Non-proportional Mixed Mode Research Articles

Effect of creepages on stress intensity factors of rolling contact fatigue cracks

In heavy-haul railway systems, the experience of high traction force of rails in curved tracks significantly accelerates the rolling contact fatigue crack growth. The overall level of traction force and its detailed distribution within a contact patch can be determined by the extent of wheel slips in different directions, also referred to as creepage. However, few studies have focused on quantifying the influence of creepage on rolling contact fatigue crack growth behaviour. In this paper, a numerical method is proposed to investigate the non-proportional mixed-mode rolling contact fatigue crack growth behaviour in the presence of severe longitudinal, lateral and spin creepages. To partially validate the proposed numerical method, the predicted crack growth directions on the rail surface are compared with the cracking patterns obtained from field observations. A parametric study is also conducted to further quantify the influence of different creepage combinations on the evolution of stress intensity factors at the crack front. Results show that both lateral and spin creepages can significantly affect the phase and maximum magnitude of KII and KIII during one complete loading cycle. Moreover, the increase of spin creepage is found to have a more detrimental effect on overall RCF crack driving force than the increase of the other two creepage components.

Prediction of the crack growth propagation direction in non-proportional mixed-mode missions

The prediction of the crack growth propagation direction due to Non-Proportional Mixed-Mode missions is still an open debate in industrial applications. This study proposes two algorithms, both based on the dominant step criterion, to evaluate the cyclic crack propagation due to missions with varying mixed-mode loading conditions. They are implemented in the commercial software FRANC3D (coupled with ABAQUS) and the MTU Aero Engines AG in house tool, CRACKTRACER3D (coupled with CalculiX). The results of the numerical simulations are validated with the experimental data from tension-torsion tests with and without phase shift. The comparisons show that both software predictions are in good agreement with the experimental results and they can be confirmed as valid tools to simulate crack propagation due to a mission.

Crack Growth Propagation Prediction in Non-Proportional Mixed-Mode Missions

The present paper focuses on the prediction of cyclic crack propagation due to missions with varying mixed-mode loading conditions. Overall, 18 different loading conditions are simulated and the numerical results are compared with the corresponding experimental results. The deflection angle prediction in correspondence of the complete mission is based on the Maximum Tensile Stress (MTS) criterion, coupled with the dominant step approach. The entire numerical simulation procedure relies on the algorithms implemented in the commercial FRacture ANalysis Code 3D, FRANC3D, and ABAQUS software. Numerical and experimental fracture surfaces, in correspondence of tension-torsion tests with and without phase shift, are compared by using an MTU in house tool, CT3D_Validator, with the aim to assess the accuracy of the numerical simulation procedure in determining the crack propagation direction.

Fatigue crack growth under non-proportional mixed-mode I + II. Role of compression while shearing

Mode I + II fatigue crack growth tests are run, following sequential or pseudo-sequential loading paths, representative of those computed for ”squat type” cracks in rails. Stereo DIC provides the near-tip displacements, from which ΔKIeffective and ΔKIIeffective are derived. Compression while shearing extends coplanar growth, by slowing down the wear-induced rise of the effective mode mixity ratio, up to the critical value for which bifurcation occurs. The consideration of plasticity and contact and friction stresses improves crack paths predictions, compared to the maximum tangential stress criterion. The coplanar crack growth rate correlates well with a combination of ΔKIeffective and ΔKIIeffective.

Fatigue Crack Growth under Non-Proportional Mixed Mode Loading in Rail and Wheel Steel Part 2: Sequential Mode I and Mode III Loading

Rolling contact fatigue cracks in rail and wheel undergo non-proportional mixed mode I/II/III loading. Fatigue tests were performed to determine the coplanar and branch crack growth rates on these materials. Sequential and overlapping mode I and III loading cycles were applied to single cracks in round bar specimens. Experiments in which this is done have been rarely performed. The fracture surface observations and the finite element analysis results suggested that the growth of long (does not branch but grown stably and straight) coplanar cracks was driven mainly by mode III loading. The cracks tended to branch when increasing the material strength and/or the degree of overlap between the mode I and III loading cycles. The equivalent stress intensity factor range that can consider the crack face contact and successfully regressed the crack growth rate data is proposed for the branch crack. Based on the results obtained in this study, the mechanism of long coplanar shear-mode crack growth turned out to be the same regardless of whether the main driving force is in-plane shear or out-of-plane shear.

Fatigue Crack Growth under Non-Proportional Mixed Mode Loading in Rail and Wheel Steel Part 1: Sequential Mode I and Mode II Loading

Fatigue tests were performed to estimate the coplanar and branch crack growth rates on rail and wheel steel under non-proportional mixed mode I/II loading cycles simulating the load on rolling contact fatigue cracks; sequential and overlapping mode I and II loadings were applied to single cracks in the specimens. Long coplanar cracks were produced under certain loading conditions. The fracture surfaces observed by scanning electron microscopy and the finite element analysis results suggested that the growth was driven mainly by in-plane shear mode (i.e., mode II) loading. Crack branching likely occurred when the degree of overlap between these mode cycles increased, indicating that such degree enhancement leads to a relative increase of the maximum tangential stress range, based on an elasto–plastic stress field along the branch direction, compared to the maximum shear stress. Moreover, the crack growth rate decreased when the material strength increased because this made the crack tip displacements smaller. The branch crack growth rates could not be represented by a single crack growth law since the plastic zone size ahead of the crack tip increased with the shear part of the loading due to the T-stress, resulting in higher growth rates.

A new type of flat-bar specimen for non-proportional mixed-mode crack tip loading under uniaxial tension

Non-proportional cyclic loading is a challenging issue for numerical crack path predictions. Principal stress axes and the vector of the J-integral rotate during loading and the mixed-mode ratio of stress intensity factors changes within a load period. Reliable criteria for crack deflection, accounting for arbitrary loading and boundary conditions under non-proportional loading, have not yet been found. Experimental research in this field nowadays requires expensive set-ups and multiaxial testing equipment such as hydraulic biaxial or tension-torsion testing machines. In this paper, a specimen is presented, suitable for uniaxial testing, however providing conditions of non-proportional loading. Finite element simulations show that the variance of mixed-mode parameters is sufficiently large to cover the range of KII/KI-ratios required for investigating crack deflection criteria.

Observations and modelling of non-proportional mixed mode cyclic loading

Under non-proportional mixed-mode loading conditions at the crack tip, some observations are reported. They are limited to cases of cylindrical specimens under normal force and torsion. The fatigue cracks have been initiated by the same loading for which the crack growth behavior has been studied. The observations are discussed against the background of a variety of influence factors on non-proportional mixed mode growth. Finally, a couple of modeling approaches are listed which are – according to the nature or the problem – not yet developed for providing prediction tools in an engineering environment.

Mixed mode II and III fatigue crack growth in a rail steel

Rolling contact fatigue cracks in rails undergo non-proportional mixed-mode I + II + III, in variable proportions along their front. In order to determine the crack growth thresholds and kinetics in mixed-mode II/III, asymmetric four point bending tests are run on a rail steel with different angles between the crack front and the shearing load, so as to vary the mode mixity ratio. For sufficiently high loading ranges, these tests give rise to coplanar shear-mode crack growth. The effective stress intensity factors (SIFs) are derived by an inverse method from the measured crack face relative displacements. It appears to be 10–70% lower than the nominal SIFs and to allow a reasonable correlation of the measured crack growth rates. The local application, ahead of the crack front, of shear-driven or tension-driven fatigue damage models – after 3D elastic-plastic computations of local stress and strain ranges – allows a successful prediction of crack fronts paths and growth rates.

A path-dependent fatigue crack propagation model under non-proportional modes I and III loading conditions

It has been well established that fatigue damage process is load-path dependent under non-proportional multi-axial loading conditions. Most of studies to date have been focusing on interpretation of S-N based test data by constructing a path-dependent fatigue damage model. This paper presents a two-parameter mixed-mode fatigue crack growth model which takes into account of crack growth dependency on both load path traversed and a maximum effective stress intensity attained in a stress intensity factor plane (e.g.,KI-KIII plane). By taking advantage of a path-dependent maximum range (PDMR) cycle definition (Dong et al., 2010; Wei and Dong, 2010), the two parameters are formulated by introducing a moment of load path (MLP) based equivalent stress intensity factor range (ΔKNP) and a maximum effective stress intensity parameter KMax incorporating an interaction term KI·KIII. To examine the effectiveness of the proposed model, two sets of crack growth rate test data are considered. The first set is obtained as a part of this study using 304 stainless steel disk specimens subjected to three combined non-proportional modes I and III loading conditions (i.e., with a phase angle of 0°, 90°, and 180°). The second set was obtained by Feng et al. (2007) using 1070 steel disk specimens subjected to similar types of non-proportional mixed-mode conditions. Once the proposed two-parameter non-proportional mixed-mode crack growth model is used, it is shown that a good correlation can be achieved for both sets of the crack growth rate test data.

Fatigue crack growth simulation under cyclic non-proportional mixed mode loading

An algorithm based on linear elastic fracture mechanics for three-dimensional fatigue crack growth simulation under non-proportional mixed-mode loading is proposed in the present paper. The crack growth behavior in thin-walled, hollow cylinders with a notch under combined non-proportional cyclic tension and torsion loadings are investigated with the finite element program ABAQUS and a 3D fracture analysis software FRANC3D. Different mixed-mode crack path prediction criteria are evaluated to estimate the crack growth direction. Fracture mode transition is observed, some influence factors are discussed. Crack growth cycles calculated based on the effective stress intensity factors satisfactorily match the experimental data for specimens tested under lower loading levels.

Fatigue crack growth of aluminium alloy 7075-T651 under proportional and non-proportional mixed mode I and II loads

This study aims to experimentally investigate fatigue growth behaviour in AA7075-T651 under proportional and non-proportional mixed mode I and II loads. Fatigue tests were performed under cyclic tension and torsion using a thin-walled tubular specimen with a key-hole style crack starter. After the generation of a single-side mode I pre-crack, varied forms of mixed mode loads were applied, which in most cases led to coplanar growth for a short distance followed by a long and stable crack path deviation. It was found that under most of the non-proportional mixed mode load cases, the direction of the deviated crack path could not be reasonably predicted using the commonly accepted maximum tangential stress criterion. Meanwhile, in some cases, the crack path directions could be approximately predicted using the maximum shear stress criterion. It was also confirmed for the first time that a long, stable and non-coplanar shear mode fatigue crack growth could be produced in AA7075-T651 under non-proportional mixed mode I and II loads. The implications of transient co-planar as well as stable shear-mode fatigue crack growth were discussed, revealing the needs of future investigations. ©2016 Commonwealth of Australia

Fatigue crack growth of aluminium alloy 7075-T651 under non-proportional mixed mode I and II loads

This study aims to investigate fatigue growth behaviour in AA7075-T651 under non-proportional mixed mode I and II loads. Fatigue tests were performed under cyclic tension and torsion using a thin-walled tubular specimen with a key-hole style crack starter. After the generation of a single-side mode I pre-crack, varied forms of mixed mode loads were applied, which in most cases led to a short distance coplanar growth followed by a long and stable crack path deviation. It was found that under most of the non-proportional mixed mode load cases, the direction of the deviated crack path could not be reasonably predicted using the commonly accepted maximum tangential stress criterion. Meanwhile, in some cases, the crack path directions could be approximately predicted using the maximum shear stress criterion. It was also confirmed for the first time that a long, stable and noncoplanar shear mode fatigue crack growth could be produced in AA7075- T651 under non-proportional mixed mode I and II loads.

Memory effects in variable amplitude and multiaxial fatigue crack growth: an incremental approach

This paper presents an incremental approach for modelling fatigue crack growth with memory effects due to the non-linear behavior of the material. This approach is developed at LMT since 2003, in collaboration with several industrial partners, mainly with Snecma, SAFRAN Group, EDF and AREVA and SNCF. The first part of this paper presents the context and the objectives, and the key assumptions on which the model is based. The second part presents some examples of applications of the model, fatigue crack growth in mode I conditions, under variable amplitude loading; non-isothermal situations; crack growth in coupled environmental and fatigue loading conditions; extension of the model to non-proportional mixed mode loading conditions, and to short cracks. The last part presents some ongoing work, possible developments and scientific challenges that remain to be overcome.

OS2139 非比例混合モード負荷による共平面き裂成長速度に及ぼす材質の影響

Experiments have been performed for the coplanar fatigue crack growth under non-proportional mixed mode loading. In the experiments, sequential and overlapping mode I and mode II, or mode I and mode III cycles were applied to the crack in the specimen made of wheel or rail steel. The growth rates were obtained and compared in terms of equivalent stress intensity factor ranges. Fracture surfaces near the crack tip region of tested specimens were observed by SEM. For the investigation of incipient crack path direction, the elasto-plastic finite element analyses were performed with the combination of the maximum tangential stress criterion.

Load path effect on fatigue crack propagation in I + II + III mixed mode conditions – Part 1: Experimental investigations

This paper is devoted to the analysis of the load path effect on mixed mode I+II+III fatigue crack growth in a 316L stainless steel. Experiments were conducted in mode I+II and in mode I+II+III. The loads were applied using the six actuators servo-hydraulic testing machine available at LMT-Cachan and the crack growth rate was measured using digital image correlation. The topographies of the crack paths were determined post-mortem using a numerical optical microscope. The fracture surfaces were also examined at high magnification using a scanning electron microscope (SEM). The load paths used in the experiments were chosen so as to be equivalent with respect to most of the fatigue crack growth criteria, in particular with those based on ΔKeq=αΔKIn+βΔKIIn+γnΔKIIIn since the same maximum, minimum and mean values of the stress intensity factors were used for each loading path. In addition, load paths were constructed by pairs, either so that the extreme values of the stress intensity factors are attained simultaneously, or so as to display the same cumulative “length”. The main result of this set of experiments is that very different crack growth rates and crack paths are observed for load paths that are however considered as equivalent in most fatigue criteria. In addition, it was shown that the load path can have a very significant effect on the crack growth rate even if the crack path is not significantly different. The comparison of the results of the experiments conducted in mode I+II and in mode I+II+III, also allowed to show that the addition of mode III loading steps to a mode I+II loading sequence is increasing the fatigue crack growth rate, even when the crack path is not significantly modified. And finally, the SEM observations of the fracture surface showed that in non-proportional mixed mode conditions, a complex system of slip bands is formed at crack tip and is used to promote crack growth.

Long co-planar mode III fatigue crack growth under non-proportional mixed mode loading in rail steel

Fatigue tests have been performed to obtain the co-planar crack growth rate in rail steel under non-proportional mixed mode I and mode III cycles. In addition, a finite element analysis (FEA) has been performed to investigate the crack growth behaviour under this loading. The experiments showed that a long co-planar crack could be produced under this loading. Based on fracture surface observations obtained using a scanning electron microscope and FEA, the long co-planar crack growth was thought to be driven mainly by mode III loading and the role of mode I was as an assistant; keeping the crack face open. It was also observed that the crack tended to branch when the degree of overlap between the mode I and mode III cycles increased. Comparing the crack growth rate data under loading with that under non-proportional mixed mode I and II cycles, it was found that the co-planar growth rates were similar regardless of whether the main driving force was in-plane shear or out-of-plane shear.

205102 レール鋼における非比例混合モードI+II負荷における疲労き裂成長方向(OS04 材料・機械・構造物における強度と寿命2,オーガナイズド・セッション)

Two criteria for the prediction of incipient crack path direction of non-proportional mixed mode I+II loading in rail steel are investigated. The analysis is based on 2D plane strain FE-simulations in which actual elasto-plastic stress distribution is used for the maximum tangential stress (MTS) criterion and maximum tangential stress range (MTSR) criterion. The purpose of the analysis is to compare the numerical results with experimental results. It is shown that both the MTS and MTSR criteria including plasticity and sophisticated work hardening law are well suited to predict incipient crack growth directions for the rail steel under non-proportional loading.

Fatigue crack propagation under non-proportional mixed mode loading

Under non-proportional mixed I+II loading, two kinds of stable crack propagation may be distinguished. An existing precrack will either kink, mode I controlled (tensile mode), or will propagate, coplanarly mode II controlled (shear mode). Shear mode growth will occur if the effective mode II range exceeds the material-specific threshold Δ K II th sm and in addition to that, the Δ K II -value on the starter crack is larger than the Δ K ∗ I (Δ ϕ)-range on the infinitesimally short additional crack. Examination under the scanning electron microscope showed that flaws are not the reason for the mode II controlled crack propagation and support the criteria introduced. If the crack opening is large enough, the crack propagation rate is higher for shear-stress controlled crack growth than for normal-stress controlled crack extension, the deviation angle of which is well predictable via the MTS criterion due to Erdogan and Sih [On the crack extension in plates under plane loading and transverse shear. J Basic Engng 1963;85:519–25].

Crack Path Prediction Under Fretting Fatigue—A Theoretical and Experimental Approach

In this paper, the direction of crack growth under fretting fatigue loading is studied through an experimental and theoretical approach. The experimental work enabled the fretting conditions to be known and the site of initiation and crack trajectory to be viewed; theoretical work permitted a prediction of those processes. Fretting wear and fretting fatigue loadings induce non-proportional mixed mode loading at the tip of the cracks initiated within the contact zone. The classical criteria predicting the direction of crack growth cannot account for the non-proportional loading. Tests were carried out to study the cracking phenomena under cumulative effects of contact and external loadings, i.e., fretting fatigue loading. The fretting contact between the two contacting bodies is modeled to evaluate the operating contact loading conditions. The response of the cracked body is determined in terms of stress intensity factors using the continuous distribution of dislocations theory coupled with a unilateral contact analysis with friction. The angle of crack extension is then predicted, at different stages of crack life, according to a new approach. The correlation of the predicted angle of crack extension with the experimental observation enables the conclusion that, under fretting fatigue loading, cracks propagate by a mode I process.