Fatigue damage modelling of adhesively bonded joints under variable amplitude loading using a cohesive zone model

In this paper, the fatigue response of adhesively bonded joints under variable amplitude (VA) cyclic loading was predicted using a numerical model. The adhesive layer was modelled using the cohesive zone model with a bi-linear traction-separation response. A damage model, incorporating fatigue load ratio effects, was utilised in conjunction with the cohesive zone model to simulate the detrimental influence of VA fatigue loading. This model was validated against published experimental results obtained from fatigue tests of adhesively bonded single lap joints subjected to various types of VA fatigue loading spectra. This model successfully predicted the damaging effect of VA fatigue loading on the adhesively bonded joints and was generally found to be a significant improvement on the other damage models considered.     

Damage modelling of adhesively bonded joints

A cohesive zone model (CZM) has been used in conjunction with both elastic and elasto– plastic continuum behaviour to predict the response of a mixed mode flexure and three different lap shear joints, all manufactured with the same adhesive. It was found that, for a specific dissipated CZM energy (Γ₀) there was a range of CZM tripping tractions (σ_u) that gave a fairly constant failure load. A value of σ_u below this range gave rise to global damage throughout the bonded region before any crack propagation initiated. A value above this range gave rise to a discontinuous process zone, which resulted in failure loads that were strongly dependent on σ_u. A discontinuous process zone gives rise to mesh dependent results. The CZM parameters used in the predictions were determined from the experimental fracture mechanics specimen test data. When damage initiated, a deviation from the linear load–displacement curve was observed. The value for σ _uwas determined by identifying the magnitude that gave rise to the experimentally observed deviation. The CZM energy (Γ ₀) was then obtained by correlating the simulated load-crack length response with corresponding experimental data. The R-curve behaviour seen with increasing crack length was successfully simulated when adhesive plasticity was included in the constitutive model of the adhesive layer. This was also seen to enhance the prediction of the lap shear specimens. Excellent correlation was found between the experimental and predicted joint strengths.     

A procedure for the simulation of fatigue crack growth in adhesively bonded joints based on the cohesive zone model and different mixed-mode propagation criteria

This work deals with the simulation of the fatigue crack growth (FCG) in bonded joints. In particular a cohesive damage model is implemented in the commercial software Abaqus, in order to take into account for the damage produced by fatigue loading. The crack growth rate is evaluated with different Paris-like power laws expressed in terms of strain energy release rate. The crack growth rate is then translated into a variation of the damage distribution over the cohesive zone setting an equivalence between the increment of crack length and the increment of damage. The model takes also into account mixed mode I/II conditions. In this work the validity of the model is tested by comparison with theoretical trends for conditions of pure mode I, pure mode II and mixed mode loading. In the case of mixed mode conditions, different models are implemented for the crack growth rate computation. The results of the model are in very good agreement with the expected trends, therefore the model is adequate to simulate the fatigue crack growth behaviour of bonded joint.     

Time-dependent cohesive zone modelling for discrete fracture simulation

The cohesive crack tip model became very popular in fracture and failure mechanics, starting with the original publications of (Dugdale, 1960) [1] and (Barenblatt, 1962) [2] and first practical applications in the early 1980s. A compact representation of the fracture mechanical basis, kinematic and constitutive issues as well as some special characteristics of the related finite element formulation are given in the first part of this contribution. The main part is dedicated to the presentation and discussion of recent developments of cohesive finite element methods for time-dependent fracture. On the basis of a rheological model assumption, a novel viscoelastic extension for cohesive traction separation laws is presented and the resultant characteristic behaviour is depicted and compared for different loading conditions. Adopting an industrial application of a peel foil specimen, the time-dependent characteristics as well as some aspects of parameter identification and application of the material model are shown.     

Assessment of debond simulation and cohesive zone length in a bonded composite joint

Finite element methods combined with cohesive elements were used to simulate progressive failure behaviour in a bonded double cantilever beam configuration. The introduced cohesive zone was represented by three cases. Responses of both global load–displacement and local cohesive traction–separation were investigated. An unexpected finding was that the overall cohesive traction stiffness was much less than the assumed input value. In addition, the local nodal separation moment was identified. Consequently, correct cohesive zone lengths were obtained using the extracted traction profile along the cohesive zone path at this moment. Information of the global load–displacement profile, traction stiffness, and cohesive zone length induced by the three zone cases was explored. Moreover, the study can explain why very small cohesive zone lengths are generated numerically, as compared to theoretical solutions. Recommendations on the application of the numerical model with cohesive elements to practical experimental analysis were suggested.     

Mixed-mode cohesive-zone models for fracture of an adhesively bonded polymer-matrix composite

As a direct extension of previous mode-I work on the adhesion of composite joints, this paper uses a cohesive-zone approach to model the mixed-mode fracture of adhesive joints made from a polymer-matrix composite. Mode-II cohesive-zone parameters were obtained using sandwich end-notch flexure specimens. These parameters were used directly with the previously determined mode-I parameters to predict the fracture and deformation of mixed-mode geometries. It was shown that numerical simulations provided quantitative predictions for these geometries, including predictions for both the strengths of the joints and for the failure mechanisms. In conjunction with the earlier work, these results demonstrate the use of cohesive-zone approaches for the design of adhesively bonded composite joints, and indicate approaches for determining the relevant material properties to describe mixed-mode fracture.     

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^®.     

Strength analysis of metallic bonded joints containing defects

Finite element simulation has been utilized to study the overall strength of metallic single lap joints with defects in their adhesive layers. Three types of defects are taken into account respectively, which are local debonding, weak bonding and void. For the first two types of defects, a developed numerical method using the cohesive zone model modified by user-defined subroutines is carried out as to consider the influences of the defect size and location. Furthermore, a modified-Gurson model is employed to simulate the adhesive layer with voids, considering the influence of the void size. The results show that the overall strength of the joints diminishes as the defect size is increased. Especially, the adhesive fracture properties and the size of the weak bonding region have combined influences on the strength of the joints.     

A parametric shear damage evolution model for combined clamped and adhesively bonded interfaces

This paper presents the development of a damage model for shear decohesion analysis of combined clamped and adhesively bonded (hybrid) interfaces. The model takes into account the influence of normal pressure on the applied shear force vs. displacement response. Decohesion finite elements were used for the setup of the cohesive zone and a damage function was set to govern the degradation. Material constants for the damage evolution model were fitted based on experiments and computational results were validated against the test data. Considerable increase in the dissipated fracture energy was found for the cases involving high normal pressure.     

Influence of hydrogen from cathodic protection on the fracture susceptibility of 25%Cr duplex stainless steel - Constant load SENT testing and FE-modelling using hydrogen influenced cohesive zone elements

Laboratory experiments and cohesive zone simulation of Hydrogen Induced Stress Cracking in SENT tests specimens of 25%Cr duplex stainless steel have been performed. A polynomial formulation of the traction separation law and hydrogen dependent critical stress was applied. Best agreement with the experiments was found for an initial critical stress of 2200 MPa and a critical separation of 0.005 mm. Proposed threshold stress intensity factor and lower bound net section stress is 20 MPa√m and 480 MPa. High crack growth rates and typical hydrogen influenced fracture topography suggest large influence of the stress and strain in the fracture process zone on the hydrogen diffusion rate.     

Rate effects in mode-II fracture of plastically deforming,adhesively bonded structures

Results from a combined experimental and numerical investigation into the effects of rate on mode-II fracture of a plastically deforming, adhesively bonded joint are presented. It is shown that a cohesive-zone model has to be modified to include coupling between normal and shear modes of deformation when there is extensive shear deformation of the adhesive layer. A suitable cohesive-zone modeling strategy is described, and the mode-II cohesive parameters determined from the model are presented as a function of loading rate. Previous studies of the same system showed that the effects of rate in mode-I were limited to the probability that a crack growing in a toughened quasi-static mode would spontaneously make a transition to a brittle mode of fracture. No such transitions were found for mode-II fracture. Crack growth always occurred in a quasi-static fashion. While there was some evidence that rate might affect the mode-II fracture parameters, these effects were very limited even up to crack velocities of about 1,000 mm/s. Any possible effects was limited to a very minor increase in toughness and strength with increased loading rates. However, the magnitude of these possible increases were comparable to the magnitude of the uncertainties in the measured values.     

Numerical evaluation for debonding failure phenomenon of adhesively bonded joints at cryogenic temperatures

In the present study, material characteristics, such as inelastic constitutive behaviour and debonding failure, of an adhesively bonded joint (ABJ) at cryogenic temperature have been evaluated using a computational approach. The modified Bodner-Partom model (BP model) has been introduced to describe the material nonlinearities of ABJ. The Gurson-Tvergaard model (GT model) has also been implemented into the constitutive model in order to analyse the phenomenon of debonding failure. An ABAQUS user-defined subroutine UMAT is developed using a damage-coupled constitutive model based on an implicit formulation. The numerical results are compared with a series of lap shear tests of ABJ at cryogenic temperature in order to verify the proposed method.     

A single-domain dual-boundary-element formulation incorporating a cohesive zone model for elastostatic cracks

A cracked elastostatic structure is artificially divided into subdomains of simpler topology such that the well-developed classic dual integral equations can be applied appropriately to each domain. Applying the continuity and equilibrium conditions along artificial boundaries and properties of the integral kernels a single-domain dual-boundary-integral equation formulation is derived for a cracked elastic structure. A cohesive zone model is used to model the crack tip processes and is coupled with the single-domain dual-boundary-integral equation formulation; the resulting nonlinear equations are solved using the iterative method of successive-over-relaxation. The constitutive law used for a crack includes three parts: a law relating cohesive force to crack displacement difference when a crack is opening, a characterization of tangential interaction between crack surfaces when the crack surfaces are in contact, and a maximum principal stress criterion of crack advance. Incorporation of local unloading effect of the cohesive zone material has enabled a simulation of fracture with initial damage, partial development of the failure process zone at structural instability and multiple crack interaction. Some of the features of the method are demonstrated by considering three examples. The first problem is a single-edge-cracked specimen that exhibits a snap-back instability. The second example is the development of wing cracks from an angled crack under compression. The last example demonstrates the capability to consider mixed-mode crack growth and interaction of cracks. Thus, the problem of crack growth has been reduced to the determination of the cohesive model for the fracture process. This revised version was published online in July 2006 with corrections to the Cover Date.     

Formulation of a cohesive zone model for a Mode III crack

Cohesive zone models have been proven effective in modeling crack initiation and propagation phenomena. In this work, a possible form for a Mode III cohesive zone model is formulated from elastic stress and displacement fields around a crack with a cohesive zone ahead of the crack tip. A traction-separation relation for the model is derived as a direct consequence of the formulation, which establishes some intrinsic connections between properties of the cohesive zone and those of the bulk material. Interestingly, this model states that the von Mises effective stress in the cohesive zone is constant, which may be related to the bulk material’s yield stress and is consistent with the assumption made in conventional strip-yield elastic-plastic solutions.     

An irreversible cohesive zone model for interface fatigue crack growth simulation

Fatigue crack growth (FCG) along an interface is studied. Instead of using the Paris equation, the actual process of material separation during FCG is described by the use of an irreversible constitutive equation for the cyclic interface traction-separation behavior within the cohesive zone model (CZM) approach. In contrast to past development of CZMs, the traction-separation behavior does not follows a predefined path. The model definition, its predicted cyclic material separation behavior and application to a numerical study of interface FCG in double-cantilever beam, end-loaded split and mixed-mode beam specimens are reported.     

The use of a cohesive zone model to study the fracture of fibre composites and adhesively-bonded joints

Analytical solutions for beam specimens used in fracture-mechanics testing of composites and adhesively-bonded joints typically use a beam on an elastic foundation model which assumes that a non-infinite, linear-elastic stiffness exists for the beam on the elastic foundation in the region ahead of the crack tip. Such an approach therefore assumes an elastic-stiffness model but without the need to assume a critical, limiting value of the stress, _max, for the crack tip region. Hence, they yield a single fracture parameter, namely the fracture energy, G _c. However, the corresponding value of _max that results can, of course, be calculated from knowledge of the value of G _c. On the other hand, fracture models and criteria have been developed which are based on the approach that two parameters exist to describe the fracture process: namely G _c and _max. Here _max is assumed to be a critical, limiting maximum value of the stress in the damage zone ahead of the crack and is often assumed to have some physical significance. A general representation of the two-parameter failure criteria approach is that of the cohesive zone model (CZM). In the present paper, the two-parameter CZM approach has been coupled mainly with finite-element analysis (FEA) methods. The main aims of the present work are to explore whether the value of _max has a unique value for a given problem and whether any physical significance can be ascribed to this parameter. In some instances, both FEA and analytical methods are used to provide a useful crosscheck of the two different approaches and the two different analysis methods.     

A new cohesive zone model for mixed mode interface fracture in bimaterials

A new two-dimensional cohesive zone model which is suitable for the prediction of mixed mode interface fracture in bimaterials is presented. The model accounts for the well known fact that the interfacial fracture toughness is not a constant, but a function of the mode mixity. Within the framework of this model, the cohesive energy and the cohesive strength are not chosen to be constant, but rather functions of the mode mixity. A polynomial cohesive zone model is derived in light of analytical and experimental observations of interface cracks. The validity of the new cohesive law is examined by analyzing double cantilever beam and Brazilian disk specimens. The methodology to determine the parameters of the model is outlined and a failure criterion for a pair of ceramic clays is suggested.