A numerical study of fatigue crack closure induced by plasticity

Crack closure delays the intrinsic mechanisms responsible for crack growth, therefore, it must be considered in fatigue crack growth modelling. The objective of this work is to develop a numerical procedure to predict crack closure induced by plasticity. First the crack closure was experimentally measured on M(T) 6082‐T6 aluminium alloy specimens of 3 mm thickness. A pin microgauge was used with the compliance technique. Then different parameters of the numerical procedure were analysed, namely the finite element mesh and the crack propagation scheme. The size of crack‐tip elements has an important influence and it is recommended to be of the same order of cyclic plastic zone. Crack‐opening levels only 10% lower than experimental results were obtained considering kinematic hardening and two load cycles in each increment.     

Plastic mismatch effect on plasticity induced crack closure: Fatigue crack propagation perpendicularly across a plastically mismatched interface

In the present work, comprehensive investigation of both theoretical analysis and numerical simulation was carried out to investigate the plastic mismatch effect on plasticity induced crack closure (PICC) behavior and effective fatigue crack tip driving force. During the process of crack tip approaching interface, crack tip load and crack tip load ratio will change, resulting in the change of PICC degree. When the crack propagates towards higher strength side, K_op/K_max increases; when the crack propagates towards lower strength side, K_op/K_max decreases firstly and then increases. The two mechanisms of “interface plastic mismatch effect on nominal fatigue crack tip driving force” and “interface plastic mismatch effect on PICC degree” were compared. The second mechanism must be considered when building crack tip driving force model for describing fatigue crack crossing plastically mismatched interface, because it is more physically factual and maybe more important than the first mechanism.     

Transient behaviour in the numerical analysis of plasticity induced crack closure

A transient behaviour is observed in the numerical analysis of plasticity induced crack closure at the beginning of crack propagation, as the residual plastic field is being formed. The extent of crack propagation prior to plasticity induced crack closure measurement has a major influence on the accuracy of numerical prediction and on computation time. The objective here is to quantify and understand the minimum propagation, Δa_stb, required to obtain stabilized crack opening values. For plane stress state, Δa_stb was found to increase with ΔK. Under plane strain conditions, a peak of closure exists at the beginning of crack propagation for relatively low ΔK values, which promotes relatively large transient periods. Two driving forces explain the stabilization behaviour, the formation of residual plastic wake and the stabilization of plastic strain, but the second seemed to control the phenomenon. Finally, two strategies are proposed to accelerate convergence. The first, consisting of a progressive increase of maximum load, is relevant in plane strain and 3D studies, in order to eliminate the initial peak. The second strategy consists of an extrapolation model and is very effective for plane stress conditions.     

Fatigue crack closure: a review of the physical phenomena

Plasticity‐induced, roughness‐induced and oxide‐induced crack closures are reviewed. Special attention is devoted to the physical origin, the consequences for the experimental determination and the prediction of the effective crack driving force for fatigue crack propagation. Plasticity‐induced crack closure under plane stress and plane strain conditions require, in principle, a different explanation; however, both types are predictable. This is even the case in the transition region from the plane strain to the plane stress state and all types of loading conditions including constant and variable amplitude loading, the short crack case or the transition from small‐scale to large‐scale yielding. In contrast, the prediction of roughness‐induced and oxide‐induced closures is not as straightforward.     

A parameter for quantitative analysis of plasticity induced crack closure

Numerical models have been successfully developed to predict plasticity induced crack closure (PICC). However, despite the large research effort a full understanding of the links between physical parameters, residual plastic wake and PICC has not been achieved yet. The plastic extension of material behind crack tip, Δy_p, obtained by the integration of vertical plastic deformation perpendicularly to crack flank, is proposed here to quantify the residual plastic field. The values of Δy_p and PICC were obtained numerically in a M(T) specimen using the finite element method. An excellent correlation was found between PICC and Δy_p which indicates that this parameter controls the phenomenon, and can be used to quantify the effect of physical parameters. An empirical model was developed to predict PICC assuming that the residual plastic field is a set of vertical plastic wedges, that the linear superposition principle applies and that the influence of a particular wedge exponentially decreases with distance to crack tip. The model was applied successfully to predict PICC for different residual plastic fields which provided an additional validation of Δy_p as the parameter controlling PICC.     

A numerical study of plasticity induced crack closure under plane strain conditions

The level of plasticity induced crack closure (PICC) is greatly affected by stress state. Under plane strain conditions, however, the level and even the existence of PICC still are controversial. The objective here is to study the influence of the main numerical parameters on plane strain PICC, namely the total crack propagation, the number of load cycles between crack increments, the finite element mesh and the parameter used to quantify PICC. The PICC predictions were included in a parallel numerical study of crack propagation, in order to quantify the impact of plane strain values on fatigue life. The results indicate that literature may be overestimating plane strain PICC due to incorrect numerical parameters. The number of load cycles usually considered is unrealistically small, and its increase was found to vanish crack closure, particularly for kinematic hardening. This effect was linked to the ratcheting effect observed at the crack tip. The total crack increment, Δa, must be large enough to obtain stabilized PICC values, but this may imply a huge numerical effort particularly for 3D models. The size of crack tip plastic zone may be overestimated in literature, which means that the meshes used may be too large. Additionally, the crack propagation study showed that the plane strain PICC has usually a dominant effect on fatigue life, and plane stress PICC is only relevant for relatively thin geometries.     

Numerical modelling of plane strain plasticity induced crack closure effects for bimaterial interfacial cracks

The effects of plane strain plasticity induced crack closure on fatigue cracks located at the interface of dissimilar steel materials are presented using finite element modelling. Based on the study, it has been observed that bimaterial cracks produced unsymmetrical residual plastic strains and crack profiles in the crack wakes. It is seen that Young’s modulus and yield stress mismatch have profound effects on the development of unsymmetrical residual plastic strain and crack profiles, whereas the effect of Poisson’s ratio is insignificant. However, it has been found that for the material properties considered, low value of crack closure levels have been identified.     

Effect of crack closure on non-linear crack tip parameters

Crack closure concept has been widely used to explain different issues of fatigue crack propagation. However, some authors have questioned the relevance of crack closure and have proposed alternative concepts. The main objective here is to check the effectiveness of crack closure concept by linking the contact of crack flanks with non-linear crack tip parameters. Accordingly, 3D-FE numerical models with and without contact were developed for a wide range of loading scenarios and the crack tip parameters usually linked to fatigue crack growth, namely range of cyclic plastic strain, crack tip opening displacement, size of reversed plastic zone and total plastic dissipation per cycle were investigated. It was demonstrated that: (i) LEFM concepts are applicable to the problem under study; (ii) the crack closure phenomenon has a great influence on crack tip parameters decreasing their values; (iii) the ΔK_eff concept is able to explain the variations of crack tip parameters produced by the contact of crack flanks; and (iv) the analysis of remote compliance is the best numerical parameter to quantify the crack opening level. Therefore the crack closure concept seems to be valid. Additionally, the curves of crack tip parameters against stress intensity factor range obtained without contact may be seen as master curves.     

Numerical simulation of plasticity induced crack closure: Identification and discussion of parameters

Numerical studies play a major role in the understanding and prediction of plasticity induced crack closure (PICC). However, the available numerical models can be considered simplifications of reality as they consider discrete crack propagations, relatively high fatigue crack growth rates (FCGR), sharp cracks, and propagation occurring at well-defined loads. Besides, there are a great number of numerical and physical parameters affecting the predictions of PICC. The aim of this paper is to discuss the numerical study of PICC. The numerical parameters affecting the accuracy of the numerical simulations, and the dependent parameters used to characterise the plastic wake and the closure level, are identified. The influence of the radial size of crack front elements and crack propagation is analysed. An extrapolation model is proposed, with excellent results. An intrinsic uncertainty is associated with the number of load cycles between crack increments and the definition of crack closure level. Finally, the effect of the stress ratio (R) on crack closure level is analysed.     

Empirical model for plasticity‐induced crack closure based on Kmax and ΔK

The mean stress has a significant effect on crack propagation life and must be included in prediction models. However, there is no consensus in the fatigue community regarding the dominant mechanism explaining the mean stress effect. The concept of crack closure has been widely used and several empirical models can be found in literature. The stress ratio, R, is usually the main parameter of these models, but present numerical results showed a significant influence of K_max. A new empirical model is therefore proposed here, dependent on K_max and ΔK, with four empirical constants. The model also includes the effect of material's yield stress, and two additional parameters were defined to account for stress state and crack closure parameter. A comparison was made with Kujawski's and Glinka's parameters, for a wide range of loading conditions. ΔK_eff lies between Kujawski's and Glinka's parameters, and some agreement is evident, although the concepts are quite different. The crack opening model was applied to literature results and was able to collapse da/dN–ΔK curves for different stress ratios to a single master curve.     

A numerical analysis of the mechanisms behind plasticity induced crack closure: Application to variable amplitude loadings

The effect of loading parameters on fatigue crack growth has been explained using the concept of crack closure. Plasticity induced crack closure (PICC) is linked to the crack tip plastic deformation, which becomes residual with crack propagation. The objective here is to identify the main mechanisms behind PICC, and for that different loading cases were considered namely overloads and load blocks. An analytical model was used to isolate the effect of residual plastic deformation on PICC, however significant differences were obtained relatively to finite element results. A second mechanism, which is crack tip blunting, was used to explain the transient behaviour observed after overloads and load blocks. For overloads and low–high load sequences there is a sudden increase of crack tip blunting with load increase which explains the sudden decrease of crack opening level. For high–low load sequences there is a sudden decrease of crack tip blunting which enhances the effect of residual plastic wake. Finally, the partial closure concept was tested looking to non-linear crack tip parameters but no evidences of Donald’s effect were found for the material studied.     

Plasticity induced crack closure in Middle‐Crack Tension specimen: numerical versus experimental

Numerical studies played a major role in the current understanding of plasticity induced crack closure (PICC), however it is well known that the numerical models simplify the reality by considering discrete crack propagations, relatively high fatigue crack growth rates, sharp cracks and that propagation occurs at a well‐defined load. The objective of this paper is to find if despite these limitations, the numerical predictions of PICC are close to the experimental results. Experimental work was developed to obtain crack opening values in 1‐mm‐thick middle‐tension (M(T)) specimens of AA6016‐T4 aluminium alloy. The opening level was determined from remote compliance data captured by a pin microgauge placed at the centre of the specimen. A numerical model was developed and optimized replicating the experimental test in terms of sample geometry, loading parameters, material behaviour and crack opening measurement technique. The adoption, in the numerical analysis, of a remote measurement technique for determining PICC was found to be robust and adequate for mesh refinement studies. A good agreement was found between experimental and numerical results.     

Application of the shadow optical method to fatigue and crack closure studies

In this paper it will be shown that crack closure and related crack shielding mechanisms can be studied successfully by applying the shadow optical method, also known as the method of caustics. It is shown that at least one mechanism - plasticity induced crack closure - can be identified and determined from measurements of the transverse diameter of the caustic in the crack tip region. For given fatigue conditions at which crack closure occurs, several experiments have been performed to investigate the effectiveness of the crack opening in a typical fatigue cycle.
Two different methods of closure determination have been used. A comparison is made between one of the most currently used methods of back face strain (BFS) compliance measurement and the shadow optical method (SOM). The underlying features of these two experimental techniques provide the key to finding the extent of the load in the fatigue cycles over which the crack is actually open. This redefines the value of K_min and the meaning of ΔK. The driving force, ΔK_eff is shown to be reduced in or near the fatigue threshold of the alloy, because crack closure is most effective in this part of fatigue loading. The introduction of SOM to fatigue crack closure provides a suitable alternative for finding the effective part of the stress intensity range between the minimum and maximum loads. Conclusions drawn from this work were that in addition to determining that part of the fatigue cycle over which the crack is actually open, the shadow optical method allowed an accurate interpretation of the entire fatigue cycle between K_max and K_min     

Numerical simulation of plasticity induced fatigue crack opening and closure for autofrettaged intersecting holes

The autofrettage of intersecting holes leads to extremely high compressive residual stress fields. These stresses in combination with the plastic deformations decelerate fatigue cracks initiated at the hole intersection notch. Simulations of plasticity induced crack closure of such cracks are presented based on the strip yield and a finite element model. The strip yield model has been extended to allow for an input of residual stresses coming from elsewhere, e.g. from a finite element calculation or measurements. The calculations are applied for constant as well as variable amplitude loading. The numerical expense of the finite element based modelling for variable amplitude loading is still too high if millions of cycles have to be considered. Therefore, a new approximation method is proposed introducing compensatory load sequences. Simulation results are compared to experimentally determined results showing good agreement. However, the accuracy of crack initiation life estimates has turned out to provide a high potential for further improvement.     

Some experimental observations on crack closure and crack-tip plasticity

This study presents a methodology for evaluating crack closure and the effect of crack-tip plasticity on stress intensity. Full-field displacement maps obtained by digital image correlation are used to obtain the mixed-mode, crack-driving force. The methodology allows the quantification of the effect of a range of contact phenomena: effects arising from interlocking, plastic deformation of crack face asperities and wedging generated as a consequence of sliding displacements of fatigue cracks have been identified. By evaluating the effective crack-tip stress intensity factor, crack opening levels can be quantified for both mode I and mode II. Moreover, the approach can take into account plasticity effects local to the crack in determining the stress intensity factor. All the information can be extracted in a non-contacting fashion with equipment that can be easily incorporated into industrial environments.     

Observations on evaluating fatigue crack closure from compliance traces

Fatigue crack closure generally has a large effect on crack growth rates, particularly in the threshold regime. Knowledge of closure values and mechanisms is important not only from a predictive viewpoint, but also in developing more fatigue-resistant materials through utilization of crack tip shielding. Closure measurement remains, however, an area of difficulty. This report considers two areas where problems can arise in compliance-based closure measurements: non-linearity of compliance traces due to plasticity and situations where the measured crack opening point does not appear to correspond with the effective crack opening point.     

Utilization of partial crack closure for fatigue crack growth modeling

Load ratio effects are of prime concern when modeling of fatigue crack growth (FCG) rate is required as a prerequisite for a reliable life prediction. The majority of research efforts regarding the load ratio effects are based on Elber's ΔK_eff approach. However, there are intrinsic difficulties encountered with its consistent application to FCG prediction. In this paper two popular crack-growth-life prediction codes FASTRAN and AFGROW are modified utilizing the enhanced partial crack closure model. The proposed utilization aggregates apparent closure mechanisms involved and demonstrates a better correlation and a significant scatter reduction of FCG data taken from literature, especially in the near-threshold region.     

A threshold fatigue crack closure model: Part I – model development

A fatigue crack closure model is developed that includes the effects of, and interactions between, the three closure mechanisms most likely to occur at threshold; plasticity, roughness, and oxide. This model, herein referred to as the CROP model (for Closure, Roughness, Oxide, and Plasticity), also includes the effects of out‐of‐plane cracking and multi‐axial loading. These features make the CROP closure model uniquely suited for, but not limited to, threshold applications. Rough cracks are idealized here as two‐dimensional sawtooths, whose geometry induces mixed‐mode crack‐tip stresses. Continuum mechanics and crack‐tip dislocation concepts are combined to relate crack face displacements to crack‐tip loads. Geometric criteria are used to determine closure loads from crack‐face displacements. Finite element results, used to verify model predictions, provide critical information about the locations where crack closure occurs. The CROP model is verified with experimental data in part II of this paper.     

The distributed dislocation technique for calculating plasticity-induced crack closure in plates of finite thickness

An analytical method for calculating plasticity-induced fatigue crack closure in plates of finite thickness is presented. The developed method utilizes the distributed dislocation technique (DDT) and Gauss-Chebyshev quadrature. Crack tip plasticity is incorporated by adopting a Dugdale type strip yield model. The finite plate thickness effects are taken into account by using a recently obtained three-dimensional solution for an edge dislocation in an infinite plate. Numerical results for the ratio of the size of the crack tip plasticity zones are presented for the cases of uniform thickness wake and linearly increasing wake for a range of plate thickness to crack length ratios and applied load ratios. The results show a very good agreement with previous analytical solutions in the limiting cases of very thick and very thin plates. Further results for the opening stress to maximum stress ratio are also provided and are compared with known three-dimensional finite element (FE) solutions. A good agreement is observed. The developed method is shown to be an effective and very powerful tool in modeling the crack closure phenomenon.     

Application of the distributed dislocation technique for calculating cyclic crack tip plasticity effects

This paper describes a method for modelling cyclic crack tip plasticity effects based on the distributed dislocation technique (DDT). A strip‐yield model is utilised to allow for the determination of the crack opening displacement, size of the plastic zones and in the case of a fatigue crack, the wake of plasticity. The DDT can be easily implemented for a wide range of cracked geometries with reliable control over the accuracy and convergence. Thickness effects can also be incorporated through a recently obtained solution for an edge dislocation in an infinite plate of finite thickness. Results for finite length cracks that have had limited growth, such that there is no plastic wake, are presented for a range of applied loads and R‐ratios. Further results are provided for a steady‐state fatigue crack in a plate of finite thickness. The present results are compared with analytical solutions and they show an excellent agreement.