Mean stress and plasticity effect prediction on notch fatigue and crack growth threshold,combining the theory of critical distances and multiaxial fatigue criteria

In the present work, we propose a robust calibration of some bi‐parametric multiaxial fatigue criteria applied in conjunction with the theory of critical distances (TCD). This is based on least‐square fitting fatigue data generated using plain and sharp‐notched specimens tested at two different load ratios and allows for the estimation of the critical distance according to the point and line method formulation of TCD. It is shown that this combination permits to incorporate the mean stress effect into the fatigue strength calculation, which is not accounted for in the classical formulation of TCD based on the range of the maximum principal stress. It is also shown that for those materials exhibiting a low fatigue‐strength‐to‐yield‐stress ratio σ_fl,R = ?1/σ_YS, such as 7075‐T6 (σ_fl,R = ?1/σ_YS = 0.30), satisfactorily accurate predictions are obtained assuming a linear‐elastic stress distribution, even at the tip of sharp notches and cracks. Conversely, for any materials characterized by higher values of this ratio, as quenched and tempered 42CrMo4 (σ_fl,R = ?1/σ_YS = 0.54), it is recommended to consider the stabilized elastic‐plastic stress/strain distribution, also for plain and blunt‐notched samples and even in the high cycle fatigue regime still with the application of the TCD.     

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.     

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.     

Multiaxial notch fatigue life prediction based on pseudo stress correction and finite element analysis under variable amplitude loading

A new computational methodology is proposed for fatigue life prediction of notched components subjected to variable amplitude multiaxial loading. In the proposed methodology, an estimation method of non‐proportionality factor (F) proposed by authors in the case of constant amplitude multiaxial loading is extended and applied to variable amplitude multiaxial loading by using Wang‐Brown's reversal counting approach. The pseudo stress correction method integrated with linear elastic finite element analysis is utilized to calculate the local elastic‐plastic stress and strain responses at the notch root. For whole local strain history, the plane with weight‐averaged maximum shear strain range is defined as the critical plane in this study. Based on the defined critical plane, a multiaxial fatigue damage model combined with Miner's linear cumulative damage law is used to predict fatigue life. The experimentally obtained fatigue data for 7050‐T7451 aluminium alloy notched shaft specimens under constant and variable amplitude multiaxial loadings are used to verify the proposed methodology and equivalent strain‐based methodology. The results show that the proposed methodology is superior to equivalent strain‐based methodology.     

A stress-based method to predict lifetime under multiaxial fatigue loadings

This paper extends to low/medium‐cycle fatigue a stress‐based method recently proposed by the same authors for high‐cycle multiaxial fatigue assessments. By considering the plane of maximum shear stress amplitude coincident with the microcrack initiation plane, the method requires the calculation both of the maximum shear stress amplitude and the maximum normal stress relative to the same plane. Multiaxial fatigue life estimates are made by means of bi‐parametric modified Wöhler curves, which take into account the mean stress effect, the influence of the out‐of‐phase angle and the presence of notches by using a generalization to multiaxial fatigue of the fatigue strength reduction factor K_f. Approximately 700 experimental data taken from the literature are used to demonstrate that the method is a useful tool to summarize fatigue strength data of both smooth and notched components, subjected to either in‐phase or out‐of‐phase loads. Finally, a simple practical rule for the calculation of the multiaxial fatigue strength reduction factor is proposed.     

Notch fatigue life prediction considering nonproportionality of local loading path under multiaxial cyclic loading

An innovative numerical methodology is presented for fatigue lifetime estimation of notched bodies experiencing multiaxial cyclic loadings. In the presented methodology, an evaluation approach of the local nonproportionality factor F for notched specimens, which defines F as the ratio of the pseudoshear strain range at 45° to the maximum shear plane and the maximum shear strain range, is proposed and discussed deeply. The proposed evaluation method is incorporated into the material cyclic stress‐strain equation for purpose of describing the nonproportional hardening behavior for some material. The comparison between multiaxial elastic‐plastic finite element analysis (FEA) and experimentally measured strains for S460N steel notched specimens shows that the proposed nonproportionality factor estimation method is effective. Subsequently, the notch stresses and strains calculated utilizing multiaxial elastic‐plastic FEA are used as input data to the critical plane‐based fatigue life prediction methodology. The prediction results are satisfactory for the 7050‐T7451 aluminum alloy and GH4169 superalloy notched specimens under multiaxial cyclic loading.     

A critical distance/plane method to estimate finite life of notched components under variable amplitude uniaxial/multiaxial fatigue loading

The present paper summarises the main features of a design technique we have devised to specifically perform, by post-processing the linear-stress fields in the vicinity of the assumed crack initiation sites, the fatigue assessment of notched components subjected to in-service variable amplitude (VA) uniaxial/multiaxial fatigue loading. In more detail, fatigue damage is estimated through the Modified Wöhler Curve Method (MWCM) applied along with the Theory of Critical Distances (TCDs), the latter being used in the form of the Point Method (PM). According to the philosophy on which the linear-elastic TCD is based, the adopted critical distance is treated as a material property whose length increases as the number of cycles to failure decreases. To correctly apply the MWCM, the orientation of the critical plane is suggested here as being calculated through that direction experiencing the maximum variance of the resolved shear stress. Further, the above direction is used also to perform the cycle counting: since, by definition, the resolved shear stress is a monodimensional stress quantity, fatigue cycles are counted by taking full advantage of the classical three-point Rain Flow method. From a philosophical point of view, the real novelty contained in the present paper is that eventually all the different pieces of theoretical work we have done over the last 15 years by investigating different aspects of the uniaxial/multiaxial fatigue issue are consistently brought together by formalising a design methodology of general validity. The accuracy and reliability of the proposed fatigue assessment technique was checked by using 124 experimental results generated by testing notched cylindrical samples of carbon steel C40. The above tests were run under three different load spectra, by exploring uniaxial as well as in- and out-of-phase biaxial situations, in the latter case the axial and torsional load signals being not only characterised by non-zero mean values, but also by different frequencies. To conclude it can be said that such a systematic validation exercise allowed us to prove that the proposed approach is highly accurate, resulting in estimates falling within the constant amplitude (CA) fully-reversed uniaxial and torsional scatter bands used to calibrate the method itself (this holding true independently of both complexity of the applied VA loading path and sharpness of the tested notch).     

Four stress analysis strategies to use the Modified Wöhler Curve Method to perform the fatigue assessment of weldments subjected to constant and variable amplitude multiaxial fatigue loading

The present paper investigates the different ways of using the Modified Wöhler Curve Method (MWCM) to perform the fatigue assessment of steel and aluminium welded joints subjected to in-service variable amplitude (VA) multiaxial load histories. Thanks to its specific features, the above critical plane approach can efficiently be applied in terms of both nominal, hot-spot, and local quantities, that is, by using any of the stress analysis strategies suggested by the Design Recommendations of the International Institute of Welding (IIW). The MWCM can efficiently be used also along with the so-called Theory of Critical Distances applied in the form of the Point Method (PM). The accuracy of the different formalisations of the MWCM investigated in the present paper was systematically checked against a large number of experimental results taken from the literature and generated by testing, under VA biaxial nominal loading, welded samples having different geometries. Such a systematic validation exercise allowed us to prove that our multiaxial fatigue criterion is successful in designing welded joints against VA multiaxial fatigue, this holding true independently from both definition adopted to calculate the necessary stress quantities and complexity of the assessed load history.     

Low‐cycle fatigue/high‐cycle fatigue interactions in notched Ti‐6Al‐4V*

Combined low‐cycle fatigue/high‐cycle fatigue (LCF/HCF) loadings were investigated for smooth and circumferentially V‐notched cylindrical Ti–6Al–4V fatigue specimens. Smooth specimens were first cycled under LCF loading conditions for a fraction of the previously established fatigue life. The HCF 10⁷ cycle fatigue limit stress after LCF cycling was established using a step loading technique. Specimens with two notch sizes, both having elastic stress concentration factors of K_t = 2.7, were cycled under LCF loading conditions at a nominal stress ratio of R = 0.1. The subsequent 10⁶ cycle HCF fatigue limit stress at both R = 0.1 and 0.8 was determined. The combined loading LCF/HCF fatigue limit stresses for all specimens were compared to the baseline HCF fatigue limit stresses. After LCF cycling and prior to HCF cycling, the notched specimens were heat tinted, and final fracture surfaces examined for cracks formed during the initial LCF loading. Fatigue test results indicate that the LCF loading, applied for 75% of total LCF life for the smooth specimens and 25% for the notched specimens, resulted in only small reductions in the subsequent HCF fatigue limit stress. Under certain loading conditions, plasticity‐induced stress redistribution at the notch root during LCF cycling appears responsible for an observed increase in HCF fatigue limit stress, in terms of net section stress.     

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.     

Intrinsic material length,Theory of Critical Distances and Gradient Mechanics: analogies and differences in processing linear‐elastic crack tip stress fields

The Theory of Critical Distances (TCD) is a bi‐parametrical approach suitable for predicting, under both static and high‐cycle fatigue loading, the non‐propagation of cracks by directly post‐processing the linear‐elastic stress fields, calculated according to continuum mechanics, acting on the material in the vicinity of the geometrical features being assessed. In other words, the TCD estimates static and high‐cycle fatigue strength of cracked bodies by making use of a critical distance and a reference strength which are assumed to be material constants whose values change as the material microstructural features vary. Similarly, Gradient Mechanics postulates that the relevant stress fields in the vicinity of crack tips have to be determined by directly incorporating into the material constitutive law an intrinsic scale length. The main advantage of such a method is that stress fields become non‐singular also in the presence of cracks and sharp notches. The above idea can be formalized in different ways allowing, under both static and high‐cycle fatigue loading, the static and high‐cycle fatigue assessment of cracked/notched components to be performed without the need for defining the position of the failure locations a priori. The present paper investigates the existing analogies and differences between the TCD and Gradient Mechanics, the latter formalized according to the so‐called Implicit Gradient Method, when such theories are used to process linear‐elastic crack tip stress fields.     

A notch critical plane approach of multiaxial fatigue life prediction for metallic notched specimens

An approach based on the local stress response is proposed to locate the fatigue critical point for metallic blunt notched specimens under multiaxial fatigue loading. According to the stress analysis, both stress gradient and gradient of loading nonproportionality exist at notch root. The plane in the vicinity of the notch that passes through the fatigue critical point and experiences the maximum shear stress amplitude is defined as the critical plane for notch specimens (CPN). Furthermore, the Susmel's fatigue damage parameter is modified to assess fatigue life of notched components by combining CPN and the theory of critical distance (TCD). The multiaxial fatigue test of the thin‐walled round tube specimens made of Ni‐base alloy GH4169 is carried out to verify the above approaches. In addition, test data of two kinds of materials are collected. The results show that the maximum absolute error of the fatigue critical point is 9.6° and the majority of the predicted life falls within the three‐time scatter band.     

Multiaxial fatigue behaviour of Sintered Steels under combined in and out of phase bending and torsion

This paper discusses the fatigue behaviour of sintered steels under multiaxial loading. These steels are the Fe-1.5% Cu and the Fe-2.0% Cu-2.5% Ni, sintered at low and high temperatures, in the densities 7.1 and 7.4 g/cm³, which are used in the production of several ready to assemble automotive parts. Fully reversed or pulsating combined loading with constant frequency and amplitudes acting in and out of phase, was applied to round notched specimens (K_tb = 1.49, K_tt = 1.24) in the finite fatigue life region (10⁴ ≤ N_f ≤ 2 · 10⁶). The mechanics of crack initiation and propagation as well as rupture were studied using fractography and microfractography. These analyses led to a mechanical model based on local normal stresses for the fatigue life evaluation. The fatigue life evaluation on the base of the local bending stress obtained under uniaxial loading describes the test results for in phase combined bending and torsion satisfactorily. But the increase of fatigue strength and life by out of phase loading is overestimated in the case of fully reversed loading. However for design purposes the out of phase loading can be neglected because of its beneficial effect in increasing fatigue life for this type of material. If the dependence of the different stress concentrations under combined in and out of phase loading on the supportable local bending stress obtained under uniaxial loadings is considered, then the calculation procedure covers all test results.     

On the prediction of high-cycle fretting fatigue strength: Theory of critical distances vs. hot-spot approach

This paper summarises the results we obtained when applying both the theory of critical distances (TCD) and the hot-spot approach to predict high-cycle fretting fatigue strength. In particular, the accuracy of such approaches was checked considering some experimental results taken from the literature and ad hoc generated to explicitly investigate the size effect in fretting fatigue. According to the well-known experimental outcome that initiation and initial growth of fretting fatigue cracks is mainly mixed-mode dominated, both the TCD and the hot-spot approach were used in conjunction with two different multiaxial fatigue criteria: the so-called modified Wöhler curve method (MWCM), that is, a conventional critical plane approach, and the well-known mesoscopic criterion due to Dang Van. Considering cylindrical-on-flat contacts tested under partial slip conditions, it was seen that the TCD is successful in predicting the size effect in fretting fatigue, resulting in more accurate predictions than those obtained by applying the classical hot-spot approach. Moreover, the present study revealed that the overall best accuracy is obtained when applying the TCD along with the MWCM. This result is very promising, especially in light of the fact that such a design methodology can be employed by simply post processing linear-elastic FE results, making it suitable for being used to assess real mechanical assemblies without the need for carrying out time-consuming elasto-plastic analyses.     

Estimating fatigue damage under variable amplitude multiaxial fatigue loading

The present paper is concerned with the use of the modified Wöhler curve method (MWCM) to estimate both lifetime and high‐cycle fatigue strength of plain engineering materials subjected to complex load histories resulting, at critical locations, in variable amplitude (VA) multiaxial stress states. In more detail, when employed to address the constant amplitude (CA) problem, the MWCM postulates that fatigue damage reaches its maximum value on that material plane (i.e. the so‐called critical plane) experiencing the maximum shear stress amplitude, fatigue strength depending on the ratio between the normal and shear stress components relative to the critical plane itself. To extend the use of the above criterion to those situations involving VA loadings, the MWCM is suggested here as being applied by defining the critical plane through that direction experiencing the maximum variance of the resolved shear stress. Such a direction is used also to perform the cycle counting: because the resolved shear stress is a monodimensional quantity, stress cycles are directly counted by the classical rain‐flow method. The degree of multiaxiality and non‐proportionality of the time‐variable stress state at the assumed critical sites instead is suggested as being measured through a suitable stress ratio which accounts for the mean value and the variance of the stress perpendicular to the critical plane as well as for the variance of the shear stress resolved along the direction experiencing the maximum variance of the resolved shear stress. Accuracy and reliability of the proposed approach was checked by using several experimental results taken from the literature. The performed validation exercise seems to strongly support the idea that the approach formalized in the present paper is a powerful engineering tool suitable for estimating fatigue damage under VA multiaxial fatigue loading, and this holds true not only in the medium‐cycle, but also in the high‐cycle fatigue regime.     

The linear‐elastic Theory of Critical Distances to estimate high‐cycle fatigue strength of notched metallic materials at elevated temperatures

This paper investigates the accuracy of the linear‐elastic Theory of Critical Distances (TCD) in estimating high‐cycle fatigue strength of notched metallic materials experiencing elevated temperatures during in‐service operations. The TCD postulates that the fatigue damage extent can be estimated by directly post‐processing the entire linear‐elastic stress field acting on the material in the vicinity of the crack initiation locations. The key feature of this theory is that the high‐cycle fatigue assessment is based on a scale length parameter that is assumed to be a material property. The accuracy of this design method was checked against a number of experimental results generated, under axial loading, by testing, at 250 °C, notched specimens of carbon steel C45. To further investigate the reliability of the TCD, its accuracy was also checked via several data taken from the literature, these experimental results being generated by testing notched samples of Inconel 718 at 500 °C as well as notched specimens of directionally solidified superalloy DZ125 at 850 °C. This validation exercise allowed us to prove that the linear‐elastic TCD is successful in estimating high‐cycle fatigue strength of notched metallic materials exposed to elevated temperature, resulting in estimates falling within an error interval of ±20%. Such a high level of accuracy suggests that, in situations of practical interest, reliable high‐cycle fatigue assessment can be performed without the need for taking into account those non‐linearities characterising the mechanical behaviour of metallic materials at high temperature, the used critical distance being still a material property whose value does not depend on the sharpness of the notch being designed.     

Long fatigue life critical crack lengths*

A model based on surface strain redistribution and crack closure is presented for prediction of the endurance or fatigue limit stress by determining the threshold stress and critical length of short cracks that develop under microstructural control. The threshold stress first decreases with crack size to a local minimum then increases to a local maximum corresponding to the fatigue limit stress. This occurs at the critical crack length corresponding to about four grain diameters. The model is capable of determining the threshold stress range and depth of propagating and non‐propagating surface cracks as a function of stress ratio, material and grain size. The microstructure is shown to be particularly significant in the very long life regime (N_f ≈ 10⁹ cycles). When the surface cracks become non‐propagating, internally initiated cracks continue growing slowly, eventually reaching the critical crack length with failure occurring after a very high number of cycles (10⁷ < N_f < 10⁹ cycles).     

FRACTURE MECHANICS ANALYSIS OF FATIGUE RESISTANCE OF SPOT WELDED COACH-PEEL JOINTS

Resistance spot welding is the most widely used joining method in automobile manufacture. The number, location, and quality of welds are some of the factors that influence the performance of welded subassemblies, and body panel structures. Therefore, design optimization requires knowledge of not only sheet metal behavior, but also weld behavior under service loadings. A linear elastic fracture mechanics approach was employed in this study to estimate the fatigue lives of spot welds subjected to tearing loads in a coach-peel specimen. Using a finite element method (FEM), the initial J-integral values for five coach-peel joints, each with different geometries, were calculated. Fatigue tests conducted on the same weld geometries provided life data. The experimental data were used to derive a relationship between the initial elastic J-integral values (ΔJ_e) and the fatigue life. It was found that the total fatigue life (N_f) of a weld at one applied stress range is related to its range of J-integral value such that a ΔJ_e vs N_f log-log plot gives a straight line relationship. This relationship can be used to evaluate the effects of geometrical variables on the fatigue life of coach-peel joints. The results show that, within the dimension range studied here, the effects of geometrical variables on the fatigue resistance can be ranked in the following decreasing order: weld eccentricity, sheet thickness and weld nugget diameter.     

The theory of critical distances and fatigue from notches in aluminium 6061

Fatigue failure of metal components containing notches, cracks and other defects has been an active research topic for many years because of its important practical and theoretical implications. Recently, Taylor and his colleagues have re‐visited this topic and proposed the theory of critical distance (TCD), which summarizes the early work by Neuber, Peterson and others in a unifying theory and predicts fatigue fracture with the use of a critical distance, L_o. In this paper, an experimental and numerical study of the fatigue of notched and un‐notched 6061 aluminium alloys is used to verify the TCD and some of the limitations of the TCD are discussed on this basis.