a crack tip and the second can simulate both the strain and traction. These elements can be easily incorporated in a standard boundary element computer programme. The elements are useful for computation of stress intensity factors (SIFs) in fracture mechanics. To improve the accuracy of such computations further, a modified crack closure integral (MCCI) based method for mechanical loading are presented. Examples of mode I and mixed mode crack problems are examined to illustrate the performance of the proposed elements and the MCCI based calculations. The effects of order of Gauss quadrature associated with such elements on the accuracy of the SIFs are also reported.

A review of SIF evaluation and modelling of singularities in BEM

Various boundary element method (BEM) based approaches to solve crack problems are discussed. The displacement method, J-integral method and the modified crack closure integral (MCCI) method for the evaluation of the stress intensity factors (SIFs) are reviewed. Effects of partial and total modelling of singularities on the accuracy of the results have been presented. Elements capable of partial and total modelling of the wellknown square root singularities, variable order singularities, neighbouring variable order singularities, etc., are also reviewed. Case studies are included to illustrate the effectiveness of the various methods of calculation of the SIFs and the performance of the special elements.

427 非定常熱応力下 IC パッケージの応力拡大係数の 3 次元有限要素法解析

An interface delamination in IC Packages is a critical issue in both structural design and material selection of plastic IC packages. The 3-D finite element analyses were carried out to examine the stress-intensity factor of delamination interface in IC packages subjected to steady and non-steady thermal stresses. The stress-intensity factor K associated with a thermal interface crack was evaluated by the modified crack closure integral. The conditions of thermal stress, which were steady and non-steady thermal stresses, were compared. Especially, it is considered that K_<III> could not be neglected at the corner part in IC Packages. Therefore, the 3-D finite element analysis is indispensable to the estimation of the stress-intensity factors at the corner part in IC packages.

Characteristics of three-dimensional stress fields in plates with a through-the-thickness crack

Three-dimensional finite element analyses were performed on plates with a through-the-thickness crack. Global-local finite element technique with sub-modeling was used to achieve the refinement required to obtain an accurate stress field. The existence of a weaker singularity was verified, and a model was proposed to explain the behavior of stresses in the boundary layer. This model is able to account for the competing interaction between the inverse square root singular term and the vertex singular term. The energy release rate was calculated using the modified crack closure method and energy balance. A simple technique without 3-D calculation was suggested for evaluating an approximate 3-D stress intensity factor at the mid-plane. The effect of plate thickness on the size of the three-dimensional region was studied, and the validity of two-dimensional linear elastic fracture mechanics was discussed.

Effect of modelling of traction and thermal singularities on accuracy of SIFS computation through modified crack closure integral in BEM

The effect of simultaneous modelling of both strain and traction singularities, and temperature derivative and heat flux singularities at the crack tip, on the accuracy of computation of stress intensity factors (SIFs) based on the modified crack closure integral in boundary element method is presented. Simple relations are given for SIF calculations. Results on mode I, mode II and mixed mode, crack subjected to mechanical and/or thermal loading are studied to illustrate the difference between partial and total modelling of the singularities. The dependence of accuracy of the SIFs on the crack tip element size is examined. The effect of order of Gaussian quadrature on the accuracy is also reported.

Opening-mode interlaminar fracture toughness of interply hybrid composite materials

In order to measure the opening-mode interlaminar fracture toughness of interply hybrid composites, double cantilever beam specimens with three different types of stacking sequence have been utilized in this work. Finite-element analysis combined with a modified crack closure integral has been applied to separate the different components of the strain-energy release rate. In addition to the finite-element method, the method of compliance calibration was used to calculate G C values. Furthermore, the effects of geometrical non-linearity, crack length, specimen width, and number of glass fiber plies were also studied. The results show that crack growth in the three types of specimens is dominated by the opening-mode and the Mode I interlaminar fracture toughness can be approximated. For hybrid composite specimens, the effect of geometrical non-linearity should be included. To obtain stable values of strain-energy release rate the specimen width needs to be carefully chosen while the fracture toughness does not change with the growth of the crack.

Decomposed crack closure integrals for estimation of SIF variations

The Modified Crack Closure Integral is a popular, convenient and very effective post-processing tool for extracting the stress intensity factors (SIF) from finite element analysis (FEA) of cracked structures in individual or mixed-mode situations. In 3D, the current practice is to estimate the strain energy release rate over the brick element and mark the corresponding SIF to the center of the brick edge representing the crack front. This suppresses the information regarding SIF variations within the element. This paper presents a novel concept of decomposing the nodal forces, nodal displacements and the strain energy release rate into their constituent parts for extracting the SIF variations consistent with the element shape functions. Decomposed Crack Closure Integral (DCCI) has been formulated for linearly varying SIF in an 8-noded brick and parabolically varying SIF in a 20-noded brick element.

Modified crack closure integral based computation of stress intensity factors for 2-D thermoelastic problems through boundary element method

A modified crack closure integral (MCCI) based computation of stress intensity factors (SIFs) for thermal loading through boundary element method (BEM) is presented. Simple relations are given for the determination of stress intensity factors (SIFs) using the BEM results for linear, quadratic and quarter point elements employed around the crack tip. Examples of crack under mode I, mode II and mixed mode thermal and/or mechanical loading are examined. The computed SIFs are compared wherever possible with solutions available in the literature. The agreement is good. The effect of crack tip element size on the accuracy of results is reported.

BEM based evaluation of SIFs using modified crack closure integral technique under remote and/or crack edge loading

This paper deals with the boundary element method based evaluation of stress intensity factors for mode I and mixed mode problems under remote and/or crack edge loading. The modified crack closure integral technique has been used to enhance the accuracy of the computed stress intensity factors. Simple, ready to use, relations for the strain energy release rate have been obtained corresponding to linear, quadratic and quarter point elements surrounding the crack tip. The boundary element method is employed by breaking a domain into two subregions for the mixed mode problems. Case studies involving remote or crack edge loading for both mode I and mixed mode problems are presented to demonstrate the accuracy of the new scheme. The stress intensity factors based on the proposed scheme show better agreement with the standard solutions available in the literature than those obtained directly through the displacement method.