An improved fatigue life prediction model based on loading sequence

With the leap-forward development of heavy-haul freight and high-speed passenger transport of railways in China, welded load-bearing parts have been widely used in various new types of rolling stock structures. However, fatigue cracking is one of the main failure modes of these welded load-bearing parts. As fatigue life prediction is related to the safety of the whole welded structure, more reasonable fatigue life prediction technology is urgently needed in the domestic railway vehicle industry to meet the actual needs of engineering practice. It is of great practical significance to study the life prediction of welded load-bearing structures (Liu et al., 2017; Kim et al., 2013), and many experts and scholars have done a lot of research work on this issue with quite a few results achieved (Zhou & Li, 2010; Yue et al., 2016; Wang, Liu, & Shang, 2013).

For some key welded parts of practically complex rolling stock structures, such as welded joints of train truck frame, a significant feature of its load-bearing state is to withstand multi-stage small loads. In fatigue damage assessment, the effect of small loads, instead of being ignored, must be taken into consideration; otherwise, the prediction results will be too conservative, which is not conducive to the safe use of welded structures. Corten-Dolan model can describe the effect of small loads on fatigue damage in a proper way, and can also deal with the interaction effect between loads. Though this model has been applied for fatigue accumulated damage assessment and life prediction in practical engineering (Meng, Wang, & Xue, 2014; Huang et al., 2011; Shang & Yao, 1998, 1999), the definition of its key parameter d, which is directly related to the prediction accuracy of fatigue life, is still controversial (Gao et al., 2015). Therefore, how to define the parameter d in Corten-Dolan model is the focus of the study, and it is necessary to carry out further analysis and discussion.

The earliest parameter d is considered to remain unchanged, and can be determined by the fatigue test of two-stage load (Marciniak, Rozumek, & Macha, 2008). Subsequently, it is found that the parameter d cannot be regarded as a constant. Although the parameter can be determined by fatigue test, the values obtained from different fatigue load tests differ a lot (Chen, 1996). With the influence of stress state and real-time damage on fatigue accumulated damage considered, Gao (2016) has redefined the function of parameter d. Good results have also been obtained in numerical verification. The prediction accuracy of the model, though getting enhanced significantly, still needs further improvement.

In the actual service environment of rolling stock, the stress states of welded parts of the vehicle structure are very complex. For the fatigue accumulated damage under complex multi-stage loads, the loading sequence and the interaction between loads should also be fully considered. Although the key parameter d in Corten-Dolan model and the existing improved model can reflect the interaction between loads, it fails to consider the influence of loading sequence on fatigue damage accumulation, which will to a large degree affect the prediction results that are highly dispersed and often can hardly meet the actual engineering requirements.

Based on small loads, damage degrees and stress states considered, this paper goes further to include the relations of loading sequence to improve the Corten-Dolan model, and it verifies the validity and feasibility of Corten-Dolan model according to the test data of the welded joints of common materials, alloy materials and Q235B steel.

According to the Corten-Dolan model, under constant-amplitude loading, after n stress cycles, the accumulated fatigue damage D_d can be expressed as,

Where, P is the number of nucleus flaw under stress; r is the damage coefficient; a' is a constant related to the material.

According to Formula (1), the critical fatigue damage is,

Where, P_max, r_max and a′_max are respectively the number of nucleus flaw, the damage coefficient and the corresponding material-related constant under the maximum stress; N_fmax is the fatigue life under the maximum stress.

Under variable amplitude loading, the fatigue accumulated damage equation can be established according to the total damage amount, that is, the sum of damage in each section should be equal to the sum of damage under constant-amplitude loading alone. Under the action of multi-stage variable amplitude loads, the accumulated damage D_d corresponding to failure is consistent with the critical fatigue damage D_c, then D_d can be expressed as follows:

Where, i is the number of stages of the stress cycle.

In fatigue failure analysis, the fatigue life prediction formula of Corten-Dolan model under multi-stage loads and multi-stage stress cycles can be expressed as follows:

Where, N_f is the fatigue life predicted by the model under the action of multi-stage stress cycles; a_i is the percentage of the i-stage stress cycles to the total cycles; σ_i is i-stage stress; σ_max is the maximum stress in multi-stage stress.

In Formula (4), parameter d is related to material properties and stress level. Gao (2016) defines parameter d as a function including cycle times ratio and stress ratio, which is as follows:

Where, n_i is the fatigue life corresponding to the actual stress cycle; N_fi is the fatigue life of the i-stage stress σ_i under the action of single cycle.

Formula (5) introduces a parameter γ, which is a constant related to material and can be determined by the test data in combination with the fatigue failure criterion. With Formula (5) substituted into Formula (4), the existing improved model of Corten-Dolan model can be expressed as follows:

According to the parameter d defined by Formula (5), the existing improved model of the corresponding Corten-Dolan model can also be written into another expression form, which is,

Compared with Miner's rule and Manson-Halford model, the existing improved model of Corten-Dolan model reflects the interaction between loads by the ratio of stress of various stages to the maximum stress while considering the contribution of small loads. Thus the prediction results of fatigue life have higher reliability, but the calculation of the only parameter d is yet to be further discussed.

The existing improved model of Corten-Dolan model does not consider the effect of real-time loading sequence of loads of various stages, and is derived based on the linear Miner accumulated damage theory where Miner accumulated damage is independent of the loading sequence. However, numerous experimental data show that the accumulated fatigue damage is related to the loading sequence. For simple two-stage test loading, the accumulated fatigue damage during failure is often not equal to 1. When the loading sequence is low first and then high, the accumulated fatigue damage Σn_i/N_fi is more than 1 and the crack initiation time is delayed. When the loading sequence is first high and then low, the accumulated fatigue damage Σn_i/N_fi is less than 1, and the high stress causes the crack to form ahead of time, while the low stress causes the crack to expand. For multi-stage loading, the same conclusion can be drawn.

According to the above analysis, when the definition of key parameter d is to be modified, besides the influence of small loads, real-time damage and stress states on fatigue accumulated damage, the influence of loading sequence also needs to be considered, so as to further improve the existing improved model of Corten-Dolan model.

Since it is often multi-stage variable amplitude loads that act in practical engineering, it is not appropriate to consider loads of all stages before loading. Therefore, only the sequential effects of the adjacent two-stage loads are taken into consideration. The parameter d can be modified with the following two methods.

• Method 1: The stress ratio coefficient is directly increased on the basis of Formula (5) to correct d, then,

Where, d* 1 is the calculation parameter when method 1 is adopted to improve the prediction model.

With Formula (8) substituted into Formula (6), the improved fatigue life prediction model 1 (prediction model 1 for short) after correction by method 1 can be obtained as,

• Method 2: As the contribution of parameter γ, a constant related to materials, to the key parameter d is fixed, the effects of loading sequence and damage degree are taken as independent influencing factors respectively to modify d, then,

Where, d* 2 is the calculation parameter when method 2 is adopted to improve the prediction model.

With Formula (10) substituted into Formula (6), the improved fatigue life prediction model 2 (prediction model 2 for short) after correction by method 2 can be obtained as,

After the key parameter d is modified by the above two models, when the adjacent loads (σ_i + 1/σ_i > 1) adopt the loading sequence of first low and then high, the predicted life is increased. When the adjacent loads (σ_i + 1/σ_i < 1) adopt the loading sequence of first high and then low, the predicted life is reduced, which is consistent with the analysis of the relationship between fatigue accumulated damage and loading sequence.

The inversion identification of the undetermined parameter γ in Formula (11) can be carried out using test measurement data with two prediction models by means of inversion optimization. In the inversion identification, parameter γ can be easily identified by the one-dimensional search method where objective function is constructed according to the least square principle based on the test measurement data and the analysis results of two prediction models.

In order to verify the fatigue prediction ability of the improved Corten-Dolan model based on loading sequence, the test data of materials such as standard 45 steel, standard 16Mn steel and hot-rolled 16Mn steel given in Gao (2016) are adopted to carry out the fatigue life prediction. By comparing the predicted value of the model with the test data, the prediction accuracy of the improved model is verified, and the prediction results of different models are compared.

The two-stage loads of standard 45 steel are 331.46 MPa and 284.40 MPa respectively, and the fatigue life of welded joints under their separate action is 5.0 × 10⁴ and 5.0 × 10⁵ times respectively. The calculation results of parameter d obtained by different methods and the comparison of fatigue life prediction results in different models under various loading modes (stress levels corresponding to different loading sequences and different stress cycles) are shown in Tables 1 and 2 respectively.

The two-stage loads of standard 16Mn steel are 562.9 MPa and 392.3 MPa respectively, and the fatigue life of welded joints under their separate action is 3.968 × 10³ and 7.8723 × 10⁴ times respectively. The calculation results of parameter d obtained by different methods and the comparison of fatigue life prediction results in different models under various loading modes are shown in Tables 3 and 4 respectively.

The two-stage loads of hot-rolled 16Mn steel are 394 MPa and 345 MPa respectively, and the fatigue life of welded joints under their separate action is 9. 35 × 10⁴ and 4.022 × 10⁵ times respectively. The calculation results of parameter d obtained by different methods and the comparison of fatigue life prediction results in different models under various loading modes are shown in Tables 5 and 6 respectively.

It can be seen from the above calculation results that under two-stage load, the Corten-Dolan improved model based on the loading sequence in this paper can accurately predict the fatigue life of welded joints for standard 45 steel, standard 16Mn steel and hot-rolled 16Mn materials. Compared with the linear Miner's rule and the improved method in GAO Huiying (2016), there is significant improvement in the prediction results of the two prediction models in this paper.

According to the test data of Q235B steel under variable amplitude loading in Gao (2016), the fatigue life of its welded joints is predicted by using different prediction models with five-stage load as an example, and the prediction comparison results are shown in Table 7.

It can be seen from the calculation results in Table 7 that the fatigue life of Q235B welded joints can also be predicted by the improved Corten-Dolan model under five-stage variable amplitude loads, and the prediction accuracy is significantly improved compared with the traditional Miner's rule and the models in references.

1. An improved Corten-Dolan model based on loading sequence is established, with sufficient consideration for the influence of small loads, damage degree, stress states and loading sequence and can effectively predict the fatigue life.

2. According to the test data verification results of the welding joints of common materials, alloy materials and Q235B steel, compared with the traditional Miner model, the two improved models in this paper have good feasibility, whether under two-stage load or multi-stage load, and the prediction accuracy is significantly improved; the prediction results are more reliable.

3. The improved model in this paper is relatively simple. There is no need for multi-layer iterative calculation, and model parameters, small in number, are easy to determine. So it is convenient to predict the fatigue life of welded structures in engineering.

Funding: The authors acknowledge the National Natural Science Foundation of China (10802015), the Joint Fund of Natural Science Foundation of Liaoning Province (2015020119), the Liaoning Province Graduate Education and Teaching Reform Research Project (2017), the Liaoning Province Transformation and the Innovation and Entrepreneurship Education Project (2017).