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Engineering components/structures with geometric discontinuities normally bear complex and variable loads, which lead to a multiaxial and random/variable amplitude stress/strain state. Hence, this study aims how to effectively evaluate the multiaxial random/variable amplitude fatigue life.

Recent studies on critical plane method under multiaxial random/variable amplitude loading are reviewed, and the computational framework is clearly presented in this paper.

Some basic concepts and latest achievements in multiaxial random/variable amplitude fatigue analysis are introduced. This review summarizes the research status of four main aspects of multiaxial fatigue under random/variable amplitude loadings, namely multiaxial fatigue criterion, method for critical plane determination, cycle counting method and damage accumulation criterion. Particularly, the latest achievements of multiaxial random/variable amplitude fatigue using critical plane methods are classified and highlighted.

This review attempts to provide references for further research on multiaxial random/variable amplitude fatigue and to promote the development of multiaxial fatigue from experimental research to practical engineering application.

Engineering components/structures are usually subjected to complex and variable loads, which result in random multiaxial stress/strain states. However, fatigue analysis methods under constant loads cannot be directly applied to fatigue life prediction analysis under random loads. Therefore, the purpose of this study is how to effectively evaluate fatigue life under multiaxial random loading.

First, the average phase difference is characterized as the ratio of the number of shear strain cycles to the number of normal strain cycles, and the new non-proportional additional hardening factor is proposed. Then, the determined random typical load spectrum is processed into a simple variable amplitude load spectrum, and the damage in each plane is calculated according to the multiaxial fatigue life prediction model and Miner theory. Meanwhile, the cumulative damage can be calculated separately by projection method. Finally, the maximum projected cumulative damage plane is defined as the critical plane of multiaxial random fatigue.

The fatigue life prediction capability of the method is verified based on test data of TC4 titanium alloy under random multiaxial loading. Most of the predicting results are within double scatter bands.

The objective of this study is to provide a reference for the determination of critical plane and non-proportional additional hardening factor under multiaxial random loading, and to promote the development of multiaxial fatigue from experimental studies to practical engineering applications.

Under multiaxial random loading, the material stress–strain response is not periodic, which makes it difficult to determine the direction of the critical plane on the material. Meanwhile, existing methods of constant loading cannot be directly applied to multiaxial random loading; this problem can be solved when an equivalent stress transformation method is used.

First, the Liu-Mahadevan critical plane is introduced into multiaxial random fatigue, which is enabled to determine the material's critical plane position under random loading. Then, an equivalent stress transformation method is proposed which can convert random load to constant load. Meanwhile, the ratio of mean stress to yield strength is defined as the new mean stress influence factor, and a new non-proportional additional strengthening factor is proposed by considering the effect of phase differences.

The proposed model is validated using multiaxial random fatigue test data of TC4 titanium alloy specimens and the results of the proposed model are compared with that based on Miner's rule and BSW model, showing that the proposed method is more accurate.

In this work, a new multiaxial random fatigue life prediction model is proposed based on equivalent stress transformation method, which considers the mean stress effect and the additional strengthening effect. Results show that the predicted fatigue lives given by the proposed model are in well accordance with the tested data.

An ideal method for predicting the fatigue life of spherical thrust elastomeric bearings has not been reported, thus far. This paper aims to present a method for predicting the fatigue life of laminated rubber spherical thrust elastomeric bearings.

First, the mechanical properties of standard rubber samples were tested; the axial stiffness, cocking stiffness, torsional stiffness and fatigue life of several full-size spherical thrust elastomeric bearings were tested. Then, the stiffness results were calculated using the neo-Hookean, Mooney–Rivlin and Yoeh models. Using a modified Mooney–Rivlin constitutive model, this paper proposes an improved method for fatigue life prediction, which considers the laminated characteristics of a spherical thrust elastomeric bearing and loads of multiple multi-axle conditions.

The Mooney–Rivlin model could accurately describe the stiffness characteristics of the spherical thrust elastomeric bearings. A comparative analysis of experimental results shows that the model can effectively predict the life of a spherical thrust elastomeric bearing within its range of use and the prediction error is within 20%.

The fatigue parameters of elastomeric bearings under multiaxial loads were fitted and corrected using experimental data and an accurate and effective multiaxial fatigue-life prediction expression was obtained. Finally, the software was redeveloped to improve the flexibility and efficiency of modeling and calculation.

A large number of data have proved that under the same von Mises equivalent strain condition, the fatigue life under multiaxial non-proportional loading is often much lower than the life under multiaxial proportional loading. This is mainly due to the influence of the non-proportional loading path and the additional hardening effect, which lead to a sharp decrease in life.

The modulus attenuation effect is used to modify the static hardening coefficient, and the predicted value obtained is closer to the additional hardening coefficient obtained from the experiment. A fatigue life model can consider non-proportional paths, and additional hardening effects are proposed. And the model uses multiaxial fatigue test data to verify the validity and adaptability of the new model. The life prediction accuracy and material application range are satisfactory.

Because loading path and additional hardening of the material affect fatigue life, a new multiaxis fatigue life model based on the critical plane approach is proposed. And introducing a non-proportional additional damage coefficient, the joint influence of the load path and the additional hardening can be considered. The model's life prediction accuracy and material applicability were verified with multiaxial fatigue test data of eight materials and nine loads compared with the prediction accuracy of the Kandil–Brown–Miller (KBM) model and Fatemi–Socie (FS) model.

The physical meaning of the new model is clear, convenient for practical engineering applications.

This paper aims to assess the fatigue life characteristics of vehicle coil spring under random strain load in the time domain. Cyclic random road loads caused fatigue failure for automotive components during their operating condition. .

The coil spring model is developed through finite element analysis software. The critical region and fatigue life cycle of coil spring is evaluated through finite element analysis. The experimental is set up to capture the random strain signal of the rural, highway and campus road. The sampling rate of the random strain signals data captured were 500 Hz in 150 s. Then, fatigue life is assessed through Goodman, Brown-Miller, Fatemi-Socie, Wang-Brown fatigue life models. Goodman model is evaluated through finite element analysis in order to compare with fatigue experimental results.

The fatigue life was estimated for Brown-Miller model is the highest (4.32E4, 4.10E4, and 3.73E4 cycles/block for rural, highway and campus respectively) followed by Goodman model, Brown-Miller, Fatemi-Socie and Wang-Brown models respectively. The conservative fatigue life 1:2 and 2:1 data scattering approach is proposed in order to determine the acceptability of the data.

Hence, the proposed fatigue life models can be used to assess multiaxial fatigue under random strain signals for the automobile coil spring.

The stress state near the notch affects fatigue damage directly, but quantifying the stress field is difficult. The purpose of this study is to provide a mathematical description method of the stress field near the notch to achieve a reliable assessment of the fatigue life of notched specimens.

Firstly, the stress distribution of notched specimens of different materials and shapes under different stress levels is investigated, and a method for calculating the stress gradient impact factor is presented. Then, the newly defined stress gradient impact factor is used to describe the stress field near the notch, and an expression for the stress at any point along a specified path is developed. Furthermore, by combining the mathematical expressions for the stress field near the notch, a multiaxial fatigue life prediction model for notched shaft specimens is established based on the damage mechanics theory and closed solution method.

The stress gradient factor for notched specimens with higher stress concentration factors (V60-notch, V90-notch) varies to a certain extent when the external load and material change, but for notched specimens with relatively lower stress concentration factors (C-notch, U-notch, stepped shaft), the stress gradient factor hardly varies with the change in load and material, indicating that the shape of the notch has a greater influence on the stress gradient. It is also found that the effect of size on the stress gradient factor is not obvious for notched specimens with different shapes, there is an obvious positive correlation between the normal stress gradient factor and the normal stress concentration factor compared with the relationship between the shear stress gradient factor and the stress concentration factor. Moreover, the predicted results of the proposed model are in better agreement with the experimental results of five kinds of materials compared with the FS model, the SWT model, and the Manson–Coffin equation.

In this paper, a new stress gradient factor is defined based on the stress distribution of a smooth specimen. Then, a mathematical description of the stress field near the notch is provided, which contains the nominal stress, notch size, and stress concentration factor which is calculated by the finite element method (FEM). In addition, a multiaxial fatigue life prediction model for shaft specimens with different notch shapes is established with the newly established expressions based on the theory of damage mechanics and the closed solution method.

The purpose of this paper is to study the influence of multiaxial loading composed with different frequencies between the axial and torsional components in 42CrMo4 concerning fatigue life and early crack growth orientation.

Biaxial fatigue tests were carried out by a biaxial servo‐hydraulic machine, considering different loading paths and different frequencies between the normal and shear stress components in 42CrMo4. Theoretical estimations for fatigue life and early crack growth orientation were performed by applying various critical plane models. In addition, fractographic analysis of the fracture surfaces was carried out. The estimated results are compared with experimental results.

Significant effects were observed of the different frequency between the axial and torsion components on fatigue life and early crack growth orientation. The critical plane models based on shear mode give better estimations when compared with experimental results.

The paper shows that improved fatigue design can be achieved by considering the influence of different frequencies in multiaxial loadings.

The purpose of this scientific work is to simulate the fatigue damage under random loading, taking into account the mean stress effect on fatigue lifetime and using the Rainflow counting technique to assess the fatigue damage by the Ansys software. The used material is aluminum alloy 6082-T6. A comparison with literature results has confirmed this investigation in this paper.

The study of fatigue under random loading is based on the same concepts as constant loading with the addition of damage summation. The proportion of damage caused by a stress cycle depends not only on the alternating stress but also on the mean stress.

Analysis of the fatigue damage shows that the number of relative damage due to each cycle.

This paper aims to simulate the fatigue damage under random loading for aluminum alloys.