Short fatigue crack growth behavior under mixed-mode loading

Mixed-mode loading represents the true loading condition in many practical situations. In addition, most of the fatigue life of many components is often spent in the short crack growth stage. The study of short crack growth behavior under mixed-mode loading has, therefore, much practical significance. This work investigated short crack growth behavior under mixed-mode loading using a common medium carbon steel. The effects of load mixity, crack closure, and load ratio on short crack growth behavior were evaluated by conducting experiments using four-point bending specimens with several initial K _II /K _I mixed-mode ratios and two load ratios. Cracks were observed to grow along the paths with very small K _II /K _I ratios (i.e. mode I). The maximum tangential stress criterion was used to predict the crack growth paths and the predictions were found to be close to the experimental observations. Several parameters including equivalent stress intensity factor range and effective stress intensity factor range were used to correlate short crack growth rates under mixed-mode loading. Threshold values for short cracks were found to be lower than those for long cracks for all the mixed-mode loading conditions. Crack closure was observed for the entire crack length regime with all load mixity conditions at R ≈ 0.05 and for short crack regime under high load mixity condition at R = 0.5. Several models were used to describe mean stress effects and to correlate crack growth rate data.     

History effect in fatigue crack growth under mixed-mode loading conditions

Plastic deformation within the crack tip region introduces internal stresses that modify subsequent behaviour of the crack and are at the origin of history effects in fatigue crack growth. Consequently, fatigue crack growth models should include plasticity-induced history effects. A model was developed and validated for mode I fatigue crack growth under variable amplitude loading conditions. The purpose of this study was to extend this model to mixed-mode loading conditions. Finite element analyses are commonly employed to model crack tip plasticity and were shown to give very satisfactory results. However, if millions of cycles need to be modelled to predict the fatigue behaviour of an industrial component, the finite element method becomes computationally too expensive. By employing a multiscale approach, the local results of FE computations can be brought to the global scale. This approach consists of partitioning the velocity field at the crack tip into plastic and elastic parts. Each part is partitioned into mode I and mode II components, and finally each component is the product of a reference spatial field and an intensity factor. The intensity factor of the mode I and mode II plastic parts of the velocity fields, denoted by dρ_I/dt and dρ_II/dt, allow measuring mixed-mode plasticity in the crack tip region at the global scale. Evolutions of dρ_I/dt and dρ_II/dt, generated using the FE method for various loading histories, enable the identification of an empirical cyclic elastic–plastic constitutive model for the crack tip region at the global scale. Once identified, this empirical model can be employed, with no need of additional FE computations, resulting in faster computations. With the additional hypothesis that the fatigue crack growth rate and direction can be determined from mixed-mode crack tip plasticity (dρ_I/dt and dρ_II/dt), it becomes possible to predict fatigue crack growth under I/II mixed-mode and variable amplitude loading conditions. To compare the predictions of this model with experiments, an asymmetric four point bend test system was setup. It allows applying any mixed-mode loading case from a pure mode I condition to a pure mode II. Initial experimental results showed an increase of the mode I fatigue crack growth rate after the application of a set of mode II overload cycles.     

Fatigue crack growth of a railway wheel

Typically, fatigue crack propagation in railway wheels is initiated at some subsurface defect and occurs under mixed mode (I–II) conditions. For a Spanish AVE train wheel, fatigue crack growth characterization of the steel in mode I, mixed mode I–II, and evaluation of crack path starting from an assumed flaw are presented and discussed.Mode I fatigue crack growth rate measurement were performed in compact tension C(T) specimens according to the ASTM E647 standard. Three different load ratios were used, and fatigue crack growth thresholds were determined according to two different procedures. Load shedding and constant maximum stress intensity factor with increasing load ratio R were used for evaluation of fatigue crack growth threshold.To model a crack growth scenario in a railway wheel, mixed mode I–II fatigue crack growth tests were performed using CTS specimens. Fatigue crack growth rates and propagation direction of a crack subjected to mixed mode loading were measured. A finite element analysis was performed in order to obtain the K_I and K_II values for the tested loading angles. The crack propagation direction for the tested mixed mode loading conditions was experimentally measured and numerically calculated, and the obtained results were then compared in order to validate the used numerical techniques.The modelled crack growth, up to final fracture in the wheel, is consistent with the expectation for the type of initial damage considered.     

Finite element analysis of a subsurface crack

The problem of a subsurface crack parallel to the surface of a half space was studied by the finite element method. Without using the interface or gap elements over the crack faces, the crack faces would penetrate into each other for the traction-free boundary condition under shear loading, which is physically impossible. Using the gap elements, this problem was avoided, and a contact zone was observed near one crack tip. The size of the contact zone decreases but the maximum contact pressure at the closed crack tip increases as the crack approaches the surface. For tensile and shear loadings, both K _I (mode I stress intensity factor) and K _II (mode II stress intensity factor) increase as the crack approaches the surface. For shear loading there is no K _I at the closed tip and the K _I and K _II at the open tip are comparable as the crack approaches the surface.     

Fatigue crack growth under non-proportional mixed-mode loading in ferritic-pearlitic steel

Mode II fatigue crack growth tests as well as tests in sequential mode I and then mode II were performed on ferritic‐pearlitic steel. For ΔK_II ranging from 7 to , bifurcation occurs after 12–450 μm of coplanar growth at a decreasing speed. By contrast, hundreds of micrometres of constant speed coplanar growth were obtained under sequential mode I and then mode II loading, for and ΔK_I ranging from 0.25 to 1.0 ΔK_II . The crack growth rate is a simple sum of the contributions of each mode for ΔK_I= 0.25 ΔK_II but above this value a synergetic effect is found. The mechanism of this fast‐propagation mode is discussed in the light of strain range maps ahead of the crack tip obtained by digital SEM image correlation and elastic–plastic finite element calculations. The stability of the crack path according to the maximum growth rate criterion is demonstrated.     

T-stress solution for DCDC specimens

A finite element study is performed in order to determine the T-stress term for DCDC specimens with holes set off from the specimen centre line. In addition, mixed-mode stress intensity factors K_I and K_II are given and the biaxiality ratio β is computed from T and K_I. The T-stress results are fitted and presented in a simple relation.     

Post-buckling and debond propagation in sandwich panels subject to in-plane compression

Nonlinear finite element analysis is conducted to predict initiation of debond propagation in compression loaded foam cored sandwich panels containing a circular face/core debond embedded at the panel center. A three-dimensional geometrically nonlinear finite element model of the debonded sandwich panel combined with linear elastic fracture mechanics is used to determine the stress intensity factors K_I and K_II and energy release rate at the debond (crack) front parallel and perpendicular to the applied load. A range of core densities and debond sizes are analyzed. The opening mode (mode I) was found to dominate the fracture process. The critical load for crack propagation predicted using fracture mechanics concepts was found to agree with measured collapse loads for smaller debonds, but fell below measured debond propagation loads for larger debonds. In all cases the predicted direction of crack propagation was perpendicular to the loading direction, in agreement with experimental observations.     

Mixed‐Mode Stress Intensity Factors for Partially Opened Cracks

Failure considerations under mixed-mode loading require knowledge about the influence of friction between partially closed crack faces in the case of a negative mode-I stress intensity factor. A simple relation is derived, which allowes to compute friction contributions to the mode-II stress intensity factor K_II for the case of negative mode-I stress intensity factors K_I. The relation is exact for the limit case of an edge crack in a half-space. It can be shown that small cracks in finite bodies can also be described with sufficient accuracy.     

The strain energy released when a crack grows normally from its tip

The strain energy release rate for a crack growing from the tip of, and perpendicular to, an initial straight crack is calculated using finite element analysis under mixed-mode conditions. The results can be represented by a simple equation involving K _I and K _II.     

A fracture mechanics model for the analysis of micro-pitting in regard to lubricated rolling–sliding contact problems

A computational model is presented for the analysis of micro-pitting in regard to lubricated rolling–sliding contact problems. This model assumes the appearance of an initial microcrack on the contact surface due to the mechanical or thermal treatment of the material, and as a consequence of an on-going process in early the stage of exploitation. The discretised model of the contacting mechanical elements is subjected to normal loading (Hertzian contact pressure), tangential loading (friction between contacting surfaces) and internal pressure to the crack surfaces. Crack propagation is predicted as follows: (1) using modified maximum tangential stress criterion, which takes into account the influence of stress intensity factors K_I and K_II, T-stress, stress on the crack’s surface caused by lubricant pressure inside the crack, and the critical distance ahead of the crack tip and (2) the classical maximum tangential stress criterion, which only takes into account the influence of the stress intensity factors K_I and K_II. The stress intensity factor based on these two criteria is then used in a short crack growth theory to determine the fatigue life of an initial crack to extent up to micro-pit. The developed model is applied to a real spur gear pair.     

Continuum shape sensitivity analysis of mixed-mode fracture using fractal finite element method

This paper presents a new fractal finite element based method for continuum-based shape sensitivity analysis for a crack in a homogeneous, isotropic, and two-dimensional linear-elastic body subject to mixed-mode (modes I and II) loading conditions. The method is based on the material derivative concept of continuum mechanics, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed in the proposed method to calculate the sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations predicts the first-order sensitivity of J-integral or mode-I and mode-II stress-intensity factors, K_I and K_II, more efficiently and accurately than the finite-difference methods. Unlike the integral based methods such as J-integral or M-integral no special finite elements and post-processing are needed to determine the first-order sensitivity of J-integral or K_I and K_II. Also a parametric study is carried out to examine the effects of the similarity ratio, the number of transformation terms, and the integration order on the quality of the numerical solutions. Four numerical examples which include both mode-I and mixed-mode problems, are presented to calculate the first-order derivative of the J-integral or stress-intensity factors. The results show that first-order sensitivities of J-integral or stress-intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study.     

Mixed mode fatigue and fracture in planar geometries: Observations on Keq and crack path modelling

The mixed mode I‐II fatigue and fracture is briefly reviewed, addressing experimental and numerical modelling aspects, and focusing on planar specimens. One major challenge concerns the determination of equivalent stress intensity factor (K_eq) in mixed mode situations. Several approaches were compared through the determination of K_eq/K_I over a wide range of values of K_I/K_II or K_II/K_I. Whereas all different approaches converge to the same value as K_I/K_II increases, the same does not happen for large K_II/K_I, where differences between values of K_eq persist. In the regions of 0 < K_I/K_II < 2 and 0 < K_II/K_I < 2, no stable trend of results can be defined. Experimental fatigue crack growth results are presented for Al alloy AA6082‐T6. Compact tension specimens, modified with holes, and four‐point bending specimens under asymmetrical loading promoting mixed mode situations, were subjected to fatigue crack growth tests, where crack path and crack growth rate were measured. The presentation of the fatigue crack growth data was made using a Paris law based upon K_eq. Differences in the Paris law constants were found for the different K_eq criteria. Recent developments in numerical techniques, as the implementation of the extended finite element method (XFEM) in finite element software packages allows to determine accurately crack paths in mixed mode fracture. This article highlights concepts for mixed‐mode fatigue and fracture and supporting data, identifying challenges still to be overcome.     

The behaviour of cracks in elastic-plastic materials under plane normal and shear loadings

Cracks in structures are often subjected to complex loading conditions. The direction of the crack extension depends on the normal and the shear components of the load. This paper is based on the kinking behaviour of cracks taking elastic-plastic behaviour of materials into account. The J-integral and the mixed-mode components J _I and J _II were determined after having performed several finite element analyses for different loading conditions. The path independence of J, J _I and J II is investigated for both, the line integral proposed by Rice and the volume integral proposed by deLorenzi. For correctly determined crack deflection angles the J _II-component vanishes when a FE-model with a kinked crack is considered. Hence, cracks propagate perpendicularly to the local mode I load.     

Spannungsfaktoren eines Risses in der Umgebung eines Kreisloches und ihr Einfluß auf das Bruchverhalten

Stress Intensity Factors in the Neighbourhood of a Circular Hole and their Influence on Crack Behavior . A photoelastic method was developed to determine the stress intensity factors K_I and K_II for cracks subjected to mixed-mode loading. The constants in the near field expansion about the crack tip were computed using a non-linear optimization program to give a best fit to the observed isochromatics. Copying the latter onto an equal density film increased their sharpness and, thus, the accuracy of the determination. The method was applied to cracks lying perpendicular to the external stress near a circular hole in a plate under uniaxial tension and the results used to describe the paths of cracks in the neighbourhood of a hole.     

Fatigue crack propagation simulation in an aircraft engine fan blade attachment

This paper discusses the computation of three-dimensional fatigue crack growth rates in a typical military aircraft engine fan blade attachment under centrifugal and aerodynamic loads. The three-dimensional crack growth simulations utilize FRANC3D, a state-of-the-art crack propagation software developed at Cornell University, which uses boundary elements and linear elastic fracture mechanics. With an existing three-dimensional finite element contact stress analysis with a prescribed coefficient of friction (COF) along the contact surface, the displacements and stress intensity factors are calculated on the crack leading edge to yield crack propagation trajectories and growth rates. Due to complex geometry of the fan blade attachment and loading conditions, all three-fracture modes are considered and the associated stress intensity factors (SIF) are calculated using the Crack Opening Displacement (COD) approach. Crack propagation trajectories under mixed-mode conditions are obtained using the planar and maximum tangential stress crack-extension criteria. The fatigue crack in the blade attachment is subjected to an over speed mission cycle that includes high cycle frequencies (i.e., spectrum load) and the crack growth rate is predicted utilizing the Forman–Newman–de Koning (FNK) model. Scanning Electron Microscope (SEM) images of a cracked component from an engine ASMET (Accelerated Simulated Mission Endurance Test) are used to evaluate and compare the simulation results. The calculated SIF's from the simulations indicate a strong Mode-I (K_I) and Mode-III (K_III) interaction at the edge of contact (EOC). However, on the free surface it is primarily a crack opening (K_I) condition only. The crack growth rates are determined using the planar extension criterion which correlates better with the test data than the maximum tangential stress extension criteria.     

The evaluation of stress intensity factors in plate bending problems using the dual boundary element method

This paper discusses the application of the boundary element method for the determination of stress intensity factors in plate bending problems. A number of case studies having a range of plan forms, with different combinations of boundary conditions, crack configurations and loading conditions are presented to illustrate the effectiveness of the boundary element method for the fracture analysis of plates. Results of K_I, K_II and K_III stress intensity factors for linear elastic fracture mechanics are presented for the case studies considered. The J-integral method, the displacement extrapolation method, the quarter point approach and the stress extrapolation method have been used to determine the stress intensity factors. The boundary element results for the case studies considered in the paper have been compared with either analytical or finite element results and in all cases good agreement has been achieved.     

Finite element analysis of axial splits in composite Iosipescu specimens

In this paper, a finite element analysis of skew-symmetric splits along the fiber direction in unidirectional composite Iosipescu specimens is performed. The energy release rates G _I, G _II, and G _total associated with axial splits in cracked Iosipescu specimens under external biaxial loading conditions are computed by four different numerical schemes: displacement correlation, displacement extrapolation, J-integral, and the modified crack closure integral. Using beam theory analysis, an analytical solution for the energy release rates is also proposed. Axial splits in Iosipescu specimen propagate under mixed mode conditions, with G _I and G _II varying with the crack length a. For short and medium crack lengths G _I>G _II, while for long cracks, G _II is dominant. The energy release rates G _I, G _II, and G _total are strongly dependent on the biaxial type of loading. The G-estimates obtained by the modified crack closure integral schemes are found to be the most accurate among all the numerical schemes chosen in this study. In the analyses of axial splits in composite Iosipescu specimens, the displacement correlation and extrapolation techniques yielded poor results. For long crack lengths, the analytical results from the beam theory analysis are in fair agreement with those from the modified crack closure integral schemes; however, for short and medium crack lengths, there is a significant difference between the analytical and numerical results. In composite Iosipescu specimens, stable crack propagation (mode I dominant) can be achieved by increasing the tension/shear ratio in the external loading boundary conditions.     

Calculation of KII in crack face contacts using X-FEM. Application to fretting fatigue

In this work, the modeling of LEFM problems that imply crack face closure and contact using the extended finite element method (X-FEM) is presented aiming at its application to fretting fatigue problems. An assessment of the accuracy in the calculation of K_II is performed for two different techniques to model crack face contacts in X-FEM: one is based on the use of additional elements to establish the contact and the other on a segment-to-segment (or mortar) approach. It is concluded that only the segment-to-segment approach can lead to optimal convergence rates of the error in K_II. The crack face contact modeling has also been applied to a fretting fatigue problem, where the estimation of K_II under crack closure conditions plays an important role in the stage I of fatigue crack propagation. The effect of the crack face friction coefficient has been studied and its influence on the range of K_II has been ascertained during loading and unloading cycles.     

Effect of mode mixity on K-dominance and three dimensionality in cracked plates

The method of Coherent Gradient Sensing (CGS) in transmission, in conjunction with two and three dimensional finite element methods, is used to study the effect of mode mixity on crack tip stress fields. Using a two dimensional finite element analysis the outer bounds of the region of K-dominance were determined. A three dimensional finite analysis was utilized to study the effect of mode mixity on the three dimensional nature of the stress field in the immediate vicinity of the crack tip and to obtain an inner bound of the region of K-dominance. It was noted that increasing mode mixity leads to an increased rotation of the three dimensional zone, keeping its shape and size unchanged. In contrast, the region of K-dominance is seen to dramatically depend on mode mixity, both in shape and size. In addition, an analysis of the CGS interferograms was conducted to obtain an estimate of the regions of K-dominance experimentally. A least squares fit data analysis technique was used to extract fracture parameters, namely the stress intensity factors K _I, K _II and subsequently the crack tip phase angle, . The data points used for the least square fitting were obtained from the determined regions of K-dominance. The same fracture parameters were also evaluated from the finite element analysis, and good agreement was found between experimental measurements and finite element predictions.