A Nonlinear Fatigue Damage Model Based on Equivalent Transformation of Stress

It is rather difficult for engineers to apply many of the fatigue damage models for requiring a knee point, material-dependent coefficient, or extensive testing, and some of them are only validated by a fatigue test of two-stage loading rather than higher-stage loading. In this paper, we propose a new model of fatigue cumulative damage in variable amplitude loading, which just requires the information of the S-N curve determined from the fatigue experiment. Specifically, the proposed model defines a stress equivalent transformation way to translate the damage of one stress to another stress through simple calculation. Experimental data of fatigue including two-, three-, and four-block loading verify the superiority of the proposedmodel by comparing it with the Miner model andMansonmodel.+e results show that the proposedmodel can be generalized to any type of loading and presents a better prediction. +erefore, the advantage of the proposed model can be easily used by an engineer.


Introduction
Fatigue fracture plays an important role in the failure of mechanical components, which accounts for 80%-90% of the failure probability [1]. Miner [2] first proposed a cumulative damage criterion that could explain the fatigue failure mechanism, which is still extensively applied in engineering owing to its simplicity. However, many fatigue experiments indicated that the Miner criterion would have a significant deviation in predicting the cumulative fatigue damage when the sequence of fatigue loading is changed [3]. erefore, relevant researchers have discussed the evolution process of fatigue cumulative damage from different perspectives.
Considering the linear damage at different stages, Manson et al. [4] proposed a double linear damage rule by setting the intersection point of two lines as a knee point. en, Manson et al. [5] improved his model and put forward a double damage curve approach. Meanwhile, some researchers have developed nonlinear models of fatigue cumulative damage. Subramanyan [6] and Hashin et al. [7] proposed the power law rule by considering a knee point in the S-N curve to predict the remaining damage, which needs the endurance limit of material. Manson et al. [5] derived the power law rule, which is a simple model relying only on the S-N curve. Shang et al. [8] proposed a new nonlinear model of fatigue cumulative damage based on the evolution law of the material ductility during fatigue damage. Yao et al. [9][10][11] constructed some nonlinear models according to the equivalent principle of between the strength degradation or stiffness degradation and fatigue damage. Djebli et al. [12] proposed a nonlinear fatigue damage accumulation model relying on the application of energy parameters. Mesmacque et al. [13] defined a damage stress to propose a new damage indicator according to sequential law in multiaxial loading, and the ultimate stress of material should be joined in the model. Despite extensive works on this area, nevertheless, many of these models require a knee point, material-dependent coefficient, or extensive testing, and some of them are only validated by a fatigue test of two-stage loading rather than higher-stage loading.
erefore, it is difficult to apply them to fatigue damage prediction because test data may not be available to an engineer.
Considering the above-mentioned situation, the present work aims to apply a new fatigue damage model in a simple way under variable amplitude loading. In Section 2, a new nonlinear fatigue damage model is proposed based on the correction and equivalent transformation of stress progression. e proposed model relies only on the S-N curve and does not need other fixed points or other material parameters, while also being applicable for fatigue analysis engineers.
en, the proposed model is applied to fatigue tests of two-, three-, and four-block loading. We compare the prediction results of the Miner model, Manson model, and the proposed model with the experimental results under different loading conditions, respectively. Finally, the effectiveness of the proposed model is verified.

Model Details
In this section, we propose a new nonlinear fatigue damage model. e proposed model only relies on equivalent transformation of stress, which just requires the information of the S-N curve. e proposed model takes the interactions between loading sequences into account.

Miner and Manson Model.
e fatigue life of the material under the action of stress amplitude S i is N i , and the damage in a cycle action is as follows: Let us define the sum of the damage D i of all extracted cycles as the cumulative fatigue damage. Let us suppose the stress S 1 , S 2 · · · S k of the material is subjected to n 1 , n 2 · · · n k , and the cumulative fatigue damage is as follows: Fatigue fracture occurs when D � 1. Manson [5] proposed a power law accumulation rule based on crack propagation for multilevel loading: where

Proposed Damage Model.
e Basquin [14] model can be applied to establish the relationship between stress amplitude and fatigue life curves of materials, which is described by the following relation: where a and b can be determined by fatigue experimental data.

Two-Block
Loading. e process of equivalent transformation relationship under two-block loading is shown in Figure 1. e assumptions of the model are as follows: (1) During the loading process, each cycle loses a certain effective life component of the sample (2) e fatigue damage of the sample is directly proportional to the work absorbed, and the work is directly proportional to the ratio of the number of stress cycles and the number of failure cycles under the stress value (3) e total amount of damage is a constant (4) e stress below the fatigue limit no longer causes damage (5) Damage is related to the action sequence of loads (6) When the sum of all damage components generated by each cyclic stress is 1, the specimen will be destroyed We can see from Figure 1 the corresponding stress amplitude S 1r under the remaining life N 1 − n 1 . erefore, the relation is as follows: In this case, ΔS 1 is the difference between S 1r and S 1 ; that is, Let us define a stress equivalent transformation way to translate the first stress S 1 to the second stress S 2 , and the expression is as follows: We can obtain the corresponding life N 2e under the stress amplitude S 2e in the S-N curve: If it is just two-block loading, the remaining cycles of stress S 2 is erefore, the equivalent damage of stress S 1 subjected to n 1 cycle transform to the second stress S 2 is e cumulative damage of two-block loading is

ree-Block
Loading. Let us translate the damage from two blocks to three blocks, and the calculation steps are as follows: We follow the same procedure until fatigue fracture.

e Remaining Life and Damage in Multiblock
Loading. Similarly, we get the remaining life and damage in multiblock loading: where N ir is the remaining life at stress level i; d ir is the remaining damage at stress level i.

Experimental Verifications
We apply the experimental data of two-, three-, and fourblock loading to validate the Miner model, Manson Table 3. e remaining damage of the Miner, Manson, and the proposed model is predicted and compared with experimental results, as illustrated in Table 4.

30CrMnSiA.
We then apply the experimental data of 30CrMnSiA [17] material under two-block loading. Fatigue life under different stress amplitudes is shown in Table 5. e remaining damage of the Miner, Manson, and proposed model is predicted and compared with experimental results in Table 6.
rough the comparative analysis of two-block loading fatigue experimental data of the three kinds of material, we can see that the proposed model can significantly improve the prediction accuracy of fatigue life compared with the Miner.

ree-Block Loading.
e fatigue test cycles of LY12CZ [18] material under different loading amplitudes are shown in Table 7. e three-block loading sequence includes lowhigh loading, low-high-low loading, high-low loading, and high-low-high loading, respectively. e remaining damage of the Miner, Manson, and the proposed model is predicted and compared in Table 8. erefore, we can see that the deviation of the Miner model and Manson model is larger than that of the proposed model in three-block loading. e Manson model tends to be aggressive or conservative when the changes between loading amplitude are large.

Four-Block Loading.
We choose Mesmacque's [13] experimental data of four-block loading to predict the fatigue life and damage. e fatigue life and applied cycles under different loading conditions are shown in Table 9. e parameters of Basquin's function are a � 856 and b � −0.08735. e loading sequence is shown in Figure 2. e cumulative fatigue life and damage of the Miner, Manson, and proposed model is predicted and compared with experimental results, shown in Table 10.

Characteristics of the Proposed Model.
e characteristics of the proposed model in fatigue damage prediction under two-block loading are shown in Figure 3.       When the ratio of second loading to first loading is greater than 1, the remaining fatigue damage tends to increase with the increase of the ratio. Because the fatigue damage is manifested as the collective crack diffusion without interaction under the action of the first-order low stress, when the damage accumulates to a certain extent, the cracks of different types and orientations interact with each other at an increasing rate, forming local stress concentration and causing local damage. erefore, the local stress concentration is very large under the added high stress, and local failure occurs very quickly.

Journal of Engineering
On the contrary, when the ratio of second loading to first loading is less than 1, the remaining fatigue damage tends to decrease with the decrease of the ratio. Because the first level of high stress quickly makes the matrix crack saturation and forms local stress concentration, the degree of local stress concentration decreases under the later addition of low stress. Only when the accumulative stress reaches a certain degree will the local damage continue to develop until destruction.
A large number of fatigue crack propagation tests [19][20][21] show that the actual crack propagation life of specimens under high-low load is generally longer than that calculated by the corresponding theoretical crack propagation model. When the specimen is subjected to low-high load, the actual crack growth life is generally shorter than the corresponding theoretical crack growth life, mainly because the sequence of loads affects the fatigue crack growth rate of the material to a certain extent. erefore, it also satisfies the fatigue test law, which shows that cumulative fatigue damage is greater than 1 at low-high loading blocks and is less than 1 at the high-low loading blocks.

Comparison of Results.
We compare the mean deviation, standard deviation, and maximum deviation of cumulative life and damage for models prediction under three kinds of block loading, shown in Tables 11 and 12, respectively. ese three statistics of cumulative life are close to each other between the

Conclusions
In this paper, we proposed a new nonlinear fatigue damage model based on equivalent transformation of stress that does not need to determine any parameter and just requires the information of the S-N curve determined from the fatigue experiments. e proposed model defines a stress equivalent transformation way to translate the damage of one stress to another stress through simple calculation. e proposed method is validated using experimental data which includes two-, three-, and four-block loading. e present model fits the experimental data well for variable amplitude loading, and the algorithm is very simple to be implemented.

Data Availability
All data generated or analyzed during this study are included in this article. Additional data can be obtained by contacting the corresponding author.

Conflicts of Interest
e authors declare no conflicts of interest.