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An Improved Nonlinear Cumulative Damage Model Considering the Influence of Load Sequence and Its Experimental Verification

An Improved Nonlinear Cumulative Damage Model Considering the Influence of Load Sequence and Its... applied sciences Article An Improved Nonlinear Cumulative Damage Model Considering the Influence of Load Sequence and Its Experimental Verification Wei Wang , Jianmin Li, Jun Pan, Huanguo Chen and Wenhua Chen * National and Local Joint Engineering Research Center of Reliability Analysis and Testing for Mechanical and Electrical Products, Zhejiang Sci-Tech University, Hangzhou 310018, China; 201610501010@mails.zstu.edu.cn (W.W.); jjm@zstu.edu.cn (J.L.); panjun@zstu.edu.cn (J.P.); hgchen@zstu.edu.cn (H.C.) * Correspondence: chenwh@zstu.edu.cn Abstract: According to the change characteristics in the toughness of the metal material during the fatigue damage process, the fatigue tests were carried out with the standard 18CrNiMo7-6 material. Scanning the fracture with an electron microscope explains the lack of linear cumulative damage in the mechanism. According to the obtained results, a nonlinear damage accumulation model which considered the loading sequence state under the toughness dissipation model was established. The recursive formula was devised under two-level. The fatigue test data verification of three metal materials showed that using this model to predict fatigue life is satisfactory and suitable for engineering applications. Keywords: nonlinear cumulative damage; loading sequence; fatigue damage; toughness dissipation; two-level loading Citation: Wang, W.; Li, J.; Pan, J.; Chen, H.; Chen, W. An Improved Nonlinear Cumulative Damage Model Considering the Influence of 1. Introduction Load Sequence and Its Experimental The failure of most engineering structures or mechanical parts is caused by the ac- Verification. Appl. Sci. 2021, 11, 6944. cumulation of fatigue damage caused by a series of cyclic loads. Factors affecting the https://doi.org/10.3390/app11156944 accumulation of fatigue damage include load size, loading sequence, load history (number of actions), and load path. The cumulative effect of fatigue damage directly determines Academic Editor: Jongwan Hu the life and reliability of mechanical parts. Scholars have done a lot of work in the field of Received: 28 June 2021 fatigue cumulative damage and have proposed many fatigue cumulative damage theories Accepted: 25 July 2021 and calculation models, which are mainly divided into linear fatigue cumulative damage Published: 28 July 2021 theory, bilinear cumulative damage theory, and nonlinear fatigue cumulative damage theory [1]. The commonly used linear fatigue cumulative damage theory [2] (Miner ’s rule) Publisher’s Note: MDPI stays neutral does not consider the influence of the load sequence, but the fact that Miner ’s rule ignores with regard to jurisdictional claims in the effects of load sequence and load interaction make lifetime estimations obtained by this published maps and institutional affil- rule unsatisfactory [3,4]. Although the bilinear cumulative damage theory considers the iations. effect of load sequence on crack growth to a certain extent, its theoretical model cannot accurately simulate the actual damage process because it is difficult to determine the inflec- tion point of crack growth. Therefore, the accuracy of life prediction is not high [5,6]. For the strain control of austenitic stainless steel, Taheri [7] proposed a conservative model of Copyright: © 2021 by the authors. fatigue damage accumulation under variable amplitude load. This model does not require Licensee MDPI, Basel, Switzerland. the constitutive law but considers plasticity through the cyclic strain stress curve. This article is an open access article In order to overcome the shortcomings of the linear damage accumulation Miner ’s rule, distributed under the terms and a wide range of nonlinear damage accumulation models have been developed. According conditions of the Creative Commons to the change characteristics of fatigue ductility, and based on the theory of continuum Attribution (CC BY) license (https:// damage mechanics, Yuan [8] proposed a modified nonlinear uniaxial fatigue damage creativecommons.org/licenses/by/ accumulation model. The model could be used to predict the failure of the specimen 4.0/). Appl. Sci. 2021, 11, 6944. https://doi.org/10.3390/app11156944 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 6944 2 of 14 and explain the whole process of fatigue damage accumulation. Biezma [9] developed a practical and simple correction factor ensuring that the linear summation of damage was conservative, so as to take the sequence effect into account in random loading. Nonlinear fatigue cumulative damage theory believes that the load sequence has a serious impact on fatigue cumulative damage [10–13]. Although these models are often capable of producing satisfying results for a specific set of experiments, Miner ’s rule remains the most widely used for fatigue design under variable amplitude loading. However, some models have recently been developed, which do not require extensive testing. Many fatigue damage accumulation theories have been proposed to remedy the drawbacks of Miner ’s rule, and a majority of these models are based on non-linear accumulation laws. Benkabouche [14] proposed a method for the prediction of the fatigue- life for different materials subjected to constant amplitude multiaxial proportional loading. The non-linear fatigue damage accumulation models can be classified into the fol- lowing categories: damage curve based models, continuum damage mechanics models, interaction between the various loadings considered models, energy-based damage meth- ods, physical properties degradation-based models, ductility exhaustion-based methods, thermodynamic entropy-based damage theories. Detailed comments on some of these models can be found in [15]. Based on the fatigue test data of the high–low loading and low–high loading of 18CrNiMo7-6 steel, an improved nonlinear cumulative fatigue damage model is proposed based on the ductile dissipation model in the nonlinear cumulative damage theory. By analyzing the damage model, the load interaction parameters can be obtained and added to the ductile dissipation model, and the value of the parameter is determined through the experimental data. This paper explains from the mechanism why Miner ’s rule has different damages under two-level loading. This paper verifies the fatigue life of several commonly used metal materials such as 18CrNiMo7-6 steel, 45 steel, and aluminum alloy under two-level loading using the proposed improved model. A comparison is made among the results calculated by the test data, the Miner ’s rule, the original model, and the modified model with little relative error, which proves the validity of the proposed model. The revised model is designed to facilitate the use of engineers. The coefficient selection is simpler than other nonlinear cumulative damage models, and the prediction results are more accurate than similar models. 2. Damage Accumulation Theory The most widely used linear damage accumulation theory is Miner ’s theory [16]. The theory defines the fatigue damage D as the ratio of the number of cycles n under a certain stress to the fatigue life N of the material under the stress: D = (1) Miner ’s theory believes that under the action of multiple levels of different stress amplitudes, fatigue failure occurs = 1 (2) f i where n is the number of cycles under the ith stress level; N is the fatigue life under the i fi ith stress. Taking into account the decrease in the material’s bearing capacity under cyclic load- ing, nonlinear fatigue damage accumulation theory introduces the concept of the material’s physical property degradation into damage accumulation [17], a typical tough dissipation model proposed by Ye Duyi [18]. According to the Griffith fracture criterion [19], San- Appl. Sci. 2021, 11, 6944 3 of 14 dor [20] established the empirical relationship between material fatigue toughness and static toughness, which was verified by a large number of experimental results [21,22]: U e = (3) W e f f U is the initial toughness without damage, W is fatigue toughness, e is the applied stress amplitude, e is the breaking strength of the material. For metal materials with certain damage, Formula (3) can be rewritten as: U e N a = (4) W e f N f N U is the toughness after N cycles of loading, W is the remaining fatigue toughness, N f N e is the residual breaking strength of the damaged material. f N From the energy consumption process of fatigue damage, the fatigue damage variable is defined as [20]: DW W DW å å i f i f N i=1 i=N+1 D = = = 1 (5) W W W f f f DW represents the plastic hysteresis energy accumulated and dissipated under a i=1 certain damage state, N is the load cycles experienced, N is the cycle of fatigue fracture. Substitute Formulas (3) and (4) into Formula (5): f N D = 1 (6) e U For materials with a power-hardening law, according to the experimental results [18], the tensile strength of the material does not show a sharp decline until it is close to fracture. The Formula (6) is further simplified as: D = 1 (7) This is the calculation formula of the damage variable defined by the material’s toughness dissipation. Its physical meaning is that the degree of fatigue damage of the material can be measured by the amount of change in the metal’s energy or the ability to absorb deformation and fracture during the fatigue process. In order to obtain the damage evolution law under the ductile dissipation model, the damage variable calculation, Formula (7), and the ductile dissipation model, Formula (8) [18], are combined together as U U N 1 f N U = U + ln(1 ) (8) ln N N f f where U is the residual toughness of the material after N 1 cycles of loading, that is, N 1 the energy absorbed by the material under the tensile load before fatigue fracture. For most fatigue problems, because the macroscopic fracture presents brittle fracture characteristics, Appl. Sci. 2021, 11, 6944 4 of 14 there is no obvious necking phenomenon; therefore. D  1 [18], the fatigue damage N 1 evolution law with toughness as a parameter is obtained as: 1U /U N 1 D = ln 1 ln N N (9) ln 1 N 1 N = ln(1 ) ln N N ln N f f f Deriving from Formula (9), the fatigue damage evolution formula can be obtained as: h  i dD = N N ln N (10) f f dN According to the principle of damage equivalence, the remaining life fraction (N /N ) 2 f 2 under the second-stage load and the occurrence of fatigue failure after the first-stage load is applied for a certain cycle of cycles N , and can be derived from Formula (9). Linear cumulative damage is expressed ln N f 2 ln N f 1 N N 2 1 = 1 (11) N N f 2 f 1 where N and N respectively, correspond to the number of fatigue fracture cycles at two f 1 f 2 , different stress levels. 3. Stress Test and Results 3.1. Test Conditions According to the design requirements of a wind-turbine gearbox, the test material is 18CrNiMo7-6 forged steel, which is surface-hardened. The chemical composition is shown in Table 1. The test uses a small sample of a smooth cylindrical shape; the shape is shown in Figure 1. The test sample after fatigue is shown in Figure 2. The fatigue test was carried out on the PWS-E100 electro-hydraulic servo universal testing machine (shown in Figure 3) manufactured by Jinan Times Tester Co., Ltd, Jinan, China. with a load–stress ratio R = 1, a test frequency of 10 Hz, and the loading form is a sine wave. Material strength limit and the yield strength  were determined by the static stretching of the test machine. Table 1. Chemical composition of 18CrNiMo7-6. Element Content C 0.15~0.21% Si 0.40% Cr 1.50~1.80% Mn 0.50~0.90% Ni 1.40~1.70% P <0.035% Mo 0.25~0.35% 3.2. Test Methods Five specimens are used for the static tensile test and the average maximum tensile strength of the material is s = 1100 MPa. Then, every five test pieces are used together to carry out the fatigue test under different stress amplitudes. The test conditions are: symmetrical cycle R = 1, sine wave loading, and a frequency of 10 Hz. The average number of cycles for each group of experiments is shown in Table 2. According to the data in Table 2, the fitting function of the material S-N curve is 4 0.002028 4 obtained, in Figure 4 (S = (2.318 10 ) N 2.208 10 ). The S-N curve is used Appl. Sci. 2021, 11, 6944 5 of 14 Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 15 Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 15 Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 15 to obtain the fatigue-fracture cycle N at various stress levels e (R = 1). According to the f a trend line, the fatigue limit of 18CrNiMo7-6 is taken as 350 MPa. Figure Figure 1. 1. T Test est piece piece size size.. Figure 1. Test piece size. Figure 1. Test piece size. Figure 2. Test sample after fatigue. Figure 2. Test sample after fatigue. Figure 2. Test sample after fatigue. Figure 2. Test sample after fatigue. Figure 3. Test machine. Figure 3. Test machine. Figure 3. Test machine. Figure 3. Test machine. 3.2. Test Methods T 3. able 2. T2. est Fatigue Methods test results. 3.2. Test Methods Five specimens are used for the static tensile test and the average maximum tensile Five specimens are used for the static tensile test and the average maximum tensile Loading Stress Amplitude/MPa N/Number of Cycles strength of the material is σ = 1100 MPa . Then, every five test pieces are used together Five specimens are used for the static tensile test and the average maximum tensile strength of the material is σ = 1100 MPa . Then, every five test pieces are used together 1100 0.5 strength of the material is σ = 1100 MPa . Then, every five test pieces are used together to carry out the fatigue test under different stress amplitudes. The test conditions are: to carry out the fatigue test under different stress amplitudes. The test conditions are: 832 471 symmetrical cycle R = −1, sine wave loading, and a frequency of 10 Hz. The average num- to carry out the fatigue test under different stress amplitudes. The test conditions are: symmetrical cycle R = −1, sine wave loading, and a frequency of 10 Hz. The average num- 728 3132 ber of cycles for each group of experiments is shown in Table 2. symmetrical cycle R = −1, sine wave loading, and a frequency of 10 Hz. The average num- 624 23,414 ber of cycles for each group of experiments is shown in Table 2. ber of cycles for each gro 520 up of experiments is shown in Table 2. 246,809 416 3,189,720 353 9,230,000 Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 15 Table 2. Fatigue test results. Loading Stress Amplitude/MPa N/Number of Cycles 1100 0.5 832 471 728 3132 624 23414 520 246,809 416 3,189,720 353 9,230,000 According to the data in Table 2, the fitting function of the material S-N curve is ob- 4 −0.002028 4 tained, in Figure 4 ( S = (2.318 × 10 ) × N − 2.208 × 10 ). The S-N curve is used to Appl. Sci. 2021, 11, 6944 6 of 14 obtain the fatigue-fracture cycle Nf at various stress levels ea (R = −1). According to the trend line, the fatigue limit of 18CrNiMo7-6 is taken as 350 MPa. Figure 4. S–N curve. Figure 4. S–N curve. According to the obtained S-N curve, the two-stage loading test plan is designed as follows: take the stress value at the inflection point of the curve 520 MPa, for high–low According to the obtained S-N curve, the two-stage loading test plan is designed as load application conditions; firstly, cycle n times with 520 MPa stress (N  240,000); and 1 f 1 follows: take the stress value at the inflection point of the curve 520 MPa, for high–low then apply a 420 MPa stress (N  3,200,000) cycle until breaking (corresponding number f 2 load application conditions; firstly, cycle n1 times with 520 MPa stress (Nf1 ≈ 240,000); and of cycles n ). For low–high load conditions, after 420 MPa stress cycle n times, apply a 2 1 then apply a 420 MPa stress (Nf2 ≈ 3,200,000) cycle until breaking (corresponding number 520 MPa stress cycle until breaking (corresponding number of cycles n ). Every five test of cycles n2). For low–high load conditions, after 420 MPa stress cycle n1 times, apply a 520 pieces are a group, and the average data from the test is shown in Table 3. MPa stress cycle until breaking (corresponding number of cycles n2). Every five test pieces are Table a group 3. 18CrNiMo7-6 , and the ave two-stage rage dat loading a from t test data. he test is shown in Table 3. Test Value Two-Stage Loading Table 3. 18CrNiMo7-6 two-stage loading test data. Stress Level/MPa n n /N n n /N n /N +n /N 1 1 f 1 2 2 f 2 1 f 1 2 f 2 Two-Stage Loading Test Value 60,000 0.25 1,480,000 0.463 0.7125 Stress Level/MPa n1 n1/Nf1 n2 n2/Nf2 n1/Nf1+n2/Nf2 120,000 0.50 850,000 0.266 0.766 520–420 60,000 0.25 1,480,000 0.463 0.7125 156,000 0.65 324,800 0.1015 0.7515 520–420 120,000 0.50 850,000 0.266 0.766 800,000 0.25 216,000 0.90 1.15 156,000 0.65 324,800 0.1015 0.7515 1,350,000 0.42 177,600 0.74 1.16 420–520 800,000 0.25 216,000 0.90 1.15 2,000,000 0.63 144,000 0.60 1.23 420–520 1,350,000 0.42 177,600 0.74 1.16 2,000,000 0.63 144,000 0.60 1.23 In the sequence of load application, the first high-stress load will have a greater impact on the overall damage as the data in Table 3. In high–low loading, the total damage value In the sequence of load application, the first high-stress load will have a greater im- n /N +n /N is less than 1; in low–high loading, the total damage value n /N +n /N is 1 1 2 2 1 1 2 2 f f f f pact on the overall damage as the data in Table 3. In high–low loading, the total damage greater than 1. Unlike in the linear damage accumulation model, the total damage value in value the cumulative n1/Nf1+n2/Nmodel f2 is less is t always han 1; in equal low to–1. high loading, the total damage value n1/Nf1+n2/Nf2 In order to further analyze the tension–compression fatigue fracture mechanism, the fatigue fracture specimen was sliced and then analyzed by the JSM-5610LV scanning electron microscope, as shown in Figures 5 and 6. After binarization, the black pixels are in the form of dimples. According to statistics, the pixels in Figure 7 occupy the total selection area of S1 (0.921%), and the pixels in Figure 8 occupy the total selection area of S2 (9.79%). S2/S1 = 10.63. High–low loading significantly promotes the formation of dimples. Binarize the middle part of Figures 5 and 6 to obtain Figures 7 and 8. Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 15 Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 15 Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 15 is greater than 1. Unlike in the linear damage accumulation model, the total damage value in the cumulative model is always equal to 1. is greater than 1. Unlike in the linear damage accumulation model, the total damage value In order to further analyze the tension–compression fatigue fracture mechanism, the is greater than 1. Unlike in the linear damage accumulation model, the total damage value in the cumulative model is always equal to 1. fatigue fracture specimen was sliced and then analyzed by the JSM-5610LV scanning elec- in the cumulative model is always equal to 1. In order to further analyze the tension–compression fatigue fracture mechanism, the tron microscope, as shown in Figures 5 and 6. Appl. Sci. 2021, 11, 6944 In order to further analyze the tension–compression fatigue fracture mechanism 7 of , the 14 fatigue fracture specimen was sliced and then analyzed by the JSM-5610LV scanning elec- fatigue fracture specimen was sliced and then analyzed by the JSM-5610LV scanning elec- tron microscope, as shown in Figures 5 and 6. tron microscope, as shown in Figures 5 and 6. Figure 5. Low–high load fracture surface (420–520 MPa, n1/N1 = 0.25). Figure 5. Low–high load fracture surface (420–520 MPa, n1/N1 = 0.25). Figure 5. Low–high load fracture surface (420–520 MPa, n /N = 0.25). Figure 5. Low–high load fracture surface (420–520 MPa, n1/N1 = 0.25). 1 1 Figure 6. High–low load fracture surface (520–420 MPa, n1/N1 = 0.25). Figure 6. Bina ri High–low loa ze the middle d fract pau rtre surface (52 of Figures 5 0–420 MPa and 6 to ob , n1t /N ain 1 = 0 Fig .25). ures 7 and 8. Figure 6. High–low load fracture surface (520–420 MPa, n1/N1 = 0.25). Figure 6. High–low load fracture surface (520–420 MPa, n /N = 0.25). 1 1 Binarize the middle part of Figures 5 and 6 to obtain Figures 7 and 8. Binarize the middle part of Figures 5 and 6 to obtain Figures 7 and 8. Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 15 Figure 7. Low–high load fracture surface (420–520 MPa, n /N = 0.25) binarization. 1 1 Figure 7. Low–high load fracture surface (420–520 MPa, n1/N1 = 0.25) binarization. Figure 7. Low–high load fracture surface (420–520 MPa, n1/N1 = 0.25) binarization. Figure 7. Low–high load fracture surface (420–520 MPa, n1/N1 = 0.25) binarization. Figure 8. High–low loading fracture surface (520–420 MPa, n1/N1 = 0.25) binarization. Figure 8. High–low loading fracture surface (520–420 MPa, n /N = 0.25) binarization. 1 1 Regardless of the loading mode of high–low or low–high, the section shows the After binarization, the black pixels are in the form of dimples. According to statistics, characteristics of multiple crack sources, and the interactive influence of different crack the pixels in Figure 7 occupy the total selection area of S1 (0.921%), and the pixels in Figure source propagation paths forming different ridge topographies. It can be clearly found 8 occupy the total selection area of S2 (9.79%). S2/S1 = 10.63. High–low loading signifi- that under low–high loading, the number of dimples in the cleavage zone formed on cantly promotes the formation of dimples. the fracture surface is less than the number of dimples formed on the fracture surface Regardless of the loading mode of high–low or low–high, the section shows the char- acteristics of multiple crack sources, and the interactive influence of different crack source propagation paths forming different ridge topographies. It can be clearly found that under low–high loading, the number of dimples in the cleavage zone formed on the fracture surface is less than the number of dimples formed on the fracture surface under high–low loading. The formation of dimples under the repeated action of normal stress accelerates the formation of microscopic voids caused by plastic deformation of the material in a small range during the high–low loading process. This difference ultimately leads to different total damage changes after low–high loading and high–low loading. That is, under the low–high loading form, the first-level low load forms the exercise effect. In the high–low loading mode, the first-level high load does not form an exercise effect and the overall fatigue life decreases rapidly with the increase in the first-level high-load cycle. Due to the presence of dislocations in the material, dislocation clusters are easily formed at grain boundaries, phase boundaries, and material defects during the tension–compression pro- cess, which leads to stress concentration and induces the initiation and growth of mi- crovoids, eventually leading to fracture. High–low loading causes a significantly higher number of micro-holes to be generated than in the case of micro-holes generated by low– high load loading, which means that the number of original microcrack sources under high–low loading is large. These microcracks are easier to connect and propagate to form cracks during cyclic loading, resulting in a decrease in the fatigue life of the material com- pared to low–high load loading. 4. Improved Cumulative Damage Model According to the Formula (11) mentioned above, the load ratio effect parameter n is introduced, so the improved cumulative damage model is expressed  σ  D =  D (12) 2 2    1  Formula (12) reflects the effect of load loading sequence on damage, and the damage relationship between the improved model and the original model as: ≥ D σ ≥ σ（Low - High） 2 2 1 ≤ D σ ≤ σ（High - Low）  2 2 1 Appl. Sci. 2021, 11, 6944 8 of 14 under high–low loading. The formation of dimples under the repeated action of normal stress accelerates the formation of microscopic voids caused by plastic deformation of the material in a small range during the high–low loading process. This difference ultimately leads to different total damage changes after low–high loading and high–low loading. That is, under the low–high loading form, the first-level low load forms the exercise effect. In the high–low loading mode, the first-level high load does not form an exercise effect and the overall fatigue life decreases rapidly with the increase in the first-level high-load cycle. Due to the presence of dislocations in the material, dislocation clusters are easily formed at grain boundaries, phase boundaries, and material defects during the tension–compression process, which leads to stress concentration and induces the initiation and growth of microvoids, eventually leading to fracture. High–low loading causes a significantly higher number of micro-holes to be generated than in the case of micro-holes generated by low– high load loading, which means that the number of original microcrack sources under high–low loading is large. These microcracks are easier to connect and propagate to form cracks during cyclic loading, resulting in a decrease in the fatigue life of the material compared to low–high load loading. 4. Improved Cumulative Damage Model According to the Formula (11) mentioned above, the load ratio effect parameter n is introduced, so the improved cumulative damage model is expressed D = D (12) Formula (12) reflects the effect of load loading sequence on damage, and the damage relationship between the improved model and the original model as: D s  s (Low High) 0 2 2 1 D s  s High Low ( ) 2 2 1 According to Formula (11), the number of cycles n under the first-stage load ampli- tude s is equivalent to the equivalent number of cycles n under the second-stage load amplitude s , expressed as: 1   2 ln 1 ln 1 N s N f 1 f 2 D = = = D (13) ln N s ln N f 1 1 f 2 ln N f 2 s ( ) ln N s f 1 2 = 1 1 (14) N N f 2 f 1 n +n ln 1 s N f 2 D = (15) s ln N 1 f 2 When the total damage degree is 1, the specimen is damaged. Formula (15) D = 1, represents the material under the action of two levels of load, after the number of cycles n of the first level load amplitude s , and the remaining life fraction of the second level load amplitude s , expressed ln N b b f 2 s s ! 1 ! 1 ( ) ( ) ln N s s 2 2 f 1 n n 1 2 1 = 1 (16) N N N f 2 f 1 f 2 Appl. Sci. 2021, 11, 6944 9 of 14 By analogy, the total damage under the multi-stage load and the remaining life fraction n /N of the last stage load s can be derived as: i f i i n +n n n ln 1 s N f i D = ( ) (17) s ln N i1 f i i=1 ln N b b s s f i i1 i1 ! ! ( ) ( ) ln N s s f i1 i i n n 1 i i1 = 1 (18) N N N f i f i1 f i According to the data obtained from the test, for 18CrNiMo7-6 material, b = 2.8. One can substitute b = 2.8 into Formula (16) as: ln N 2.8 2.8 f 2 s s ! 1 ! 1 ( ) ( ) ln N s s f 1 2 2 n n 1 2 1 = 1 (19) N N N f 2 f 1 f 2 5. Test Results and Analysis In order to verify the effectiveness of the proposed improved model, based on the fatigue test data of the material 18CrNiMo7-6 listed in Table 3, the life prediction results of this model, the linear damage accumulation model, and the ductile dissipation model, are compared. The results are shown in Table 4. In addition, for 45# steel and Al-2024 aluminum alloy, commonly used in mechanical engineering, calculations and comparisons are also made based on the experimental data of the literature [23,24]. The results are listed in Tables 5 and 6. For each material in the corresponding loading mode, the relative error between the experimental value and the theoretical calculation is shown in Figures 9–14. Table 4. 18CrNiMo7-6 two-stage loading test data and the remaining life prediction results of each model. Miner’s Rule Ductile Dissipation The Proposed Model Test Value Two-Stage (Formula (1)) Model (Formula (11)) (Formula (19)) Loading Stress Relative Relative Relative Level/MPa n n /N n n /N n /N n /N n /N 1 1 1 2 2 2 2 2 2 2 2 2 f f f f f Error/% Error/% Error/% 60,000 0.25 1,480,000 0.583 0.75 28.64% 0.7062 21.13% 0.5168 11.35% 520–420 120,000 0.50 850,000 0.266 0.50 87.97% 0.4325 62.59% 0.2038 23.38% 156,000 0.65 324,800 0.1015 0.35 244.8% 0.2810 176.8% 0.0899 11.43% 800,000 0.25 216,000 0.9 0.75 16.67% 0.7883 12.41% 0.8818 2.02% 420–520 1,350,000 0.42 177,600 0.74 0.55 25.67% 0.6356 14.10% 0.7872 6.38% 2,000,000 0.63 144,000 0.60 0.35 41.67% 0.4443 25.95% 0.6518 8.63% Table 5. The 45# steel two-stage loading test data and remaining life prediction results of each model. Miner’s Rule Ductile Dissipation The Proposed Model Test Value Two-Stage (Formula (1)) Model (Formula (11)) (Formula (19)) Loading Stress Relative Relative Relative Level/MPa n n /N n n /N n /N n /N n /N 1 1 1 2 2 2 2 2 2 2 2 f f f f2 f Error/% Error/% Error/% 12,500 0.25 250,400 0.5008 0.7500 49.76% 0.7055 40.87% 0.5755 14.91% 25,000 0.50 168,300 0.3366 0.5000 48.54% 0.4314 28.17% 0.2671 20.64% 331.5–284.4 37,500 0.75 64,500 0.1290 0.2500 93.80% 0.1861 44.29% 0.0698 45.89% 125,000 0.25 37,900 0.7580 0.7500 1.06% 0.7888 4.07% 0.8608 13.56% 284.4–331.5 250,000 0.50 38,900 0.7780 0.5000 35.73% 0.5646 27.42% 0.6970 10.41% 375,000 0.75 43,400 0.8680 0.2500 71.20% 0.3188 63.27% 0.4858 44.03% Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 15 Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 15 listed in Tables 5 and 6. For each material in the corresponding loading mode, the relative listed in Tables 5 and 6. For each material in the corresponding loading mode, the relative error between the experimental value and the theoretical calculation is shown in Figures error between the experimental value and the theoretical calculation is shown in Figures 9–14. 9–14. Appl. Sci. 2021, 11, 6944 10 of 14 Table 4. 18CrNiMo7-6 two-stage loading test data and the remaining life prediction results of each model. Table 4. 18CrNiMo7-6 two-stage loading test data and the remaining life prediction results of each model. Miner’s Rule (For- Ductile Dissipation The Proposed Model Two-Stage Two-Stage Miner’s Rule (For- Ductile Dissipation The Proposed Model Test Value Table 6. Al-2024 aluminum alloy two-stage Test Value loading test data and remaining life prediction results of each model. Loading mula (1)) Model (Formula (11)) (Formula (19)) Loading mula (1)) Model (Formula (11)) (Formula (19)) Stress Relative Relative Relative Miner’s Rule Ductile Dissipation The Proposed Model Stress Relative Relative Relative n1 n1/Nf1 n2 n2/Nf2 n2/Nf2 n2/Nf2 n2/Nf2 Test Value n1 n1/Nf1 n2 n2/Nf2 n2/Nf2 n2/Nf2 n2/Nf2 Two-Stage (Formula (1)) Model (Formula (11)) (Formula (19)) Level/MPa Error/% Error/% Error/% Level/MPa Error/% Error/% Error/% Loading Stress Relative Relative Relative 60,000 0.25 1,480,000 0.583 0.75 28.64% 0.7062 21.13% 0.5168 11.35% Level/MPa n n /N60,000 n 0.25 n /N1,480,00n0 /N 0.583 0.75 n /N28.64% 0.7062 2 n /N1.13% 0.5168 11.35% 1 1 f 1 2 2 f 2 2 f 2 2 f 2 2 f 2 Error/% Error/% Error/% 520–420 120,000 0.50 850,000 0.266 0.50 87.97% 0.4325 62.59% 0.2038 23.38% 520–420 120,000 0.50 850,000 0.266 0.50 87.97% 0.4325 62.59% 0.2038 23.38% 30,000 0.2000 228,700 0.5319 0.8000 50.40% 0.7844 47.47% 0.5623 5.71% 156,000 0.65 324,800 0.1015 0.35 244.8% 0.2810 176.8% 0.0899 11.43% 60,000 0.4000156,000 101,0500.65 0.2350324,800.6000 0 0.1015 155.32%0.35 0.5735244.8% 144.05%0.2810 1 0.267776.8% 0 13.91%.0899 11.43% 200–150 90,000 0.6000 76,050 0.1769 0.4000 126.12% 0.3689 108.53% 0.0941 46.80% 800,000 0.25 216,000 0.9 0.75 16.67% 0.7883 12.41% 0.8818 2.02% 800,000 0.25 216,000 0.9 0.75 16.67% 0.7883 12.41% 0.8818 2.02% 86,000 420–520 0.20001,350,00 144,500 0 0.42 1 0.963377,60 0.80000 0.74 16.95% 0.55 0.814625.67% 15.43% 0.6356 0.915314.10% 4.98%0.7872 6.38% 420–520 1,350,000 0.42 177,600 0.74 0.55 25.67% 0.6356 14.10% 0.7872 6.38% 172,000 0.4000 133,500 0.8900 0.6000 32.58% 0.6254 29.73% 0.8186 8.02% 150–200 2,000,000 0.63 144,000 0.60 0.35 41.67% 0.4443 25.95% 0.6518 8.63% 2,000,000 0.63 144,000 0.60 0.35 41.67% 0.4443 25.95% 0.6518 8.63% 258,000 0.6000 81,700 0.5447 0.4000 26.57% 0.4309 20.89% 0.6992 28.36% Formula 1 Formula 1 Formula 11 Formula 11 Formula 19 Formula 19 Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 15 0.25 0.5 0.65 0.25 0.5 0.65 n /N n /N 1 f1 case of high–low loading. This 1 also f1shows that, from the side, different loading sequences have different effects on the fatigue life. Figure 9. (520–420 MPa) The relative error of each model under different n1/Nf1 (18CrNiMo7-6). Figure 9. (520–420 MPa) The relative error of each model under different n1/Nf1 (18CrNiMo7-6). Figure 9. (520–420 MPa) The relative error of each model under different n /N (18CrNiMo7-6). 1 f 1 Table 5. The 45# steel two-stage loading test data and remaining life prediction results of each model. Two-Stage Miner’s Rule (For- Ductile Dissipation The Proposed Model Formula 1 Test Value Formula 1 mula (1)) Model (Formula (11)) (Formula (19)) Loading Formula 11 Formula 11 Stress Relative Relative Relative n1 n1/Nf1 n2 n2/Nf2 n2/Nf2 n2/Nf2 n2/Nf2 Formula 19 Formula 19 Level/MPa Error/% Error/% Error/% 12,500 0.25 250,400 0.5008 0.7500 49.76% 0.7055 40.87% 0.5755 14.91% 331.5–284.4 25,000 0.50 168,300 0.3366 0.5000 48.54% 0.4314 28.17% 0.2671 20.64% 37,500 0 0.75 64,500 0.1290 0.2500 93.80% 0.1861 44.29% 0.0698 45.89% 0.25 0.42 0.63 0.25 0.42 0.63 125,000 0.25 37,900 0.7580 0.7500 1.06% 0.7888 4.07% 0.8608 13.56% n /N 284.4–331.5 250,000 0.50 38,900 0.7780 0.5000 35.73% 0.5646 27.42% 0.6970 10.41% n /N 1 f1 1 f1 375,000 0.75 43,400 0.8680 0.2500 71.20% 0.3188 63.27% 0.4858 44.03% Figure 10. (420–520 MPa) The relative error of each model under different n /N (18CrNiMo7-6). Figure 10. (420–520 MPa) The relative error of each model under different n1/Nf1 (18CrNiMo7-6). 1 f 1 Figure 10. (420–520 MPa) The relative error of each model under different n1/Nf1 (18CrNiMo7-6). It can be seen from Figures 9 and 10, for 18CrNiMo7-6 material, under high–low load It can be seen from Figures 9 and 10, for 18CrNiMo7-6 material, under high–low load loadingFo , asrm thu e proport la 1 ion of high load continues to expand, the relative error of Miner’s loading, as the proportion of high load continues to expand, the relative error of Miner’s rule increases significantly. The relative error of the tough dissipation model is also grad- Formula 11 rule increases significantly. The relative error of the tough dissipation model is also grad- ually increasing, but the overall value is smaller than Miner’s rule. The relative error be- ually incr Fo easing, but the o rmula 19 verall value is smaller than Miner’s rule. The relative error be- tween the result obtained by the improved model and the actual value is the smallest tween the result obtained by the improved model and the actual value is the smallest among these three models. In the case of low–high loading, the error between the results among these three models. In the case of low–high loading, the error between the results obtained by the three models and the actual value is smaller than the error obtained in the obtained by the three models and the actual value is smaller than the error obtained in the 0.25 0.50 0.75 n /N 1 f1 Figure 11. (331.5–284.4 MPa) The relative error of each model under different n1/Nf1 (45# steel). Figure 11. (331.5–284.4 MPa) The relative error of each model under different n /N (45# steel). 1 f 1 Formula 1 Formula 11 Formula 19 0.25 0.50 0.75 n /N 1 f1 Figure 12. (284.4–331.5 MPa) The relative error of each model under different n1/Nf1 (45# steel). It can be seen from Figures 11 and 12, for 45# steel, under high–low load loading, as the proportion of high load continues to increases, the relative error of Miner’s rule in- creases significantly. The relative error fluctuation of the toughness dissipation model is relatively stable, and the overall value is smaller than Miner’s rule. The relative error be- tween the result obtained by the improved model and the actual value is the smallest among these three models. In the case of low–high loading, the error between the results obtained by the three models and the actual value is smaller than the error obtained in the case of high–low loading. Like the results of 18CrNiMo7-6, both reflect that under high– Relative error/% Relative error/% Relative error/% Relative error/% Relative error/% Relative error/% Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 15 case of high–low loading. This also shows that, from the side, different loading sequences have different effects on the fatigue life. Table 5. The 45# steel two-stage loading test data and remaining life prediction results of each model. Two-Stage Miner’s Rule (For- Ductile Dissipation The Proposed Model Test Value mula (1)) Model (Formula (11)) (Formula (19)) Loading Stress Relative Relative Relative n1 n1/Nf1 n2 n2/Nf2 n2/Nf2 n2/Nf2 n2/Nf2 Level/MPa Error/% Error/% Error/% 12,500 0.25 250,400 0.5008 0.7500 49.76% 0.7055 40.87% 0.5755 14.91% 331.5–284.4 25,000 0.50 168,300 0.3366 0.5000 48.54% 0.4314 28.17% 0.2671 20.64% 37,500 0.75 64,500 0.1290 0.2500 93.80% 0.1861 44.29% 0.0698 45.89% 125,000 0.25 37,900 0.7580 0.7500 1.06% 0.7888 4.07% 0.8608 13.56% 284.4–331.5 250,000 0.50 38,900 0.7780 0.5000 35.73% 0.5646 27.42% 0.6970 10.41% 375,000 0.75 43,400 0.8680 0.2500 71.20% 0.3188 63.27% 0.4858 44.03% Formula 1 Formula 11 Formula 19 Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 15 Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 15 0.25 0.50 0.75 n /N 1 f1 Appl. Sci. 2021, 11, 6944 11 of 14 low loading, as the proportion of high load continues to increase, the relative e rror value low loading, as the proportion of high load continues to increase, the relative error value is higher than under low–high loading, and the proportion of low load continues to in- Figure 11. (331.5–284.4 MPa) The relative error of each model under different n1/Nf1 (45# steel). is higher than under low–high loading, and the proportion of low load continues to in- crease. crease. Table 6. Al-2024 aluminum alloy two-stage loading test data and remaining life prediction results of each model. Table 6. Al-2024 aluminum alloy two-stage loading test data and remaining life prediction results of each model. Formula 1 Two-Stage Miner’s Rule (For- Ductile Dissipation The Proposed Model Two-Stage Miner’s Rule (For- Ductile Dissipation The Proposed Model Formula 11 Test Value Test Value Loading 100 mula (1)) Model (Formula (11)) (Formula (19)) mula (1)) Model (Formula (11)) (Formula (19)) Loading Formula 19 Stress Relative Relative Relative Stress Relative Relative Relative n1 n1/Nf1 n2 n2/Nf2 n2/Nf2 n2/Nf2 n2/Nf2 n1 n1/Nf1 n2 n2/Nf2 n2/Nf2 n2/Nf2 n2/Nf2 Level/MPa Error/% Error/% Error/% Level/MPa Error/% Error/% Error/% 30,000 0.2000 228,700 0.5319 0.8000 50.40% 0.7844 47.47% 0.5623 5.71% 30,000 0.2000 228,700 0.5319 0.8000 50.40% 0.7844 47.47% 0.5623 5.71% 200–150 60,000 0.4000 101,050 0.2350 0.6000 155.32% 0.5735 144.05% 0.2677 13.91% 200–150 60,000 0.4000 101,050 0.2350 0.6000 155.32% 0.5735 144.05% 0.2677 13.91% 90,000 0.6000 76,050 0.1769 0.4000 126.12% 0.3689 108.53% 0.0941 46.80% 90,000 0.6000 76,050 0.1769 0.4000 126.12% 0.3689 108.53% 0.0941 46.80% 86,000 0.2000 1 0.25 44,500 0.9633 0.050.8000 16.95% 0.8146 0.75 15.43% 0.9153 4.98% 86,000 0.2000 144,500 0.9633 0.8000 16.95% 0.8146 15.43% 0.9153 4.98% 150–200 172,000 0.4000 133,500 0.8900 n0/N.6000 32.58% 0.6254 29.73% 0.8186 8.02% 150–200 172,000 0.4000 133,500 0.8900 0.6000 32.58% 0.6254 29.73% 0.8186 8.02% 1 f1 258,000 0.6000 81,700 0.5447 0.4000 26.57% 0.4309 20.89% 0.6992 28.36% 258,000 0.6000 81,700 0.5447 0.4000 26.57% 0.4309 20.89% 0.6992 28.36% Figure 12. (284.4–331.5 MPa) The relative error of each model under different n1/Nf1 (45# steel). Figure 12. (284.4–331.5 MPa) The relative error of each model under different n /N (45# steel). 1 f 1 It can be seen from Figures 11 and 12, for 45# steel, under high–low load loading, as Formula 1 the proportion of high load continues to increases, the relative error of Miner’s rule in- Formula 1 150 creases significantly. The relative error fluctuation of the toughness dissipation model is Formula 11 Formula 11 relatively stable, and the overall value is smaller than Miner’s rule. The relative error be- Formula 19 Formula 19 tween the result obtained by the improved model and the actual value is the smallest among these three models. In the case of low–high loading, the error between the results obtained by the three models and the actual value is smaller than the error obtained in the case of high–low loading. Like the results of 18CrNiMo7-6, both reflect that under high– 0.20 0.40 0.60 0.20 0.40 0.60 n /N n /N 1 f1 1 f1 Figure 13. (200–150 MPa) The relative error of each model under different n1/Nf1 (Al-2024). Figure 13. (200–150 MPa) The relative error of each model under different n1/Nf1 (Al-2024). Figure 13. (200–150 MPa) The relative error of each model under different n /N (Al-2024). 1 1 Formula 1 Formula 1 150 Formula 11 Formula 11 Formula 19 Formula 19 0.20 0.40 0.60 0.20 0.40 0.60 n /N n /N 1 f1 1 f1 Figure 14. (150–200 MPa) The relative error of each model under different n1/Nf1 (Al-2024). Figure 14. (150–200 MPa) The relative error of each model under different n /N (Al-2024). Figure 14. (150–200 MPa) The relative error of each model under different 1 1 n1/Nf1 (Al-2024). It can be seen from It ca Figur n be es seen f 9 and rom 10 Fi , gures 13 for 18CrNiMo7-6 and 14, for ( material, Al-2024) under , under high–l high–low ow loading, as the It can be seen from Figures 13 and 14, for (Al-2024), under high–low loading, as the proportion of high load continues to increase, the relative error stability of the results ob- load loading, as the proportion of high load continues to expand, the relative error of proportion of high load continues to increase, the relative error stability of the results ob- Miner ’s rule increases tained by the du significantly.ctil The e dissipa relative tierr on model i or of thestough better tha dissipation n that of model the Mi isner’s rul also e. Under the tained by the ductile dissipation model is better than that of the Miner’s rule. Under the gradually increasing, same con but the ditions, the overall value accur isasmaller cy of the than results Miner from the ’s rule.proposed mo The relativedel error is higher than that same conditions, the accuracy of the results from the proposed model is higher than that between the result of M obtained iner’s ru by lethe and t impr he oved ductile model dissip and ation model the actual . Un value der low is the –hismallest gh loading, the results of of Miner’s rule and the ductile dissipation model. Under low–high loading, the results of among these three each mode models. In l athe re re case lative ofly low–high stable. The loading, accurac the y o err f th or e between improved model the results is the highest. each model are relatively stable. The accuracy of the improved model is the highest. obtained by the three models and the actual value is smaller than the error obtained in the case of high–low loading. This also shows that, from the side, different loading sequences have different effects on the fatigue life. It can be seen from Figures 11 and 12, for 45# steel, under high–low load loading, as the proportion of high load continues to increases, the relative error of Miner ’s rule increases significantly. The relative error fluctuation of the toughness dissipation model Relative error/% Relative error/% Relative error/% Relative error/% Relative error/% Relative error/% Appl. Sci. 2021, 11, 6944 12 of 14 is relatively stable, and the overall value is smaller than Miner ’s rule. The relative error between the result obtained by the improved model and the actual value is the smallest among these three models. In the case of low–high loading, the error between the results obtained by the three models and the actual value is smaller than the error obtained in the case of high–low loading. Like the results of 18CrNiMo7-6, both reflect that under high–low loading, as the proportion of high load continues to increase, the relative error value is higher than under low–high loading, and the proportion of low load continues to increase. It can be seen from Figures 13 and 14, for (Al-2024), under high–low loading, as the proportion of high load continues to increase, the relative error stability of the results obtained by the ductile dissipation model is better than that of the Miner ’s rule. Under the same conditions, the accuracy of the results from the proposed model is higher than that of Miner ’s rule and the ductile dissipation model. Under low–high loading, the results of each model are relatively stable. The accuracy of the improved model is the highest. From the data in Tables 4–6, the linear damage accumulation model (Miner ’s rule) assumes that damage is not related to the load state, damage accumulation is similarly not related to the load sequence, the interaction between loads cannot be considered, and the deviation from the test results is the largest. The relative error stability of the results obtained by the ductile dissipation model is better than that of the Miner ’s rule. The improved model, proposed in this paper, increases the influence factors of the sequential stress sequence and magnitude and the error is smaller than the original ductile dissipation model. In this paper, the improved model is extended and applied to commonly used 45# steel and Al-2024 aluminum alloy. The error is larger than that of 18CrNiMo7-6, but it is still smaller than the original toughness dissipation model, indicating that the model in this paper has better material applicability. 6. Conclusions (1) Based on the ductile dissipation theory, a nonlinear fatigue cumulative damage model considering the loading sequence is established, that is, an improved toughness dissipation model, which can consider the impact of loading sequence on damage with parameters which are simple and suitable for engineering applications; (2) The fracture sections of the 18CrNiMo7-6 specimens, which were scanned by electron microscope, explain from the mechanism why Miner ’s rule has different damages under two-level loading. The results of the electron microscope showed that the number of dimples formed on the fracture surface under low–high load was less than the number of dimples formed on the fracture surface under high–low load. This indicated that the number of micro-crack sources in the cross-section was relatively small, and the probability of micro-cracks connecting and expanding into cracks was relatively small, resulting in a small totally effective level of damage under low–high load cycles and a longer fatigue life; (3) The improved model proposed in this paper is based on the test data of 18CrNiMo7-6 forged steel. This paper uses #45 steel, AL-2024 aluminum alloy and the nonlinear damage accumulation processes of other common materials, to predict that the life under two-stage load has smaller errors and better accuracy than the classic ductile dissipation model, indicating that the improved model has a good material appli- cability. Although this article uses three kinds of metal materials to test and verify the established model and has achieved good results, for other types of materials and test environments, much test verification and further research are needed. Fur- ther in-depth research can be done on issues such as the quantification of toughness dissipation, the improvement of model accuracy, and material verification. Author Contributions: W.W. and J.L. designed experiment verification and validation schemes. J.P. and H.C. gathered experimental data and analyzed experimental results. W.C. was supervision. W.W. wrote the manuscript. All authors have read and agreed to the published version of the manuscript. Appl. Sci. 2021, 11, 6944 13 of 14 Funding: This research was funded by International S&T Cooperaion Program of China, grantnum- ber (2015DFA71400), NSFC(U1709210), National Natural Science Foundation of Chna: (51975535) and “The APC was funded by International S&T Cooperaion Program of China, grant number (2015DFA71400)”. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Publicly available datasets were analyzed in this study. This data can be found here: [http://en.cnki.com.cn/Article_en/CJFDTOTAL-GCHE201804014.htm] (accessed on 25 July 2021) and [http://en.cnki.com.cn/Article_en/CJFDTOTAL-GCHE201804014.htm] (accessed on 25 July 2021). Acknowledgments: The present research is supported by the International S&T Cooperaion Program of China (2015DFA71400), NSFC (U1709210), National Natural Science Foundation of China: (51975535). Conflicts of Interest: The authors declare no conflict of interest. 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An Improved Nonlinear Cumulative Damage Model Considering the Influence of Load Sequence and Its Experimental Verification

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Abstract

applied sciences Article An Improved Nonlinear Cumulative Damage Model Considering the Influence of Load Sequence and Its Experimental Verification Wei Wang , Jianmin Li, Jun Pan, Huanguo Chen and Wenhua Chen * National and Local Joint Engineering Research Center of Reliability Analysis and Testing for Mechanical and Electrical Products, Zhejiang Sci-Tech University, Hangzhou 310018, China; 201610501010@mails.zstu.edu.cn (W.W.); jjm@zstu.edu.cn (J.L.); panjun@zstu.edu.cn (J.P.); hgchen@zstu.edu.cn (H.C.) * Correspondence: chenwh@zstu.edu.cn Abstract: According to the change characteristics in the toughness of the metal material during the fatigue damage process, the fatigue tests were carried out with the standard 18CrNiMo7-6 material. Scanning the fracture with an electron microscope explains the lack of linear cumulative damage in the mechanism. According to the obtained results, a nonlinear damage accumulation model which considered the loading sequence state under the toughness dissipation model was established. The recursive formula was devised under two-level. The fatigue test data verification of three metal materials showed that using this model to predict fatigue life is satisfactory and suitable for engineering applications. Keywords: nonlinear cumulative damage; loading sequence; fatigue damage; toughness dissipation; two-level loading Citation: Wang, W.; Li, J.; Pan, J.; Chen, H.; Chen, W. An Improved Nonlinear Cumulative Damage Model Considering the Influence of 1. Introduction Load Sequence and Its Experimental The failure of most engineering structures or mechanical parts is caused by the ac- Verification. Appl. Sci. 2021, 11, 6944. cumulation of fatigue damage caused by a series of cyclic loads. Factors affecting the https://doi.org/10.3390/app11156944 accumulation of fatigue damage include load size, loading sequence, load history (number of actions), and load path. The cumulative effect of fatigue damage directly determines Academic Editor: Jongwan Hu the life and reliability of mechanical parts. Scholars have done a lot of work in the field of Received: 28 June 2021 fatigue cumulative damage and have proposed many fatigue cumulative damage theories Accepted: 25 July 2021 and calculation models, which are mainly divided into linear fatigue cumulative damage Published: 28 July 2021 theory, bilinear cumulative damage theory, and nonlinear fatigue cumulative damage theory [1]. The commonly used linear fatigue cumulative damage theory [2] (Miner ’s rule) Publisher’s Note: MDPI stays neutral does not consider the influence of the load sequence, but the fact that Miner ’s rule ignores with regard to jurisdictional claims in the effects of load sequence and load interaction make lifetime estimations obtained by this published maps and institutional affil- rule unsatisfactory [3,4]. Although the bilinear cumulative damage theory considers the iations. effect of load sequence on crack growth to a certain extent, its theoretical model cannot accurately simulate the actual damage process because it is difficult to determine the inflec- tion point of crack growth. Therefore, the accuracy of life prediction is not high [5,6]. For the strain control of austenitic stainless steel, Taheri [7] proposed a conservative model of Copyright: © 2021 by the authors. fatigue damage accumulation under variable amplitude load. This model does not require Licensee MDPI, Basel, Switzerland. the constitutive law but considers plasticity through the cyclic strain stress curve. This article is an open access article In order to overcome the shortcomings of the linear damage accumulation Miner ’s rule, distributed under the terms and a wide range of nonlinear damage accumulation models have been developed. According conditions of the Creative Commons to the change characteristics of fatigue ductility, and based on the theory of continuum Attribution (CC BY) license (https:// damage mechanics, Yuan [8] proposed a modified nonlinear uniaxial fatigue damage creativecommons.org/licenses/by/ accumulation model. The model could be used to predict the failure of the specimen 4.0/). Appl. Sci. 2021, 11, 6944. https://doi.org/10.3390/app11156944 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 6944 2 of 14 and explain the whole process of fatigue damage accumulation. Biezma [9] developed a practical and simple correction factor ensuring that the linear summation of damage was conservative, so as to take the sequence effect into account in random loading. Nonlinear fatigue cumulative damage theory believes that the load sequence has a serious impact on fatigue cumulative damage [10–13]. Although these models are often capable of producing satisfying results for a specific set of experiments, Miner ’s rule remains the most widely used for fatigue design under variable amplitude loading. However, some models have recently been developed, which do not require extensive testing. Many fatigue damage accumulation theories have been proposed to remedy the drawbacks of Miner ’s rule, and a majority of these models are based on non-linear accumulation laws. Benkabouche [14] proposed a method for the prediction of the fatigue- life for different materials subjected to constant amplitude multiaxial proportional loading. The non-linear fatigue damage accumulation models can be classified into the fol- lowing categories: damage curve based models, continuum damage mechanics models, interaction between the various loadings considered models, energy-based damage meth- ods, physical properties degradation-based models, ductility exhaustion-based methods, thermodynamic entropy-based damage theories. Detailed comments on some of these models can be found in [15]. Based on the fatigue test data of the high–low loading and low–high loading of 18CrNiMo7-6 steel, an improved nonlinear cumulative fatigue damage model is proposed based on the ductile dissipation model in the nonlinear cumulative damage theory. By analyzing the damage model, the load interaction parameters can be obtained and added to the ductile dissipation model, and the value of the parameter is determined through the experimental data. This paper explains from the mechanism why Miner ’s rule has different damages under two-level loading. This paper verifies the fatigue life of several commonly used metal materials such as 18CrNiMo7-6 steel, 45 steel, and aluminum alloy under two-level loading using the proposed improved model. A comparison is made among the results calculated by the test data, the Miner ’s rule, the original model, and the modified model with little relative error, which proves the validity of the proposed model. The revised model is designed to facilitate the use of engineers. The coefficient selection is simpler than other nonlinear cumulative damage models, and the prediction results are more accurate than similar models. 2. Damage Accumulation Theory The most widely used linear damage accumulation theory is Miner ’s theory [16]. The theory defines the fatigue damage D as the ratio of the number of cycles n under a certain stress to the fatigue life N of the material under the stress: D = (1) Miner ’s theory believes that under the action of multiple levels of different stress amplitudes, fatigue failure occurs = 1 (2) f i where n is the number of cycles under the ith stress level; N is the fatigue life under the i fi ith stress. Taking into account the decrease in the material’s bearing capacity under cyclic load- ing, nonlinear fatigue damage accumulation theory introduces the concept of the material’s physical property degradation into damage accumulation [17], a typical tough dissipation model proposed by Ye Duyi [18]. According to the Griffith fracture criterion [19], San- Appl. Sci. 2021, 11, 6944 3 of 14 dor [20] established the empirical relationship between material fatigue toughness and static toughness, which was verified by a large number of experimental results [21,22]: U e = (3) W e f f U is the initial toughness without damage, W is fatigue toughness, e is the applied stress amplitude, e is the breaking strength of the material. For metal materials with certain damage, Formula (3) can be rewritten as: U e N a = (4) W e f N f N U is the toughness after N cycles of loading, W is the remaining fatigue toughness, N f N e is the residual breaking strength of the damaged material. f N From the energy consumption process of fatigue damage, the fatigue damage variable is defined as [20]: DW W DW å å i f i f N i=1 i=N+1 D = = = 1 (5) W W W f f f DW represents the plastic hysteresis energy accumulated and dissipated under a i=1 certain damage state, N is the load cycles experienced, N is the cycle of fatigue fracture. Substitute Formulas (3) and (4) into Formula (5): f N D = 1 (6) e U For materials with a power-hardening law, according to the experimental results [18], the tensile strength of the material does not show a sharp decline until it is close to fracture. The Formula (6) is further simplified as: D = 1 (7) This is the calculation formula of the damage variable defined by the material’s toughness dissipation. Its physical meaning is that the degree of fatigue damage of the material can be measured by the amount of change in the metal’s energy or the ability to absorb deformation and fracture during the fatigue process. In order to obtain the damage evolution law under the ductile dissipation model, the damage variable calculation, Formula (7), and the ductile dissipation model, Formula (8) [18], are combined together as U U N 1 f N U = U + ln(1 ) (8) ln N N f f where U is the residual toughness of the material after N 1 cycles of loading, that is, N 1 the energy absorbed by the material under the tensile load before fatigue fracture. For most fatigue problems, because the macroscopic fracture presents brittle fracture characteristics, Appl. Sci. 2021, 11, 6944 4 of 14 there is no obvious necking phenomenon; therefore. D  1 [18], the fatigue damage N 1 evolution law with toughness as a parameter is obtained as: 1U /U N 1 D = ln 1 ln N N (9) ln 1 N 1 N = ln(1 ) ln N N ln N f f f Deriving from Formula (9), the fatigue damage evolution formula can be obtained as: h  i dD = N N ln N (10) f f dN According to the principle of damage equivalence, the remaining life fraction (N /N ) 2 f 2 under the second-stage load and the occurrence of fatigue failure after the first-stage load is applied for a certain cycle of cycles N , and can be derived from Formula (9). Linear cumulative damage is expressed ln N f 2 ln N f 1 N N 2 1 = 1 (11) N N f 2 f 1 where N and N respectively, correspond to the number of fatigue fracture cycles at two f 1 f 2 , different stress levels. 3. Stress Test and Results 3.1. Test Conditions According to the design requirements of a wind-turbine gearbox, the test material is 18CrNiMo7-6 forged steel, which is surface-hardened. The chemical composition is shown in Table 1. The test uses a small sample of a smooth cylindrical shape; the shape is shown in Figure 1. The test sample after fatigue is shown in Figure 2. The fatigue test was carried out on the PWS-E100 electro-hydraulic servo universal testing machine (shown in Figure 3) manufactured by Jinan Times Tester Co., Ltd, Jinan, China. with a load–stress ratio R = 1, a test frequency of 10 Hz, and the loading form is a sine wave. Material strength limit and the yield strength  were determined by the static stretching of the test machine. Table 1. Chemical composition of 18CrNiMo7-6. Element Content C 0.15~0.21% Si 0.40% Cr 1.50~1.80% Mn 0.50~0.90% Ni 1.40~1.70% P <0.035% Mo 0.25~0.35% 3.2. Test Methods Five specimens are used for the static tensile test and the average maximum tensile strength of the material is s = 1100 MPa. Then, every five test pieces are used together to carry out the fatigue test under different stress amplitudes. The test conditions are: symmetrical cycle R = 1, sine wave loading, and a frequency of 10 Hz. The average number of cycles for each group of experiments is shown in Table 2. According to the data in Table 2, the fitting function of the material S-N curve is 4 0.002028 4 obtained, in Figure 4 (S = (2.318 10 ) N 2.208 10 ). The S-N curve is used Appl. Sci. 2021, 11, 6944 5 of 14 Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 15 Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 15 Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 15 to obtain the fatigue-fracture cycle N at various stress levels e (R = 1). According to the f a trend line, the fatigue limit of 18CrNiMo7-6 is taken as 350 MPa. Figure Figure 1. 1. T Test est piece piece size size.. Figure 1. Test piece size. Figure 1. Test piece size. Figure 2. Test sample after fatigue. Figure 2. Test sample after fatigue. Figure 2. Test sample after fatigue. Figure 2. Test sample after fatigue. Figure 3. Test machine. Figure 3. Test machine. Figure 3. Test machine. Figure 3. Test machine. 3.2. Test Methods T 3. able 2. T2. est Fatigue Methods test results. 3.2. Test Methods Five specimens are used for the static tensile test and the average maximum tensile Five specimens are used for the static tensile test and the average maximum tensile Loading Stress Amplitude/MPa N/Number of Cycles strength of the material is σ = 1100 MPa . Then, every five test pieces are used together Five specimens are used for the static tensile test and the average maximum tensile strength of the material is σ = 1100 MPa . Then, every five test pieces are used together 1100 0.5 strength of the material is σ = 1100 MPa . Then, every five test pieces are used together to carry out the fatigue test under different stress amplitudes. The test conditions are: to carry out the fatigue test under different stress amplitudes. The test conditions are: 832 471 symmetrical cycle R = −1, sine wave loading, and a frequency of 10 Hz. The average num- to carry out the fatigue test under different stress amplitudes. The test conditions are: symmetrical cycle R = −1, sine wave loading, and a frequency of 10 Hz. The average num- 728 3132 ber of cycles for each group of experiments is shown in Table 2. symmetrical cycle R = −1, sine wave loading, and a frequency of 10 Hz. The average num- 624 23,414 ber of cycles for each group of experiments is shown in Table 2. ber of cycles for each gro 520 up of experiments is shown in Table 2. 246,809 416 3,189,720 353 9,230,000 Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 15 Table 2. Fatigue test results. Loading Stress Amplitude/MPa N/Number of Cycles 1100 0.5 832 471 728 3132 624 23414 520 246,809 416 3,189,720 353 9,230,000 According to the data in Table 2, the fitting function of the material S-N curve is ob- 4 −0.002028 4 tained, in Figure 4 ( S = (2.318 × 10 ) × N − 2.208 × 10 ). The S-N curve is used to Appl. Sci. 2021, 11, 6944 6 of 14 obtain the fatigue-fracture cycle Nf at various stress levels ea (R = −1). According to the trend line, the fatigue limit of 18CrNiMo7-6 is taken as 350 MPa. Figure 4. S–N curve. Figure 4. S–N curve. According to the obtained S-N curve, the two-stage loading test plan is designed as follows: take the stress value at the inflection point of the curve 520 MPa, for high–low According to the obtained S-N curve, the two-stage loading test plan is designed as load application conditions; firstly, cycle n times with 520 MPa stress (N  240,000); and 1 f 1 follows: take the stress value at the inflection point of the curve 520 MPa, for high–low then apply a 420 MPa stress (N  3,200,000) cycle until breaking (corresponding number f 2 load application conditions; firstly, cycle n1 times with 520 MPa stress (Nf1 ≈ 240,000); and of cycles n ). For low–high load conditions, after 420 MPa stress cycle n times, apply a 2 1 then apply a 420 MPa stress (Nf2 ≈ 3,200,000) cycle until breaking (corresponding number 520 MPa stress cycle until breaking (corresponding number of cycles n ). Every five test of cycles n2). For low–high load conditions, after 420 MPa stress cycle n1 times, apply a 520 pieces are a group, and the average data from the test is shown in Table 3. MPa stress cycle until breaking (corresponding number of cycles n2). Every five test pieces are Table a group 3. 18CrNiMo7-6 , and the ave two-stage rage dat loading a from t test data. he test is shown in Table 3. Test Value Two-Stage Loading Table 3. 18CrNiMo7-6 two-stage loading test data. Stress Level/MPa n n /N n n /N n /N +n /N 1 1 f 1 2 2 f 2 1 f 1 2 f 2 Two-Stage Loading Test Value 60,000 0.25 1,480,000 0.463 0.7125 Stress Level/MPa n1 n1/Nf1 n2 n2/Nf2 n1/Nf1+n2/Nf2 120,000 0.50 850,000 0.266 0.766 520–420 60,000 0.25 1,480,000 0.463 0.7125 156,000 0.65 324,800 0.1015 0.7515 520–420 120,000 0.50 850,000 0.266 0.766 800,000 0.25 216,000 0.90 1.15 156,000 0.65 324,800 0.1015 0.7515 1,350,000 0.42 177,600 0.74 1.16 420–520 800,000 0.25 216,000 0.90 1.15 2,000,000 0.63 144,000 0.60 1.23 420–520 1,350,000 0.42 177,600 0.74 1.16 2,000,000 0.63 144,000 0.60 1.23 In the sequence of load application, the first high-stress load will have a greater impact on the overall damage as the data in Table 3. In high–low loading, the total damage value In the sequence of load application, the first high-stress load will have a greater im- n /N +n /N is less than 1; in low–high loading, the total damage value n /N +n /N is 1 1 2 2 1 1 2 2 f f f f pact on the overall damage as the data in Table 3. In high–low loading, the total damage greater than 1. Unlike in the linear damage accumulation model, the total damage value in value the cumulative n1/Nf1+n2/Nmodel f2 is less is t always han 1; in equal low to–1. high loading, the total damage value n1/Nf1+n2/Nf2 In order to further analyze the tension–compression fatigue fracture mechanism, the fatigue fracture specimen was sliced and then analyzed by the JSM-5610LV scanning electron microscope, as shown in Figures 5 and 6. After binarization, the black pixels are in the form of dimples. According to statistics, the pixels in Figure 7 occupy the total selection area of S1 (0.921%), and the pixels in Figure 8 occupy the total selection area of S2 (9.79%). S2/S1 = 10.63. High–low loading significantly promotes the formation of dimples. Binarize the middle part of Figures 5 and 6 to obtain Figures 7 and 8. Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 15 Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 15 Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 15 is greater than 1. Unlike in the linear damage accumulation model, the total damage value in the cumulative model is always equal to 1. is greater than 1. Unlike in the linear damage accumulation model, the total damage value In order to further analyze the tension–compression fatigue fracture mechanism, the is greater than 1. Unlike in the linear damage accumulation model, the total damage value in the cumulative model is always equal to 1. fatigue fracture specimen was sliced and then analyzed by the JSM-5610LV scanning elec- in the cumulative model is always equal to 1. In order to further analyze the tension–compression fatigue fracture mechanism, the tron microscope, as shown in Figures 5 and 6. Appl. Sci. 2021, 11, 6944 In order to further analyze the tension–compression fatigue fracture mechanism 7 of , the 14 fatigue fracture specimen was sliced and then analyzed by the JSM-5610LV scanning elec- fatigue fracture specimen was sliced and then analyzed by the JSM-5610LV scanning elec- tron microscope, as shown in Figures 5 and 6. tron microscope, as shown in Figures 5 and 6. Figure 5. Low–high load fracture surface (420–520 MPa, n1/N1 = 0.25). Figure 5. Low–high load fracture surface (420–520 MPa, n1/N1 = 0.25). Figure 5. Low–high load fracture surface (420–520 MPa, n /N = 0.25). Figure 5. Low–high load fracture surface (420–520 MPa, n1/N1 = 0.25). 1 1 Figure 6. High–low load fracture surface (520–420 MPa, n1/N1 = 0.25). Figure 6. Bina ri High–low loa ze the middle d fract pau rtre surface (52 of Figures 5 0–420 MPa and 6 to ob , n1t /N ain 1 = 0 Fig .25). ures 7 and 8. Figure 6. High–low load fracture surface (520–420 MPa, n1/N1 = 0.25). Figure 6. High–low load fracture surface (520–420 MPa, n /N = 0.25). 1 1 Binarize the middle part of Figures 5 and 6 to obtain Figures 7 and 8. Binarize the middle part of Figures 5 and 6 to obtain Figures 7 and 8. Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 15 Figure 7. Low–high load fracture surface (420–520 MPa, n /N = 0.25) binarization. 1 1 Figure 7. Low–high load fracture surface (420–520 MPa, n1/N1 = 0.25) binarization. Figure 7. Low–high load fracture surface (420–520 MPa, n1/N1 = 0.25) binarization. Figure 7. Low–high load fracture surface (420–520 MPa, n1/N1 = 0.25) binarization. Figure 8. High–low loading fracture surface (520–420 MPa, n1/N1 = 0.25) binarization. Figure 8. High–low loading fracture surface (520–420 MPa, n /N = 0.25) binarization. 1 1 Regardless of the loading mode of high–low or low–high, the section shows the After binarization, the black pixels are in the form of dimples. According to statistics, characteristics of multiple crack sources, and the interactive influence of different crack the pixels in Figure 7 occupy the total selection area of S1 (0.921%), and the pixels in Figure source propagation paths forming different ridge topographies. It can be clearly found 8 occupy the total selection area of S2 (9.79%). S2/S1 = 10.63. High–low loading signifi- that under low–high loading, the number of dimples in the cleavage zone formed on cantly promotes the formation of dimples. the fracture surface is less than the number of dimples formed on the fracture surface Regardless of the loading mode of high–low or low–high, the section shows the char- acteristics of multiple crack sources, and the interactive influence of different crack source propagation paths forming different ridge topographies. It can be clearly found that under low–high loading, the number of dimples in the cleavage zone formed on the fracture surface is less than the number of dimples formed on the fracture surface under high–low loading. The formation of dimples under the repeated action of normal stress accelerates the formation of microscopic voids caused by plastic deformation of the material in a small range during the high–low loading process. This difference ultimately leads to different total damage changes after low–high loading and high–low loading. That is, under the low–high loading form, the first-level low load forms the exercise effect. In the high–low loading mode, the first-level high load does not form an exercise effect and the overall fatigue life decreases rapidly with the increase in the first-level high-load cycle. Due to the presence of dislocations in the material, dislocation clusters are easily formed at grain boundaries, phase boundaries, and material defects during the tension–compression pro- cess, which leads to stress concentration and induces the initiation and growth of mi- crovoids, eventually leading to fracture. High–low loading causes a significantly higher number of micro-holes to be generated than in the case of micro-holes generated by low– high load loading, which means that the number of original microcrack sources under high–low loading is large. These microcracks are easier to connect and propagate to form cracks during cyclic loading, resulting in a decrease in the fatigue life of the material com- pared to low–high load loading. 4. Improved Cumulative Damage Model According to the Formula (11) mentioned above, the load ratio effect parameter n is introduced, so the improved cumulative damage model is expressed  σ  D =  D (12) 2 2    1  Formula (12) reflects the effect of load loading sequence on damage, and the damage relationship between the improved model and the original model as: ≥ D σ ≥ σ（Low - High） 2 2 1 ≤ D σ ≤ σ（High - Low）  2 2 1 Appl. Sci. 2021, 11, 6944 8 of 14 under high–low loading. The formation of dimples under the repeated action of normal stress accelerates the formation of microscopic voids caused by plastic deformation of the material in a small range during the high–low loading process. This difference ultimately leads to different total damage changes after low–high loading and high–low loading. That is, under the low–high loading form, the first-level low load forms the exercise effect. In the high–low loading mode, the first-level high load does not form an exercise effect and the overall fatigue life decreases rapidly with the increase in the first-level high-load cycle. Due to the presence of dislocations in the material, dislocation clusters are easily formed at grain boundaries, phase boundaries, and material defects during the tension–compression process, which leads to stress concentration and induces the initiation and growth of microvoids, eventually leading to fracture. High–low loading causes a significantly higher number of micro-holes to be generated than in the case of micro-holes generated by low– high load loading, which means that the number of original microcrack sources under high–low loading is large. These microcracks are easier to connect and propagate to form cracks during cyclic loading, resulting in a decrease in the fatigue life of the material compared to low–high load loading. 4. Improved Cumulative Damage Model According to the Formula (11) mentioned above, the load ratio effect parameter n is introduced, so the improved cumulative damage model is expressed D = D (12) Formula (12) reflects the effect of load loading sequence on damage, and the damage relationship between the improved model and the original model as: D s  s (Low High) 0 2 2 1 D s  s High Low ( ) 2 2 1 According to Formula (11), the number of cycles n under the first-stage load ampli- tude s is equivalent to the equivalent number of cycles n under the second-stage load amplitude s , expressed as: 1   2 ln 1 ln 1 N s N f 1 f 2 D = = = D (13) ln N s ln N f 1 1 f 2 ln N f 2 s ( ) ln N s f 1 2 = 1 1 (14) N N f 2 f 1 n +n ln 1 s N f 2 D = (15) s ln N 1 f 2 When the total damage degree is 1, the specimen is damaged. Formula (15) D = 1, represents the material under the action of two levels of load, after the number of cycles n of the first level load amplitude s , and the remaining life fraction of the second level load amplitude s , expressed ln N b b f 2 s s ! 1 ! 1 ( ) ( ) ln N s s 2 2 f 1 n n 1 2 1 = 1 (16) N N N f 2 f 1 f 2 Appl. Sci. 2021, 11, 6944 9 of 14 By analogy, the total damage under the multi-stage load and the remaining life fraction n /N of the last stage load s can be derived as: i f i i n +n n n ln 1 s N f i D = ( ) (17) s ln N i1 f i i=1 ln N b b s s f i i1 i1 ! ! ( ) ( ) ln N s s f i1 i i n n 1 i i1 = 1 (18) N N N f i f i1 f i According to the data obtained from the test, for 18CrNiMo7-6 material, b = 2.8. One can substitute b = 2.8 into Formula (16) as: ln N 2.8 2.8 f 2 s s ! 1 ! 1 ( ) ( ) ln N s s f 1 2 2 n n 1 2 1 = 1 (19) N N N f 2 f 1 f 2 5. Test Results and Analysis In order to verify the effectiveness of the proposed improved model, based on the fatigue test data of the material 18CrNiMo7-6 listed in Table 3, the life prediction results of this model, the linear damage accumulation model, and the ductile dissipation model, are compared. The results are shown in Table 4. In addition, for 45# steel and Al-2024 aluminum alloy, commonly used in mechanical engineering, calculations and comparisons are also made based on the experimental data of the literature [23,24]. The results are listed in Tables 5 and 6. For each material in the corresponding loading mode, the relative error between the experimental value and the theoretical calculation is shown in Figures 9–14. Table 4. 18CrNiMo7-6 two-stage loading test data and the remaining life prediction results of each model. Miner’s Rule Ductile Dissipation The Proposed Model Test Value Two-Stage (Formula (1)) Model (Formula (11)) (Formula (19)) Loading Stress Relative Relative Relative Level/MPa n n /N n n /N n /N n /N n /N 1 1 1 2 2 2 2 2 2 2 2 2 f f f f f Error/% Error/% Error/% 60,000 0.25 1,480,000 0.583 0.75 28.64% 0.7062 21.13% 0.5168 11.35% 520–420 120,000 0.50 850,000 0.266 0.50 87.97% 0.4325 62.59% 0.2038 23.38% 156,000 0.65 324,800 0.1015 0.35 244.8% 0.2810 176.8% 0.0899 11.43% 800,000 0.25 216,000 0.9 0.75 16.67% 0.7883 12.41% 0.8818 2.02% 420–520 1,350,000 0.42 177,600 0.74 0.55 25.67% 0.6356 14.10% 0.7872 6.38% 2,000,000 0.63 144,000 0.60 0.35 41.67% 0.4443 25.95% 0.6518 8.63% Table 5. The 45# steel two-stage loading test data and remaining life prediction results of each model. Miner’s Rule Ductile Dissipation The Proposed Model Test Value Two-Stage (Formula (1)) Model (Formula (11)) (Formula (19)) Loading Stress Relative Relative Relative Level/MPa n n /N n n /N n /N n /N n /N 1 1 1 2 2 2 2 2 2 2 2 f f f f2 f Error/% Error/% Error/% 12,500 0.25 250,400 0.5008 0.7500 49.76% 0.7055 40.87% 0.5755 14.91% 25,000 0.50 168,300 0.3366 0.5000 48.54% 0.4314 28.17% 0.2671 20.64% 331.5–284.4 37,500 0.75 64,500 0.1290 0.2500 93.80% 0.1861 44.29% 0.0698 45.89% 125,000 0.25 37,900 0.7580 0.7500 1.06% 0.7888 4.07% 0.8608 13.56% 284.4–331.5 250,000 0.50 38,900 0.7780 0.5000 35.73% 0.5646 27.42% 0.6970 10.41% 375,000 0.75 43,400 0.8680 0.2500 71.20% 0.3188 63.27% 0.4858 44.03% Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 15 Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 15 listed in Tables 5 and 6. For each material in the corresponding loading mode, the relative listed in Tables 5 and 6. For each material in the corresponding loading mode, the relative error between the experimental value and the theoretical calculation is shown in Figures error between the experimental value and the theoretical calculation is shown in Figures 9–14. 9–14. Appl. Sci. 2021, 11, 6944 10 of 14 Table 4. 18CrNiMo7-6 two-stage loading test data and the remaining life prediction results of each model. Table 4. 18CrNiMo7-6 two-stage loading test data and the remaining life prediction results of each model. Miner’s Rule (For- Ductile Dissipation The Proposed Model Two-Stage Two-Stage Miner’s Rule (For- Ductile Dissipation The Proposed Model Test Value Table 6. Al-2024 aluminum alloy two-stage Test Value loading test data and remaining life prediction results of each model. Loading mula (1)) Model (Formula (11)) (Formula (19)) Loading mula (1)) Model (Formula (11)) (Formula (19)) Stress Relative Relative Relative Miner’s Rule Ductile Dissipation The Proposed Model Stress Relative Relative Relative n1 n1/Nf1 n2 n2/Nf2 n2/Nf2 n2/Nf2 n2/Nf2 Test Value n1 n1/Nf1 n2 n2/Nf2 n2/Nf2 n2/Nf2 n2/Nf2 Two-Stage (Formula (1)) Model (Formula (11)) (Formula (19)) Level/MPa Error/% Error/% Error/% Level/MPa Error/% Error/% Error/% Loading Stress Relative Relative Relative 60,000 0.25 1,480,000 0.583 0.75 28.64% 0.7062 21.13% 0.5168 11.35% Level/MPa n n /N60,000 n 0.25 n /N1,480,00n0 /N 0.583 0.75 n /N28.64% 0.7062 2 n /N1.13% 0.5168 11.35% 1 1 f 1 2 2 f 2 2 f 2 2 f 2 2 f 2 Error/% Error/% Error/% 520–420 120,000 0.50 850,000 0.266 0.50 87.97% 0.4325 62.59% 0.2038 23.38% 520–420 120,000 0.50 850,000 0.266 0.50 87.97% 0.4325 62.59% 0.2038 23.38% 30,000 0.2000 228,700 0.5319 0.8000 50.40% 0.7844 47.47% 0.5623 5.71% 156,000 0.65 324,800 0.1015 0.35 244.8% 0.2810 176.8% 0.0899 11.43% 60,000 0.4000156,000 101,0500.65 0.2350324,800.6000 0 0.1015 155.32%0.35 0.5735244.8% 144.05%0.2810 1 0.267776.8% 0 13.91%.0899 11.43% 200–150 90,000 0.6000 76,050 0.1769 0.4000 126.12% 0.3689 108.53% 0.0941 46.80% 800,000 0.25 216,000 0.9 0.75 16.67% 0.7883 12.41% 0.8818 2.02% 800,000 0.25 216,000 0.9 0.75 16.67% 0.7883 12.41% 0.8818 2.02% 86,000 420–520 0.20001,350,00 144,500 0 0.42 1 0.963377,60 0.80000 0.74 16.95% 0.55 0.814625.67% 15.43% 0.6356 0.915314.10% 4.98%0.7872 6.38% 420–520 1,350,000 0.42 177,600 0.74 0.55 25.67% 0.6356 14.10% 0.7872 6.38% 172,000 0.4000 133,500 0.8900 0.6000 32.58% 0.6254 29.73% 0.8186 8.02% 150–200 2,000,000 0.63 144,000 0.60 0.35 41.67% 0.4443 25.95% 0.6518 8.63% 2,000,000 0.63 144,000 0.60 0.35 41.67% 0.4443 25.95% 0.6518 8.63% 258,000 0.6000 81,700 0.5447 0.4000 26.57% 0.4309 20.89% 0.6992 28.36% Formula 1 Formula 1 Formula 11 Formula 11 Formula 19 Formula 19 Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 15 0.25 0.5 0.65 0.25 0.5 0.65 n /N n /N 1 f1 case of high–low loading. This 1 also f1shows that, from the side, different loading sequences have different effects on the fatigue life. Figure 9. (520–420 MPa) The relative error of each model under different n1/Nf1 (18CrNiMo7-6). Figure 9. (520–420 MPa) The relative error of each model under different n1/Nf1 (18CrNiMo7-6). Figure 9. (520–420 MPa) The relative error of each model under different n /N (18CrNiMo7-6). 1 f 1 Table 5. The 45# steel two-stage loading test data and remaining life prediction results of each model. Two-Stage Miner’s Rule (For- Ductile Dissipation The Proposed Model Formula 1 Test Value Formula 1 mula (1)) Model (Formula (11)) (Formula (19)) Loading Formula 11 Formula 11 Stress Relative Relative Relative n1 n1/Nf1 n2 n2/Nf2 n2/Nf2 n2/Nf2 n2/Nf2 Formula 19 Formula 19 Level/MPa Error/% Error/% Error/% 12,500 0.25 250,400 0.5008 0.7500 49.76% 0.7055 40.87% 0.5755 14.91% 331.5–284.4 25,000 0.50 168,300 0.3366 0.5000 48.54% 0.4314 28.17% 0.2671 20.64% 37,500 0 0.75 64,500 0.1290 0.2500 93.80% 0.1861 44.29% 0.0698 45.89% 0.25 0.42 0.63 0.25 0.42 0.63 125,000 0.25 37,900 0.7580 0.7500 1.06% 0.7888 4.07% 0.8608 13.56% n /N 284.4–331.5 250,000 0.50 38,900 0.7780 0.5000 35.73% 0.5646 27.42% 0.6970 10.41% n /N 1 f1 1 f1 375,000 0.75 43,400 0.8680 0.2500 71.20% 0.3188 63.27% 0.4858 44.03% Figure 10. (420–520 MPa) The relative error of each model under different n /N (18CrNiMo7-6). Figure 10. (420–520 MPa) The relative error of each model under different n1/Nf1 (18CrNiMo7-6). 1 f 1 Figure 10. (420–520 MPa) The relative error of each model under different n1/Nf1 (18CrNiMo7-6). It can be seen from Figures 9 and 10, for 18CrNiMo7-6 material, under high–low load It can be seen from Figures 9 and 10, for 18CrNiMo7-6 material, under high–low load loadingFo , asrm thu e proport la 1 ion of high load continues to expand, the relative error of Miner’s loading, as the proportion of high load continues to expand, the relative error of Miner’s rule increases significantly. The relative error of the tough dissipation model is also grad- Formula 11 rule increases significantly. The relative error of the tough dissipation model is also grad- ually increasing, but the overall value is smaller than Miner’s rule. The relative error be- ually incr Fo easing, but the o rmula 19 verall value is smaller than Miner’s rule. The relative error be- tween the result obtained by the improved model and the actual value is the smallest tween the result obtained by the improved model and the actual value is the smallest among these three models. In the case of low–high loading, the error between the results among these three models. In the case of low–high loading, the error between the results obtained by the three models and the actual value is smaller than the error obtained in the obtained by the three models and the actual value is smaller than the error obtained in the 0.25 0.50 0.75 n /N 1 f1 Figure 11. (331.5–284.4 MPa) The relative error of each model under different n1/Nf1 (45# steel). Figure 11. (331.5–284.4 MPa) The relative error of each model under different n /N (45# steel). 1 f 1 Formula 1 Formula 11 Formula 19 0.25 0.50 0.75 n /N 1 f1 Figure 12. (284.4–331.5 MPa) The relative error of each model under different n1/Nf1 (45# steel). It can be seen from Figures 11 and 12, for 45# steel, under high–low load loading, as the proportion of high load continues to increases, the relative error of Miner’s rule in- creases significantly. The relative error fluctuation of the toughness dissipation model is relatively stable, and the overall value is smaller than Miner’s rule. The relative error be- tween the result obtained by the improved model and the actual value is the smallest among these three models. In the case of low–high loading, the error between the results obtained by the three models and the actual value is smaller than the error obtained in the case of high–low loading. Like the results of 18CrNiMo7-6, both reflect that under high– Relative error/% Relative error/% Relative error/% Relative error/% Relative error/% Relative error/% Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 15 case of high–low loading. This also shows that, from the side, different loading sequences have different effects on the fatigue life. Table 5. The 45# steel two-stage loading test data and remaining life prediction results of each model. Two-Stage Miner’s Rule (For- Ductile Dissipation The Proposed Model Test Value mula (1)) Model (Formula (11)) (Formula (19)) Loading Stress Relative Relative Relative n1 n1/Nf1 n2 n2/Nf2 n2/Nf2 n2/Nf2 n2/Nf2 Level/MPa Error/% Error/% Error/% 12,500 0.25 250,400 0.5008 0.7500 49.76% 0.7055 40.87% 0.5755 14.91% 331.5–284.4 25,000 0.50 168,300 0.3366 0.5000 48.54% 0.4314 28.17% 0.2671 20.64% 37,500 0.75 64,500 0.1290 0.2500 93.80% 0.1861 44.29% 0.0698 45.89% 125,000 0.25 37,900 0.7580 0.7500 1.06% 0.7888 4.07% 0.8608 13.56% 284.4–331.5 250,000 0.50 38,900 0.7780 0.5000 35.73% 0.5646 27.42% 0.6970 10.41% 375,000 0.75 43,400 0.8680 0.2500 71.20% 0.3188 63.27% 0.4858 44.03% Formula 1 Formula 11 Formula 19 Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 15 Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 15 0.25 0.50 0.75 n /N 1 f1 Appl. Sci. 2021, 11, 6944 11 of 14 low loading, as the proportion of high load continues to increase, the relative e rror value low loading, as the proportion of high load continues to increase, the relative error value is higher than under low–high loading, and the proportion of low load continues to in- Figure 11. (331.5–284.4 MPa) The relative error of each model under different n1/Nf1 (45# steel). is higher than under low–high loading, and the proportion of low load continues to in- crease. crease. Table 6. Al-2024 aluminum alloy two-stage loading test data and remaining life prediction results of each model. Table 6. Al-2024 aluminum alloy two-stage loading test data and remaining life prediction results of each model. Formula 1 Two-Stage Miner’s Rule (For- Ductile Dissipation The Proposed Model Two-Stage Miner’s Rule (For- Ductile Dissipation The Proposed Model Formula 11 Test Value Test Value Loading 100 mula (1)) Model (Formula (11)) (Formula (19)) mula (1)) Model (Formula (11)) (Formula (19)) Loading Formula 19 Stress Relative Relative Relative Stress Relative Relative Relative n1 n1/Nf1 n2 n2/Nf2 n2/Nf2 n2/Nf2 n2/Nf2 n1 n1/Nf1 n2 n2/Nf2 n2/Nf2 n2/Nf2 n2/Nf2 Level/MPa Error/% Error/% Error/% Level/MPa Error/% Error/% Error/% 30,000 0.2000 228,700 0.5319 0.8000 50.40% 0.7844 47.47% 0.5623 5.71% 30,000 0.2000 228,700 0.5319 0.8000 50.40% 0.7844 47.47% 0.5623 5.71% 200–150 60,000 0.4000 101,050 0.2350 0.6000 155.32% 0.5735 144.05% 0.2677 13.91% 200–150 60,000 0.4000 101,050 0.2350 0.6000 155.32% 0.5735 144.05% 0.2677 13.91% 90,000 0.6000 76,050 0.1769 0.4000 126.12% 0.3689 108.53% 0.0941 46.80% 90,000 0.6000 76,050 0.1769 0.4000 126.12% 0.3689 108.53% 0.0941 46.80% 86,000 0.2000 1 0.25 44,500 0.9633 0.050.8000 16.95% 0.8146 0.75 15.43% 0.9153 4.98% 86,000 0.2000 144,500 0.9633 0.8000 16.95% 0.8146 15.43% 0.9153 4.98% 150–200 172,000 0.4000 133,500 0.8900 n0/N.6000 32.58% 0.6254 29.73% 0.8186 8.02% 150–200 172,000 0.4000 133,500 0.8900 0.6000 32.58% 0.6254 29.73% 0.8186 8.02% 1 f1 258,000 0.6000 81,700 0.5447 0.4000 26.57% 0.4309 20.89% 0.6992 28.36% 258,000 0.6000 81,700 0.5447 0.4000 26.57% 0.4309 20.89% 0.6992 28.36% Figure 12. (284.4–331.5 MPa) The relative error of each model under different n1/Nf1 (45# steel). Figure 12. (284.4–331.5 MPa) The relative error of each model under different n /N (45# steel). 1 f 1 It can be seen from Figures 11 and 12, for 45# steel, under high–low load loading, as Formula 1 the proportion of high load continues to increases, the relative error of Miner’s rule in- Formula 1 150 creases significantly. The relative error fluctuation of the toughness dissipation model is Formula 11 Formula 11 relatively stable, and the overall value is smaller than Miner’s rule. The relative error be- Formula 19 Formula 19 tween the result obtained by the improved model and the actual value is the smallest among these three models. In the case of low–high loading, the error between the results obtained by the three models and the actual value is smaller than the error obtained in the case of high–low loading. Like the results of 18CrNiMo7-6, both reflect that under high– 0.20 0.40 0.60 0.20 0.40 0.60 n /N n /N 1 f1 1 f1 Figure 13. (200–150 MPa) The relative error of each model under different n1/Nf1 (Al-2024). Figure 13. (200–150 MPa) The relative error of each model under different n1/Nf1 (Al-2024). Figure 13. (200–150 MPa) The relative error of each model under different n /N (Al-2024). 1 1 Formula 1 Formula 1 150 Formula 11 Formula 11 Formula 19 Formula 19 0.20 0.40 0.60 0.20 0.40 0.60 n /N n /N 1 f1 1 f1 Figure 14. (150–200 MPa) The relative error of each model under different n1/Nf1 (Al-2024). Figure 14. (150–200 MPa) The relative error of each model under different n /N (Al-2024). Figure 14. (150–200 MPa) The relative error of each model under different 1 1 n1/Nf1 (Al-2024). It can be seen from It ca Figur n be es seen f 9 and rom 10 Fi , gures 13 for 18CrNiMo7-6 and 14, for ( material, Al-2024) under , under high–l high–low ow loading, as the It can be seen from Figures 13 and 14, for (Al-2024), under high–low loading, as the proportion of high load continues to increase, the relative error stability of the results ob- load loading, as the proportion of high load continues to expand, the relative error of proportion of high load continues to increase, the relative error stability of the results ob- Miner ’s rule increases tained by the du significantly.ctil The e dissipa relative tierr on model i or of thestough better tha dissipation n that of model the Mi isner’s rul also e. Under the tained by the ductile dissipation model is better than that of the Miner’s rule. Under the gradually increasing, same con but the ditions, the overall value accur isasmaller cy of the than results Miner from the ’s rule.proposed mo The relativedel error is higher than that same conditions, the accuracy of the results from the proposed model is higher than that between the result of M obtained iner’s ru by lethe and t impr he oved ductile model dissip and ation model the actual . Un value der low is the –hismallest gh loading, the results of of Miner’s rule and the ductile dissipation model. Under low–high loading, the results of among these three each mode models. In l athe re re case lative ofly low–high stable. The loading, accurac the y o err f th or e between improved model the results is the highest. each model are relatively stable. The accuracy of the improved model is the highest. obtained by the three models and the actual value is smaller than the error obtained in the case of high–low loading. This also shows that, from the side, different loading sequences have different effects on the fatigue life. It can be seen from Figures 11 and 12, for 45# steel, under high–low load loading, as the proportion of high load continues to increases, the relative error of Miner ’s rule increases significantly. The relative error fluctuation of the toughness dissipation model Relative error/% Relative error/% Relative error/% Relative error/% Relative error/% Relative error/% Appl. Sci. 2021, 11, 6944 12 of 14 is relatively stable, and the overall value is smaller than Miner ’s rule. The relative error between the result obtained by the improved model and the actual value is the smallest among these three models. In the case of low–high loading, the error between the results obtained by the three models and the actual value is smaller than the error obtained in the case of high–low loading. Like the results of 18CrNiMo7-6, both reflect that under high–low loading, as the proportion of high load continues to increase, the relative error value is higher than under low–high loading, and the proportion of low load continues to increase. It can be seen from Figures 13 and 14, for (Al-2024), under high–low loading, as the proportion of high load continues to increase, the relative error stability of the results obtained by the ductile dissipation model is better than that of the Miner ’s rule. Under the same conditions, the accuracy of the results from the proposed model is higher than that of Miner ’s rule and the ductile dissipation model. Under low–high loading, the results of each model are relatively stable. The accuracy of the improved model is the highest. From the data in Tables 4–6, the linear damage accumulation model (Miner ’s rule) assumes that damage is not related to the load state, damage accumulation is similarly not related to the load sequence, the interaction between loads cannot be considered, and the deviation from the test results is the largest. The relative error stability of the results obtained by the ductile dissipation model is better than that of the Miner ’s rule. The improved model, proposed in this paper, increases the influence factors of the sequential stress sequence and magnitude and the error is smaller than the original ductile dissipation model. In this paper, the improved model is extended and applied to commonly used 45# steel and Al-2024 aluminum alloy. The error is larger than that of 18CrNiMo7-6, but it is still smaller than the original toughness dissipation model, indicating that the model in this paper has better material applicability. 6. Conclusions (1) Based on the ductile dissipation theory, a nonlinear fatigue cumulative damage model considering the loading sequence is established, that is, an improved toughness dissipation model, which can consider the impact of loading sequence on damage with parameters which are simple and suitable for engineering applications; (2) The fracture sections of the 18CrNiMo7-6 specimens, which were scanned by electron microscope, explain from the mechanism why Miner ’s rule has different damages under two-level loading. The results of the electron microscope showed that the number of dimples formed on the fracture surface under low–high load was less than the number of dimples formed on the fracture surface under high–low load. This indicated that the number of micro-crack sources in the cross-section was relatively small, and the probability of micro-cracks connecting and expanding into cracks was relatively small, resulting in a small totally effective level of damage under low–high load cycles and a longer fatigue life; (3) The improved model proposed in this paper is based on the test data of 18CrNiMo7-6 forged steel. This paper uses #45 steel, AL-2024 aluminum alloy and the nonlinear damage accumulation processes of other common materials, to predict that the life under two-stage load has smaller errors and better accuracy than the classic ductile dissipation model, indicating that the improved model has a good material appli- cability. Although this article uses three kinds of metal materials to test and verify the established model and has achieved good results, for other types of materials and test environments, much test verification and further research are needed. Fur- ther in-depth research can be done on issues such as the quantification of toughness dissipation, the improvement of model accuracy, and material verification. Author Contributions: W.W. and J.L. designed experiment verification and validation schemes. J.P. and H.C. gathered experimental data and analyzed experimental results. W.C. was supervision. W.W. wrote the manuscript. All authors have read and agreed to the published version of the manuscript. Appl. Sci. 2021, 11, 6944 13 of 14 Funding: This research was funded by International S&T Cooperaion Program of China, grantnum- ber (2015DFA71400), NSFC(U1709210), National Natural Science Foundation of Chna: (51975535) and “The APC was funded by International S&T Cooperaion Program of China, grant number (2015DFA71400)”. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Publicly available datasets were analyzed in this study. This data can be found here: [http://en.cnki.com.cn/Article_en/CJFDTOTAL-GCHE201804014.htm] (accessed on 25 July 2021) and [http://en.cnki.com.cn/Article_en/CJFDTOTAL-GCHE201804014.htm] (accessed on 25 July 2021). Acknowledgments: The present research is supported by the International S&T Cooperaion Program of China (2015DFA71400), NSFC (U1709210), National Natural Science Foundation of China: (51975535). Conflicts of Interest: The authors declare no conflict of interest. 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Journal

Applied Sciences – Multidisciplinary Digital Publishing Institute

Published: Jul 28, 2021

Keywords: nonlinear cumulative damage; loading sequence; fatigue damage; toughness dissipation; two-level loading

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