Obtaining mode mixity for a bimaterial interface crack using the virtual crack closure technique

We review, unify and extend work pertaining to evaluating mode mixity of interfacial fracture utilizing the virtual crack closure technique (VCCT). From the VCCT, components of the strain energy release rate (SERR) are obtained using the forces and displacements near the crack tip corresponding to the opening and sliding contributions. Unfortunately, these components depend on the crack extension size, Δ, used in the VCCT. It follows that a mode mixity based upon these components also will depend on the crack extension size. However, the components of the strain energy release rate can be used for determining the complex stress intensity factors (SIFs) and the associated mode mixity. In this study, we show that several—seemingly different—suggested methods presented in the literature used to obtain mode mixity based on the stress intensity factors are indeed identical. We also present an alternative, simpler quadratic equation to this end. Moreover, a Δ-independent strain energy release based mode mixity can be defined by introducing a “normalizing length parameter.” We show that when the reference length (used for the SIF-based mode mixity) and the normalizing length (used for Δ-independent SERR-based mode mixity) are equal, the two mode mixities are only shifted by a phase angle, depending on the bimaterial parameter ε.     

有限变形理论能量释放率增率形式

基于有限变形理论中的能量原理和变分原理,考虑以有裂纹的瞬时位形为参考,建立增量变形引起的裂纹扩展方程能够更真实的描述裂纹尖端的扩展机制,在含有裂纹物体的瞬时变形的基础上,推导了裂纹体的能量释放率和增率的形式,提出了裂纹扩展判据.该判据反映了单位时间内裂纹扩展单位面积可以提供的能量与单位时间内裂纹扩展单位面积所需要的能量...     

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.     

Evaluation of energy release rate-based approaches for predicting delamination growth in laminated composites

A variety of energy release rate-based approaches are evaluated for their accuracy in predicting delamination growth in unidirectional and multidirectional laminated composites. To this end, a large number of unidirectional and multidirectional laminates were tested in different bending and tension configurations. In all cases, the critical energy release rate was determined from the tests in the most accurate way possible, such as by compliance calibration or the area method of data reduction. The mode mix from the tests, however, was determined by a variety of different approaches. These data were then examined to determine whether any of the approaches yielded the result that toughness was a single-valued function of mode mix. That is, for an approach to have accurate predictive capabilities, different test geometries that are predicted to be at the same mode mix must display the same toughness. It was found that variously proposed singular field-based mode mix definitions, such as the =0 approach or basing energy release rate components on a finite amount of crack extension, had relatively poor predictive capabilities. Conversely, an approach that used a previously developed crack tip element analysis and which decomposed the total energy release rate into non-classical components was found to have excellent predictive capabilities. It is postulated that this approach is more appropriate for many present-day laminated composites.     

A new fracture mechanics theory for orthotropic materials like wood

A new fracture mechanics theory is derived based on a new orthotropic-isotropic transformation of the Airy stress function, making the derivation of the Wu-“mixed mode I-II” fracture criterion possible, based on the failure criterion of a flat elliptic crack. As a result of this derivation, the right fracture energy and theoretical relation between mode I and II stress intensities and energy release rates are obtained.     

Analysis of mixed-mode dynamic crack propagation by interface element based on virtual crack closure technique

An interface element tailored for the virtual crack closure technique (VCCT) was used to study an example of dynamic crack propagation under mixed mode loading. Through this interfacial element approach, VCCT can be implemented into a commercial finite element analysis (FEA) code having user subroutines without interrupting the main code. Further, with the implementation of relevant fracture criteria, this interface element can be used to simulate a wide range of fracture problems by utilizing the enhanced capabilities available by the commercial FEA codes. For illustration, this element has been implemented with the commercial FEA software ABAQUS^® through the user defined element (UEL). One example of fast crack propagation at constant speed and under mixed-mode loading was examined by comparison to the other’s numerical results using singular moving elements. No convergence difficulty was encountered for all the cases with different values of crack velocity. Neither singular element, nor the collapsed element was required. Therefore, due to its simplicity, the VCCT interface element as demonstrated could be a potential tool for engineers to practice dynamic fracture analysis in conjunction with commercial FEA codes.     

双悬臂梁试件裂纹动态扩展的准静态数值分析

本文采用双悬臂梁（DCB）试件研究了复合材料层合板层间插入韧性胶膜（Interleaf）层的Ⅰ型断裂行为。试验结果表明,含和不含Interleaf层试件分别呈现脆性非稳态和脆性稳态分层扩展特性。针对非稳定裂纹扩展问题,依据动态断裂力学中应变能释放率与动能变化率的关系,提出了以断裂韧性值G_IC变化来抵消动能变化对裂纹扩展过程影响的准静态分析方法,根据试验中裂纹扩展的韧性变化,推导出适用于准静态裂纹扩展模拟的等效韧性G_IC*,利用ABAQUS平台和虚裂纹闭合技术（VCCT）建立了三维有限元计算模型;实现了从起裂到止裂的整个裂纹动态扩展过程的数值模拟,揭示了非稳定裂纹扩展过程中一些复杂的力学现象。      

Analysis of beam-type fracture specimens with crack-tip deformation

A novel bi-layer beam model is developed to account for local effects at the crack tip of a bimaterial interface by modeling a bi-layer composite beam as two separate shear deformable beams. The effect of interface stresses on the deformations of sub-layers, which is referred to as the elastic foundation effect in the literature, is considered in this model by introducing two interface compliance coefficients; thus a flexible joint condition at the crack tip is considered in contrast to the rigid joint condition used in the conventional bi-layer model. An elastic crack tip deformable model is presented, and the closed-form solutions of local deformation at the crack tip are then obtained. By applying this novel crack tip deformation model, the new terms due to the local deformations at the crack tip, which are missing in the conventional composite beam solutions of compliance and energy release rate (ERR) of beam-type fracture specimens, are recovered. Several commonly used beam-type fracture specimens are examined under the new light of the present model, and the improved solutions for ERR and mode mixity are thus obtained. A remarkable agreement achieved between the present and available solutions illustrates the validity of the present study. The significance of local deformation at the crack tip is demonstrated, and the improved solutions developed in this study provide highly accurate predictions of fracture properties which can actually substitute the full continuum elasticity analysis such as the finite element analysis. The new and improved formulas derived for several specimens provide better prediction of ERR and mode mixity of beam-type fracture experiments.*Author for correspondence (E-mail address: qiao@uakron.edu)     

Determination of energy release rate and mode mix in three-dimensional layered structures using plate theory

A plate theory-based method for determining energy release rates is presented for general loadings of three dimensional layered structures. Mode decomposition is performed for cases that exhibit an inverse square root singularity and for which certain other restrictions apply. Predictions for energy release rate and mode mix for typical problems are presented and verified by comparison with results obtained by three dimensional finite element analyses.     

Face/core interface fracture characterization of mixed mode bending sandwich specimens

Debonding of the core from the face sheets is a critical failure mode in sandwich structures. This paper presents an experimental study on face/core debond fracture of foam core sandwich specimens under a wide range of mixed mode loading conditions. Sandwich beams with E‐glass fibre face sheets and PVC H45, H100 and H250 foam core materials were evaluated. A methodology to perform precracking on fracture specimens in order to achieve a sharp and representative crack front is outlined. The mixed mode loading was controlled in the mixed mode bending (MMB) test rig by changing the loading application point (lever arm distance). Finite element analysis was performed to determine the mode‐mixity at the crack tip. The results showed that the face/core interface fracture toughness increased with increased mode II loading. Post failure analysis of the fractured specimens revealed that the crack path depends on the mode‐mixity at the crack tip, face sheet properties and core density.     

Prediction of initiation and growth of single level delaminations in a transversely loaded composite specimen using fracture mechanics

The behaviour of a composite test specimen with an embedded delamination subjected to transverse tension has been investigated through experimental testing and finite element (FE) analyses. The testing program consisted of specimens in two geometrical configurations; square and rectangular delamination. The initiation and growth of the delamination was numerically predicted by fracture mechanics. FE models were analysed with both MSC.Nastran and Abaqus FE codes. The MSC.Nastran model was used to calculate strain energy release rates employing a crack tip element methodology. The Abaqus model was evaluated using the virtual crack closure technique. Both approaches accurately predicted failure initiation locations as observed in the test specimens. Failure loads were also well predicted. The mode mix at the crack tip in the proposed specimen was found to be similar to the mode mix expected in a conventional in-plane compression specimen.     

湿热环境下复合材料的混合型层间断裂特性研究

采用混合型挠曲(MMF)试件,研究了材料吸湿和环境温度对T300/5405复合材料混合型层间断裂韧性的影响。给出了在不同温度下,不同吸湿含量试件分层临界扩展时的Ⅰ型分量和Ⅱ型能量释放率分量散点图。结果表明:在吸湿和温度的综合作用下,分层尖端存在塑性变形;常温下,吸湿对材料的层间断裂韧性影响不明显,在高温环境中,随吸湿量增加,层间断裂韧性显著增加;温度对干态材料的断裂韧性影响较小,试件吸湿后,随温度升高,韧性增强。      

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.     

Study of Debond Fracture Toughness of Sandwich Composites with Metal Foam Core

Two types of experiments were designed and performed to evaluate the adhesive bond in metal foam composite sandwich structures.The tensile bond strength of face/core was determined through the flatwise tensile test (FWT).The test results show that the interfacial peel strength is lower than the interlaminar peel strength in FWT test.The mode I interfacial fracture toughness(GIC)of sandwich structures containing a pre-crack on the upper face/core interface is determined by modified cracked sandwich beam(MCSB...     

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.     

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.     

Delamination tolerance studies in laminated composite panels

Determination of levels of tolerance in delaminated composite panels is an important issue in composite structures technology. The primary intention is to analyse delaminated composite panels and estimate Strain Energy Release Rate (SERR) parameters at the delamination front to feed into acceptability criteria. Large deformation analysis is necessary to cater for excessive rotational deformations in the delaminated sublaminate. Modified Virtual Crack Closure Integral (MVCCI) is used to estimate all the three SERR components at the delamination front from the finite element output containing displacements, strains and stresses. The applied loading conditions are particularly critical and compressive loading on the panel could lead to buckling of the delaminated sublaminate and consequent growth of delamination. Numerical results are presented for circular delamination of various sizes and delamination at various interfaces (varying depth-wise location) between the base- and the sub-laminates. Numerical data are also presented on the effect of bi-axial loading and in particular on compressive loading in both directions. The results can be used to estimate delamination tolerance at various depths (or at various interfaces) in the laminate.     

On the Method of Determination of Strain Energy Release Rate During Fatigue Delamination in Composite Materials

In this paper, the problem of calculation of the energy release rate for a fatigue test on composite material has been investigated. The application of the Linear Elastic Failure Mechanics (LEFM) leads to the use of varation of the energy release rate ( G). As the energy release rate is a function of the load squared, the variation of G becomes either a function of variation of the load squared ( G = f((P²))) or a function of the square of the load variation ( G = f(( P)²)).In this paper, we determine, by different fatigue tests, which of the two theoretical results is the best to describe the experiments. These fatigue tests have been made on DCB test-specimen in mode I with different R ratios (R = P_max / P_min) and different maximum loads. The material was a unidirectionnal glass-epoxy.The results show that considering G as a function of ( P)² seems more appropriated to describe a cracking test in fatigue.     

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 sandwich three-point bend specimen for testing mode-I interlaminar fracture toughness for fiber-reinforced composite materials

A sandwich three-point bend specimen has recently been proposed to test mode-I interlaminar fracture toughness for fiber-reinforced composite materials. The test composite consist of a thin layer bonded by two lateral reusable steel bars (Sohn et al. 1995). Some time earlier this specimen configuration was used to test fracture toughness of adhesives (Zdaniewsk et al. 1987). However, formulae for analysing its fracture mechanics parameters such as stress intensity factor and energy release rate can not be found in the literature. The lack of adequate formulae may explain why suitable quantitative analysis using this specimen configuration has not been achieved. In this paper, a simple and effective homogenisation method is used to change the bi-material system, which represents the specimen, into single uniform test material. This physical homogenisation is carried out by geometric change of the cross section of lateral steel parts based on equal deflection rigidity. For the transformed specimen configuration of single uniform material, the corresponding stress intensity factor solution from handbooks is available. Two formulae of stress intensity factor for the sandwich three-point bend specimen are given as upper limit and lower limit respectively, they are plotted with varying elastic tensile modulus mismatch. Then the relation between stress intensity factor and energy release rate, with special consideration of orthotropy of the tested composite material, is used to derive its energy release rate. The specimen and its formulae can also be applied to test other materials such as wood, welded joints (Burstow and Ainsworth, 1995), as well as to test dynamic fracture toughness. This revised version was published online in July 2006 with corrections to the Cover Date.