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.     

Comparison of using the crack mouth displacement (CMOD) and load line displacement (LLD) methods in the determination of critical J integral in SENB specimens

Various methods for direct and indirect determination of LLD and CMOD were used to determine J from SENB specimens in three different steels. The influence of the displacement measurement on J is discussed, and shows that the values of J using LLD determined from clip gauge methods to the ASTM E1820 or ISO 12135 standards are consistent with values of J determined from CMOD (either directly or using clip gauge methods), as defined in ASTM E1820. From this work it is recommended that standard methods such as ISO 12135 should permit load‐CMOD and load‐LLD as alternative methods to determine J. Methods to determine LLD by corrections to the ram displacement were also shown to be effective in determining J, for applications where the use of clip gauges may be challenging, such as fracture toughness testing in sour environments, dynamic tests, or testing at very high temperature.     

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.     

A numerical study of crack shielding and deflection under extensive plasticity

Experimentally observed crack deflection events in multi-layered material systems, occurring even under pure mode-I loading, are here simulated and explained through elasto-plastic finite element modelling. The crack tip opening displacement is adopted as the crack driving force and estimated along crack paths whose deflection is predicted using the maximum tangential strain criterion. Shielding conditions that promote deflection and bifurcation are evaluated. It is shown that, under conditions of extended plasticity, CTOD evolution as a crack approaches an interface can reveal crack shielding and amplification, and that crack deflection and growth can be assessed from the variation of tangential strains around the crack tip.     

Elastic–plastic finite element analysis of the effect of compressive loading on crack tip parameters and its impact on fatigue crack propagation rate

The effects of applied compressive stress on the crack tip parameters and their implications on fatigue crack growth have been studied. Four center-cracked panel specimens with different crack lengths are analysed using finite element method. The results show that under tension–compression loading the local crack tip parameters are determined by two loading parameters. The two loading parameters are the maximum stress intensity and the maximum compressive stress in the applied stress cycle. Predictions of fatigue crack propagation behaviour based on the maximum stress intensity and maximum compressive stress agree well with experimental observations.     

Effect of underloads or overloads in fatigue crack growth by crack-tip blunting

Crack-tip blunting under tensile loads and re-sharpening of the crack-tip during unloading is one of the basic mechanisms for fatigue crack growth in ductile metals. Here, based on an elastic-perfectly plastic material model, crack growth computations are continued up to 700 full cycles by using re-meshing at several stages of the plastic deformation. A compressive underload in one of the cycles tends to increase the rate of cyclic crack growth, and this effect is studied in detail for a single underload, based on the blunting re-sharpening mechanism. Subsequently, the increased rate of crack growth due to periodically occurring underloads is analysed. A single overload has the opposite effect of giving a significant delay in the subsequent fatigue crack growth. An analysis is carried out to compare the effect of a small overload to that of a larger overload.     

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.     

Correlation between crack length and load drop for low-cycle fatigue crack growth in Ti-6242

For low-cycle fatigue tests with smooth bars the number of cycles to initiation is commonly defined from a measured relative drop in maximum load. This criterion cannot be directly related to the crack length, which is the actual measure of interest. In order to establish a relation between load drop and crack length for the high strength titanium alloy Ti-6242, this investigation compares data from controlled low-cycle fatigue crack growth tests and numerical simulations of these tests. To achieve sufficient accuracy in this relation, focus is given to modelling of mean stress relaxation. Three constitutive models, the Chaboche, the Ohno–Wang and the Chaboche with threshold, are evaluated with respect to experiments. Furthermore, a straightforward method with cycle-scaling of the material parameters are used to efficiently reduce calculation cost. It is shown that it is possible to determine the relationship between load drop and crack length from numerical simulations, provided that care is taken to relevant aspects of the materials stress–strain response. These results are also used to numerically evaluate the effect on load drop of the extensometer position relative to the crack.     

A crystal plasticity study of cyclic constitutive behaviour, crack-tip deformation and crack-growth path for a polycrystalline nickel-based superalloy

Crystal plasticity has been applied to model the cyclic constitutive behaviour of a polycrystalline nickel-based superalloy at elevated temperature using finite element analyses. A representative volume element, consisting of randomly oriented grains, was considered for the finite element analyses under periodic boundary constraints. Strain-controlled cyclic test data at 650 °C were used to determine the model parameters from a fitting process, where three loading rates were considered. Model simulations are in good agreement with the experimental results for stress–strain loops, cyclic hardening behaviour and stress relaxation behaviour. Stress and strain distributions within the representative volume element are of heterogeneous nature due to the orientation mismatch between neighbouring grains. Stress concentrations tend to occur within “hard” grains while strain concentrations tend to locate within “soft” grains, depending on the orientation of grains with respect to the loading direction. The model was further applied to study the near-tip deformation of a transgranular crack in a compact tension specimen using a submodelling technique. Grain microstructure is shown to have an influence on the von Mises stress distribution near the crack tip, and the gain texture heterogeneity disturbs the well-known butterfly shape obtained from the viscoplasticity analysis at continuum level. The stress–strain response near the crack tip, as well as the accumulated shear deformation along slip system, is influenced by the orientation of the grain at the crack tip, which might dictate the subsequent crack growth through grains. Individual slip systems near the crack tip tend to have different amounts of accumulated shear deformation, which was utilised as a criterion to predict the crack growth path.     

Prediction of energy release rates for crack growth using FEM-based energy derivative technique

This paper presents a new method, named energy derivative technique, to calculate energy release rate for a variety of crack growth scenarios. The new method is based on energy conservation principle for crack growth, and is applicable to crack development in any quasi-static condition in which dynamic energy for crack growth is negligible. The method has the advantage over existing finite element-based methods in that the former does not require an elaborate fine mesh in the vicinity of a crack tip, and is not limited to linear deformation behaviour. Several case studies are presented to demonstrate validity of the method, which are (i) growth of penny-shaped crack for linear elastic fracture behaviour, (ii) crack growth in rubber sheet under tension for nonlinear elastic fracture behaviour, (iii) delamination in end-notched flexure specimen with friction, and (iv) crack growth with plastic deformation in double-edge-notched plate under tension. Results from these case studies show excellent agreement with data available in the literature, which were determined using either analytical or other FEM-based techniques.     

Determination of SIFs, crack-tip coordinates and crack angle of anisotropic materials

This paper investigates the accuracy of a numerical method incorporating an image-correlation experiment for finding the crack-tip coordinates, the axes of the measurement and the mixed-mode stress intensity factors (SIFs) of anisotropic materials. Finite element and experimental simulations in this paper indicate that the proposed linear search method incorporating Powell's method is considerably stable for finding the unknown crack-tip coordinates and measurement axes, even though the initial input is far away from the actual data. Moreover, the displacement field very near the crack tip is not necessary. In the experimental validation, the maximum SIF error of the image-correlation experiment is about 9%, which should be acceptable for the mixed-mode fracture problems.     

Analysis of crack tip plasticity for microstructurally small cracks using crystal plasticity theory

Crack tip plastic zone sizes and crack tip opening displacements (CTOD) for stationary microstructurally small cracks are calculated using the finite element method. To simulate the plastic deformation occurring at the crack tip, a two-dimensional small strain constitutive relationship from single crystal plasticity theory is implemented in the finite element code ANSYS as a user-defined plasticity subroutine. Small cracks are modeled in both single grains and multiple grains, and different crystallographic conditions are considered. The computed plastic zone sizes and CTOD are compared with those found using conventional isotropic plasticity theory, and significant differences are observed.     

Effect of anisotropic plasticity on mixed mode interface crack growth

Crack growth along an interface between a solid with plastic anisotropy and an elastic substrate is modelled by representing the fracture process in terms of a traction-separation law specified on a crack plane. A phenomenological elastic-viscoplastic material model is applied, using one of two different anisotropic yield criteria to account for the plastic anisotropy. Conditions of small-scale yielding are assumed, and due to the mismatch of elastic properties across the interface the corresponding oscillating stress singularity field is applied as boundary conditions on the outer edge of the region analyzed. Crack growth resistance curves are calculated numerically, and based on these results the dependence of the steady-state fracture toughness on the near-tip mode mixity is determined. Different initial orientations of the principal axes relative to the interface are considered and it is found that the steady-state fracture toughness is quite sensitive to this orientation of the anisotropy.     

Effect of overload on fatigue crack retardation of aerospace Al-alloy laser welds using crack-tip plasticity analysis

Investigations on fatigue crack growth retardation due to single tensile and periodic multiple over load in strength undermatched laser beam welded 3.2 mm thick aerospace grade aluminium alloy 2139-T8 sheets are conducted. The effect of overload on the fatigue crack propagation behaviours of the homogenous base metal and welded panels (200 mm wide, centre cracked) was compared using experimental and FE analysis methods. The effective crack tip plasticity has been determined in homogeneous M(T) specimens using Irwin’s method and in both homogeneous and laser welded specimen by calculating crack tip plastic strain using FE analysis for single tensile overload. The crack retardation due to the overload in welded specimens is described by the Wheeler Model. The crack tip plastic zone size in the welded specimen was determined by FE analysis using maximum plastic zone extension at the mid sheet thickness. The results show that the Wheeler Model can be implemented to the highly heterogeneous undermatched weld to describe the crack retardation in fatigue following single tensile overload. Fatigue crack growth retardation due to single overload is found to be larger than the base metal. However, after periodic multiple overload, shorter crack retardation has occurred for undermatched welds than the base metal.     

Coupling of the meshfree and finite element methods for determination of the crack tip fields

This paper develops a new concurrent simulation technique to couple the meshfree method with the finite element method (FEM) for the analysis of crack tip fields. In the sub-domain around a crack tip, we applied a weak-form based meshfree method using the moving least squares approximation augmented with the enriched basis functions, but in the other sub-domains far away from the crack tip, we employed the finite element method. The transition from the meshfree to the finite element (FE) domains was realized by a transition (or bridge region) that can be discretized by transition particles, which are independent of both the meshfree nodes and the FE nodes. A Lagrange multiplier method was used to ensure the compatibility of displacements and their gradients in the transition region. Numerical examples showed that the present method is very accurate and stable, and has a promising potential for the analyses of more complicated cracking problems, as this numerical technique can take advantages of both the meshfree method and FEM but at the same time can overcome their shortcomings.     

Comparison of virtual crack extension and strain energy density methods applied to contact surface crack growth

The paper is concerned with comparison of two crack propagation methods applied to a two-dimensional computational model of the surface initiated crack growth in the lubricated contact area of meshing gears. The virtual crack extension method and the minimum strain energy density criterion are used for simulation of the crack propagation in the framework of the finite element analysis. The discretised equivalent contact model, with the assumed size and orientation of the initial crack, is subjected to contact loading conditions, accounting for the elasto-hydro-dynamic lubrication effects, tangential loading due to sliding and the influence of lubricating fluid, driven into the crack by hydraulic mechanism. The computational results show that both crack propagation methods give comparable results, although the virtual crack extension method has some clear advantages due to its theoretical superiority in dealing with mixed-mode short crack propagation close to the loading boundary.     

Numerical and experimental analysis of residual stresses for fatigue crack growth

In this paper computational and experimental results are presented concerning residual stress effects on fatigue crack growth in a Compact Tension Shear (CTS) specimen under cyclic mode I loading. For a crack of constant length it is found that hardly any compressive residual stresses or crack closure effects are generated along the crack surfaces behind the crack tip through the considered cyclic mode I loading with a load ratio of R=0.1. Only if fatigue crack growth is modelled during the simulation of the cyclic loading process these well-known effects are found. On the other hand it is shown that they have hardly any influence on the residual stresses ahead of the crack tip and thus on further fatigue crack growth. For all cases considered the computational finite element results agree well with the experimental findings obtained through X-ray diffraction techniques.     

Combined plasticity and creep analysis in 2D by means of the boundary element method

This work presents a formulation to make a combined analysis of plasticity and creep in two-dimensional (2D) using the Boundary Element Method. This new approach is developed to combine the constitutive equation for time hardening creep and the constitutive equation for plasticity, the latter based on the Von Misses criterion and the Prandtl–Reuss flow. The implementation of creep strain in the formulation is achieved through domain integrals. The creep phenomenon takes place in the domain which is discretized into quadratic quadrilateral continuous and discontinuous cells. The creep analysis is applied to metals with a power law creep for the secondary creep stage. The results, obtained with reference to three models, show a good agreement when compared to those published in the literature. This finding shows that the Boundary Element Method is a suitable tool to deal with combined nonlinear problems.     

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.     

Effect of electric boundary conditions on crack energy density and its derivatives for piezoelectric material

This paper deals with crack energy density (hereafter CED) as a possible fracture parameter in piezoelectricity under arbitrary electric boundary conditions on a notch-like crack surface. The definitions of CED and its derivatives are given first under exact boundary condition. Next, their path independent integrals are also derived and their approximate expressions are discussed under some restrictions on the crack surfaces. It is found that electrical terms along the notch-like crack surface do not vanish unlike in the case of impermeable crack. Then, we introduce evaluation methods of CED, and, with the help of the results of finite element analyses (FEA), we closely examine how electric boundary conditions along the notch surface and initial notch width influence CED and its derivatives. It is shown from the FEA results that because of the difficulties of computing path integral terms along the notch-like crack tip in the path independent expressions, the evaluation by the definitions of CED and its derivatives is preferable and more convenient than the evaluation of their path independent expressions. It is also found that all the parameters are significantly affected by both permittivity inside the electric inclusion and root radius of the notch. Finally, the possibility of mechanical CED as a governing fracture parameter is discussed.