A virtual crack closure-integral method (VCCM) to compute the energy release rates and stress intensity factors based on quadratic tetrahedral finite elements

This paper proposes a new virtual crack closure-integral method (VCCM) for quadratic tetrahedral finite element to compute the energy release rates/stress intensity factors. The formulations, numerical implementations and some numerical results of proposed VCCM are presented in this paper. Proposed VCCM enables us to adopt the tetrahedral finite element in 3D crack problems and us to use automatic mesh generation programs. Therefore process time to perform 3D crack analysis drastically reduces compared with the case of hexahedral elements.     

Application of virtual crack closure integral method for interface cracks in low-k integrated circuit devices under thermal load

The virtual crack closure integral (VCCI) method is used to evaluate the stress intensity factor (SIF) and energy release rate (ERR) of an interface crack under thermal load. The VCCIs used in this work include the traditionally known “Mode I” and “Mode II” VCCIs and an additional coupling VCCI. The singularity element is used in the finite element method (FEM) implementation. The SIF and ERR calculated by the FEM are compared to the exact solution in the case of a joint dissimilar semi-infinite plates with double edge crack under thermal loading. The FEM result agrees well with the exact solution for relatively coarse meshes. The contribution of the mesh density and material mismatch to the FEM error is also explored. The VCCI method is used with the multi-scale FEM in a delamination risk assessment of a low-k integrated circuits device in flip-chip plastic ball grid array packages. The ERR is calculated for different package configurations and the prediction of the delamination risk is confirmed by reliability tests.     

Interaction effect crack–interfacial crack using finite element method

The interaction effect of an interfacial crack–microcrack modifies considerably the fracture behaviour of S45C/Si₃N₄ bimaterial. This work aims at studying the interaction effect of a crack located in one of the materials constituting the assembly near the interface, and that between an interfacial crack and a microcrack parallel to the interface by using the finite element method. The effect of transverse and longitudinal interaction distances between the interfacial crack and the microcrack are highlighted. The stress intensity factor of the interacting cracks and the bimaterial mechanical properties influence on the conditions of deviation and propagation of crack by interface and intercrack are examined.     

Calculation of transient strain energy release rates under impact loading based on the virtual crack closure technique

This paper describes an interface element to calculate the strain energy release rates based on the virtual crack closure technique (VCCT) in conjunction with finite element analysis (FEA). A very stiff spring is placed between the node pair at the crack tip to calculate the nodal forces. Dummy nodes are introduced to extract information for displacement openings behind the crack tip and the virtual crack jump ahead of the crack tip. This interface element leads to a direct calculation of the strain energy release rate (both components G_I and G_II) within a finite element analysis without extra post-processing. Several examples of stationary cracks under impact loading were examined. Dynamic stress intensity factors were converted from the calculated transient strain energy release rate for comparison with the available solutions by the others from numerical and experimental methods. The accuracy of the element is validated by the excellent agreement with these solutions. No convergence difficulty has been encountered for all the cases studied. Neither special singular elements nor the collapsed element technique is used at the crack tip. Therefore, the fracture interface element for VCCT is shown to be simple, efficient and robust in analyzing crack response to the dynamic loading. This element has been implemented into commercial FEA software ABAQUS^® with the user defined element (UEL) and should be very useful in performing fracture analysis at a structural level by engineers using ABAQUS^®.     

A modified zigzag approach to approximate moving crack front with arbitrary shape

This paper proposed a modified zigzag approach to approximate crack front with arbitrary shape based on virtual crack closure-integral technique. The important features of this approach are that: (i) more detailed consideration to determine the required virtually closed area and displacement opening were presented and (ii) adaptive re-meshing technique was avoid in studying crack growth. Three cases were presented to assess the validity of this approach by comparison to the results obtained from other approaches. Reasonable agreement between the present study and the analytical solutions were obtained. Also, the shape changes during crack propagation can be tracked with ease.     

Influence of bimaterial interface on kinking behaviour of a crack growth emanating from notch

The mechanism of crack deviation by an interface modifies considerably the behaviour of bimaterials fracture. Their fracture resistance is highly affected by the difference of the elastic properties of the bonded materials. In this work, the finite element method is applied to analyze the behaviour of a crack emanating from semicircular notch root growing in interface ceramic/metal composites and perpendicularly to this interface. The obtained results showed that the crack grew to interface from harder material, its energy decreased at the approach of the interface, in this case was retarded; an inverse phenomenon occurs if the crack is propagated towards a lower strength material and its energy increases, it has tendency to accelerate. The effects of geometry on the crack deflection near the interface are also discussed.     

An Abaqus implementation of the extended finite element method

In this paper, we introduce an implementation of the extended finite element method for fracture problems within the finite element software ABAQUS^TM. User subroutine (UEL) in Abaqus is used to enable the incorporation of extended finite element capabilities. We provide details on the data input format together with the proposed user element subroutine, which constitutes the core of the finite element analysis; however, pre-processing tools that are necessary for an X-FEM implementation, but not directly related to Abaqus, are not provided. In addition to problems in linear elastic fracture mechanics, non-linear frictional contact analyses are also realized. Several numerical examples in fracture mechanics are presented to demonstrate the benefits of the proposed implementation.     

Effect of electric boundary conditions on crack energy density and its derivatives for piezoelectric material

This paper deals with crack energy density (hereafter CED) as a possible fracture parameter in piezoelectricity under arbitrary electric boundary conditions on a notch-like crack surface. The definitions of CED and its derivatives are given first under exact boundary condition. Next, their path independent integrals are also derived and their approximate expressions are discussed under some restrictions on the crack surfaces. It is found that electrical terms along the notch-like crack surface do not vanish unlike in the case of impermeable crack. Then, we introduce evaluation methods of CED, and, with the help of the results of finite element analyses (FEA), we closely examine how electric boundary conditions along the notch surface and initial notch width influence CED and its derivatives. It is shown from the FEA results that because of the difficulties of computing path integral terms along the notch-like crack tip in the path independent expressions, the evaluation by the definitions of CED and its derivatives is preferable and more convenient than the evaluation of their path independent expressions. It is also found that all the parameters are significantly affected by both permittivity inside the electric inclusion and root radius of the notch. Finally, the possibility of mechanical CED as a governing fracture parameter is discussed.     

Fatigue crack propagation behavior in four-points bending specimens with multiple parallel edge notches at regular intervals

Fatigue crack propagation tests were performed using specimens with multiple parallel edge notches at regular intervals. Fatigue pre-cracks of uniform length were successfully introduced by eccentric tension-compression loading. Fatigue crack propagation tests were carried out under four-points bending loading. Obtained crack propagation behavior was simulated by using newly developed stress intensity factor equations for multiple parallel edge cracks with alternately different lengths. Simulated results showed a good agreement with the experiment for specimens with relatively broad crack intervals, while for specimens with narrow crack intervals showed a different tendency from the experiment when crack lengths became long.     

Strain energy release rate calculation for a moving delamination front of arbitrary shape based on the virtual crack closure technique. Part I: Formulation and validation

This paper proposes a simple, efficient algorithm to trace a moving delamination front with an arbitrary and changing shape so that delamination growth can be analyzed by using stationary meshes. Based on the algorithm, a delamination front can be defined by two vectors that pass through any point on the front. The normal vector and the tangent vector for the local coordinate system can then be obtained based on the two delamination front vectors. An important feature of this approach is that it does not require the use of meshes that are orthogonal to the delaminations front. Therefore, the approach avoids adaptive re-meshing techniques that may create a large computational burden in delamination growth analysis. An interface element that can trace the instantaneous delamination front, determine the local coordinate system, approximate strain energy release rate components and apply fracture mechanics criteria has been developed and implemented into ABAQUS^® with its user-defined element (UEL) feature. In this Part I of a two-part paper, the approach and its implementation are described and validated by comparison to results from existing cases having analytical solutions or other established FEA predictions.     

Growth evaluation of multiple interacting surface cracks. Part I: Experiments and simulation of coalesced crack

In order to develop a procedure for assessing the growth of interacting surface cracks, the relationship between the area of the crack face and fatigue crack growth behavior was investigated. Fatigue crack growth tests were conducted using stainless steel plate specimens with surface notches. Then, finite element analyses were performed to simulate the growth behavior obtained by the experiment. It was shown that the change in area can be predicted by assuming the extension of crack front based on evaluated stress intensity factor at each position along the front. Based on experimental and analysis results, it was revealed that the growth of interacting surface cracks as well as independent cracks can be represented well by change in area and showed good correlation with the driving force based on area. It was also shown that, in the case of parallel cracks, the area on the projected plane was dominant. It was concluded that, when the magnitude of the interaction is sufficiently large, by replacing the two cracks with a semi-elliptical crack of the same area on the projected plane, the growth in area can be predicted precisely.     

Strain energy release rate calculation for a moving delamination front of arbitrary shape based on the virtual crack closure technique. Part II: Sensitivity study on modeling details

The interface element and VCCT process described in Part I of this two-part paper, developed to compute strain energy release rates of an arbitrary delamination front using non-orthogonal finite element meshes, are further investigated in this paper for robustness and ease of use in tracking delamination growth. Standard 3-D elements are used in conjunction with the interface elements. No special singularity elements are required. Stationary meshes that are independent of the shape of the delamination front can be used. Three cases having different initial delamination shapes are examined. The process is shown to be insensitive to the values used for the interfacial spring stiffness, the orientation of the interface element, or even the mesh pattern if the mesh has a reasonable degree of refinement. Therefore, the method can be used with ease and confidence in general-purpose delamination growth analysis for engineering applications.     

Numerical analysis of 2-D crack propagation problems using the numerical manifold method

The numerical manifold method is a cover-based method using mathematical covers that are independent of the physical domain. As the unknowns are defined on individual physical covers, the numerical manifold method is very suitable for modeling discontinuities. This paper focuses on modeling complex crack propagation problems containing multiple or branched cracks. The displacement discontinuity across crack surface is modeled by independent cover functions over different physical covers, while additional functions, extracted from the asymptotic near tip field, are incorporated into cover functions of singular physical covers to reflect the stress singularity around the crack tips. In evaluating the element matrices, Gaussian quadrature is used over the sub-triangles of the element, replacing the simplex integration over the whole element. First, the method is validated by evaluating the fracture parameters in two examples involving stationary cracks. The results show good agreement with the reference solutions available. Next, three crack propagation problems involving multiple and branched cracks are simulated. It is found that when the crack growth increment is taken to be 0.5h≤da≤0.75h, the crack growth paths converge consistently and are satisfactory.     

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.     

On the through-thickness crack with a curve front in center-cracked tension specimens

Three-dimensional finite element analyses are performed on through-thickness cracks with slightly wavy front in center-cracked plates. Considering there is an inherent relationship between the crack shape and the corresponding stress intensity factor (SIF) distribution of a crack, the curved configuration of the crack is determined using a heuristically derived iterative procedure if the SIF distribution function is known. Several simple SIF distribution functions, for instance the constant SIF distribution along the crack front, are assumed to determine the crack shape. Under the assumption that the rate of fatigue crack growth depends on the SIF range or the effective SIF range, possible effects of plate thickness, crack length and crack closure level gradient on the behaviour of crack tunneling are investigated. The stability of the curved shape of a through-thickness crack in fatigue is also discussed, i.e. whether a crack can maintain its shape satisfying the conditions of constant SIF distribution or other distribution along the crack front during fatigue growth. This study will be useful for a better understanding of the behaviour of crack tunneling and help to evaluate the validity of the two-dimensional linear elastic fracture mechanics in cracked plates.     

界面具有垂直裂纹的热障涂层的失效模拟

采用有限元分析软件ANSYS构造了一个在热生长氧化层（TGO）与陶瓷层界面具有一个垂直裂纹的纳米结构热障涂层的有限元模型。并计算了在热震过程中裂纹处的应力分布图,及裂纹尖端的应力场强度因子K1变化图。计算结果表明：裂纹处存在应力集中现象,且裂纹尖端的应力场强度因子K1在热障涂层热循环的冷却过程中随着时间的延长而减小,且在冷却最开始阶段,温度梯度变化最大,K1值也变化最大,裂纹在冷却的初始具有最大的扩展可能性。且涂层最有可能发生开裂失效。     

An inverse method for evaluating weld residual stresses via fatigue crack growth test data

This paper presents an inverse method for calculating the thermal residual stresses in welded specimens via measured fatigue crack growth rates. Firstly, fracture-mechanics superposition law has been used to extract the stress intensity factor due to residual stress contribution from measured crack growth rate. Secondly, a so-called B matrix has been established by performing finite element analysis. Residual stress distribution is then determined by solving linear algebraic equations relating the B matrix and residual stress intensity factors obtained from crack growth test data. The inverse method has been validated by a well-established residual stress distribution and corresponding stress intensity factor, and then applied to an M(T) sample in 2024-T3 alloy with a longitudinal weld. Agreement with the measured residual stresses is reasonably good and reasons for certain differences between the calculated and measured are discussed.     

Extended finite element method and fast marching method for three-dimensional fatigue crack propagation

A numerical technique for planar three-dimensional fatigue crack growth simulations is proposed. The new technique couples the extended finite element method (X-FEM) to the fast marching method (FMM). In the X-FEM, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modeled by finite elements with no explicit meshing of the crack surfaces. The initial crack geometry is represented by level set functions, and subsequently signed distance functions are used to compute the enrichment functions that appear in the displacement-based finite element approximation. The FMM in conjunction with the Paris crack growth law is used to advance the crack front. Stress intensity factors for planar three-dimensional cracks are computed, and fatigue crack growth simulations for planar cracks are presented. Good agreement between the numerical results and theory is realized.     

Thermal stress intensities at an interface crack between two elastic layers

The thermal stress intensities (energy release rate and stress intensity factors) due to temperature changes are derived in closed-form for an interface crack between two elastic layers of dissimilar materials. The solutions are two-dimensional and tabulated over a wide range of material and layer thickness combinations. The tables serve as rapid evaluations of the thermal stress intensities for given temperature changes. A strain gauge technique is given for determining constraint coefficients which reflect the constraint conditions during the temperature changes. The solutions are compared with results from the literature. The stress intensities due to thermal and mechanical loads are generally superimposed. As an example of application, the solutions are utilized to obtain the complete thermal and mechanical stress intensities for a four-point bend specimen.     

Application of the crack compliance method to long axial cracks in pipes with allowance for geometrical nonlinearity and shape imperfections (dents)

Application of the crack compliance method to the analysis of thin-walled rings with a radial crack has two features: a crack is considered as a concentrated angular compliance and the deformation of all other sections of the rings is calculated as for a curvilinear beam. The latter can be most conveniently found by the method of initial parameters where the values of generalized forces and displacements at the end of some zone are determined as a linear combination of their values at the beginning of the zone. The goal of the study is to derive and apply the method of initial parameters equations taking into account the influence of circumferential stresses on the ring curvature. As far as the authors know, this is the first time that the stress intensity factor has been derived for an elastic thin-walled pipe with a radial crack in a geometrically nonlinear formulation. Here, an increase in pressure leads to a somewhat slowed increase in the stress intensity factor. In addition, a number of problems for dents are considered. The effect of the dent shape on the stress-strain state is analyzed. An expression for the stress intensity factor for a complex defect, a crack emanating from the dent apex, is presented.