A modification of Smith–Watson–Topper damage parameter for fatigue life prediction under non‐proportional loading

In this paper, the shortcomings of the Smith–Watson–Topper (SWT) damage parameter are analysed on the basis of the critical plane concept. It is found that the SWT model usually overestimates the fatigue lives of materials since it only takes into account the fatigue damage caused by the tensile components. To solve this problem, Chen et al. (CXH) modified the SWT model through considering the shear components. However, there are at least two problems present in CXH model: (1) the mean stress is not considered and (2) the different influence of the normal and shear components on fatigue life is not included. Besides, experimental validations show that the modification by Chen et al. usually leads to conservative fatigue life predictions during non‐proportional loading. In order to overcome the shortcomings of SWT and CXH models, a damage parameter as the effective strain energy density (ESED) is proposed. Experimental validations by using eight kinds of materials show that the ESED model can give satisfactory fatigue life predictions under the non‐proportional loading.     

A mean stress correction model for tensile and compressive mean stress fatigue loadings

A new mean stress fatigue model based on the distortional strain energy is proposed to account for the mean stress effects on fatigue life. The proposed model is compared with the Morrow and the Smith‐Watson‐Topper (SWT) mean stress correction models using a number of experimental data sets for one cast iron, two steels and two aluminium alloys under tensile and compressive mean stress loadings. It is found that both the proposed mean stress correction model and the SWT model yield similar results and provide very good correlation for positive mean stress data and moderate negative mean stress data. For high compressive mean stresses, the proposed model shows reasonably good correlations, while the SWT model fails to correlate the fatigue data. The Morrow model was found to give poor correlations for all fatigue data analysed by yielding conservative results for compressive mean stresses and non‐conservative results for tensile mean stresses.     

Initiation and short crack growth in austenitic–ferritic duplex steel‐effect of positive mean stress

The effect of the mean stress on the crack initiation and short crack growth of austenitic–ferritic duplex steel has been studied. High mean stresses and stress amplitudes result in appreciable mean strain relaxation and long‐term hardening. Mean stress produces unidirectional slip bands and slip steps that serve as nuclei for persistent slip bands and persistent slip markings. It leads to the acceleration of the crack initiation and production of a high density of cracks. Crack linkage contributes to the growth of short cracks. The concept of equivalent crack was used to describe the crack growth. The kinetics of short crack growth with positive mean stress is similar to that in symmetric loading, that is, exponential growth is observed. Positive mean stress results in earlier crack initiation and in the acceleration of the crack growth rate. Both factors contribute to the decrease of the fatigue life.     

Modelling the dependency of the Smith–Watson–Topper parameter on the cycles‐to‐failure using serial hybrid neural networks

Calculating the fatigue damage with a strain‐based approach requires an ?–N durability curve that links the strain amplitude to the corresponding number of cycles‐to‐failure. This ?–N curve is usually modelled by the Coffin–Manson relationship. If a loading mean‐level also needs to be considered, the original Coffin–Manson relationship is modified using a Smith–Watson–Topper parameter. In this article a methodology for modelling the dependence of the Smith–Watson–Topper parameter on the number of cycles‐to‐failure is presented. The core of the presented methodology represents a multilayer perceptron neural network combined with the Smith–Watson–Topper analytical model. The article presents the theoretical background of the methodology, which is applied for the case of the experimental fatigue data. The results show that it is possible to model ?–N curves for different influential parameters, such as the specimen's diameter and the testing temperature. The results further show that it is possible to predict ?–N curves even for those combinations of the influential parameters for which no experimental data about the material endurance is available. This fact makes the presented model very suitable for the application in an R&D process when a durability of a product should be estimated on the basis of a very limited set of experimental data about the material endurance characteristics.     

A new algorithm for estimating fatigue life under mean value of stress

This paper discusses two problems: allowing for mean value of torsional stress and the variability of material properties with out of‐parallel fatigue characteristics. The effect of normal mean stress and shear mean stress is modified by reduction coefficients, which, to a large extent, depend on the value of existing loads. These coefficients have been developed experimentally on the basis of an analysis of the findings from fatigue tests on 2017A‐T4 and 6082‐T6 aluminium alloys and S355 alloy steel. The methods of calculation, suggested in this paper, are applicable to the materials in elastic–plastic state. The suggested algorithm for estimating fatigue life for the combination of bending or tension and compression and torsion under shear stress is based on Kluger's stress criterion. The usability of the algorithm was verified by comparing the calculation results with the results of own experimental tests on 2017A‐T4 and 6082‐T6 aluminium alloys, which have been noted to indicate sensitivity to shear mean stress, and the tests found in the professional literature (tests on S355, 30CrNiMo8 and 30NCD16 steel and Ti‐6Al‐4 V titanium alloy). A comparative analysis of the calculation and experimental results proved that there is a satisfactory correlation between them.     

The effect of mean stress and stress biaxiality in high‐cycle fatigue

This article presents a review of selected multiaxial high‐cycle fatigue criteria with an emphasis on their ability to take into account the mean stress effect and the effect of a biaxial stress state. It is shown that the predictions of the various criteria are very different for the case of biaxial tensile loads. This is in contrast to the case of combined tension‐torsion loads, where the predictions are very similar. The second part of the article investigates which mechanical parameter (eg, the hydrostatic stress or the normal stress) is the most appropriate to take into account these cyclic stress states.     

Fatigue limit and life evaluation formulae for compressive mean stress states

Considering the difference of stress concentration behaviour near micro-defects in materials, a unified fatigue limit evaluation formula has been developed, which can be applied to compressive mean stress states without the necessity of introducing additional material’s properties except for the modification parameter of mean stress. A generalised life evaluation formula has also been proposed and the dependence of fatigue life on mean stress can be simply expressed by the effect of fatigue limit corresponding to the mean stress. By considering the effect of fatigue limit corresponding to the compressive mean stress, the fatigue life can be evaluated by the same generalised formula as that developed for tensile mean stress state, without the necessity of carrying out additional S–N curve fatigue tests.     

Ratcheting behaviour and mean stress considerations in uniaxial low-cycle fatigue of Inconel 718 at 649 °C

The ratcheting behaviour of Inconel 718 was investigated at 649 °C under uniaxial cyclic loading. Stress-control tests have been conducted at various combinations of stress amplitude and mean stress. The ratcheting strain at failure increases with increasing mean stress for a given stress amplitude and with decreasing stress amplitude for a given mean stress. Fatigue lives were correlated using three mean stress models: the Goodman equation, the Smith–Watson–Topper (SWT) parameter and the Walker parameter. It has been shown that the Goodman equation and the SWT parameter do not correlate life data, while the Walker parameter yields acceptable correlation. The SWT parameter was modified to incorporate the ratcheting effect. The new parameter is found to yield correlation similar to that of the Walker parameter.     

Development of the positive mean stress diagrams using genetic algorithm approach

In this numerical study, a new optimum positive mean stress fatigue failure equation is developed using previous experimental data and genetic algorithm. Two independent curve fitting coefficients are implemented in the equation to supply better correlations with experimental data of the failure strength envelope. In the literature, Gerber, Goodman, Soderberg, Morrow, Bagci, ASME (elliptic line), Clemson, Sekercioglu and so on suggested different equations for estimating fatigue strength envelope under mean stress condition. In these models, the effect of the materials was not considered in details. Some of these mean stress linear expressions are very conservative or have stress area bigger than the yield limit. The yield strength and the effect of materials are considered in the proposed model. The values of the positive mean stress, which correspond to fatigue failure, are obtained, and a minimum average absolute error among the models presented in the literature is remarked. The proposed model, which has less conservative structure, can be effectively used in the mechanical design process.     

Effects of mean stress and stress concentration on fatigue behavior of short fiber reinforced polymer composites

An experimental study was conducted to evaluate the effect of mean stress on fatigue behavior of two short glass fiber reinforced thermoplastic composites and the effect of stress concentration on fatigue behavior of an unreinforced and a short glass fiber reinforced thermoplastic. Load‐controlled fatigue tests were conducted on unnotched (smooth) specimens at R ratios of ?1, 0.1, and 0.3 in different mold flow directions or fiber orientations and at a range of temperatures between ?40 and 125 °C. Effect of mean stress on fatigue life was found to be significant at all temperatures. Several mean stress parameters including modified Goodman, Walker, and Smith–Watson–Topper were evaluated for their ability to correlate mean stress data. A general fatigue life prediction model was also used to account for the effect of mean stress, temperature, and fiber orientation. Notched fatigue tests of an unreinforced polymer and a short glass fiber thermoplastic composite were also conducted using plate type specimens with a central circular hole and with or without the presence of mean stress. Effect of stress concentration was found to be considerable, with or without mean stress and in both the longitudinal and transverse directions. The commonly used Neuber's rule for metallic materials, nonlinear finite element analysis, as well as critical distance approaches were utilized for notch deformation and fatigue life analyses.     

Mean stress and ratcheting corrections in fatigue life prediction of metals

Ratcheting occurs easily because of the presence of mean stress during the stress‐control fatigue of engineering components. For ductility exhaustion dominated fatigue failure, a new fatigue life prediction model is developed by introducing the mean ratcheting strain rate to incorporate the effects of ratcheting and mean stress on fatigue life. The prediction accuracy of the proposed model was compared with that of the generalised damage parameter, Xia–Kujawski–Ellyin, Walker and Goswami models. Specifically, the model predictions and tested lives were compared using nine sets of experimental data from the literature. In the statistical analysis of these five models, the proposed model provides the highest accuracy and robust life predictions with the lowest model prediction errors.     

Evaluation of three current methods for including the mean stress effect in fatigue crack growth rate prediction

The fatigue crack growth rates curves of engineering materials depend on two parameters. In addition to the dependence on the classical stress intensity factor (SIF) range ΔK, there is a dependence on the mean load (or mean SIF), mainly in the near‐threshold region. The present paper provides some useful suggestions and good practices for using three of the current available methods to reduce this second dependence through the use of tuning constants. The methods considered here are the Elber, Walker and Vasudevan (or unified approach). For each approach, multiple regression analyses are performed on experimental data from the literature, and the correlations in two and three dimensions are graphically analyzed. Numerical examples of crack growth analysis for cracks growing under nominal stresses of constant amplitude in single‐edge and notch/hole geometries are performed, assuming an identical material component to that of the available experimental data. The resulting curves of crack size versus number of cycles (a versus N) are then compared. All three models gave approximately the same (a versus N) curves in both geometries. Differences between the behaviors of the (a versus N) curves in both geometries are highlighted, and the reasons for these particular behaviors are discussed.     

High‐cycle and very‐high‐cycle fatigue behaviour of a titanium alloy with equiaxed microstructure under different mean stresses

The fatigue behaviour of a titanium alloy Ti‐6Al‐4V with equiaxed microstructure (EM) under different values of tensile mean stress or stress ratio (R) was investigated from high‐cycle fatigue (HCF) to very‐high‐cycle fatigue (VHCF) regimes via ultrasonic axial cycling. The effect of mean stress or R on the fatigue strength of HCF and VHCF was addressed by Goodman, Gerber, and Authors' formula. Three types of crack initiation, namely, surface‐with‐RA (rough area), surface‐without‐RA, and interior‐with‐RA, were classified. The maximum value of stress intensity factor (SIF) at RA boundary for R < 0 keeps constant regardless of R in HCF and VHCF regimes. The SIF range at RA boundary for R > 0 also keeps constant regardless of R in VHCF regime, but this value decreases linearly with the increase of R for surface RA cases. The microstructure observation at RA regions gives a new result of nanograin formation only in the cases of negative stress ratios for the titanium alloy with EM, which is explained by the mechanism of numerous cyclic pressing.     

Fatigue behaviour and modelling of talc‐filled and short glass fibre reinforced thermoplastics,including temperature and mean stress effects

Effects of temperature and mean stress on fatigue behaviour of talc‐filled polypropylene (PP‐T) and short glass fibre reinforced polypropylene (PP‐G), polyamide‐66 (PA66), and a blend of polyphenylene ether and polystyrene (PPE/PS) were investigated. Load‐controlled fatigue tests were conducted under positive stress ratios (R = 0.1 and 0.3) and at several temperatures (T = 23, 85 and 120 °C). Larson–Miller parameter was used and a shift factor of Arrhenius type was developed to correlate fatigue data at various temperatures. Effect of mean stress on fatigue life was significant for some of the studied materials; however, for the PPE/PS blend no effect of mean stress was observed. Modified Goodman and Walker mean stress equations were evaluated for their ability to correlate mean stress data. A general fatigue life prediction model was also used to account for the effects of mean stress, temperature, anisotropy and frequency.     

The effect of material hardness and mean stress on the fatigue limit of steels containing small defects

Fatigue tests on material containing small defects were performed under a wide range of mean stress for three grades of steels with different hardness. The ΔK_th of small defects had a peculiar dependency on material hardness and mean stress, which was quite different from those of long cracks or plain specimens. The crack closure of short cracks was measured. It was shown that the formation of the crack closure was affected by the material hardness and mean stress. This behaviour of crack closure resulted in characteristic fatigue limit properties of materials containing small defects.     

Effect of mean stress on fretting fatigue of Ti-6Al-4V on Ti-6Al-4V

Fretting fatigue tests of Ti‐6Al‐4V on Ti‐6Al‐4V have been conducted to determine the influence of stress amplitude and mean stress on life. The stress ratio was varied from R=−1 to 0.8. Both flat and cylindrical contacts were studied using a bridge‐type fretting fatigue test apparatus operating either in the partial slip or mixed fretting regimes. The fretting fatigue lives were correlated to a Walker equivalent stress relation. The influence of mean stress on fretting fatigue crack initiation, characterized by the value of the Walker exponent, is smaller compared with plain fatigue. The fretting fatigue knockdown factor based on the Walker equivalent stress is 4. Formation of fretting cracks is primarily associated with the tangential force amplitude at the contact interface. A simple fretting fatigue crack initiation metric that is based on the strength of the singular stress field at the edge of contact is evaluated. The metric has the advantage in that it is neither dependent on the coefficient of friction nor the location of the stick/slip boundary, both of which are often difficult to define with certainty a priori.     

Study on material property changes of mild steel S355 caused by block loads with varying mean stress

The paper presents the results of research on material property and structure changes of S355 steel samples induced by fatigue of material subjected to block bending loads, with varying mean load value. In the tests, the mean load was increased and decreased in subsequent blocks, where the amplitude varies in accordance with the accepted maximum bending moment value. The changes of strains recorded during the tests show higher maximum mean strain for the case, where the mean load increases in block. Otherwise, the maximum mean strain is three times less, than before. Some hardening occurs, which affects the mechanical properties of the material. This is visible in the results of tensile tests, where ultimate strength increases for load path with increased mean load value. Metallographic examinations revealed no fatigue cracks at this level of fatigue life.     

Uniaxial ratcheting behavior and fatigue life models of commercial pure titanium

The uniaxial fatigue and ratcheting behavior of commercial pure titanium (CP‐Ti) was investigated by asymmetric cyclic stress‐controlled experiments at room temperature. The effects of mean stress, stress amplitude, stress ratio, and peak stress on ratcheting behavior and fatigue life were discussed. It was found that increasing mean stress, stress amplitude, and peak stress or decreasing stress ratio reduced fatigue life and promoted ratcheting behavior. The applicability of different fatigue life models was analyzed, and a new stress ratio‐related failure model was proposed based on the exponential increase of fatigue life with stress ratio. Among all the models investigated in this study, the exponential stress ratio‐related model has more advantage in fatigue life predictions for CP‐Ti under ratcheting‐fatigue interaction.     

Mean stress effects in strain–life fatigue

A mean stress equation can be incorporated into the strain–life curve in a manner that is consistent with the stress-based use of the same equation. Doing so for the Walker mean stress relationship gives excellent results for a number of strain–life data sets with non-zero mean stresses, including data on steels, one titanium alloy and aluminium alloys. This approach has a number of advantages: All data at all mean stresses can be combined into a single fitting procedure to determine the constants for the stress–life curve, which values also apply to the elastic strain term of the strain–life curve. The Walker parameter γ that also arises from this fitting is related to the sensitivity of the material to mean stress, giving this approach a versatility that is not possessed by other common mean stress methods. Where non-zero mean stress data are not available to obtain γ from fitting, an equation based on existing fitted values can be used to make estimates for steels. For precipitation-hardened aluminium alloys in the 2000 and 7000 series, an estimate of  γ= 0.5  may be applied, so that the method becomes similar to that of Smith, Watson and Topper. For other metals, a default estimate of  γ= 0.5  is suggested. For life estimates using the strain-based approach, it is recommended that the Walker mean stress method, incorporated into the strain–life curve, should be employed as an alternative to other methods, or perhaps to even replace them entirely.