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.     

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 of vulcanized natural rubber under proportional and non‐proportional loading

Two natural rubber (NR) compounds of hardness shore A 60 and 70 with 10 different biaxial displacement/twist paths were investigated using an axisymmetric cylindrical hollow dumbbell specimen. The effects of the proportional and non‐proportional loading modes on the fatigue life are discussed. In total, 52 fatigue test results are reported with fatigue life results of 3,918 to 488,000 cycles. The effect of the channel phase on the fatigue life is discussed.     

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.     

Evaluation of fatigue life for titanium alloy TC4 under variable amplitude multiaxial loading

Fatigue tests under variable amplitude multiaxial loading were conducted on titanium alloy TC4 tubular specimens. A method to estimate the fatigue life under variable amplitude multiaxial loading has been proposed. Multiaxial fatigue parameter based on Wu–Hu–Song approach and rainflow cycle counting and Miner–Palmgren rule were applied in this method. The capability of fatigue life prediction for the proposed method was checked against the test data of TC4 alloy under variable amplitude multiaxial loading. The prediction results are all within a factor of two scatter band of the test results.     

A modification of Morrow and Smith–Watson–Topper mean stress correction models

A modification of the Morrow and the Smith, Watson and Topper (SWT) mean stress correction models is proposed to account for the mean stress effect on fatigue life. The capability and accuracy of the proposed model are compared to those of the original Morrow and the SWT model using published mean stress fatigue test data. The proposed mean stress correction model was found to be superior to both the SWT and the Morrow model in the case of the Incoloy 901 superalloy and the ASTM A723 steel. On the other hand both the proposed and the original SWT model provided equally good correlation with experimental data in the case of 7075‐T561 aluminium alloy and 1045 HRC 55 steel. The Morrow model was found to give the least accurate predictions for all four materials analysed.     

Fatigue life prediction of vulcanized natural rubber under proportional and non-proportional loading

To investigate the multiaxial fatigue properties of vulcanized natural rubber (NR), a series of tests including both proportional and non-proportional loading paths on small specimens were performed. The existing fatigue life prediction approaches are evaluated with life data obtained in the tests. It is shown that the equivalent strain approach presents a good prediction of the fatigue life although it has a certain shortcoming. Compared with the strain energy density (SED) model, the cracking energy density (CED) model represents the portion of SED that is available to be released by virtue of crack growth on a given material plane, so it gives better results in the life prediction. Some of the approaches based on critical plane which are widely used for metal fatigue are also tested in this paper, and the results show that the Chen-Xu-Huang (CXH) model gives a better prediction, compared with the Smith-Watson-Topper (SWT) and Wang–Brown (WB) model. A modified Fatemi–Socie's model has also been introduced, and the results show that the modified model can be used to predict the fatigue life of rubber material well.     

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 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 simple energy‐based model for nonproportional low‐cycle multiaxial fatigue life prediction under constant‐amplitude loading

From the literature concerning the traditional nonproportional (NP) multiaxial cyclic fatigue prediction, special attentions are usually paid to multiaxial constitutive relations to quantify fatigue damage accumulation. As a result, estimation of NP hardening effect decided by the entire history path is always proposed, which is a challenging and complex task. To simplify the procedure of multiaxial fatigue life prediction of engineering components, in this paper, a novel effective energy parameter based on simple material properties is proposed. The parameter combines uniaxial cyclic plastic work and NP hardening effects. The fatigue life has been assessed based on traditional multiaxial fatigue criterion and the proposed parameter, which has been validated by experimental results of 316 L stainless steel under different low‐cycle loading paths.     

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 hybrid frequency–time domain method for predicting multiaxial fatigue life of 7075‐T6 aluminium alloy under random loading

A hybrid frequency–time domain method for predicting multiaxial fatigue life under random loading is developed on the basis of combination of the frequency domain and time domain analysis. The critical damage point of the structure is determined by the frequency domain equivalent stress method. Then, the fatigue life prediction is made in time domain by generating random load‐time histories from the power spectral density of the critical point. The method is validated with the random vibration fatigue test of 7075‐T6 aluminium alloy. It has been shown that the results of fatigue life calculated by hybrid method are well correlated with the experiment.     

A multi‐axial low‐cycle fatigue life prediction model considering effects of additional hardening

It is generally accepted that the additional hardening of materials could largely shorten multi‐axis fatigue life of engineering components. To consider the effects of additional hardening under multi‐axial loading, this paper summarizes a new multi‐axial low‐cycle fatigue life prediction model based on the critical plane approach. In the new model, while critical plane is adopted to calculate principal equivalent strain, a new plane, subcritical plane, is also defined to calculate a correction parameter due to the effects of additional hardening. The proposed fatigue damage parameter of the new model combines the material properties and the angle of the loading orientation with respect to the principal axis and can be established with Coffin‐Manson equation directly. According to experimental verification and comparison with other traditional models, it is clear that the new model has satisfactory reliability and accuracy in multi‐axial fatigue life prediction.     

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.     

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.     

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.     

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.     

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.     

A computerized procedure for long-life fatigue assessment under complex multiaxial loading

A computerized procedure is presented and evaluated for application examples of long-life fatigue analyses of metallic materials under complex multiaxial loading. The method is based on the stress invariants and uses the minimum circumscribed ellipse approach for evaluating the effective shear stress amplitude under complex multiaxial loading. The applicability of the procedure for handling non-proportional loading is examined through typical examples such as combined normal/shear stresses and combined bi-axial normal stresses with complex stress time histories. The effects of phase shift angles, frequency ratios and waveforms on fatigue endurance were re-analysed and compared with available experimental results from the literature. The comparison shows that the presented procedure based on stress invariants is a potential conservative engineering approach, very suitable for fast fatigue evaluation in the integrated computer aided fatigue design.