Simulation of hydrogen assisted stress corrosion cracking using the cohesive model

This paper investigates the effect of hydrogen diffusion on stable crack propagation by using numerical finite element simulations based on the cohesive model. The model with its two common parameters, cohesive strength, T₀, and critical separation, δ₀, and its two additional parameters for stress corrosion cracking, the effective diffusivity, D_eff, and a material parameter, α, which represents the reduction of the cohesive strength, is described. This model is then employed to predict the stable crack propagation in C(T) specimens made from a high strength structural steel which were tested under hydrogen charging conditions in rising displacement tests using various deformation rates. It is shown that, in general, the prediction of stable crack propagation is promising, but may be further improved by the use of a more sophisticated diffusion equation. Finally, the influence of variations of the effective diffusivity and the cohesive strength reduction on the thus simulated crack growth resistance curves is studied.     

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^®.     

Comparison of cohesive zone model and linear elastic fracture mechanics for a mode I crack near a compliant/stiff interface

Cohesive zone model has been widely applied to simulate crack growth along interfaces, but its application to crack growth perpendicularly across the interface is rare. In this paper, the cohesive zone model is applied to a crack perpendicularly approaching a compliant/stiff interface in a layered material model. One aim is to understand the differences between the cohesive zone model and linear elastic fracture mechanics in simulating mode I crack growth near a compliant/stiff interface. Another aim is to understand the effects of elastic modulus mismatch and cohesive strength of the stiff layer on the crack behavior near the interface. To simulate crack growth approaching an interface, the cohesive zone model which incorporates both the energy criterion and the strength criterion is an effective method.     

A procedure for superposing linear cohesive laws to represent multiple damage mechanisms in the fracture of composites

The relationships between a resistance curve (R-curve), the corresponding fracture process zone length, the shape of the traction-displacement softening law, and the propagation of fracture are examined in the context of the through-the-thickness fracture of composite laminates. A procedure for superposing linear cohesive laws to approximate an experimentally-determined R-curve is proposed. Simple equations are developed for determining the separation of the critical energy release rates and the strengths that define the independent contributions of each linear softening law in the superposition. The proposed procedure is demonstrated for the longitudinal fracture of a fiber-reinforced polymer-matrix composite. It is shown that the R-curve measured with a Compact Tension Specimen test cannot be predicted using a linear softening law, but can be reproduced by superposing two linear softening laws.     

Interfacial fracture characteristic and crack propagation of thermal barrier coatings under tensile conditions at elevated temperatures

Thermal barrier coatings (TBCs) have been extensively used in aircraft engines for improved durability and performance for more than fifteen years. In this paper, thermal barrier coating system with plasma sprayed zirconia bonded by a MCrAlY layer to SUS304 stainless steel substrate was performed under tensile tests at 1000°C. The crack nucleation, propagation behavior of the ceramic coatings in as received and oxidized conditions were observed by high-performance camera and discussed in detail. The relationship of the transverse crack numbers in the ceramic coating and tensile strain was recorded and used to describe crack propagation mechanism of thermal barrier coatings. It was found that the fracture/spallation locations of air plasma sprayed (APS) thermal barrier coating system mainly located within the ceramic coating close to the bond coat interface by scanning electron microscope (SEM) and energy dispersive X-Ray (EDX). The energy release rate and interface fracture toughness of APS TBCs system were evaluated by the aid of Suo–Hutchinson model. The calculations revealed that the energy release rate and fracture toughness ranged, respectively, from 22.15 J m⁻² to 37.8 J m⁻² and from 0.9 MPa m^1/2 to 1.5 MPa m^1/2. The results agree well with other experimental results.     

An embedded cohesive crack model for finite element analysis of brickwork masonry fracture

This paper presents a numerical procedure for fracture of brickwork masonry based on the strong discontinuity approach. The model is an extension of the cohesive model prepared by the authors for concrete, and takes into account the anisotropy of the material. A simple central-force model is used for the stress versus crack opening curve. The additional degrees of freedom defining the crack opening are determined at the crack level, thus avoiding the need of performing a static condensation at the element level. The need for a tracking algorithm is avoided by using a consistent procedure for the selection of the separated nodes. Such a model is then implemented into a commercial code by means of a user subroutine, consequently being contrasted with experimental results. Fracture properties of masonry are independently measured for two directions on the composed masonry, and then input in the numerical model. This numerical procedure accurately predicts the experimental mixed-mode fracture records for different orientations of the brick layers on masonry panels.     

A discrete cohesive model for fractal cracks

The fractal crack model described here incorporates the essential features of the fractal view of fracture, the basic concepts of the LEFM model, the concepts contained within the Barenblatt-Dugdale cohesive crack model and the quantized (discrete or finite) fracture mechanics assumptions proposed by Pugno and Ruoff [Pugno N, Ruoff RS. Quantized fracture mechanics. Philos Mag 2004;84(27):2829-45] and extended by Wnuk and Yavari [Wnuk MP, Yavari A. Discrete fractal fracture mechanics. Engng Fract Mech 2008;75(5):1127-42]. The well-known entities such as the stress intensity factor and the Barenblatt cohesion modulus, which is a measure of material toughness, have been re-defined to accommodate the fractal view of fracture.For very small cracks or as the degree of fractality increases, the characteristic length constant, related to the size of the cohesive zone is shown to substantially increase compared to the conventional solutions obtained from the cohesive crack model. In order to understand fracture occurring in real materials, whether brittle or ductile, it seems necessary to account for the enhancement of fracture energy, and therefore of material toughness, due to fractal and discrete nature of crack growth. These two features of any real material appear to be inherent defense mechanisms provided by Nature.     

Strain dependence electrical resistance and cohesive strength of ITO thin films deposited on electroactive polymer

Transparent, conducting, indium tin oxide (ITO) films have been deposited, by pulsed dc magnetron sputtering, on glass and electroactive polymer (poly(vinylidene fluoride)—PVDF) substrates. Samples have been prepared at room temperature by varying the oxygen partial pressure. Electrical resistivity around 8.4 × 10^− 4 Ω cm has been obtained for films deposited on glass, while a resistivity of 1.7 × 10^− 3 Ω cm has been attained in similar coatings on PVDF. Fragmentation tests were performed on PVDF substrates with thicknesses of 28 μm and 110 μm coated with 40 nm ITO layer. The coating's fragmentation process was analyzed and the crack onset strain and cohesive strength of ITO layers were evaluated.     

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.     

Continuum coupled cohesive zone elements for analysis of fracture in solid bodies

Embedding cohesive surfaces into finite element models is a widely used technique for the numerical simulation of material separation (i.e. crack propagation). Typically, a traction-separation law is specified that relates the magnitude of the cohesive traction to the distance between the separating surfaces. Thus the characterization of fracture in such models is not directly coupled to the bulk constitutive response, in the sense that the cohesive traction does not explicitly depend on material stretching in the plane of the fracture surface. In this work, an initially-rigid cohesive-traction formulation that is coupled to the surrounding continuum is introduced as a further development of the cohesive zone idea. In this model, the traction-separation law - and therefore the fracture phenomenology - derives directly from the bulk constitutive law. The immediate goal is an improved cohesive zone framework that naturally and logically initiates cohesive separation behavior, and couples its evolution to the material state in the region of the crack tip. A cohesive element based on this model is implemented in an explicit three-dimensional finite element code. Proof-of-concept analyses using both linear elastic and Gurson void growth constitutive relations are presented. A three-point bend simulation is found to give good agreement with experimental results.     

Crack propagation analyses with CTOA and cohesive model: Comparison and experimental validation

Two numerical models, namely an R-curve approach based on the crack tip opening angle (CTOA) and a cohesive model, are compared regarding their ability to predict ductile crack extension in thin aluminium sheets, which can be simulated under the assumption of plane stress. The experimental database is presented, the measuring techniques for the various quantities (optically and with clip gauges) are shown and the identification and validation of the respective model parameters are explained. A general concept for their identification is then derived for the case of thin walled structures under Mode I conditions.In order to investigate the performance of the models under different constraint conditions and the transferability of their parameters, C(T) specimens are used for parameter identification and M(T) specimens for validation. It is shown that for both models a single set of parameters describes the mechanical behaviour of both types of specimens. Cross-checking the two models, the crack tip opening angle is determined from the cohesive model calculations and compared with the experimental values.     

On the practical application of the cohesive model

The cohesive model has been formulated such that it can be used for practical application. A specific traction-separation law is proposed which is mainly given by the cohesive stress, T₀ and the cohesive energy, Γ₀. Experimental procedures have been developed which allow the determination of these material parameters. By means of experiments on three different materials and different specimen geometries it has been demonstrated that the proposed procedure provides very good predictions.     

Numerical prediction of slant fracture with continuum damage mechanics

Ductile specimens always exhibit an inclined fracture surface with an angle relative to the loading axis. This paper reports a numerical study on the cup-cone fracture mode in round bar tensile tests and the slant fracture in plane-strain specimens based on continuum damage mechanics. A combined implicit-explicit numerical scheme is first developed within ABAQUS through user defined material subroutines, in which the implicit solver: Standard, and the explicit solver: Explicit, are sequentially used to predict one single damage/fracture process. It is demonstrated that this numerical approach is able to significantly reduce computational cost for the simulation of fracture tests under quasi-static or low-rate loading. Comparison with various tensile tests on 2024-T351 aluminum alloy is made showing good correlations in terms of the load-displacement response and the fracture patterns. However, some differences exist in the prediction of the critical displacement to fracture.     

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.     

Triaxiality dependent cohesive zone model

A versatile cohesive zone model to predict ductile fracture at different states of stresses is proposed. The formulation developed for mode-I plane strain accounts for triaxiality of the stress-state explicitly by using basic elastic-plastic constitutive relations combined with two stress-state independent new model parameters. Comparison with available predictions of cohesive models based on porous plasticity damage models is used to demonstrate the efficacy of the proposed model. The effect of triaxiality on conventional cohesive parameters is well predicted as peak stresses are shown to increase while the cohesive energy decreases with triaxiality.     

The cohesive zone model: advantages, limitations and challenges

This paper reviews the cohesive process zone model, a general model which can deal with the nonlinear zone ahead of the crack tip--due to plasticity or microcracking--present in many materials. Furthermore, the cohesive zone model is able to adequately predict the behaviour of uncracked structures, including those with blunt notches, and not only the response of bodies with cracks--a usual drawback of most fracture models. The cohesive zone model, originally applied to concrete and cementitious composites, can be used with success for other materials. More powerful computer programs and better knowledge of material properties may widen its potential field of application. In this paper, the cohesive zone model is shown to provide good predictions for concrete and for different notched samples of a glassy polymer (PMMA) and some steels. The paper is structured in two main sections: First, the cohesive model is reviewed and emphasis is on determination of the softening function, an essential ingredient of the cohesive model, by inverse analysis procedures. The second section is devoted to some examples of the predictive capability of the cohesive zone model when applied to different materials; concrete, PMMA and steel.     

Determination of fracture energy and tensile cohesive strength in Mode I delamination of angle-ply laminated composites

The energy release rate in delamination of angle-ply laminated double cantilever composite beam specimens was calculated using the compliance equation, and interlaminar cohesive strengths were obtained. Instead of the traditional approach of a beam on an elastic foundation, a second-order shear-thickness deformation beam theory (SSTDBT) was considered. The equilibrium equations were obtained using the principle of minimum total potential energy and the system of ordinary differential equations were solved analytically. The problem was solved for [0°]₆ , [±30°]₅, and [±45°]₅ laminates with mid-plane delaminations and the results were verified using experimental evidence available in the literature.     

Constitutive models for cohesive zones in mixed-mode fracture of plain concrete

Discrete mixed-mode fracture (modes I and II) of plain concrete is investigated using a coupled and an uncoupled cohesive zone constitutive model in a finite element context. Fracture surfaces are confined to inter-element boundaries that are not necessarily coincident with the actual fracture surfaces. For this reason, traction components on the cohesive zone do not correspond to actual values either. In this work is demonstrated that only the coupled model is able to cope with these spurious traction components, that must decrease with crack opening. It is shown also that, in this regard, the key variable is the plastic potential adopted in the integration of tractions. Three mixed-mode fracture examples were tested in this work: a three-point single-edge notched beam, double-edge notched plates under variable lateral and normal deformation and four-point double-edge notched beams. A good fitting with experiments was obtained only for the coupled model. Mode II parameters can change in a large range without noticeable change in results, at least in the tested examples.     

Derivation of separation laws for cohesive models in the course of ductile fracture

The paper addresses the determination of the traction-separation law of the cohesive model on a micromechanical basis. For this task, a specific failure mechanism, i.e. ductile damage consisting of void nucleation, growth and coalescence, is investigated. An approach already described in the literature is to transfer the deformation behaviour of the simplest representative volume element, i.e. a single voided unit cell, to the cohesive interface. After reviewing the existing approach, its main drawback, namely that the unit cell contains both, deformation and damage of a material point whereas the cohesive model should contain the material separation only, is addressed. A new approach is presented, in which the behaviour of a unit cell is partitioned in its elasto-plastic deformation and damage, and only the damage contribution is applied as the traction-separation law for the cohesive model. Instead of modelling the voided unit cell, a single element with Gurson type plastic potential for the damage has been employed as a reference for the behaviour at the microscale. A study with fracture specimens, C(T) and M(T), made of an engineering Aluminium alloy shows that the new approach exhibits a better transferability than the existing one.