The theory of critical distances: a review of its applications in fatigue

This paper attempts to review the most interesting findings in the use of the theory of critical distances (TCD) to predict fatigue strength of notched mechanical components. Initially, the most modern formalisations of the TCD are considered, showing their peculiarities and differences. An ad-hoc section is then focused on the multiaxial high-cycle fatigue problem, considering all the open questions arising in the presence of complex stress fields damaging the fatigue process zone in the vicinity of the stress concentrator apex. Subsequently, the physical idea on the structural volume concept is briefly investigated showing some peculiar results generated in the high-cycle fatigue regime under both uniaxial and biaxial fatigue loading. Finally, our idea to extend the use of the TCD down to the low-medium cycle fatigue regime is briefly explained.Working in collaboration with Prof. David Taylor, we have spent the last five years investigating this theory both to better understand its physical meaning and to systematically check its accuracy in predicting notch fatigue strength under different loading conditions. After so much work done in this area we feel so confident to proudly and loudly say that the TCD is a powerful engineering tool suitable for assessing real mechanical components in situations of practical interest. Finally, it can be highlighted also that the best TCD formalisations were seen to be those based on the use of linear-elastic stresses. This suggests that such a theory can successfully be used to post-process simple linear-elastic finite element (FE) models reducing time and costs of the design process.     

A novel formulation of the theory of critical distances to estimate lifetime of notched components in the medium-cycle fatigue regime

In the present paper, the theory of critical distances (TCD) is reformulated in order to make it suitable for predicting fatigue lifetime of notched components in the medium-cycle fatigue regime. This extension of the TCD takes as its starting point the idea that the material characteristic length, L, changes as the number of cycles to failure, N_f, changes. In order to define the L versus N_f relationship two different strategies were investigated. Initially, we attempted to determine it by using the L values calculated considering material properties defined at the two extremes, namely static failure and the fatigue limit. This strategy, though correct from a philosophical point of view, contained some problems in its practical application. We subsequently attempted to determine the L versus N_f relationship by means of two calibration fatigue curves; (one generated by testing plain specimens and the second one generated by testing notched specimens). This second strategy was found to be much more simple to apply to practical problems, resulting in estimations characterized by a higher accuracy. The reliability of the devised method was systematically checked by using experimental results generated by testing notched specimens of low-carbon steel containing different geometrical features and tested using various loading types, stress ratios and specimen thicknesses. The accuracy of the method was further verified by using several data sets taken from the literature. Our method was seen to be successful giving predictions falling always within the scatter band of the data from the parent material. These results are very interesting, especially considering that the TCD is very easy to use because it requires only a linear-elastic stress analysis.     

On the application of multiaxial high-cycle fatigue criteria using the theory of critical distances

This paper proposes that the application of multiaxial fatigue criteria in terms of the theory of critical distances requires the use of a distance which may be different from the widely adopted value given by half of the El Haddad’s intrinsic crack length. Three criteria (Modified Wöhler Curve Method, Crossland and Dang Van) are evaluated at the appropriate critical distance and compared with experimental data obtained from specimens containing small and/or sharp notches under proportional loading. The Modified Wöhler Curve Method provided the best estimates. It is also shown that this theory can be extended to the prediction of the loading ratio, R, effect on the threshold stress intensity factor range for propagation of long cracks.     

Recent developments in the understanding of fretting fatigue

Considerable progress has been made in the understanding of fretting fatigue over the last decade. Experiments have become more standardised and carefully controlled and this has provided the data necessary for development of methods for predicting fretting fatigue performance. This paper reviews a number of recent developments, starting with attempts to apply multiaxial initiation criteria to the fretting problem. The importance of the size effect is highlighted and an analogy is made between fretting and notch fatigue. Methods for characterising crack initiation using asymptotic analysis are discussed, together with short crack arrest concepts which provide a means of predicting fretting fatigue limits from plain fatigue data.     

A unifying approach to estimate the high-cycle fatigue strength of notched components subjected to both uniaxial and multiaxial cyclic loadings

This paper proposes an engineering method suitable for predicting the fatigue limit of both plain and notched components subjected to uniaxial as well as to multiaxial fatigue loadings. Initially, some well‐known concepts formalized by considering the cracking behaviour of metallic material under uniaxial cyclic loads have been extended to multiaxial fatigue situations. This theoretical extension allowed us to form the hypothesis that fatigue limits can be estimated by considering the linear–elastic stress state calculated at the centre of the structural volume. This volume was assumed to be the zone where all the main physical processes take place in fatigue limit conditions. The size of the structural volume was demonstrated to be constant, that is, independent from the applied loading type, but different for different materials. Predictions have been made by Susmel and Lazzarin's multiaxial fatigue criterion, applied using the linear–elastic stress state determined at the centre of the structural volume. The accuracy of this method has been checked by using a number of data sets taken from the literature and generated by testing notch specimens both under uniaxial and multiaxial fatigue loadings. Our approach is demonstrated to be a powerful engineering tool for predicting the fatigue limit of notch components, independently of material, stress concentration feature and applied load type. In particular, it allowed us to perform predictions within an error interval of about ±25% in stress, even though some material mechanical properties were either estimated or taken from different sources.     

Probabilistic approach in high-cycle multiaxial fatigue: volume and surface effects

The stress gradient and the size of a component are known to influence the fatigue strength of metallic components. Indeed, in high‐cycle fatigue, experiments prove that the stress distribution as well as the size of the loaded specimen can be responsible for changes in the fatigue limit (for instance, the fatigue limits in tension and bending are different, and decrease with the size of the specimen). When dealing with multiaxial load conditions, those effects still act but a relevant criterion must be used to account for the complex state of stress. The weakest‐link concept together with a multiaxial endurance criterion based on a microplasticity analysis are then combined to describe the fatigue limit distribution of different metallic materials. Several load conditions are analysed: tension–compression, torsion, rotating bending and plane bending. By means of the proposed model, all the known effects on fatigue strength can be reflected. First, the endurance probability can be adequately predicted for any complex load conditions knowing some reference data from uniaxial fatigue tests. It can be linked to the probability of finding a defect with a critical size. The weakest‐link theory also accounts for the decrease of multiaxial fatigue limit with the stressed volume. For the same load condition (i.e. for the same stress distribution in the volume), the probability of finding a critical defect increases with the component size and then according to the weakest‐link theory the fatigue strength drops. A second model, based only on the damage developed at the surface, is also proposed. While the original Weibull theory makes no distinction between potential initiation sites at the free surface and in the volume and can lead to unsatisfactory predictions when applied to materials containing defects such as nodular cast iron, the new surface approach distinguishes between surface and volume effects.     

Multiaxial fatigue limits and material sensitivity to non-zero mean stresses normal to the critical planes

This paper is concerned with an attempt to reformulate the so-called Modified Wöhler Curve Method (MWCM) in order to more efficiently account for the detrimental effect of non-zero mean stresses perpendicular to the critical planes. In more detail, by taking as a starting point the well-established experimental evidence that engineering materials exhibit different sensitivities to superimposed tensile static stresses, an effective value of the normal mean stress relative to the critical plane was attempted to be calculated by introducing a suitable correction factor. Such a mean stress sensitivity index was assumed to be a material constant, i.e. a material parameter to be determined by running appropriate experiments. The accuracy of the novel reformulation of the MWCM proposed here was systematically checked by using several experimental data taken from the literature. In particular, in order to better explore the main features of the improved MWCM, its accuracy in estimating multiaxial high-cycle fatigue damage was evaluated by considering fatigue results generated not only under non-zero mean stresses but also under non-proportional loading. Such a validation exercise allowed us to prove that the systematic use of the mean stress sensitivity index resulted in estimates falling within an error interval equal to about ±10%, and this held true independently of considered material and complexity of the investigated loading path. Finally, such a novel reformulation of the MWCM was also applied along with the Theory of Critical Distances (TCD) to predict the high-cycle fatigue strength of notched samples tested under in-phase bending and torsion with superimposed tensile and torsional static stresses: again our method was seen to be highly accurate, correctly predicting high-cycle multiaxial fatigue damage also in the presence of stress concentration phenomena.     

Influence of fretting on the fatigue strength at the vise clamp-specimen interface

Fretting fatigue is one of the most important phenomena for inducing a significant reduction of fatigue strength and consequently, leading to unexpected failure accidents of the engineering structures even at very low stresses. In the present study, both plain and fretting fatigue tests with zero mean stress were carried out on two different types of steel, low-carbon steel and martensitic stainless steel, by means of a reversed bending fatigue testing machine. The drop in the fatigue strengths through fretting at vise clamp-specimen interface were significant for both tested steels. The fretting processes produced a reduction in fatigue strength of about 27% for low-carbon steel and 16% for martensitic stainless steel.     

The use of multiaxial fatigue models to predict fretting fatigue life of components subjected to different contact stress fields

This work describes the application of multiaxial fatigue criteria based on critical plane and mesoscopic (Dang Van, 1973, Sciences et Techniques de lÁrmement, 47 , 647—722) approaches to predict the fatigue initiation life of fretted components. To validate the analysis, several tests under closely controlled laboratory conditions are carried out in a Ti‐6Al‐4V alloy. These classical Hertzian tests reveal a size effect where fretting fatigue lives vary with contact size. Experimentally available data for fretting fatigue of an Al‐4Cu alloy are also used to assess the models. Neither the critical plane models nor the mesoscopic criterion considered can account for the effects of different contact stress fields on the initiation life, if the calculation is based only on highly stressed points on the surface. It is shown, however, that satisfactory results can be achieved if high values of the fatigue parameters are sustained over a critical volume.     

A probabilistic approach to predict the very high-cycle fatigue behaviour of spheroidal graphite cast iron structures

A new probabilistic approach is developed to study structures made of spheroidal graphite cast iron and subjected to very high-cycle fatigue. Until now, the probabilistic approach was based on S–N curves obtained from experiments carried out only until 10⁷ cycles. To validate this approach, failure predictions relating to the safety of components are computed and compared to experimental results. In addition to this development, an extension is proposed in order to improve the very long life assessment of complex structures. An extrapolation of the previous fatigue results to 10⁹–10¹¹ cycles illustrates the error made on cumulative failure probabilities. Finally, the respective influence of the casting flaw distribution, volume and stress field heterogeneity within specimens and industrial components is studied.     

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.     

On the estimation of fatigue failure under fretting conditions using notch methodologies

ABSTRACT According to experimental evidence, the early stages of fatigue crack propagation under fretting conditions are strongly influenced by the stress gradient generated in the material near the contact zone. This suggests that the crack growth process can be analysed using methodologies similar to those employed to predict the fatigue behaviour of notched elements. This paper assesses the applicability of a number of models originally developed for notched components to fretting fatigue problems. The ability of such models to predict fatigue failure is discussed and compared with experimental results for Al 7075‐T6 specimens that were subjected to fretting fatigue under spherical contact.     

On the application of the Kitagawa-Takahashi diagram to foreign-object damage and high-cycle fatigue

The role of foreign-object damage (FOD) and its effect on high-cycle fatigue (HGF) failures in a turbine engine Ti-6Al-4V alloy is examined in the context of the use of the Kitagawa-Takahashi diagram to describe the limiting conditions for such failures. Experimentally, FOD is simulated by firing 1 and 3.2 mm diameter steel spheres onto the flat specimen surface of tensile fatigue specimens at velocities of 200 and 300 m/s. Such damage was found to markedly reduce the fatigue strength of the alloy, primarily due to four factors: stress concentration, microcrack formation, impact-induced plasticity and tensile residual stresses associated with the impact damage. Two groups of fatigue failures could be identified. The first group initiated directly at the impact site, and can be readily described through the use of a fatigue-crack growth threshold concept. Specifically, a Kitagawa-Takahashi approach is presented where the limiting threshold conditions are defined by the stress-concentration corrected smooth-bar fatigue limit (at microstructurally small crack sizes) and a “worst-case” fatigue-crack growth threshold (at larger crack sizes). The second group of failures was caused by fatigue cracks that initiated at locations far from the impact site in regions of high tensile residual stresses, the magnitude of which was computed numerically and measured experimentally using synchrotron X-ray diffraction. Specifically, these failures could be rationalized due to the superposition of the residual stresses on the far-field applied mean stress, leading to a locally elevated load ratio (ratio of minimum to maximum loads). The effects of residual stress, stress concentration, and microstructurally small cracks are combined in a modified Kitagawa-Takahashi approach to provide a mechanistic basis for evaluating the detrimental effect of FOD on HCF failures in Ti-6Al-4V blade alloys.     

The critical distance theory for fatigue analysis of notched aluminium specimens subjected to repeated bending

This paper provides new insights in the use of the critical distance method for fatigue analysis of notched aluminium components subjected to constant amplitude bending loading. A straightforward test setup was developed to load test samples with different stress concentrations in repeated bending at high test frequency. The mean values of the local endurable stress amplitudes are determined with the staircase method and the Dixon and Mood theory using a minimum amount of test samples. The critical distance is determined using these fatigue limits and the corresponding stress gradients determined by means of finite element analysis. The results indicate a unique critical distance of 0.22 mm for fatigue crack initiation. Consequently, the critical distance theory can be successfully applied for fatigue analysis of notched specimens or engineering components of aluminium EN AW 7075 T7351 with geometrical features of various size and shape subjected to fluctuating loading in bending.     

Probabilistic prediction of high cycle fatigue reliability of high strength steel butt‐welded joints

The aim of this paper is to develop a probabilistic approach of high cycle fatigue (HCF) behaviour prediction of welded joints taking into account the surface modifications induced by welding and the post‐welding shot peening treatment. In this work, the HCF Crossland criterion has been used and adopted to the case of welded and shot peened welded parts, by taking into account the surface modifications which are classified as follows: (i) the compressive residual stresses, (ii) the surface work‐hardening, (iii) the geometrical irregularities and (iv) the superficial defects. The random effects due to the dispersions of: (i) the HCF Crossland criterion material characteristics (ii) the applied loading and (iii) the surface modifications parameters are introduced in the proposed model. The HCF reliability has been computed by using the ‘strength load’ method with Monte Carlo simulation. The reliability computation results lead to obtain interesting and useful iso‐probabilistic Crossland diagrams (PCD) for different welding and shot peening surface conditions. To validate the proposed method, the approach has been applied to a butt‐welded joint made of S550MC high strength steel (HSS). Four types of specimens are investigated: (i) base metal (BM), (ii) machined and grooved (MG) condition, (iii) As welded (AW) condition and (iv) as welded and shot peened (AWSP) condition. The comparison between the computed reliabilities and the experimental investigations reveals good agreement leading to validate the proposed approach. The effects of the different welded and post‐weld shot peened specimen's surface properties are analysed and discussed using the design of experiments (DoE) techniques.     

Effect of hole lubrication on the fretting fatigue life of double shear lap joints: An experimental and numerical study

In this research the affect that lubrication at a hole and pin connection has on the fatigue life of a double shear lap joint is studied both experimentally and numerically. The study focuses on the joint middle plate item, which is connected via a central hole to the outer plates by means of a clearance fitting pin, thereby placing the hole in double shear. In the experimental work three identical batches of fatigue specimens, which are made from aluminum alloy 2024-T3, were fatigue tested. In the first batch the surface of the fastener hole was not lubricated whilst the hole in the other two batches was lubricated – each batch using a different lubricant. The three batches of double shear lap joint specimens were fatigue tested and their S–N curves established. The results show that the specimens in which the holes were lubricated have better fatigue lives than the non-lubricated hole specimens. In the numerical study, FE simulations were performed to include hole lubrication effect on the stress distribution by using different friction coefficient at the interface of the hole and its fastener (pin). The FE results have helped to gain an understanding of the reasons for fatigue life improvement and also have helped to quantify the level of improvement.     

A review of multiaxial fatigue of weldments: experimental results, design code and critical plane approaches

A survey of biaxial (bending or tension and torsion) constant amplitude fatigue of welded connections is presented. Re-analysis of 233 experimental results from eight different studies has been performed based on hot spot stresses and three potential damage parameters: maximum principal stress range; maximum shear stress range; and a modified critical plane model for welds. Of the three methods, the critical plane model was most successful in resolving the data to a single S – N line. The design curve for all toe failures based on the critical plane model was FAT 97 with a slope of 3. By excluding butt welds and including only fillet welds that failed at the weld toe, the design curve was increased to FAT 114 with a slope of 3. However, observed scatter was 70–100% larger than that observed in uniaxial loaded specimens analysed using the hot spot approach.