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.     

Multiaxial fatigue damage parameter and life prediction for medium-carbon steel based on the critical plane approach

The tension–torsion fatigue characteristics were investigated under proportional and non-proportional loading in this paper. The fatigue cracks on the surface of multiaxial fatigue specimens were observed and analyzed by a scan electron microscope. On the basis of the investigation on the Kindil–Brown–Miller and Fatemi–Socie’s critical plane approaches, a shear strain based multiaxial fatigue damage parameter was proposed by von Mises criterion based on combining the maximum shear strain and the normal strain excursion between adjacent turning points of the maximum shear strain on the critical plane. The proposed multiaxial fatigue damage parameter does not include the weight constants. According to the proposed multiaxial fatigue damage parameter, the multiaxial fatigue life prediction model was established with the Coffin–Manson equation, which is used to predict the multiaxial fatigue life of medium-carbon steel. The results showed that the proposed multiaxial fatigue damage parameter could be used under either multiaxial proportional or non-proportional loading.     

Assessment of notched structural steel components using failure assessment diagrams and the theory of critical distances

When the structural integrity of notched components is analysed, it is generally assumed that notches behave as cracks, something which generally provides overconservative results. The proposal of this paper consists, on the one hand, in the application of the theory of critical distances for the estimation of the notch fracture toughness and, therefore, for the conversion of the notched situation into an equivalent cracked situation in which the material develops a higher fracture resistance. On the other hand, once the notch fracture toughness has been defined, the assessment is performed using the failure assessment diagram methodology, and assuming that the notch effect on the limit load is negligible. The methodology has been applied to 336 CT notched fracture specimens made of two different structural steels, covering temperatures from the corresponding lower shelf up to the upper shelf, providing satisfactory results and a noticeable reduction in the overconservatism derived from the analyses in which the notch effect is not considered.     

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.     

A study on the combined effect of notch and fretting on the fatigue life behaviour of Al 7075-T6

In this study, an investigation was conducted on the fatigue performance of Al 7075-T6 plates in the presence of stress raisers (notch, fretting, and a combination of notch and fretting). Fretting situation was induced on the surface of the aluminium plate through steel contacting pads under two different clamping forces of 2 kN and 5.6 kN. The fatigue tests revealed a more dominant effect from stress concentrators originating from geometrical discontinuities such as the tested notch compared to the fretting wear conditions. Therefore, no noticeable differences were found between the fatigue lives of the notched specimens and the combined notch and fretting condition. A finite element stress analyses of the notched model under the contacting fretting pads agreed with the experimental results. The stress distribution at the clamped area introduced tensile stresses at the edge of the contact region, however, the stress at the notch tip was observed to be higher when an axial tensile load was applied to the end of the plate. Fractographic analyses confirmed the presence of cracks initiating from the fretting damaged surface for most of the combined notch and fretting fatigue test specimens particularly at the high cycle fatigue (HCF) zone.     

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 new approach of fatigue life prediction for metallic materials under multiaxial loading

Based on experimental data found in literatures, four traditionally multiaxial fatigue life criteria are analyzed and verified. It is discovered that these conventional criteria cannot reflect well the combined effect both under tension and torsion loadings for some materials, such as 6082-T6 and AlCu4Mg1, due to lack of enough consideration about the influence of stress amplitude ratio and stress level on fatigue life even under proportional loading. In order to solve this problem, a new approach of fatigue life prediction, based on the equal-life curve, is proposed and it is composed of three parts: the multiaxial fatigue life surface, a new path-dependent factor for multiaxial high-cycle fatigue and a material parameter describing material sensitivity to non-proportional loading. Finally, the precision of the presented approach is systematically checked against the experimental data found in literatures for four different materials under proportional and non-proportional loadings.     

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.     

Experimental investigation and numerical prediction of fatigue crack growth of 2024-T4 aluminum alloy

Compact specimens were employed to study fatigue crack growth of 2024-T4 aluminum alloy under constant/variable amplitude loading. Apparent R-ratio effect under constant amplitude loading was identified with the nominal stress intensity factor range. Fatigue crack growth rates predicted by a unified model agreed with the experimental data well. Single tensile overload resulted in significant retardation of crack growth which was fully recovered after propagating out of overload-affected zone. Retarded crack growth induced by three-step sequence loading was heavily dependent on two sequence loading parameters. The influence of variable amplitude loading on crack growth was reasonably characterized by Wheeler’s model.     

A critical assessment of methods for the determination of the shear stress amplitude in multiaxial fatigue criteria belonging to critical plane class

Multiaxial high cycle fatigue criteria based on the critical plane approach necessitate unambiguous definitions of the amplitude and mean value of the shear stress (τ_a and τ_m) acting on the material planes. Four of the existing definitions relate the values of τ_a and τ_m to a geometrical element of the curve described by the tip of the shear stress vector (curve Ψ), respectively, the radius of the Minimum Circumscribed Circle, the Longest Chord, the Longest Projection, the diagonal of the Maximum Rectangular Hull (MRH).In this paper a critical assessment of the above definitions is proposed, focusing on that based on the concept of MRH, which is the most recently developed. The main issues of the comparison are the uniqueness of the solution in the determination of τ_a and τ_m, the ability to differentiate proportional and non-proportional stresses, the differences of the values of τ_a obtained by each of the 4 methods for differently shaped curves Ψ.     

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.