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.     

Comparison of HCF life prediction methods based on different critical planes under multiaxial variable amplitude loading

High‐cycle fatigue life prediction methods based on different critical planes, including the maximum shear stress (MSS) plane, the weighted average shear stress plane and the Maximum Variance shear stress plane, are compared by two multiaxial cycle counting methods, i.e. the main and auxiliary channels (MAC) counting and the relative equivalent stress counting. A modified damage model is used to calculate the multiaxial fatigue damage. Compared with the experimental lives for 7075‐T651 aluminium alloy, the predicted results show that the MSS method together with MAC counting is suitable for the multiaxial fatigue life prediction.     

A high-cycle fatigue life model for variable amplitude multiaxial loading

A high‐cycle fatigue life model for structures subjected to variable amplitude multiaxial loading is presented in this paper. It treats any kind of repeated blocks of variable amplitude multiaxial loading without using a cycle counting method. This model based on a mesoscopic approach is characterized by the following features: (i) the choice of a damage factor related to the accumulated mesoscopic plastic strain per stabilised cycle; (ii) the use of a mesoscopic mechanical behaviour taking into account the fatigue mechanisms such as plasticity and void growth. This behaviour is a von Mises elastoplastic model with linear kinematic hardening and hydrostatic stress dependent yield stress. The fatigue life model has six parameters identified with one SN curve and two fatigue limits. In‐phase and out‐of‐phase experimental tests from the literature are simulated. The predicted fatigue lives are compared to experimental ones.     

Multiaxial notch fatigue life prediction based on the dominated loading control mode under variable amplitude loading

A new calculation approach is suggested to the fatigue life evaluation of notched specimens under multiaxial variable amplitude loading. Within this suggested approach, if the computed uniaxial fatigue damage by the pure torsional loading path is larger than that by the axial tension–compression loading path, a shear strain‐based multiaxial fatigue damage parameter is assigned to calculate multiaxial fatigue damage; otherwise, an axial strain‐based multiaxial fatigue damage parameter is assigned to calculate multiaxial fatigue damage. Furthermore, the presented method employs shear strain‐based and axial strain‐based multiaxial fatigue damage parameters in substitution of equivalent strain amplitude to consider the influence of nonproportional additional hardening. The experimental data of GH4169 superalloy and 7050‐T7451 aluminium alloy notched components are used to illustrate the presented multiaxial fatigue lifetime estimation approach for notched components, and the results reveal that estimations are accurate.     

Fatigue assessment of metallic components under uniaxial and multiaxial variable amplitude loading

In the present paper, the fatigue lifetime of metallic structural components subjected to variable amplitude loading is evaluated by applying 2 different multiaxial high‐cycle fatigue criteria. Such criteria, proposed by some of the present authors, are based on the critical plane approach and aim at reducing a given multiaxial stress state to an equivalent uniaxial stress condition. In particular, the procedure employed by both criteria consists of the following 3 steps: (1) definition of the critical plane; (2) counting of loading cycles; and (3) estimation of fatigue damage. Finally, the previous criteria are validated by comparing the theoretical results with experimental data related to smooth metallic specimens subjected to uniaxial and multiaxial variable amplitude loading.     

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.     

Life prediction based on weight‐averaged maximum shear strain range plane under multiaxial variable amplitude loading

Based on Wang and Brown's reversal counting method, a new approach to the determination of the critical plane is proposed by the defined plane with a weight‐averaged maximum shear strain range under multiaxial variable amplitude loading. According to the determined critical plane, a detailed procedure of multiaxial fatigue life prediction is introduced to predict lives in the low‐cycle multiaxial fatigue regime. The proposed approach is verified by two multiaxial fatigue damage models and Miner's linear cumulative damage law. The results showed that the proposed approach can effectively predict the orientation of the failure plane under multiaxial variable amplitude loading and give a satisfactory life prediction.     

A critical plane‐energy model for multiaxial fatigue life prediction

A new critical plane‐energy model is proposed in this paper for multiaxial fatigue life prediction of metals. A brief review of existing methods, especially on the critical plane‐based and energy‐based methods, is given first. Special focus is on the Liu–Mahadevan critical plane approach, which has been shown to work for both brittle and ductile metals. One potential drawback of the Liu–Mahadevan model is that it needs an empirical calibration parameter for non‐proportional multiaxial loadings because only the strain terms are used and the out‐of‐phase hardening cannot be explicitly considered. An energy‐based model using the Liu–Mahadevan concept is proposed with the help of the Mróz–Garud plasticity model. Thus, the empirical calibration for non‐proportional loading is not needed because the out‐of‐phase hardening is naturally included in the stress calculation. The model predictions are compared with experimental data from open literature, and the proposed model is shown to work for both proportional and non‐proportional multiaxial loadings without the empirical calibration.     

An improved critical plane and cycle counting method to assess damage under variable amplitude multiaxial fatigue loading

The plane with the maximum variance of the resolved shear stress is taken as the critical plane. Two algorithms are used along with the maximum variance method (MVM) to determine the orientation of the critical plane. The maximum variance of the normal stress on the potential critical planes is calculated to determine the one experiencing the maximum extent of fatigue damage. A new multiaxial cycle counting method is proposed to count cycles on the critical plane. The modified Wöhler curve method is used to assess fatigue damage. About 200 experimental results were collected from the technical literature to validate the approaches being proposed. The results show that the improved design technique being proposed is successful in assessing fatigue damage under variable amplitude multiaxial cyclic loading.     

Multiaxial low‐cycle fatigue life evaluation under different non‐proportional loading paths

This paper presents analytical and experimental investigations for fatigue lives of structures under uniaxial, torsional, multiaxial proportional, and non‐proportional loading conditions. It is known that the rotation of principal stress/strain axes and material additional hardening due to non‐proportionality of cycle loading are the 2 main causes resulting in shorter fatigue lives compared with those under proportional loading. This paper treats these 2 causes as independent factors influencing multiaxial fatigue damage and proposes a new non‐proportional influencing parameter to consider their combined effects on the fatigue lives of structures. A critical plane model for multiaxial fatigue lives prediction is also proposed by using the proposed non‐proportional influencing factor to modify the Fatemi‐Socie model. The comparison between experiment results and theoretical evaluation shows that the proposed model can effectively predict the fatigue life due to multiaxial non‐proportional loading.     

Fatigue analysis of ductile and brittle behaving steels under variable amplitude multiaxial loading

The analysis of fatigue behavior under multiaxial variable amplitude stress states, despite its wide applicability, has not been fully studied. Issues such as varying degrees of nonproportionality of the load history, cycle counting, damage accumulation, failure behavior of the material, and mean stress fluctuations which can significantly affect the results of these analyses have not been well understood. In this study, a methodology for the analysis of fatigue behavior under multiaxial variable amplitude loading conditions is employed which accounts for the aforementioned issues. At its core, the applied methodology uses critical plane analysis based on the failure behavior of each material to assess the fatigue damage. In order to evaluate the performance of the analysis method, axial, torsional, and combined axial‐torsional variable amplitude tests were performed on one ductile and one brittle behaving steel. The applied methodology resulted in close estimation of the experimental fatigue life for both ductile and brittle behaving steels.     

Low‐cycle fatigue life prediction of various metallic materials under multiaxial loading

This paper proposed a simple life prediction model for assessing fatigue lives of metallic materials subjected to multiaxial low‐cycle fatigue (LCF) loading. This proposed model consists of the maximum shear strain range, the normal strain range and the maximum normal stress on the maximum shear strain range plane. Additional cyclic hardening developed during non‐proportional loading is included in the normal stress and strain terms. A computer‐based procedure for multiaxial fatigue life prediction incorporating critical plane damage parameters is presented as well. The accuracy and reliability of the proposed model are systematically checked by using about 300 test data through testing nine kinds of material under both zero and non‐zero mean stress multiaxial loading paths.     

A weight function-critical plane approach for low-cycle fatigue under variable amplitude multiaxial loading

Low‐cycle fatigue data of type 304 stainless steel obtained under axial‐torsional loading of variable amplitudes are analyzed using four multiaxial fatigue parameters: SWT, KBM, FS and LKN. Rainflow cycle counting and Morrow's plastic work interaction rule are used to calculate fatigue damage. The performance of a fatigue model is dependent on the fatigue parameter, the critical plane and the damage accumulation rule employed in the model. The conservatism and non‐conservatism of predicted lives are examined for some combinations of these variables. A new critical plane called the weight function‐critical plane is introduced for variable amplitude loading. This approach is found to improve the KBM‐based life predictions.     

An improved multiaxial high‐cycle fatigue criterion based on critical plane approach

Several groups of fatigue damage parameters are discussed and then an improved multiaxial high‐cycle fatigue criterion based on critical plane defined by the plane of maximum shear stress range is presented in this paper. A compromising solution to consider the mean normal stress acting on the critical plane is also proposed. The new fatigue criterion extends the range of metallic materials which is valid for the ratio 1.25 < f_?1/t_?1 < 2. The predictions based on the presented model show a good agreement with test data.     

A review of multiaxial fatigue criteria for random variable amplitude loads

Nowadays, the estimation of fatigue life under multiaxial random loading is still an extremely complex task. In this paper, a comprehensive review of the multiaxial random fatigue criteria available in the literature is presented. Such a review is mainly devoted to stress‐based criteria for the evaluation of fatigue life in high‐cycle regime. Time and frequency domain approaches are examined. The focus of this paper is related to uniform stress/strain distribution, but also the effect of stress/strain gradient is tangentially addressed. More than 200 references are cited.     

A multiaxial fatigue criterion for random loading

ABSTRACT A multiaxial fatigue criterion for random loading is proposed. Firstly, the orientation of the critical plane, where fatigue life estimation is carried out, is determined from the weighted mean position of the principal stress directions. Then, the scalar value of the normal stress vector N (t) perpendicular to the critical plane is taken as the cycle counting variable since the direction of such a vector is fixed with respect to time (conversely to the time‐varying direction of the shear stress vector C (t)), and a nonlinear combination of normal and shear stress components acting on the critical plane is used to define an equivalent stress amplitude. Finally, a damage accumulation model is employed to process such an equivalent stress amplitude and to determine fatigue endurance. This criterion is herein applied to some relevant random fatigue tests (proportional bending and torsion).     

A fatigue driving energy approach to high‐cycle fatigue life estimation under variable amplitude loading

In this paper, a concept of fatigue driving energy is formulated to describe the process of fatigue failure. The parameter is taken as a combination of the fatigue driving stress and strain energy density. By assessing the change of this parameter, a new non‐linear damage model is proposed for residual life estimation within high‐cycle fatigue regime under variable amplitude loading. In order to consider the effects of loading histories on damage accumulation under such condition, the load interaction effects are incorporated into the new model, and a modified version is thus developed. Life predictions by these two models and Miner rule are compared using experimental data from literature. The results show that the proposed model gives lower deviations than the Miner rule, while the modified model shows better prediction performances than the others. Moreover, the proposed model and its modifications are ease of implementation with the use of S–N curve.     

A novel engineering method based on the critical plane concept to estimate the lifetime of weldments subjected to variable amplitude multiaxial fatigue loading

This paper summarizes an attempt at proposing a new engineering method suitable for estimating the fatigue lifetime of steel‐ and aluminium‐welded connections subjected to variable amplitude multiaxial fatigue loading. In particular, the proposed approach is based on the use of the so‐called Modified Wöhler Curve Method (MWCM), i.e. a bi‐parametrical critical plane approach, whose accuracy has been checked so far solely in addressing the constant amplitude multiaxial fatigue problem. In order to extend the use of our criterion to variable amplitude situations, the critical plane is suggested here as being determined by taking full advantage of the maximum variance concept, that is, such a plane is assumed to be the one containing the direction along which the variance of the resolved shear stress reaches its maximum value. The main advantage of such a strategy is that the cycle counting can directly be performed by considering the shear stress resolved along the maximum variance direction: by so doing, the problem is greatly simplified, allowing those well‐established cycle counting methods specifically devised to address the uniaxial variable amplitude problem to be extended to those situations involving multiaxial fatigue loading. The validity of the proposed methodology was checked by using two different datasets taken from the literature and generated by testing both steel and aluminium tube‐to‐plate welded connections subjected to in‐phase and 90° out‐of‐phase variable amplitude bending and torsion. This new fatigue life assessment technique was seen to be highly accurate allowing the estimates to fall within the calibration scatter bands not only when the constants in the governing equations were calculated by using the experimental uniaxial and torsional fully reversed fatigue curves, but also when they were determined by using the reference curves supplied, for the investigated geometry, by the available standard codes. These results seem to strongly support the idea that, thanks to its peculiar features, our method can be considered as an effective engineering approach capable of performing multiaxial fatigue assessment under variable amplitude loading which fully complies with the recommendations of the available standard codes.     

A critical plane-strain energy density criterion for multiaxial low-cycle fatigue life under non-proportional loading

A series of multiaxial low-cycle fatigue experiments was performed on 45 steel under non-proportional loading. The present evaluations of multiaxial low-cycle fatigue life were systematically analysed. A combined energy density and critical plane concept is proposed that considers different failure mechanisms for a shear-type failure and a tensile-type failure, and from which different damage parameters for the critical plane-strain energy density are proposed. For tensile-type failures in material 45 steel and shear-type failures in material 42CrMo steel, the new damage parameters permit a good prediction for multiaxial low-cycle fatigue failure under non-proportional loading. The currently used critical plane models are a special and simple form of the new model.     

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.