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.     

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.     

A new path‐dependent multiaxial fatigue model for metals under different paths

Based on the critical plane approach, a new path‐dependent multiaxial fatigue model in low‐cycle fatigue is proposed. The proposed model includes damage contribution from four sources: the normal strain amplitude, the shear strain amplitude on the critical plane, the hydrostatic mean strain and a new path‐dependent factor. The effect of mean strain is considered by the hydrostatic mean strain. The experimental data of 11 kinds of materials are used to demonstrate the effectiveness of this new model under both zero and non‐zero mean strain multiaxial loading path.     

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.     

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.     

Biaxial fatigue for proportional and non‐proportional loading paths

In order to study the use of a local approach to predict crack‐initiation life on notches in mechanical components under multiaxial fatigue conditions, the study of the local cyclic elasto‐plastic behaviour and the selection of an appropriate multiaxial fatigue model are essential steps in fatigue‐life prediction. The evolution of stress–strain fields from the initial state to the stabilized state depends on the material type, loading amplitude and loading paths. A series of biaxial tension–compression tests with static or cyclic torsion were carried out on a biaxial servo‐hydraulic testing machine. Specimens were made of an alloy steel 42CrMo4 quenched and tempered. The shear stress relaxations of the cyclic tension–compression with a steady torsion angle were observed for various loading levels. Finite element analyses were used to simulate the cyclic behaviour and good agreement was found. Based on the local stabilized cyclic elastic–plastic stress–strain responses, the strain‐based multiaxial fatigue damage parameters were applied and correlated with the experimentally obtained lives. As a comparison, a stress‐invariant‐based approach with the minimum circumscribed ellipse (MCE) approach for evaluating the effective shear stress amplitude was also applied for fatigue life prediction. The comparison showed that both the equivalent strain range and the stress‐invariant parameter with non‐proportional factors correlated well with the experimental results obtained in this study.     

A novel accumulative fatigue damage model for multiaxial step spectrum considering the variations of loading amplitude and loading path

In this paper, based on the process of the fatigue crack initiation and the critical plane theory, a continuous stress parameter was proposed to quantify the driving force of the fatigue crack initiation for the fully reversed multiaxial fatigue loading. In this stress parameter, the shear stress amplitude and normal stress amplitude on the critical plane were combined with the variable coefficients which were affected by the normalized fatigue life and the loading non‐proportionality. Owing to these coefficients, for the multiaxial loadings with different non‐proportionalities, the driving force of the fatigue crack initiation during the whole life could be described. After that, a novel accumulative fatigue damage model was established for the multiaxial two‐stage step spectrum. In this model, the accumulative damage was calculated according to the variation of the proposed stress parameter on the critical plane. Considering the directionality of the multiaxial fatigue damage, for the spectrum in which the loading path was variable, the damage accumulation was carried out on the critical planes of the both loadings, and the larger one was chosen as the final accumulative fatigue damage. In order to verify the new model, up to 41 different multiaxial two‐stage step spectrum loading tests on 2024‐T4 aluminium alloy were collected. The new model, as well as other five commonly used models, was applied to calculate the accumulative fatigue damage. The final results showed that, compared with other commonly used models, the new model had the most accurate results with the smallest scatters.     

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.     

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.     

Use of an energy‐based/critical plane model to assess fatigue life under low‐cycle multiaxial cycles

For engineering components subjected to multiaxial loading, fatigue life prediction is crucial for guaranteeing their structural security and economic feasibility. In this respect, energy‐based models, integrating the stress and strain components, are widely used because of their availability in fatigue prediction. Through employing the plastic strain energy concept and critical plane approach, a new energy‐based model is proposed in this paper to evaluate the low‐cycle fatigue life, in which the critical plane is defined as the maximum damage plane. In the proposed model, a newly defined NP factor κ^*  is used to quantify the nonproportional (NP) effect so that the damage parameter can be conveniently calculated. Moreover, a simple estimation method of weight coefficient is developed, which can reflect different contributions of shear and normal plastic strain energy on total fatigue damage. Experimental data of 10 kinds of materials are employed to assess the effectiveness of this model as well as three other energy‐based models.     

一种基于临界平面法的多轴疲劳寿命预测模型

在多轴交变应力作用下,由于非比例循环附加强化效应导致疲劳寿命降低。针对这一问题,以薄壁圆管疲劳试件为研究对象,在分析临界平面上剪应变和正应变随相位角变化特征的基础上,引入了一个新的有效循环变量———临界平面上的等效应力,提出了一种新的多轴疲劳预测模型。新的损伤参量不含经验常数,便于工程实际的运用。通过和铝合金7075-T651多轴疲劳实验数据比较,结果表明,所提出的多轴寿命预测模型具有更好的预测精度,适用于比例与非比例加载条件。     

Strain-based multiaxial fatigue damage modelling

A new multiaxial fatigue damage model named characteristic plane approach is proposed in this paper, in which the strain components are used to correlate with the fatigue damage. The characteristic plane is defined as a material plane on which the complex three‐dimensional (3D) fatigue problem can be approximated using the plane strain components. Compared with most available critical plane‐based models for multiaxial fatigue problem, the physical basis of the characteristic plane does not rely on the observations of the fatigue crack in the proposed model. The cracking information is not required for multiaxial fatigue analysis, and the proposed model can automatically adapt for different failure modes, such as shear or tensile‐dominated failure. Mean stress effect is also included in the proposed model by a correction factor. The life predictions of the proposed fatigue damage model under constant amplitude loading are compared with a wide range of metal fatigue results in the literature.     

Multiaxial fatigue life prediction method based on path‐dependent cycle counting under tension/torsion random loading

A path‐dependent cycle counting method is proposed by applying the distance formula between two points on the tension‐shear equivalent strain plane for the identified half‐cycles first. The Shang–Wang multiaxial fatigue damage model for an identified half‐cycle and Miner's linear accumulation damage rule are used to calculate cumulative fatigue damage. Therefore, a multiaxial fatigue life prediction procedure is presented to predict conveniently fatigue life under a given tension and torsion random loading time history. The proposed method is evaluated by experimental data from tests on cylindrical thin‐walled tubes specimens of En15R steel subjected to combined tension/torsion random loading, and the prediction results of the proposed method are compared with those of the Wang–Brown method. The results showed that both methods provided satisfactory 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.     

Fatigue life prediction for metals using an improved strain energy density model

Abstract

An improved strain energy density model is proposed on the basis of critical plane concept to better predict the multiaxial fatigue life of metals, especially during nonproportional loadings. This approach is based on the normal and shear strain energy densities on maximum principal strain range plane. Procedures used to determine the normal and shear strain energy densities are also presented. Experimental data taken from the literature are used to validate the capabilities of the improved model, including 4 different metals and 24 different loading paths. The results show that the proposed model gives good predictions for most of these materials and loading paths.     

Online multiaxial fatigue damage evaluation method by real‐time cycle counting and energy‐based critical plane criterion

To realize online multiaxial fatigue damage assessment for the mechanical components in service, an online multiaxial cycle counting method is proposed coupled with the segment processing technique and Wang‐Brow's relative equivalent strain concept. Meanwhile, considering all the stress and strain components, which contribute to the fatigue damage on the critical plane, a multiaxial fatigue damage model without any weight coefficients is also proposed in an equivalent form of shear strain energy. Then, an online fatigue damage evaluation method for multiaxial random loading is developed by combining with the proposed damage model and online cycle counting method. The experimental results showed that the proposed online cycle counting method can be successfully applied to the calculation of multiaxial fatigue damage under random loading. Moreover, the proposed online multiaxial fatigue damage evaluation method can provide satisfactory predictions.     

Multiaxial fatigue of rubber: Part I: equivalence criteria and theoretical aspects

This paper investigates commonly used approaches for fatigue crack nucleation analysis in rubber, including maximum principal strain (or stretch), strain energy density and octahedral shear strain criteria. The ability of these traditional equivalence criteria, as well as a recent equivalence criterion (the cracking energy density) to predict multiaxial fatigue behaviour is explored. Theoretical considerations are also introduced relating to the applicability of various fatigue life analysis approaches. These include the scalar nature of traditional equivalence criteria, robustness of the criteria investigated for a wide range of multiaxial loadings, effects of crack closure and applications to non‐proportional multiaxial loadings. It is shown that the notion of a stress or strain amplitude tensor used for the analysis of multiaxial loading in metals is not appropriate in the analysis of rubber due to nonlinearity associated with finite strains and near incompressibility. Taken together, these considerations illustrate that traditional criteria are not sufficiently consistent or complete to permit confident analysis of arbitrary multiaxial loading histories, and that an analysis approach specific to the failure plane is needed. Of the three traditional criteria, maximum principal strain is shown to match most closely to the cracking energy density criterion, in terms of a failure locus in principal stretch space.     

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.     

A multiaxial high-cycle fatigue life evaluation model for notched structural components

A model for multiaxial high-cycle fatigue life evaluation of notched structural components is proposed, which considers the impact of the stress field on fatigue life by utilizing the Theory of Critical Distances (TCD) and Finite Element Method (FEM). The maximum shear stress range plane is defined as the critical plane, and the damage parameters are the maximum effective shear stress amplitude and the maximum effective normal stress, which are obtained by averaging the stress in the hemisphere volume around the maximum stress point. To validate the accuracy of the model, multiaxial fatigue tests are carried out for both smooth and notched specimens of Aluminum–Silicon alloy. The results indicate that the evaluated life and experimental life have a good agreement.     

Stochastic modelling of low-cycle fatigue damage in 316L stainless steel under variable multiaxial loading

In the present study, a stochastic model is developed for the low-cycle fatigue life prediction and reliability assessment of 316L stainless steel under variable multiaxial loading. In the proposed model, fatigue phenomenon is considered as a Markov process, and damage vector and reliability are defined on every plane. Any low-cycle fatigue damage evaluating method can be included in the proposed model. The model enables calculation of statistical reliability and crack initiation direction under variable multiaxial loading, which are generally not available. In the present study, a critical plane method proposed by Kandil et al . ( Metals Soc., London 280, 203–210, 1982) maximum tensile strain range, and von Mises equivalent strain range are used to calculate fatigue damage. When the critical plane method is chosen, the effect of multiple critical planes is also included in the proposed model. Maximum tensile strain and von Mises strain methods are used for the demonstration of the generality of the proposed model. The material properties and the stochastic model parameters are obtained from uniaxial tests only. The stochastic model made of the parameters obtained from the uniaxial tests is applied to the life prediction and reliability assessment of 316L stainless steel under variable multiaxial loading. The predicted results show good aggreement with experimental results.