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.     

层压复合材料分层扩展分析的虚拟裂纹闭合技术及其应用

提出一种基于虚拟裂纹闭合技术的界面元模型,用以模拟复合材料的分层破坏和预测结构的承载能力。界面元被嵌入在模型分层扩展路径上,计算结构的能量释放率,结合幂指数破坏准则,模拟复合材料的分层扩展。对由于裂尖单元长度不同所带来的分析误差进行了适当的修正,以降低网格粗细变化所带来的不利影响。为了检验该界面元的可靠性,分别将其应用于对双悬臂梁(DCB) 模型、端边切口(ENF) 模型和混合模式弯曲(MMB) 模型的分层扩展分析中。计算结果与解析解基本吻合,从而验证了采用该界面元模拟复合材料分层破坏的可行性。用该方法对3个含有不同初始损伤复合材料T型接头的界面拉脱分层破坏进行数值模拟,计算结果与试验数据吻合良好。      

Fracture criterion for kinking cracks in a tri-material adhesively bonded joint under mixed mode loading

The fracture behavior of a composite/adhesive/steel bonded joint was investigated by using double cantilever beam specimens. A starter crack is embedded at the steel/adhesive interface by inserting Teflon tape. The composite adherend is a random carbon fiber reinforced vinyl ester resin composite while the other adherend is cold rolled steel. The adhesive is a one-part epoxy that is heat cured. The Fernlund-Spelt mixed mode loading fixture was employed to generate five different mode mixities. Due to the dissimilar adherends, crack turning into the adhesive (or crack kinking) associated with joint failure, was observed. The bulk fracture toughness of the adhesive was measured separately by using standard compact tension specimens. The strain energy release rates for kinking cracks at the critical loads were calculated by a commercial finite element analysis software ABAQUS in conjunction with the virtual crack closure technique. Two fracture criteria related to strain energy release rates were examined. These are (1) maximum energy release rate criterion (G_max) and, (2) mode I facture criterion (G_II = 0). They are shown to be equivalent in this study. That is, crack kinking takes place at the angle close to maximum G or G_I (also minimum G_II, with a value that is approximately zero). The average value of G_IC obtained from bulk adhesive tests using compact tension specimens is shown to be an accurate indicator of the mode I fracture toughness of the kinking cracks within the adhesive layer. It is concluded that the crack in tri-material adhesively bonded joint tends to initiate into the adhesive along a path that promotes failure in pure mode I, locally.     

Discrete cohesive zone model for mixed-mode fracture using finite element analysis

The discrete cohesive zone model (DCZM) is implemented using the finite element (FE) method to simulate fracture initiation and subsequent growth when material non-linear effects are significant. Different from the widely used continuum cohesive zone model (CCZM) where the cohesive zone model is implemented within continuum type elements and the cohesive law is applied at each integral point, DCZM uses rod type elements and applies the cohesive law as the rod internal force vs. nodal separation (or rod elongation). These rod elements have the provision of being represented as spring type elements and this is what is considered in the present paper. A series of 1D interface elements was placed between node pairs along the intended fracture path to simulate fracture initiation and growth. Dummy nodes were introduced within the interface element to extract information regarding the mesh size and the crack path orientation. To illustrate the DCZM, three popular fracture test configurations were examined. For pure mode I, the double cantilever beam configuration, using both uniform and biased meshes were analyzed and the results show that the DCZM is not sensitive to the mesh size. Results also show that DCZM is not sensitive to the loading increment, either. Next, the end notched flexure for pure mode II and, the mixed-mode bending were studied to further investigate the approach. No convergence difficulty was encountered during the crack growth analyses. Therefore, the proposed DCZM approach is a simple but promising tool in analyzing very general two-dimensional crack growth problems. This approach has been implemented in the commercial FEA software ABAQUS^® using a user defined subroutine and should be very useful in performing structural integrity analysis of cracked structures by engineers using ABAQUS^®.     

Mixed-mode fracture load prediction in lead-free solder joints

Double cantilever beam (DCB) fracture specimens were made by joining copper bars with both continuous and discrete SAC305 solder layers of different lengths under standard surface mount (SMT) processing conditions. The specimens were then fractured under mode-I and various mixed-mode loading conditions. The loads corresponding to crack initiation in the continuous joints were used to calculate the critical strain energy release rate, J_ci, at the various mode ratios using elastic–plastic finite element analysis (FEA). It was found that the J_ci from the continuous joint DCBs provided a lower bound strength prediction for discrete 2 mm and 5 mm long joints at the various mode ratios. Additionally, these J_ci values calculated from FEA using the measured fracture loads agreed reasonably with J_ci estimated from measured crack opening displacements at crack initiation in both the continuous and discrete joints. Therefore, the critical strain energy release rate as a function of the mode ratio of loading is a promising fracture criterion that can be used to predict the strength of solder joints of arbitrary geometry subject to combined tensile and shear loads.     

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.     

Fracture load prediction of lead-free solder joints

Continuous and discrete SAC305 solder joints of different lengths were made between copper bars under standard surface mount (SMT) processing conditions, and then fractured under mode-I loading. The load-displacement behavior corresponding to crack initiation and the subsequent toughening before ultimate failure were recorded and used to calculate the critical strain energy release rates. The fracture of the discrete solder joints was then simulated using finite elements with two different failure criteria: one in terms of the critical strain energy release rate at initiation, G_ci, and another based on a cohesive zone model at the crack tip (CZM). Both criteria predicted the fracture loads reasonably well. In addition, the CZM was able to predict accurately the overall load-displacement behavior of the discrete joint specimen. It could also predict the load sharing that occurred between neighboring solder joints as a function of joint pitch and adherend stiffness. This has application in the modeling of the strength of solder joint arrays such as those found in ball grid array packages.     

Examination of the 4-ENF test for measuring the mode III R-curve of wood

The four-point bend end-notched flexure (4-ENF) test, which was originally developed for measuring the mode II R-curve, is thought to be applicable for measuring the mode III R-curve. In this study, a 4-ENF fracture test of spruce was conducted for obtaining the mode III R-curve, and the test method was numerically and experimentally analyzed. In the numerical analysis, three-dimensional finite element calculations were conducted to determine the distribution of the strain energy release rate along the delamination front by the virtual crack closure technique (VCCT). In the experimental analysis, the mode III R-curve was examined by the modified beam theory and compliance calibration methods of data reduction, which have been conventionally used for analyzing the mode I or mode II R-curve. In addition to these conventional data reduction methods, the strain at each loading point was measured, as was the loading-line displacement and critical load for crack propagation, and the R-curve was obtained by the combination of loading-line compliance, load-longitudinal strain compliance, and critical load for crack propagation, which is named the “compliance combination method”. The finite element analyses suggested that the pure mode III fracture state existed in the mid-section of the specimen in spite of the existence of a small mode II component at the free edges of the delamination front, and the mode III strain energy release rate component calculated by the VCCT coincided well with those obtained by the data reduction methods examined here. The actual R-curve obtained by the compliance combination method coincided well with those by the modified beam theory and compliance calibration methods when the strain was appropriately measured. From these results, therefore, the 4-ENF fracture test is a promising means for obtaining the mode III R-curve of wood.     

Fracture Parameters for Natural Rubber Under Dynamic Loading

Abstract: An experimental study was conducted to evaluate the tear energy of unfilled and 25 phr carbon black‐filled natural rubber with varying loading rates. The variation of the tear energy with far‐field sample strain rate between 0.01 to 10 s^?1 was found to be different from tensile strip and pure shear specimens. Above a sample strain rate of 10 s^?1, the tear energy calculated from either specimen was comparable. The differences in the tear energy derived from the tensile strip and pure shear specimens were attributed to differences in the local crack tip stress state and strengthening of the material due to strain‐induced crystallisation. Both of these factors resulted in crack speeds 3–4 times higher in the pure shear specimen as compared to the tensile strip specimen. Finite element analysis (FEA) indicated that fracture would initiate at the crack tip either when the strain energy density approached the material toughness or when the maximum principal stress and strain approached the material tensile strength and fracture strain, respectively. It was concluded that these parameters would be better than the tear energy in predicting fracture of natural rubber under dynamic loading.     

Mode III interlaminar fracture of carbon/epoxy laminates using a four-point bending plate test

A new four-point bending plate (4PBP) test was used for characterising the mode III interlaminar fracture of carbon/epoxy laminates. The specimen has a cross-ply lay-up and two edge delaminations whose propagation becomes visible at the edges. Although the test setup is very simple, determination of the mode III critical strain energy release rate G_IIIc requires finite element analyses (FEA). The virtual crack closure technique with an assumed initiation region was first proposed for computing G_IIIc. This scheme was subsequently validated by crack growth simulations with a cohesive zone model. The results showed an average G_IIIc = 1550 J/m², which is significantly higher than the G_IIIc = 850–1100 J/m² and G_IIc = 800 J/m² measured in previous studies.     

Crack growth analysis at adhesive–adherent interface in bonded joints under mixed mode I/II

The propagation of an interface crack subjected to mixed mode I/II was investigated for two 2024-T351 aluminum thin layers joined by means of DP760 epoxy adhesive produced by 3M^©. On the basis of beam theory, an analytical expression for computing the energy release rate is presented for the mixed-mode end loaded split (MMELS) test. The analytical strain energy release rate was compared by finite element (FE) analysis using the virtual crack closure technique (VCCT). Several fatigue crack growth tests were carried out in a plane bending machine to compare the experimental energy release rates to those of the analytical and FE solutions. Experimental results showed the relationship between the delamination modality and initial crack length rather than the applied load. The crack growth behavior showed stable crack growth followed by rapid propagation at the interface with the adhesive layer.     

Crack growth simulation for arbitrarily shaped cracks based on the virtual crack closure technique

In this paper, crack growth simulation for arbitrarily shaped cracks was investigated based on the virtual crack closure technique. During simulations, the crack front was represented by an approximated zigzag line which had the same general shape as the given crack. For this approximated zigzag crack front, a modified approach was developed to determine the required nodal forces, virtually closed area and displacement opening. After the strain energy release rate G was calculated, crack growth was governed by the fracture criterion G/G _C = 1 at all the crack tip nodes. The important features of the proposed approach are that (i) a simple stationary finite element mesh can be used for arbitrarily shaped cracks and (ii) adaptive re-meshing technique is avoided in studying crack growth. Three cases having different initial crack shapes are presented to assess the validity of this approach and to evaluate the ease of use in tracking crack growth. Reasonable agreement between the present study and other approaches are obtained. The shape changes during crack propagation can also be tracked with ease.     

Analysis of the fracture behaviour of a stitched single-lap joint

The effect of stitching on the fracture response of single-lap composite joints was studied by a combined experimental and numerical analysis. Unstitched and Kevlar stitched joints were tested under static and fatigue loading to characterize damage progression and failure modes; a three-dimensional finite element analysis was carried out to evaluate the influence of stitches on strain energy release rates as a function of damage and to identify the role of various stitching parameters on the fracture behaviour of joints.It was observed that the failure of the joints occurs as a consequence of the propagation of delamination at the interface between the adherends; the propagation is stable under fatigue loads and unstable under static loads. Stitching does not improve the static strength of joints but significantly prolongs the duration of the crack propagation phase under fatigue loading.The results of finite element modelling indicate that the incorporation of stitches reduce G_I to zero after the delamination front passes the stitch line, but it is not effective in reducing mode II energy release rate. They also show that strain energy release rates are not greatly affected by the length of stitch-laminate debonding, which, conversely, does influence stitch tensioning. Moreover, 3D analysis reveals that stitches become less efficient in reducing the crack driving force with increasing stitching steps.     

A numerical analysis of the elastic-plastic peel test

The adhesive fracture energy, G_c, is determined from two types of elastic-plastic peel tests (i.e. the single-arm 90° and T-peel methods) and a linear-elastic fracture-mechanics (LEFM) test method (i.e. the tapered double-cantilever beam, TDCB method). A rubber-toughened epoxy adhesive, with both aluminium-alloy and steel substrates, has been used in the present work to manufacture the bonded joints. The peel tests are then modelled using numerical methods. The overall approach to modelling the elastic-plastic peel tests is to employ a finite-element analysis (FEA) approach and to model the crack advance through the adhesive layer via a node-release technique, based upon attaining a critical plastic strain in the element immediately ahead of the crack tip. It is shown that this ‘critical plastic strain fracture model (CPSFM)’ results in predicted values of the steady-state peel loads which are in excellent agreement with the experimentally-measured values. Also, the resulting values of G_c, as determined using the FEA CPSFM approach, have been found to be in excellent agreement with values from previously-reported analytical and direct-measurement methods. Further, it has been found that the calculated values of G_c are independent of whether a standard LEFM test or an elastic-plastic peel test method is employed. Therefore, it has been demonstrated that the value of the adhesive fracture energy, G_c, is independent of the geometric parameters studied and the value of G_c is indeed a characteristic of the joint, in this case for cohesive fracture through the adhesive layer. Finally, it is noted that the FEA CPSFM approach promises considerable potential for the analysis of peel tests which involve very extensive plastic deformation of the peeling arm and for analysing, and predicting, the performance of more complex adhesively-bonded geometries which involve extensive plastic deformation of the substrates.     

Fracture behaviour of composite maritime T-joints

This paper outlines a study on the fracture behaviour of a glass fibre reinforced polymer T-joint commonly used in composite marine vessels. Finite element analysis was conducted using the virtual crack closure technique (VCCT) to investigate the fracture behaviour of the structure. The structure analysed contained initial disbond in various locations with various sizes under a straight pull-off load. The strain energy release rate (SERR) at the disbond tips were used to predict the failure loads and crack growth mechanism of the structure. The experimental results validated the VCCT as a tool for assessing the fracture behaviour and damage criticality of such structures. It was also discovered that skewed loading affected the SERR at the crack tips which altered the fracture behaviour of such structures, therefore sensitivity analysis is recommended to enhance the prediction accuracy.     

Orientation and loading condition dependence of fracture toughness in cortical bone

The fracture toughness at crack initiation were determined for bovine cortical bone under tension (mode I), shear (mode II), and tear (mode III). A total of 140 compact tension specimens, compact shear specimens and triple pantleg (TP) specimens were used to measure fracture toughness under tension, shear, and tear, respectively. Multiple-sample compliance method was utilized to measure the critical strain energy release rate (G_c) at the a/W=0.55 (crack length, a, to specimen width, W, ratio). The critical stress intensity factor (K_c) was also calculates from the critical loading (P_c) of the specimens at the a/W=0.55. The effect of the anisotropy of bone on its resistance to crack initiation under shear and tear loading was investigated as well. Fracture toughness of bone with precrack orientations parallel (designed as longitudinal fracture) and vertical (designed as transverse fracture) to the longitudinal axis of bone were compared. In longitudinal fracture, the critical strain energy release rate (G_c) of cortical bone under tension, shear, and tear was 644±102, 2430±836, and 1723±486 N/m, respectively. In transverse fracture, the critical strain energy release rate (G_c) of cortical bone under tension, shear, and tear was 1374±183, 4710±1284, and 4016±948 N/m, respectively. An unpaired t-test analysis demonstrated that the crack initiation fracture toughness of bone under shear and tear loading were significantly greater than that under tensile loading in both longitudinal and transverse fracture (P<0.0001 for all). Our results also suggest that cortical bone has been “designed” to prevent crack initiation in transverse fracture under tension, shear, and tear.     

Finite element analysis of the end notched flexure specimen for measuring mode II fracture toughness

This paper presents a finite element analysis of the End Notched Flexure (ENF) test specimen for Mode II interlaminar fracture testing of composite materials. Virtual crack closure and compliance techniques employed to calculate strain energy release rates from linear elastic two-dimensional analysis indicate that the ENF specimen is a pure Mode II fracture test within the constraints of small deflection theory. Furthermore, the ENF fracture specimen is shown to be relatively insensitive to process induced cracks offset from the laminate midplane. Frictional effects are investigated by including the contact problem in the finite element model. A parametric study investigating the influence of delamination length, span, thickness and material properties is presented to assess the accuracy of beam theory expressions for compliance and strain energy release rate, G_II. Finite element results indicate that data reduction schemes based upon beam theory underestimate G_II by approximately 20–40% for typical unidirectional graphite fiber composite test specimen geometries. Consequently, an improved data reduction scheme which retains the simplicity of beam theory but also includes the accuracy of the finite element method is proposed.     

含面\芯开裂损伤复合材料夹层板考虑几何非线性的能量释放率数值分析

基于一阶剪切变形理论,zig-zag变形假定和von Karman大挠度理论,提出了含不同形状面\芯开裂损伤复合材料夹层板在受压缩载荷作用下的开裂前缘能量释放率研究的有限元分析方法,研究了在轴向应变作用下,具有面\芯开裂损伤复合材料夹层板的分层断裂力学行为,并讨论了在大变形下几何非线性对能量释放率分布规律的影响。通过典型算例分析表明:具有面\芯开裂损伤复合材料夹层板的分层前缘能量释放率的大小和分布规律与开裂面积、开裂形状和受载方向有关。     

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.     

Analysis of three-dimensional interface cracks using enriched finite elements

Many important interface crack problems are inherently three-dimensional in nature, e.g., debonding of laminated structures at corners and holes. In an effort to accurately analyze three-dimensional interface fracture problems, an efficient computational technique was developed that utilizes enriched crack tip elements containing the correct interface crack tip asymptotic behavior. In the enriched element formulation, the stress intensity factors K _I, K _II, and K _III are treated as additional degrees of freedom and are obtained directly during the finite element solution phase. In this study, the results that should be of greatest interest are obtained for semi-circular surface and quarter-circular corner cracks. Solutions are generated for uniform remote tension and uniform thermal loading, over a wide range of bimaterial combinations. Of particular interest are the free surface effects, and the influence of Dundurs’ material parameters on the strain energy release rate magnitudes and corresponding phase angles.