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.     

Predicting fracture of solder joints with different constraint factors

Double cantilever beam (DCB) specimens of 2.5‐mm‐long SAC305 solder joints were prepared with thickness of copper adherends varying from 8 to 21 mm each. The specimens were tested under mode I loading conditions (ie, pure opening mode with no shear component of loading) with a strain rate of 0.03 second^?1. The measured fracture load was used to calculate the critical strain energy release rate for crack initiation, J_ci, in each case. Fracture behaviour showed a significant dependence on the adherend thickness; the J_ci and plastic deformation of the solder at crack initiation decreased significantly with increase in adherend thickness. This behaviour was attributed to changes in stress distribution along the solder layer when the adherend thickness was varied. The capability of J_ci as a property was then assessed to predict the fracture load of solder joints in specimens with different constraint levels caused by variations in adherend thicknesses. In light of the results obtained, a cohesive zone model (CZM) was developed to predict the fracture load of solder joints as a function of adherend thickness. Finally, a CZM with a single set of parameters was established to predict the fracture loads for all the cases. It was concluded that CZM was a better methodology to account for changes in degree of joint constraint imposed by bonding adherends.     

Mixed-mode quasi-static failure criteria for adhesively-bonded pultruded GFRP joints

The strain energy release rates of adhesively-bonded pultruded GFRP joints were determined experimentally. The crack propagated in the adherend along paths outside the symmetry plane accompanied by fiber bridging. A new method, designated the “extended global method”, was introduced to facilitate mode partitioning in the mixed-mode experiments. Non-linear finite element models were developed in order to quantify the effect of the observed fiber bridging on crack propagation. An exponential traction-separation cohesive law was used to model the fiber bridging zone and calculate the energy release rate due to the fiber bridging, while the virtual crack closure technique was used for calculation of the fracture components at the crack tip. Experimental, analytical and numerical analyses were used to establish quasi-static mixed-mode failure criteria for crack initiation and propagation. The derived mixed-mode failure criteria can be used for simulating progressive crack propagation in other joint configurations comprising the same adhesive and adherends.     

Crack path selection in adhesively bonded joints: the roles of external loads and speciment geometry

This paper investigates the roles of external loads and specimen geometry on crack path selection in adhesively bonded joints. First, the effect of mixed mode fracture on crack path selection is studied. Using epoxy as an adhesive and aluminum as the adherends, double cantilever beam (DCB) specimens with various T-stress levels are prepared and tested under mixed mode fracture loading. Post-failure analyses on the failure surfaces using X-ray photoelectron spectroscopy (XPS) suggest that the failure tends to be more interfacial as the mode II fracture component in the loading increases. This fracture mode dependence of the locus of failure demonstrates that the locus of failure is closely related to the direction of crack propagation in adhesive bonds. Through analyzing the crack trajectories in failed specimens, the effect of mixed mode fracture on the directional stability of cracks is also investigated. The results indicate that the direction of the crack propagation is mostly stabilized when more than 3% of mode II fracture component is present at the crack tip regardless of the T-stress levels in the specimens for the material system studied. Second, using a high-speed camera to monitor the fracture sequence in both quasi-static and low-speed impact tests, the effect of debond rate on the locus of failure and directional stability of cracks is investigated. Post-failure analyses including XPS, Auger electron spectroscopic depth profile, and scanning electron microscopy indicate that as the crack propagation rate increases, the failure tends to be more cohesive and the cracks tend to be directionally unstable. Last, as indicated by the finite element analyses results, the T-stresses, and therefore the directional stability of cracks in adhesive bonds, are closely related to the thickness of the adhesive layer and also the thickness of adherend. This specimen geometry dependence of crack path selection is studied analytically and is verified experimentally.     

Recrystallization-etch approach to study the plastic energy absorbtion at crack initiation and extension of pressure-vessel steel A533B-1

To evaluate the elastic-plastic fracture toughness parameter of nuclear pressure-vessel steel A533B-1, a newly developed technique (the recrystallization-etch technique) for plastic strain measurement was applied to different sizes of compact tension specimens with a crack length/specimen width of 0.6–0.5 that were tested to generate resistance curves for stable crack extensions. By means of the recrystallization-etch technique, the plastic energy dissipation or work done within an intense strain region at the crack tip during crack initiation and extension was measured experimentally. Furthermore, the thickness effects on this crack tip energy dissipation rate were examined in comparison with other fracture-parameter J integrals. Thickness effects on critical energy dissipation and energy dissipation rate during crack extension were obtained and the energy dissipation rate dW _p/da in the mid-section shows a constant value irrespective of specimen geometry and size, which can be used as a fracture parameter or crack resistance property.     

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 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.     

Energy rates and crack stability in small scale yielding

Crack stability in small scale yielding is traditionally analysed using the R-curve approach with toughness indexed by either of the linear elastic fracture mechanics parameters K or G. In ductile materials stable tearing commences well before crack instability and progresses under increasing G_R. This is often assumed to mean that toughness is increasing with crack growth. It is shown in this paper that a rising G_R curve is generated even when a crack propagates with constant toughness (constant energy dissipation rate). The paper demonstrates that this apparent anomaly occurs because G does not represent the energy input rate for a crack advancing under increasing load in an elastic-plastic material. The constant energy dissipation rate model is consistent with a size independent G_R curve; also crack instability predictions are identical with both theories. The G_R curve approach has practical advantages, but use of energy dissipation rate offers better physical insight and greater versatility when analysing tough materials.     

Numerical study of geometric constraint and cohesive parameters in steady-state viscoelastic crack growth

This paper presents a finite element study of cohesive crack growth in a thin infinite viscoelastic strip to investigate the effects of viscoelastic properties, strip height, and cohesive model parameters on the crack growth resistance. The results of the study show that the dependence of the fracture energy on the viscoelastic properties for the strip problem is similar to that obtained for the infinite body problem even when the cohesive zone length is large compared to the height of the strip. The fracture energy also depends on the crack speed v through the dimensionless parameter v τ/L _∞ where L _∞ is the characteristic length of the cohesive zone and τ is the characteristic relaxation time of the bulk material. This relationship confirms that at least two properties of the fracture process must be prescribed accurately to model viscoelastic crack growth. In contrast, the fracture energy and crack speed are insensitive to the strip height even in situations where the growth of the dissipation zone is severely constrained by the strip boundaries. We observe that at high speeds, where the fracture energy asymptotically approaches the maximum value, the material surrounding the cohesive zone is in the rubbery (equilibrium) state and not the glassy state.     

The effect of mode ratio and bond interface on the fatigue behavior of a highly-toughened epoxy

The fatigue threshold and the cyclic crack growth of a highly-toughened epoxy adhesive were studied under mode I and several mixed-mode loading cases and compared with the quasi-static critical fracture energies. Four different adhesive systems were examined using steel and aluminum substrates having different surface roughness, and surface treatment. The effect of increasing the amount of mode II (increasing the phase angle) on the fatigue threshold strain energy release rate and the cyclic crack growth rate was found to be insignificant at low phase angles. However, a significant increase in the fatigue threshold and decrease in the cyclic crack growth rate was observed at higher phase angles. These trends were similar to that seen in adhesive joint fracture. Adherend surface roughness and surface preparation affected the fatigue behavior significantly, particularly at low crack speeds and high phase angles. The fatigue properties were essentially the same for both steel and aluminum adherends provided that the crack paths were cohesive. A general observation was that the fatigue crack path moved progressively closer to the more highly strained adherend under mixed-mode loading as the applied strain energy release rate and hence the crack speed, decreased. This caused mixed-mode cracks to be nearly interfacial in the threshold region.     

Prediction of the energy dissipation rate in ductile crack propagation

In this paper, energy dissipation rate D vs. Δa curves in ductile fracture are predicted using a ‘conversion’ between loads, load‐point displacements and crack lengths predicted by NLEFM and those found in real ELPL propagation. The NLEFM/ELPL link was recently discovered for the DCB testpiece, and we believe it applies to other cracked geometries. The predictions for D agree with experimental results. The model permits a crack tip toughness R(Δa) which rises from J_c and saturates out when (if) steady state propagation is reached after a transient stage in which all tunnelling, crack tip necking and shear lip formation is established. J_R is always greater than the crack tip R(Δa) and continues to rise even after R(Δa) levels off. The analysis is capable of predicting the usual D vs. Δa curves in the literature which have high initial values and fall monotonically to a plateau at large Δa. It also predicts that D curves for CCT testpieces should be higher than those for SENB/CT, as found in practice. The possibility that D curves at some intermediate Δa may dip to a minimum below the levelled‐off value at large Δa is predicted and confirmed by experiment. Recently reported D curves that have smaller initial D than the D‐values after extensive propagation can also be predicted. The testpiece geometry and crack tip R(Δa) conditions required to produce these different‐shaped D vs. Δa curves are established and confirmed by comparison with experiment. The energy dissipation rate D vs. Δa is not a transferable property as it depends on geometry. The material characteristic R(Δa) may be the ‘transferable property’ for scaling problems in ELPL fracture. How it can be deduced from D vs. Δa curves (and by implication, J_R vs. Δa curves) is established.     

Modeling of crack tip high inertia zone in dynamic brittle fracture

A phenomenological modification is proposed to the existing cohesive constitutive law of Roy and Dodds to model the crack tip high inertia region proposed by Gao. The modification involves addition of a term which is attributed to fracture mechanisms that result in high energy dissipation around the crack tip. This term is assumed to be a function of external energy per unit volume input into the system. Finite element analysis is performed on PMMA with constant velocity boundary conditions and mesh discretizations based on the work of Xu and Needleman. The cohesive model with the proposed dissipative term is only applied in the high inertia zone and the traditional Roy and Dodds model is applied on cohesive elements in the rest of the domain. The results show that crack propagates in three phases with a speed of 0.35c_R before branching, confirming experimental observations. The modeling of high inertia zone is one of the key aspects to understanding brittle fracture.     

The cohesive zone model in a problem of delayed hydride cracking of zirconium alloys

The crack tip model with the cohesive zone ahead of a finite crack tip has been presented. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration, according to Westergaard's theory, with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. Local criterion of brittle fracture and modelling of the fracture process zone by cohesive zone were used to describe fracture initiation at the hydride platelet in the process zone ahead of the crack tip. It was shown that the theoretical K _IH-estimation applied to the case of mixed plane condition within the process zone is qualitatively consistent with experimental data for unirradiated Zr-2.5Nb alloy. In the framework of the proposed model, the theoretical value of K ^H _IC for a single hydride platelet at the crack tip has been also estimated.     

On the fracture of constrained layers

The problem of a crack embedded in a layer sandwiched between two elastic adherends is analysed accounting for the influence material property mismatch on the crack tip plastic deformation, which is contained in the layer. The cohesive crack model developed by Dugdale and Barrenblatt is adopted to model the strip yielding behaviour in a constrained layer. It is found that, due to the constraint imparted by elastic adherends with higher moduli, the near tip plastic deformation exhibits a sharp transition (plastic zone grows faster than the square of stress intensity factor) from small scale to large scale yielding. Because the region of singularity dominance for a crack embedded in a layer is generally much smaller than the layer thickness when the layer has a modulus much lower than the adherends, the prevailing failure mode of most bonded joints should be under large scale yielding conditions. A model based on energy balance is proposed to determine the fracture energy of bonded joints under such condition, taking into account of the plastic dissipation in the constrained layer. Comparison with experimental results demonstrates that the theory correctly predicts the dependence of fracture toughness on layer thickness as observed in experiments. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of peel arm curvature for the determination of fracture toughness in metal-polymer laminates

Variable angle fixed arm peel and mandrel peel tests were performed on four metal-polymer laminate systems. In total, four polymeric adhesives and three grades of aluminium alloy (AA) substrates were used, enabling a wide range of material properties to be encompassed in the study. Mandrel peel tests provided a direct determination of the plastic bending energy (G_p) and adhesive fracture toughness (G_a). For the fixed arm tests, a global energy-balance analysis (ICPeel software) was used to determine G_a and G_p analytically. This was done via the calculation of the maximum curvature of the peel arm (1/R₀) and the root rotation angle (θ₀) from a beam on elastic foundation model. In order to investigate the accuracy of the analytical approach, an experimental method based on high resolution digital photography enabled 1/R₀ and θ₀ to be measured independently. It was then possible to compare these parameters by measurement and by analytical approach (ICPeel software). θ₀ and R₀ relate to the slope and curvature of the peel arm at the debonding front, respectively. In order to measure these parameters, the coordinates of the edge of the peel arm were extracted from each digital photograph, and the slope and curvature were calculated numerically from these curves. The crack tip was then defined as the point of maximum curvature 1/R₀, in accordance with traditional beam theory. It was found that the smoothing in the calculation of first and second derivatives could generate significant errors in the value of θ₀. On the other hand, R₀ was found to be a more robust measurement, with little dependence on smoothing. Nevertheless, on most occasions, the measured values of θ₀ and R₀, as well as the resulting G_a were shown to be in good agreement with the analytical model. Since the peel fractures were generally cohesive, G_a was compared with the cohesive fracture toughness (G_c) obtained from Tapered Double Cantilever Beam (TDCB) tests with a fracture mechanics analysis. Good agreement was observed, confirming that G_a is likely to be a geometry-independent fracture parameter.     

The toughening of cyanate-ester polymers Part I Physical modification using particles, fibres and woven-mats

A fracture mechanics approach has been used to investigate the effects of the addition of physical modifiers on the fracture energy, G _c, of brittle cyanate-ester polymers. Tests were performed using adhesive joint specimens at –55, 21 and 150°C, with all the specimens exhibiting cohesive failure in the cyanate-ester adhesive layer. The fracture energies of systems modified using a range of inorganic and thermoplastic particles, fibres and woven-mats have been measured, and scanning electron microscopy has been used to determine the toughening micromechanisms involved. Firstly, it is shown that the addition of 10% by weight of particulate modifiers can increase the fracture energy of the cyanate-ester polymer by 100%, due to a combination of toughening micromechanisms such as crack deflection, pinning and matrix cavitation around the second-phase particles. These experimental data have been compared to predictions from an analytical model. Secondly, it is demonstrated that the use of long fibres or woven-mats can give an a major increase in the value of the fracture energy, G _c, at initiation, and a further increase with increasing crack length, i.e. a significant R-curve effect is observed. At relatively long crack lengths, the measured fracture energy may be six times greater than that of the unmodified polymer value, due to fibres debonding and bridging across the fracture surfaces. Finally, it is shown that several of the physically-modified polymers developed in the present work have fracture energies that are greater than a typical commercially-available cyanate-ester film adhesive.     

Influence of root curvature on the fracture energy of adhesive layers

Previously performed experiments to study the mode I behavior of an adhesive layer revealed an apparent increase in the fracture toughness when the adherends deformed plastically. Attempts to simulate the experiments are made; both with elastically and plastically deforming adherends. Thus, effects of the size of the process zone and the deformation of the adherends are revealed. The adhesive layer is modeled using finite elements with different approaches; cohesive elements and representative volume elements. The adherends are modeled with solid elements. With a long process zone, all models give good results as compared to the experiments. However, only the model with representative volume elements gives good agreement for large root curvatures and correspondingly short process zones. The results are interpreted by analyzing the deformation and mechanisms of crack propagation in the representative volume elements. It is shown that with large root curvature of the adherends, the in-plane stretching of the adhesive layer gives a substantial contribution to the fracture energy. A simple formula is derived and shown to give an accurate prediction of the effects of the root curvature. This result indicates the limits of conventional cohesive zone modeling of an adhesive layer of finite thickness.     

Measurement of the fracture energy using three-point bend tests: Part 2—Influence of bulk energy dissipation

Avialable measures of the fracture energy G_F obtained with the procedure proposed by RILEM TC-50 provide values that appear to change with sample size, calling into question whether G_F can be considered as a material parameter. In a previous paper, possible sources of energy dissipation from the testing equipment and lateral supports were considered. In this paper new possible sources of energy dissipation in the sample, apart from the fracture crack itself, are considered. Such dissipation will take place inside the bulk of the most stressed regions of the specimen and, if it is not taken into account, higher values of G_F will be recorded than that strictly due to surface fracture energy. When this constribution and the possible energy dissipation analysed in previous work are considered, they are not enough to account for the measured size effect. If G_F is to be considered a material parameter, the evaluation of the results from the RILEM method should be analysed more carefully. In any case, the dissipated energy reported here represents a non-negligible amount of G_F and should be taken into account when performing measurements.     

Effect of adhesive thickness on fatigue and fracture of toughened epoxy joints – Part I: Experiments

The effect of bondline thickness, from 130 μm to 790 μm, on the fatigue and quasi-static fracture behavior of aluminum joints bonded using a toughened epoxy adhesive was studied experimentally under mode-I (DCB) and mixed-mode (ADCB) loading. Under mode-I loading, it was found that the fatigue threshold energy release rate, G_th, decreased for very thin bondlines, while under mixed-mode loading, the G_th changed very little with bondline thickness. In both cases, the effect of bondline thickness was more pronounced at higher crack growth rates. For quasi-static fracture, the effect of adhesive thickness on the energy release rate for the onset of fracture from the fatigue threshold, G_c0, was similar to that found for the fatigue threshold; however, the steady-state energy release rate, , increased linearly with increasing bondline thickness.     

Effect of bond angle on mixed-mode adhesive fracture

A maximum in mixed-mode adhesive fracture energy has been observed at bond angles of 45° using scarf-joint test specimens. It is shown here that by reducing the adherend surface roughness from 1.2m CLA roughness (milled surfaces) to 0.08m CLA roughness (polished surfaces) the fracture energy becomes a linear function of bond angle (no maximum at 45°) and there is an overall crease in fracture energy at all bond angles. These results are discussed in terms of crack initiation being focused into the interfacial region and a pinning of crack-tip shear displacements by the surface roughness of the milled adherends which does not occur for the polished adherends.