# Evaluation of Methods for Estimation of Cyclic Stress-Strain Parameters from Monotonic Properties of Steels

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## Abstract

**:**

## 1. Introduction

_{f}′, ε

_{f}′, b and c are fatigue strength and ductility parameters obtained from fully reversed tension-compression fatigue tests.

_{e}′ and R-O parameters (K′ and n′) various monotonic properties and their combinations are used, with ultimate strength R

_{m}and yield stress R

_{e}being the most common since they are readily available. Detailed overview of monotonic properties used for estimation of cyclic parameters of metallic materials and systematic study of their relevance for estimation purposes is provided in [14].

_{e}′ and cyclic stress-strain parameters K′ and n′ from monotonic properties. For this purpose, a large and independent set of material data was collected from relevant sources. Since previous investigations [15,16,17] confirmed that dividing steels into different subgroups might improve estimation accuracy, this will also be taken into consideration. One-way Analysis of Variance (one-way ANOVA) and post hoc Tukey’s test will be performed in order to check whether individual steel groups are statistically different regarding their cyclic parameters R

_{e}′, K′ and n′. If such differences are confirmed to exist, in addition to evaluation of existing methods for all steels together, partial evaluations for each steel subgroup will be performed as well.

## 2. Overview of Existing Methods for Estimation of Cyclic Stress-Strain Parameters

#### 2.1. Methods for Estimation of Cyclic Yield Stress R_{e}′

_{e}′ of steels from ultimate strength R

_{m}and reduction of area at fracture RA. Equation (3) was developed using monotonic and cyclic properties of 27, mostly unalloyed and low-alloy steels:

_{e}′ deviate at most 14% from their experimental counterparts.

_{e}′ of steels. These were developed and validated on a relatively large number of steels consisting mostly of unalloyed and low-alloy steels, covering a wide variation of chemical composition and mechanical properties, with ultimate stress R

_{m}ranging from 279 to 2450 MPa and hardness from 80 to 595 HB. Materials were divided according to ultimate strength to yield stress ratio R

_{m}/R

_{e}, since it was shown that such division improves the accuracy of cyclic parameters estimation. Ratio R

_{m}/R

_{e}was originally proposed by Smith et al. [18] to be used for prediction of cyclic behavior (hardening, softening, stable behavior) of materials. Correspondingly, authors proposed a number of separate expressions for estimation of R

_{e}′ depending on value of R

_{m}/R

_{e}of which the most successful ones are:

_{m}/R

_{e}is also proposed:

^{2}for expressions (4a), (4b) and (5) were 0.88, 0.99, and 0.94 respectively. Evaluation was performed on a single dataset comprising data used for developing expressions and additional data (all together 121 materials, mostly unalloyed and low-alloy steels). It was established that 84% of estimated values of R

_{e}′ from yield stress R

_{e}(Equations (4a) and (4b)) deviate up to ±20% from experimental values while 79% of values of R

_{e}′ estimated from ultimate strength R

_{m}(Equation (5)) deviated up to ±20% from experimental values.

_{e}′ when experimental value of R

_{e}′ exceeds 900 MPa, Li et al. [13] recently modified Equation (3) to:

^{2}= 0.961. Analysis was performed on the majority of data used in [12]. For evaluation, data used for developing Equation (6) was complemented with additional data. Results showed that most values of R

_{e}′ estimated from Equation (6) deviate up to 20% from their experiment-based counterparts. It must be noted that [11] and [13] suggest that values of true fracture strength σ

_{f}can be calculated using the expression:

_{m}and true fracture stress σ

_{f}, recommended by Manson [5,9] is:

_{e}′, but also cyclic parameters K′ and n′ that will be discussed later in Section 2.2.

#### 2.2. Methods for Estimation of Cyclic Parameters K′and n′

_{f}/R

_{p0.2}< 1.6 and β = −1 for σ

_{f}/R

_{p0.2}> 1.6. As most successful expressions authors proposed estimation of K′ based on strength coefficient K (Equation (10)) and estimation of n′ based on ultimate strength R

_{m}, yield stress R

_{e}, true fracture stress σ

_{f}and strain hardening exponent n (Equation (13a) through Equation (13c), depending on value of α). For steels, values of K′ and n′ estimated in such a way deviated up to 27% and 34%, respectively, from their experiment-based counterparts. Data tables with percentage deviation for aluminium and titanium alloys suggest even larger deviations of estimates of n′ (up to 65%). They also suggested that, for stress amplitudes Δσ/2 calculated from estimated values of K′ and n′, besides percentage deviation of particular parameter, sign of deviation is also significant. If sign of deviations of K′ and n′ is the same, calculated and experimental cyclic stress-strain curves are in good agreement.

_{e}′, Lopez and Fatemi developed several relationships between Brinell hardness HB or monotonic properties and cyclic parameters K′ and n′ of steels. Steels are divided into two subgroups according to the value of the R

_{m}/R

_{e}ratio (as was the case for estimation of cyclic yield stress R

_{e}′) and different expressions are proposed accordingly. Equations (14a) and (14b) are denoted as most successful:

^{2}only for expressions (14a) and (14b). It is worth noting that R

^{2}of expressions proposed for estimation of K′ for steels with R

_{m}/R

_{e}> 1.2 is 0.75 which is significantly lower than 0.90 obtained for steels with R

_{m}/R

_{e}≤ 1.2. About 73% values of K′ estimated using Equations (14a) and (14b) deviate less than ±20% from their experimental values. As for n′, percentage of values estimated from Equations (15a) and (15b) that deviate less than ±20% from their experiment-based counterparts is around 60%.

^{2}obtained was 0.79. Percentage of values of n′ estimated from Equation (16) that deviated up to ±20% from experimental values was 68%.

_{e}′ is estimated using Equation (6). However, Equations (17) and (18) can be used only when either K′ or n′ are available, so in the same paper an alternative method for estimation of these parameters was proposed. Cyclic strength coefficient K′ should be estimated using Equations (19a), (19b) or (19c) first, then cyclic strain hardening n′ exponent is calculated from estimated values of K′.

^{2}for Equation (19a) through Equation (19c) decrease with higher values of R

_{m}/R

_{e}, which is in accordance with findings from [12]. R

^{2}obtained for steels with R

_{m}/R

_{e}≤ 1.2 is 0.921, while for steels with 1.2 < R

_{m}/R

_{e}< 1.4 and R

_{m}/R

_{e}≥ 1.4 coefficients of determination are R

^{2}= 0.813 and R

^{2}= 0.712, respectively. Again, caution is advised when using Equation (18) due to the suggested way of estimating R

_{e}′ that was already discussed at the end of Section 2.1.

#### 2.3. Conclusions

_{e}′ proposed in [10] which was developed using only 27 steel datasets.

_{m}/R

_{e}.

_{f}relationships) of unalloyed, low-alloy and high-alloy steels. Also, preliminary investigations on cyclic parameters in [16,17] showed that dividing steels by alloying content could result in more accurate estimations of cyclic parameters and hence, more accurate estimations of cyclic stress-strain curves of materials.

## 3. Methods and Data

#### 3.1. Methods for Statistical Analysis

_{e}′ and cyclic stress-strain parameters K′ and n′ of unalloyed, low-alloy and high-alloy steels, one-way Analysis of Variance (one-way ANOVA) is performed. One-way ANOVA is a technique that provides a statistical test of whether or not means of several (typically three or more) groups are all equal. If results obtained by one-way ANOVA show that statistically significant difference exists between cyclic parameters of analyzed groups, post hoc analysis by Tukey’s multiple comparison method will be performed in order to determine pairwise differences between groups. Significance level α for one-way ANOVA is set to 0.05, while overall significance (family error rate) in Tukey’s multiple comparison test is set to 0.05 to counter type I error for a series of comparisons. Procedure for both one-way ANOVA and Tukey’s multiple comparison test are given in [21]. Statistical analyses were performed in statistical package MINITAB [22].

#### 3.2. Data to Be Analyzed

_{f}which is necessary for calculation of parameters by Zhang et al. method [10], values were calculated by their relationship between ultimate strength R

_{m}and true fracture strain ε

_{f}, according to Equation (8). Also, if a dataset contained only reduction of area at fracture RA, true fracture strain ε

_{f}was calculated by the relationship between these two properties:

_{f}. Additionally, data for high-alloy steels were complemented with values of true fracture stress σ

_{f}, strength coefficient K and strain hardening exponent n since those are required so that evaluations of particular existing methods could be performed.

## 4. Analysis and Results

#### 4.1. Results of One-Way ANOVA and Tukey’s Multiple Comparison Test

_{e}′, K′ and n′) of unalloyed, low- and high-alloy steel subgroups showed that statistically significant differences exist between steel subgroups regarding cyclic yield stress R

_{e}′ (F(2, 113) = 32.25; p < 0.05), cyclic strength coefficient K′ (F(2, 113) = 22.61; p < 0.05), and cyclic strain hardening exponent n′ (F(2, 113) = 72.00; p < 0.05).

_{e}′, K′ and n′, post hoc Tukey’s test was performed to determine which subgroups are mutually different. Results showed that unalloyed and low-alloy steels as well as low-alloy and high-alloy steels differ significantly regarding the cyclic yield stress R

_{e}′. No such difference was determined between unalloyed and high-alloy steels. Statistically significant difference was also found for cyclic strength coefficient K′ of unalloyed and high-alloy steels, as well as low-alloy and high-alloy steels, while no such difference was found between unalloyed and low-alloy steels. Cyclic strain hardening exponent n′ differs between pairs of all three groups.

#### 4.2. Evaluation of Methods for Estimation of Cyclic Yield Stress R_{e}′ and Ramberg-Osgood Parameters K′ and n′ of Steels

_{e}′ and R-O parameters K′ and n′ of steels whose predictive accuracy will be evaluated are listed in Table 1. For every material, expressions for estimation of cyclic yield stress R

_{e}′ and R-O parameters K′ and n′ will be used according to ranges defining the applicability of the models regarding criteria for grouping of materials in original papers (new fracture ductility coefficient α, ultimate strength to yield stress ratio R

_{m}/R

_{e}).

#### 4.2.1. Evaluation of Methods for Estimation of Cyclic Yield Stress R_{e}′

_{e}′ estimated according to selected methods (Table 1) that deviate up to 10%, 20% and up to 30% from experiment-based values were calculated and are given in diagrams on Figure 1.

_{e}′ deviate up to 20%, while all estimates fall within ±30% deviation from corresponding experimental values. Highest percentage of data that deviate only up to 10% is obtained by Lopez and Fatemi 2 (about 70%).

_{e}′ of low-alloy steels are obtained using Lopez and Fatemi 1 method, for which all estimates deviate 20% or less from their experiment-based counterparts.

#### 4.2.2. Evaluation of Methods for Estimation of Ramberg-Osgood Parameters K′ and n′ of Steels

_{e}′. Lopez and Fatemi 1 method is the most accurate while both Zhang et al. methods are least successful, as was the case for individual steel subgroups.

## 5. Discussion

_{m}/R

_{e}ratio. In [10], grouping criteria used was the new fracture ductility parameter α, which is cumbersome to use since true fracture stress ε

_{f}or reduction of area RA needed for its calculation are often unavailable.

_{e}′ and cyclic parameters K′ and n′ of mentioned group of steels. According to these findings, authors propose evaluation of existing methods for estimation of R

_{e}′, K′ and n′ to be performed for each group individually, in addition to all steels together.

_{e}′, K′ and n′ of both unalloyed and low-alloy steels, methods by Lopez and Fatemi [12], and by Li et al. [13] provide very good results. However, estimations for high-alloy steels are notably worse, especially those obtained using the Li et al. method which is not surprising since both methods are developed on the same set of data, consisting mostly of unalloyed and low-alloy steels.

## 6. Conclusions

_{e}′ and cyclic stress-strain parameters K′ and n′ of steels and their applicability to individual steel subgroups and to steels as a general group were studied. A large, independent set of steel data was collected in order to perform the study as it was shown that number and type of materials used for development of estimation methods have significant influence on their performance and evaluation results.

_{e}′, K′ and n′.

_{e}′ of unalloyed and low-alloy steels methods proposed by Li et al. and Lopez and Fatemi were found to provide very good results, while for high-alloy steels, only the method dividing steels by R

_{e}/R

_{m}ratio proposed by Lopez and Fatemi provides reasonably accurate estimates. The method for estimations of K′ and n′ of unalloyed steels proposed by Li et al. gives the best estimates followed closely by the Lopez and Fatemi method which considers the R

_{e}/R

_{m}ratio. For low-alloy steels, both methods by Lopez and Fatemi and the method by Li et al. provide excellent results. Of all methods, only the method proposed by Lopez and Fatemi considering the R

_{e}/R

_{m}ratio can be considered for use with the high-alloy steels group.

_{e}′, K′ and n′ was notably lower for high-alloy steels in comparison to other two subgroups, which can be attributed to the fact that high-alloy steels were found to be underrepresented in material datasets used for development of estimation methods.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

Material Designation | Monotonic Properties | Cyclic Parameters | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

DIN or SAE/other | E (MPa) | R_{e} or R_{p0.2} (MPa) | R_{m} (MPa) | R_{m}/R_{e} (-) | RA (%) | K (MPa) | n (-) | σ_{f} (MPa) | ε_{f} (-) | R_{e}′ (MPa) | K′ (MPa) | n′ (-) |

1038 (SAE) | 207,000 | 347 | 610 | 1.758 | 55.5 | 511 | 0.071 | 956 | 0.590 | 332 | 1207 | 0.208 |

Armco (other) | 210,000 | 207 | 359 | 1.734 | 64 | 675 | 0.285 | 653 | 1.030 | 280 | 858 | 0.18 |

C 20 | 190,000 | 224 | 414 | 1.848 | 70 | 330 | 0.061 | 953 | 1.190 | 239 | 1050 | 0.238 |

C 10 | 217,510 | 435 | 566 | 1.301 | 68 | 659 | 0.073 | 1205 | 1.130 | 463 | 1381 | 0.176 |

Ck 15 | 196,793 | 263 | 392 | 1.490 | 55 | 711 | 0.224 | 746 | 0.806 | 249 | 824 | 0.193 |

Ck 15 | 204,500 | 320 | 434 | 1.356 | 67.5 | 394 | 0.067 | 848.7 | 1.126 | 269 | 813 | 0.178 |

Ck 15 | 202,000 | 431.3 | 615.2 | 1.426 | 54 | 598 | 0.045 | 1011.7 | 0.776 | 492 | 1296 | 0.156 |

Ck 15 | 203,000 | 660 | 828 | 1.255 | 2.6 | 863 | 0.042 | 850.5 | 0.026 | 687 | 1165 | 0.085 |

Ck 25 | 210,000 | 346 | 507 | 1.465 | 63 | 926 | 0.264 | 1027 | 0.994 | 280 | 1345 | 0.252 |

Ck 25 | 210,000 | 307 | 464 | 1.511 | 65 | 924 | 0.276 | 982 | 1.050 | 278 | 1111 | 0.223 |

Ck 25 | 210,000 | 366 | 527 | 1.440 | 60 | 1033 | 0.264 | 997 | 0.916 | 303 | 1217 | 0.224 |

Ck 35 | 210,000 | 414 | 617 | 1.490 | 58 | 1216 | 0.258 | 1150 | 0.868 | 328 | 1355 | 0.229 |

Ck 35 | 210,000 | 394 | 593 | 1.505 | 62 | 1168 | 0.257 | 1169 | 0.968 | 333 | 1460 | 0.238 |

Ck 35 | 210,000 | 396 | 565 | 1.427 | 63 | 1134 | 0.264 | 1134 | 0.994 | 316 | 1534 | 0.254 |

Ck 35 | 210,000 | 587 | 780 | 1.329 | 67 | 1356 | 0.186 | 1514 | 1.109 | 463 | 1106 | 0.14 |

Ck 35 | 210,000 | 480 | 656 | 1.367 | 74 | 1196 | 0.207 | 1468 | 1.347 | 393 | 1033 | 0.156 |

Ck 35 | 210,000 | 596 | 733 | 1.230 | 71 | 1170 | 0.152 | 1541 | 1.238 | 447 | 1027 | 0.134 |

Ck 35 | 210,000 | 542 | 730 | 1.347 | 68 | 1311 | 0.2 | 1473 | 1.139 | 430 | 1087 | 0.149 |

Ck 35 | 210,000 | 513 | 669 | 1.304 | 70 | 1121 | 0.18 | 1417 | 1.204 | 387 | 1081 | 0.165 |

Ck 45 | 206,000 | 540 | 790 | 1.463 | 60 | 730 | 0.047 | 1400 | 0.916 | 481 | 980 | 0.115 |

Ck 45 | 210,500 | 531 | 790 | 1.488 | 60 | 1219 | 0.0151 | 1271 | 0.777 | 462 | 1078 | 0.133 |

Ck 45 | 199,700 | 622 | 915 | 1.471 | 59 | 1606 | 0.18 | 1784 | 0.900 | 591 | 2407 | 0.226 |

Ck 45 | 199,700 | 622 | 915 | 1.471 | 59 | 1606 | 0.18 | 1784 | 0.900 | 538 | 1762 | 0.191 |

Ck 45 | 201,500 | 380 | 684 | 1.800 | 36.8 | 735 | 0.092 | 987 | 0.460 | 336 | 1414 | 0.231 |

Ck 45 | 205,000 | 760 | 1018 | 1.339 | 0 | 1141 | 0.059 | 1018 | 0.000 | 722 | 2075 | 0.17 |

Ck 45 | 199,000 | 466 | 737 | 1.582 | 54 | 1469 | 0.248 | 1296 | 0.777 | 368 | 1486 | 0.225 |

Ck 45 | 207,000 | 462 | 672 | 1.455 | 61 | 1288 | 0.235 | 1298 | 0.942 | 354 | 1391 | 0.22 |

Ck 45 | 208,000 | 588 | 730 | 1.241 | 70 | 1154 | 0.148 | 1540 | 1.204 | 420 | 1194 | 0.168 |

Ck 45 | 207,000 | 551 | 774 | 1.405 | 68 | 1297 | 0.166 | 1559 | 1.139 | 464 | 1235 | 0.158 |

Ck 45 | 206,000 | 728 | 844 | 1.159 | 64 | 1208 | 0.108 | 1582 | 1.022 | 516 | 1217 | 0.138 |

Ck 45 | 210,000 | 652 | 787 | 1.207 | 68 | 1200 | 0.129 | 1568 | 1.139 | 472 | 1285 | 0.161 |

Ck 45 | 204,000 | 702 | 863 | 1.229 | 66 | 1268 | 0.118 | 1651 | 1.079 | 526 | 1243 | 0.138 |

St 37 | 210,000 | 295 | 435 | 1.475 | 64 | 829 | 0.275 | 835 | 1.020 | 273 | 988 | 0.207 |

St 52-3 | 210,000 | 400 | 597 | 1.493 | 63 | 1061 | 0.225 | 1083 | 0.980 | 389 | 1228 | 0.185 |

Material Designation | Monotonic Properties | Cyclic Parameters | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

DIN | E (MPa) | R_{e} or R_{p0.2} (MPa) | R_{m} (MPa) | R_{m}/R_{e} (-) | RA (%) | K (MPa) | n (-) | σ_{f} (MPa) | ε_{f} (-) | R_{e}′ (MPa) | K′ (MPa) | n′ (-) |

100 Cr 6 | 207,000 | 1927 | 2016 | 1.046 | 12 | 2281 | 0.031 | 2230 | 0.120 | 1341 | 3328 | 0.146 |

11 NiMnCrMo 55 | 210,000 | 745 | 852 | 1.144 | 57 | 1277 | 0.124 | 1327 | 0.834 | 663 | 1145 | 0.088 |

14 Mn 5 | 206,000 | 580 | 697 | 1.202 | 68 | 858 | 0.067 | 1222 | 1.150 | 537 | 1436 | 0.158 |

16 NiCrMo 3 2 | 209,000 | 891 | 939 | 1.054 | 63 | 963 | 0.011 | 1491 | 0.994 | 617 | 1080 | 0.09 |

17 MnCrMo 33 | 214,000 | 833 | 929 | 1.115 | 58 | 1285 | 0.099 | 1446 | 0.867 | 663 | 1252 | 0.102 |

20 Mn 3 | 206,000 | 910 | 960 | 1.055 | 43 | 1190 | 0.06 | 1090 | 0.561 | 675 | 1313 | 0.107 |

22 MnCrNi 3 | 198,000 | 1200 | 1510 | 1.258 | 42 | 2447 | 0.114 | 2034 | 0.549 | 1046 | 2149 | 0.112 |

22 MnCrNi 3 | 195,000 | 1200 | 1586 | 1.322 | 3 | 2586 | 0.115 | 1669 | 0.026 | 1193 | 2759 | 0.135 |

23 Mn 4 | 207,000 | 1008 | 1091 | 1.082 | 61 | 1185 | 0.026 | 1616 | 0.950 | 656 | 1616 | 0.145 |

23 NiCr 4 | 208,531 | 725 | 808 | 1.114 | 66 | 762 | 0.007 | 1215 | 1.080 | 541 | 1221 | 0.131 |

25 Mn 3 | 200,000 | 351 | 540 | 1.538 | 67 | 992 | 0.236 | 1173 | 1.100 | 322 | 1219 | 0.214 |

25 Mn 5 | 207,000 | 904 | 1008 | 1.115 | 49 | 1138 | 0.033 | 1284 | 0.680 | 608 | 1900 | 0.183 |

28 MnCu 6 | 204,000 | 330 | 580 | 1.758 | 64 | 938 | 0.19 | 950 | 1.030 | 347 | 1151 | 0.193 |

30 CrMo 2 | 221,000 | 780 | 898 | 1.151 | 67 | 1117 | 0.063 | 1692 | 1.120 | 579 | 1366 | 0.138 |

30 CrMo 2 | 200,250 | 1360 | 1429 | 1.051 | 55 | 1661 | 0.033 | 2085 | 0.790 | 814 | 1758 | 0.124 |

30 CrMoNiV 5 11 | 212,000 | 605 | 773 | 1.278 | 62 | 717 | 0.027 | 1332 | 0.968 | 497 | 894 | 0.094 |

30 CrNiMo 8 | 206,000 | 700 | 910 | 1.300 | 66 | 1128 | 0.079 | 1168 | 0.708 | 573 | 972 | 0.085 |

30 CrNiMo 8 | 206,000 | 700 | 910 | 1.300 | 66 | 1128 | 0.079 | 1168 | 0.708 | 522 | 995 | 0.095 |

30 MnCr 5 | 206,000 | 820 | 950 | 1.159 | 64 | 1250 | 0.097 | 1445 | 1.068 | 576 | 1618 | 0.166 |

34 CrMo 4 | 193,000 | 1017 | 1088 | 1.070 | 65 | 1344 | 0.056 | 1903 | 1.050 | 692 | 1310 | 0.103 |

34 CrMo 4 | 188,000 | 847 | 939 | 1.109 | 69 | 1215 | 0.074 | 1795 | 1.171 | 624 | 1008 | 0.077 |

34 CrMo 4 | 190,000 | 893 | 978 | 1.095 | 67 | 1338 | 0.089 | 1787 | 1.109 | 650 | 987 | 0.067 |

34 CrMo 4 | 197,000 | 980 | 1078 | 1.100 | 61 | 1382 | 0.07 | 1818 | 0.942 | 711 | 1373 | 0.106 |

34 CrMo 4 | 194,000 | 780 | 881 | 1.129 | 71 | 1299 | 0.116 | 1740 | 1.238 | 556 | 1198 | 0.124 |

4 NiCrMn 4 | 206,000 | 454 | 623 | 1.372 | 76 | 753 | 0.081 | 1229 | 1.450 | 505 | 1111 | 0.127 |

40 CrMo 4 | 208,780 | 840 | 940 | 1.119 | 64 | 1300 | 0.094 | 1440 | 1.035 | 583 | 1307 | 0.13 |

40 NiCrMo 6 | 201,000 | 1084 | 1146 | 1.057 | 59 | 1549 | 0.083 | 1857 | 0.890 | 758 | 1550 | 0.115 |

40 NiCrMo 6 | 190,000 | 910 | 1015 | 1.115 | 62 | 1372 | 0.089 | 1808 | 0.970 | 660 | 1392 | 0.12 |

40 NiCrMo 6 | 202,000 | 953 | 1029 | 1.080 | 62 | 1448 | 0.1 | 1724 | 0.970 | 659 | 1628 | 0.145 |

40 NiCrMo 6 | 193,000 | 998 | 1067 | 1.069 | 62 | 1474 | 0.092 | 1761 | 0.970 | 716 | 1292 | 0.095 |

40 NiCrMo 6 | 205,000 | 810 | 884 | 1.091 | 67 | 1378 | 0.142 | 1680 | 1.110 | 586 | 1303 | 0.129 |

40 NiCrMo 7 | 193,500 | 1374 | 1471 | 1.071 | 38 | 1796 | 0.04 | 1920 | 0.480 | 905 | 1890 | 0.118 |

40 NiCrMo 7 | 193,500 | 635 | 829 | 1.306 | 43 | 1175 | 0.098 | 1201 | 0.570 | 474 | 1332 | 0.167 |

41 MnCr 3 4 | 207,280 | 800 | 930 | 1.163 | 62 | 1350 | 0.112 | 1390 | 0.960 | 551 | 1340 | 0.143 |

42 Cr 4 | 195,000 | 903 | 1006 | 1.114 | 62 | 1293 | 0.068 | 1716 | 0.968 | 679 | 1153 | 0.085 |

42 Cr 4 | 194,000 | 813 | 921 | 1.133 | 65 | 1249 | 0.086 | 1674 | 1.050 | 613 | 1147 | 0.101 |

42 Cr 4 | 194,000 | 845 | 952 | 1.127 | 62 | 1288 | 0.086 | 1689 | 0.968 | 619 | 1207 | 0.107 |

42 Cr 4 | 192,000 | 833 | 943 | 1.132 | 65 | 1289 | 0.09 | 1690 | 1.050 | 621 | 1192 | 0.105 |

42 Cr 4 | 193,000 | 717 | 840 | 1.172 | 69 | 1240 | 0.118 | 1617 | 1.171 | 543 | 1161 | 0.122 |

42 CrMo 4 | 211,400 | 998 | 1111 | 1.113 | 60 | 1469 | 0.069 | 1525 | 0.496 | 716 | 1367 | 0.104 |

49 MnVS 3 | 210,200 | 566 | 840 | 1.484 | 19 | 1428 | 0.194 | 1152 | 0.380 | 520 | 1396 | 0.159 |

50 CrMo 4 | 205,000 | 970 | 1086 | 1.120 | 48.6 | 1132 | 0.026 | 1609 | 0.665 | 700 | 1568 | 0.13 |

50 CrMo 4 | 205,000 | 947 | 983 | 1.038 | 14.6 | 1042 | 0.018 | 926 | 0.157 | 774 | 1754 | 0.132 |

8 Mn 6 | 198,000 | 862 | 965 | 1.119 | 57 | 1227 | 0.054 | 1579 | 0.850 | 580 | 1256 | 0.125 |

8 Mn 6 | 198,000 | 821 | 869 | 1.058 | 53 | 1085 | 0.046 | 1434 | 0.750 | 674 | 1258 | 0.101 |

80 Mn 4 | 187,500 | 502 | 931 | 1.855 | 16 | 1100 | 0.127 | 1060 | 0.174 | 459 | 1859 | 0.225 |

WStE 460 | 210,000 | 560 | 667 | 1.191 | 61 | 1096 | 0.153 | 1171 | 0.932 | 514 | 1194 | 0.128 |

Material Designation | Monotonic Properties | Cyclic Parameters | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

DIN | E (MPa) | R_{e} or R_{p0.2} (MPa) | R_{m} (MPa) | R_{m}/R_{e} (-) | RA (%]) | K (MPa) | n (-) | σ_{f} (MPa) | ε_{f} (-) | R_{e}′ (MPa) | K′ (MPa) | n′ (-) |

X 10 CrNi 18 8 | 204,000 | 245 | 635 | 2.592 | 79 | 1416 | 0.362 | 1908 | 1.563 | 307 | 2397 | 0.331 |

X 10 CrNiNb 18 9 | 210,000 | 237 | 615 | 2.595 | 72 | 1398 | 1.273 | 271 | 1967 | 0.319 | ||

X 10 CrNiNb 18 9 | 210,000 | 237 | 615 | 2.595 | 72 | 1398 | 1.273 | 276 | 1667 | 0.289 | ||

X 10 CrNiTi 18 9 | 210,000 | 211 | 677 | 3.209 | 67 | 1428 | 1.109 | 455 | 8384 | 0.469 | ||

X 10 CrNiTi 18 9 | 210,000 | 182 | 668 | 3.670 | 68 | 1429 | 1.139 | 414 | 6179 | 0.435 | ||

X 10 CrNiTi 18 9 | 210,000 | 211 | 677 | 3.209 | 69 | 1470 | 1.171 | 496 | 3647 | 0.321 | ||

X 10 CrNiTi 18 9 | 210,000 | 177 | 516 | 2.915 | 74 | 1211 | 1.347 | 220 | 2264 | 0.375 | ||

X 10 CrNiTi 18 9 | 210,000 | 177 | 516 | 2.915 | 74 | 1211 | 1.347 | 250 | 1535 | 0.292 | ||

X 10 CrNiTi 18 9 | 210,000 | 214 | 529 | 2.472 | 74 | 1242 | 1.347 | 228 | 2086 | 0.357 | ||

X 10 CrNiTi 18 9 | 210,000 | 214 | 529 | 2.472 | 74 | 1242 | 1.347 | 251 | 1682 | 0.306 | ||

X 10 CrNiTi 18 9 | 210,000 | 177 | 535 | 3.023 | 77 | 1321 | 1.470 | 220 | 3080 | 0.424 | ||

X 10 CrNiTi 18 9 | 210,000 | 177 | 535 | 3.023 | 77 | 1321 | 1.470 | 241 | 2097 | 0.348 | ||

X 15 Cr 13 | 210,000 | 598 | 736 | 1.231 | 70 | 1622 | 1.204 | 475 | 1056 | 0.128 | ||

X 15 Cr 13 | 210,000 | 598 | 736 | 1.231 | 70 | 1622 | 1.204 | 497 | 987 | 0.11 | ||

X 15 CrNiSi 25 20 | 210,000 | 271 | 630 | 2.325 | 69 | 1368 | 1.171 | 289 | 2302 | 0.334 | ||

X 15 CrNiSi 25 20 | 210,000 | 271 | 630 | 2.325 | 69 | 1368 | 1.171 | 284 | 2242 | 0.332 | ||

X 2 CrNi 18 9 | 192,000 | 280 | 601 | 2.146 | 46 | 455 | 0.097 | 971 | 0.616 | 207 | 2807 | 0.419 |

X 20 CrMo 12 1 | 210,000 | 795 | 1013 | 1.274 | 47 | 1656 | 0.635 | 716 | 1325 | 0.099 | ||

X 20 CrMo 12 1 | 210,000 | 795 | 1013 | 1.274 | 47 | 1656 | 0.635 | 730 | 1301 | 0.093 | ||

X 25 CrNiMn 25 20 | 193,340 | 220 | 642 | 2.918 | 63 | 754 | 0.228 | 1360 | 1.010 | 421 | 2267 | 0.271 |

X 3 CrNi 19 9 | 172,625 | 746 | 953 | 1.277 | 69 | 1114 | 0.063 | 2037 | 1.160 | 882 | 2313 | 0.155 |

X 3 CrNi 19 9 | 186,435 | 255 | 746 | 2.925 | 74 | 548 | 0.136 | 1920 | 1.370 | 678 | 4634 | 0.309 |

X 5 CrNi 18 9 | 210,000 | 207 | 611 | 2.952 | 75 | 1458 | 1.386 | 197 | 3331 | 0.455 | ||

X 5 CrNi 18 9 | 210,000 | 207 | 611 | 2.952 | 83 | 1694 | 1.772 | 203 | 3001 | 0.434 | ||

X 5 CrNiMo 18 10 | 210,000 | 230 | 587 | 2.552 | 78 | 1476 | 1.514 | 256 | 1644 | 0.299 | ||

X 5 CrNiMo 18 10 | 210,000 | 231 | 587 | 2.541 | 78 | 1476 | 1.514 | 247 | 2755 | 0.388 | ||

X 5 CrNiMo 18 10 | 210,000 | 257 | 606 | 2.358 | 79 | 1830 | 1.561 | 313 | 2000 | 0.298 | ||

X 5 CrNiMo 18 10 | 210,000 | 228 | 665 | 2.917 | 81 | 1769 | 1.661 | 259 | 2081 | 0.336 | ||

X 5 CrNiMo 18 10 | 210,000 | 228 | 665 | 2.917 | 81 | 1769 | 1.661 | 259 | 2674 | 0.376 | ||

X 5 NiCrTi 26 15 | 210,000 | 777 | 1158 | 1.490 | 52 | 2008 | 0.734 | 713 | 1617 | 0.132 | ||

X 5 NiCrTi 26 15 | 210,000 | 777 | 1158 | 1.490 | 52 | 2008 | 0.734 | 711 | 1543 | 0.125 | ||

X 6 CrNi 19 11 | 183,000 | 325 | 650 | 2.000 | 80 | 1210 | 0.193 | 1400 | 1.610 | 267 | 1628 | 0.291 |

X 8 CrNiTi 18 10 | 204,000 | 222 | 569 | 2.563 | 76 | 349 | 0.062 | 1381 | 1.427 | 383 | 5234 | 0.421 |

X2 CrNiMo 18 10 | 210,000 | 373 | 700 | 1.877 | 75 | 1670 | 1.386 | 295 | 1232 | 0.23 | ||

X5 CrNi 18 9 | 198,000 | 242 | 666 | 2.752 | 82 | 484 | 0.113 | 2407 | 1.715 | 275 | 2872 | 0.378 |

## References

- Williams, C.R.; Lee, Y.-L.; Rilly, J.T. A practical method for statistical analysis of strain-life fatigue data. Int. J. Fatigue
**2003**, 25, 427–436. [Google Scholar] [CrossRef] - Blackmore, P.A. A critical review of Baumel-Seeger method for estimating strain-life fatigue properties of metallic materials. Eng. Integr.
**2009**, 27, 6–11. [Google Scholar] - Ramberg, W.; Osgood, W.R. Description of Stress-Strain Curves by Three Parameters; Technical Note No. 902; National Advisory Committee for Aeronautics (NACA): Washington, DC, USA, 1943.
- Manson, S.S.; Halford, G.R. Fatigue and Durability of Structural Materials, 1st ed.; ASM International: Materials Park, OH, USA, 2005. [Google Scholar]
- Manson, S.S. Fatigue: A complex subject—Some simple approximations. Exp. Mech. SESA
**1965**, 5, 193–226. [Google Scholar] [CrossRef] - Bäumel, A., Jr.; Seeger, T. Materials Data for Cyclic Loading—Supplement 1, 1st ed.; Elsevier: Amsterdam, The Netherlands, 1990. [Google Scholar]
- Roessle, M.L.; Fatemi, A. Strain-controlled fatigue properties of steels and some simple approximations. Int. J. Fatigue
**2000**, 22, 495–511. [Google Scholar] [CrossRef] - Hatscher, A.; Seeger, T.; Zenner, H. Abschätzung von zyklischen Werkstoffkennwerten—Erweiterung und Vergleich bisheriger Ansätze. Materialprufung
**2007**, 49, 2–14. [Google Scholar] - Park, J.H.; Song, J.H. Detailed evaluation of methods for estimation of fatigue properties. Int. J. Fatigue
**1995**, 17, 365–373. [Google Scholar] [CrossRef] - Zhang, Z.; Qiao, Y.; Sun, Q.; Li, C.; Li, J. Theoretical estimation to the cyclic strength coefficient and the cyclic strain-hardening exponent for metallic materials: Preliminary study. J. Mater. Eng. Perform.
**2009**, 18, 245–254. [Google Scholar] [CrossRef] - Li, J.; Sun, Q.; Zhang, Z.; Li, C.; Qiao, Y. Theoretical estimation to the cyclic yield strength and fatigue limit for alloy steels. Mech. Res. Commun.
**2009**, 36, 316–321. [Google Scholar] [CrossRef] - Lopez, Z.; Fatemi, A. A method of predicting cyclic stress-strain curve from tensile properties for steels. Mater. Sci. Eng. A Struct.
**2012**, 556, 540–550. [Google Scholar] [CrossRef] - Li, J.; Zhang, Z.; Li, C. An improved method for estimation of Ramberg-Osgood curves of steels from monotonic tensile properties. Fatigue Fract. Eng. Mater. Struct.
**2016**, 39, 412–426. [Google Scholar] [CrossRef] - Marohnić, T.; Basan, R. Study of Monotonic Properties’ Relevance for Estimation of Cyclic Yield Stress and Ramberg–Osgood Parameters of Steels. J. Mater. Eng. Perform.
**2016**, 25, 4812–4823. [Google Scholar] [CrossRef] - Basan, R.; Franulović, M.; Prebil, I.; Črnjarić-Žic, N. Analysis of strain-life fatigue parameters and behavior of different groups of metallic materials. Int. J. Fatigue
**2011**, 33, 484–491. [Google Scholar] [CrossRef] - Basan, R.; Marohnić, T.; Prebil, I.; Franulović, M. Preliminary investigation of the existence of correlation between cyclic Ramberg-Osgood parameters and monotonic properties of low-alloy steels. In Proceedings of the 3rd International Conference Mechanical Technologies and Structural Materials MTSM 2013, Split, Croatia, 26–27 September 2013; Živković, D., Ed.; Croatian society for mechanical technologies: Split, Croatia, 2013. [Google Scholar]
- Marohnić, T.; Basan, R.; Franulović, M. Evaluation of the Possibility of Estimating Cyclic Stress-strain Parameters and Curves from Monotonic Properties of Steels. Procedia Eng.
**2015**, 101, 277–284. [Google Scholar] [CrossRef] - Smith, R.W.; Hirschberg, M.H.; Manson, S.S. Fatigue Behavior of Materials under Strain Cycling in Low and Intermediate Life Range; Technical Note D-1574; National Aeronautics and Space Administration (NASA): Washington, DC, USA, 1963.
- Zhang, Z.; Wu, W.; Chen, D.; Sun, Q.; Zhao, W. New Formula Relating the Yield Stress-Strain with the Strength Coefficient and the Strain-Hardening Exponent. J. Mater. Eng. Perform.
**2004**, 13, 509–512. [Google Scholar] - Landgraf, R.W.; Morrow, J.; Endo, T. Determination of cyclic stress-strain curve. J. Mater.
**1969**, 4, 176–188. [Google Scholar] - Devore, J. Probability and Statistics for Engineering and the Sciences; Brooks/Cole Cengage Learning: Boston, MA, USA, 2011. [Google Scholar]
- Minitab 17 Statistical Software, Product version 17.3.1; Computer Software; Minitab, Inc.: State College, PA, USA, 2010.
- Boller, C.; Seeger, T. Materials Data for Cyclic Loading, Part A–D, 1st ed.; Elsevier: Amsterdam, The Netherlands, 1987. [Google Scholar]
- Basan, R. MATDAT Materials Properties Database, Version 1.1. 2011. Available online: http://www.matdat.com/ (accessed on 15 January 2016).

**Figure 1.**Percentage of R

_{e}′ values estimated by selected methods that deviate up to 10%, 20% and 30% from experiment-based values.

**Figure 2.**Percentage of Δσ/2 values estimated by selected methods that deviate up to 10%, 20% and 30% from experiment-based values.

Evaluated Value | Method | Estimated Parameters | Originally Proposed for | Equation Number |
---|---|---|---|---|

R_{e}′ | Lopez and Fatemi 1 [12] | R_{e}′ | steels divided by R_{m}/R_{e} | (4a,b) |

Lopez and Fatemi 2 [12] | R_{e}′ | all steels | (5) | |

Li et al. [13] | R_{e}′ | all steels | (6) | |

Δσ/2 | Zhang et al. 1 [10] (K and n available) | K′ | steels, Al and Ti alloys | (10) |

n′ | steels, Al and Ti alloys divided by α | (11a,b,c) | ||

Zhang et al. 2 [10] (K and n not available) | K′ | steels, Al and Ti alloys divided by α | (12a,b) | |

n′ | steels, Al and Ti alloys divided by α | (13a,b,c) | ||

Lopez and Fatemi 1 [12] | K′ | steels divided by R_{m}/R_{e} | (14a,b) | |

n′ | steels divided by R_{m}/R_{e} | (15a,b) | ||

Lopez and Fatemi 2 [12] | K′ | steels divided by R_{m}/R_{e} | (14a,b) | |

n′ | all steels | (16) | ||

Li et al. [13] | K′ | steels divided by R_{m}/R_{e} | (19a,b,c) | |

n′ | steels divided by R_{m}/R_{e} | (18) |

**Table 2.**Recommended methods for estimation of cyclic yield stress R

_{e}′ and cyclic parameters K′ and n′ of steels.

Steel Subgroup | Estimation of R_{e}′ | Estimation of Δσ/2 (K′, n′) |
---|---|---|

Unalloyed steels | Li et al. | 1. Li et al. |

Lopez and Fatemi 2 | 2. Lopez and Fatemi 1 | |

Low-alloy steels | 1. Lopez and Fatemi 1 | Lopez and Fatemi 1 |

2. Li et al. | Lopez and Fatemi 2 | |

3. Lopez and Fatemi 2 | Li et al. | |

High-alloy steels | Lopez and Fatemi 1 | Lopez and Fatemi 1 |

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**MDPI and ACS Style**

Marohnić, T.; Basan, R.; Franulović, M.
Evaluation of Methods for Estimation of Cyclic Stress-Strain Parameters from Monotonic Properties of Steels. *Metals* **2017**, *7*, 17.
https://doi.org/10.3390/met7010017

**AMA Style**

Marohnić T, Basan R, Franulović M.
Evaluation of Methods for Estimation of Cyclic Stress-Strain Parameters from Monotonic Properties of Steels. *Metals*. 2017; 7(1):17.
https://doi.org/10.3390/met7010017

**Chicago/Turabian Style**

Marohnić, Tea, Robert Basan, and Marina Franulović.
2017. "Evaluation of Methods for Estimation of Cyclic Stress-Strain Parameters from Monotonic Properties of Steels" *Metals* 7, no. 1: 17.
https://doi.org/10.3390/met7010017