# Statistical Estimation of Resistance to Cyclic Deformation of Structural Steels and Aluminum Alloy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experiment and Materials

#### 2.2. Low Cycle Deformation Diagram and Its Characteristics

_{1}and A

_{2}define the loop widths ${\overline{\delta}}_{1}$ and ${\overline{\delta}}_{2}$ of the first and second semi-cycles, respectively. The parameter α describes the cyclic properties of materials, i.e., hardening, softening, and stability.

_{1f}and A

_{2f}define the loop widths ${\overline{\delta}}_{1f}$ and ${\overline{\delta}}_{2f}$ of the first and second semi-cycles respectively.

## 3. Results and Discussion

#### 3.1. Statistical Assessment of the Low-Cycle Fatigue Curves Parameters

#### 3.2. Probability Evaluation of Low-Cycle Fatigue Curves

_{k}resulting from accumulated unilateral plastic deformation and fatigue damage d

_{N}resulting from cyclic plastic deformation, characterized by the width of the hysteresis loop ${\overline{\delta}}_{k}$:

_{2}and m were derived from the probability curves $lg{\overline{\delta}}_{1}-lg{k}_{c}$ for strain-controlled loading and are presented in Table 4. Hysteresis loop width ${\overline{\delta}}_{1}$ was determined using Equation (30). It can be seen that the probability curves reach a suitable order, i.e., the 1% curve is the lowest and the 99% curve the highest. It can be seen that there is a large scatter (Figure 8b), which indicates a high dependence of durability N on the loading level ${\overline{\sigma}}_{0}.$ The range of curves becomes narrower with increasing durability. When comparing the experimental results for steel 15Cr2MoVa and C45 with the analytical results, the opposite effect can be observed: the experimental curves have a much tighter range and scatter than the analytically estimated ones. At loading level ${\overline{\sigma}}_{0}=1.0$ the experimentally estimated durability of 1% to 99% is distributed between the analytical curves of 15% and 90%.

## 4. Conclusions

- Statistical investigations of the elastic-plastic parameters of the low-cycle fatigue curves showed that these parameters vary more than the statistical parameters of the mechanical characteristics. This is probably due to the sensitivity of the cyclic strain characteristic to variations in chemical composition and thermal and technological processing conditions, as well as to the effect of the considerable relative experimental error on the hysteresis loop width. As the low cycle load level decreases, the scatter of low cycle deformation characteristics increases. This is apparently due to the lower stability of the hardening and softening processes in the fatigue and transient failure zones compared to the quasi-static failure zone.
- It has been established that relative coordinates $\overline{\sigma}-\overline{\epsilon}$ should be used under stress-controlled loading. The use of absolute values, due to the significant dispersion of the proportionality limit ${e}_{pr}$, leads to a large dispersion of the deformation and failure parameters of the low-cycle deformation.
- Analysis of the histograms shows that for all materials and loading levels investigated, there is a positive asymmetry, but as the sampling rate increases, the histograms become smoother and approximate to a normal distribution law.
- Calculation of cyclic characteristics for all materials using mean absolute deviation, variation amplitude, and compatibility criteria has shown that the results could be represented by the normal distribution law.
- The scattering of the parameters of cyclic elastic-plastic deformation curves is satisfactorily described by the laws of normal and log-normal distribution, with obvious improvement in fit with the increasing sample size, or when tested material groups are grouped by chemical composition, surface hardening, and heat treatment.
- The low-cycle deformation diagram becomes flatter as the load level increases, so that small stress variations lead to larger hysteresis loop width variations. This increases the dispersion of durability.
- The analysis performed on the low-cycle fatigue curves under controlled stress loading shows only fatigue damage. For anisotropic materials steel 15Cr2MoVa and steel C45, satisfactory agreement between the experimental and analytical low cycle fatigue curves is only possible at a loading level ${\overline{\sigma}}_{0}=1.0$ and ${\sigma}_{pr}$. The influence of damage increases at higher values of the quasi-static load level.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${a}_{1}$ | tabular function value |

${A}_{1},{A}_{2}$ | constant describing the first and second semi-cycle form respectively |

${A}_{1f},{A}_{2f}$ | fictitious constants of the first and second semi-cycle form respectively |

$c$ | materials hardening or softening equation coefficient |

$C,m$ | constants of the Coffin–Manson equation |

${C}_{1},{C}_{2},{C}_{3}$ | constants of the fatigue curve under strain-controlled loading |

${d}_{k},{d}_{N}$ | quasi-static and fatigue damage respectively |

D | dispersion |

$e$ | monotonous strain (%) |

${e}_{0}$ | strain if initial (0 semi-cycle) loading (%) |

${\overline{e}}_{0}$ | strain if initial (0 semi-cycle) loading normalized to proportional limit strain (%) |

${e}_{f}$ | fracture strain (%) |

${e}_{k}$ | cyclic strain of k semi-cycle (%) |

${e}_{pk}$ | accumulated plastic strain after loading semi-cycles in the direction of tension (%) |

${e}_{pr}$ | proportional limit strain (%) |

${e}_{u}$ | static loading monotonous strain (%) |

E | modulus of elasticity (MPa) |

k | number of loading semi-cycle under controlled stress |

${k}_{c}$ | number of loading semi-cycle under controlled strain |

${m}_{1},{m}_{2},{m}_{3},$ | constants of the fatigue curve under strain-controlled loading |

P | probability |

q, l | equation exponent |

$s$ | standard deviation |

$S$ | cyclic stress at cyclic loading (MPa) |

$Sk$ | skewness |

${S}_{k}$ | cyclic stress of k semi-cycle (MPa) |

${\overline{S}}_{k}$ | cyclic stress of k semi-cycles normalized respectively to proportional limit stress (MPa) |

${S}_{T}$ | stress of cyclic proportionality limit (MPa) |

${\overline{S}}_{T}$, ${\overline{S}}_{Tk}$ | stress normalized respectively to proportional limit stress (MPa) |

$\overline{x}$ | sample mean |

${x}_{p}^{U}$ | the upper endpoint of the confidence interval |

${x}_{p}^{L}$ | the lower endpoint of the confidence intervals |

$V$ | coefficient of variation |

Greek symbols | |

$\alpha ,\beta $ | constants characterizing materials hardening or softening |

${\alpha}_{1}$ | level of significance |

${\delta}_{1}$ | width of hysteresis loop for first semi-cycle (%) |

${\delta}_{1f},{\delta}_{2f}$ | fictitious width of hysteresis loop of the first and second semi-cycle respectively (%) |

${\delta}_{k}$ | width of hysteresis loop for k semi-cycle (%) |

${\delta}_{k4},{\delta}_{k4f},$ | width of hysteresis loop for fourth semi-cycle (%) |

$\overline{\delta}$ | average width of hysteresis loop (%) |

${\overline{\delta}}_{k}$ | width of hysteresis loop normalized to proportional limit strain |

$\psi $ | percent area reduction (%) |

${\psi}_{u}$ | percent area reduction at failure (%) |

$\epsilon $ | cyclic strain at cyclic loading (%) |

${\epsilon}_{k}$ | cyclic strain of k semi-cycle (%) |

${\overline{\epsilon}}_{k}$ | cyclic strain of k semi-cycles normalized respectively to proportional limit strain (%) |

${\overline{\epsilon}}_{pk}$ | cyclic accumulated plastic strain normalized respectively to proportional limit strain (%) |

$\sigma $ | monotonous stress (MPa) |

${\sigma}_{0}$ | stress of initial (0 semi-cycle) loading (MPa) |

${\overline{\sigma}}_{0}$ | stress of initial (0 semi-cycle) loading normalized to proportional limit stress (MPa) |

${\sigma}_{0.2}$ | elastic limit or yield strength, the stress at which 0.2% plastic strain occurs (MPa) |

${\sigma}_{f}$ | fracture stress (MPa) |

${\sigma}_{k}$ | cyclic stress of k semi-cycle (MPa) |

${\sigma}_{pr}$ | proportional limit stress (MPa); |

${\sigma}_{u}$ | ultimate tensile stress (MPa) |

$\overline{\sigma}$ | normalized to proportional limit cyclic stress (MPa) |

${\omega}^{2}$ | Smirnov compatibility criterion |

${W}_{a}^{2}$ | critical value of Smirnov criterion (${W}_{0.1}^{2}=0.104$; ${W}_{0.2}^{2}=0.126$; ${W}_{0.01}^{2}=0.178$) |

$\Phi ({\widehat{z}}_{i})$ | Laplace function value |

## Appendix A

**Figure A1.**Histograms of low cyclic load diagram characteristics of steel 15Cr2MoVa; loading level ${\overline{\sigma}}_{0}=1.00$—5, 8; loading level ${\overline{\sigma}}_{0}=1.12$—1, 3, 6, 9; loading level ${\overline{\sigma}}_{0}=1.25$—2, 4, 7, 10 (1, 2—A

_{1}; 3, 4—A

_{1f}; 5–7—A

_{2}; 8–10—A

_{2f}).

**Figure A2.**Histograms of low cyclic load diagram characteristics of steel C45; loading level ${\overline{\sigma}}_{0}=1.00$—5, 8; loading level ${\overline{\sigma}}_{0}=1.12$—1, 3, 6, 9; loading level ${\overline{\sigma}}_{0}=1.25$—2, 4, 7, 10 (1, 2—A

_{1}; 3, 4—A

_{1f}; 5–7—A

_{2}; 8–10—A

_{2f}).

**Figure A3.**Histograms of low cyclic load diagram characteristics of steel 15Cr2MoVa C45; loading level ${\overline{\sigma}}_{0}=1.00$—1, 4; loading level ${\overline{\sigma}}_{0}=1.12\u2014$ 2, 5; loading level ${\overline{\sigma}}_{0}=1.25\u2014$ 3, 6.

## Appendix B

**Table A1.**Statistical properties for normal, log-normal, and Weibull distribution of cyclic properties.

Cyclic Properties | Material | ${\overline{\mathit{\sigma}}}_{0}$ | n | Normal | Log-Normal | Weibull | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\overline{\mathit{x}}$ | $\mathit{s}$ | $\mathit{D}$ | $\mathit{S}\mathit{k}$ | $\mathit{V}$ | $\overline{\mathit{x}}$ | $\mathit{s}$ | $\mathit{D}$ | $\mathit{S}\mathit{k}$ | $\mathit{V}$ | $\overline{\mathit{x}}$ | $\mathit{s}$ | $\mathit{S}\mathit{k}$ | $\mathit{V}$ | ||||

${A}_{1}$ | 15Cr2MoVa | 1.00 | 20 | 1.8175 | 1.1206 | 1.2558 | 1.329 | 0.616 | 1.5293 | 0.2676 | 0.0716 | −0.135 | 1.451 | 1.8067 | 1.1206 | 1.555 | 0.620 |

1.12 | 40 | 1.2237 | 0.4515 | 0.2038 | 1.151 | 0.365 | 1.1636 | 0.1481 | 0.0219 | 0.284 | 2.251 | 1.1884 | 0.4514 | 1.243 | 0.379 | ||

1.25 | 20 | 0.8915 | 0.2146 | 0.0460 | 0.008 | 0.241 | 0.8649 | 0.1132 | 0.0128 | −0.713 | −1.796 | 0.8915 | 0.2145 | −0.009 | 0.241 | ||

C45 | 1.00 | 20 | 1.0477 | 0.2201 | 0.0484 | −0.470 | 0.210 | 1.0215 | 0.1063 | 0.0113 | −1.491 | 1.148 | 1.0477 | 0.2201 | −0.550 | 0.210 | |

1.25 | 40 | 0.9135 | 0.1149 | 0.0132 | −0.965 | 0.126 | 0.9056 | 0.0598 | 0.0036 | −1.412 | −1.390 | 0.9135 | 0.1143 | −1.041 | 0.126 | ||

1.50 | 20 | 0.8865 | 0.0974 | 0.0095 | 1.004 | 0.109 | 0.8817 | 0.0459 | 0.0021 | 0.617 | −0.839 | 0.8503 | 0.0974 | 1.175 | 0.114 | ||

D16T1 | 1.15 | 20 | 0.6092 | 0.2212 | 0.0489 | 0.475 | 0.363 | 0.5714 | 0.1618 | 0.0262 | −0.147 | −0.666 | 0.6092 | 0.2212 | 0.558 | 0.363 | |

${A}_{1f}$ | 15Cr2MoVa | 1.00 | 20 | 1.3849 | 0.4455 | 0.1985 | −0.088 | 0.322 | 1.3034 | 0.1672 | 0.279 | −1.227 | 1.453 | 1.3849 | 0.4455 | −0.103 | 0.322 |

1.12 | 40 | 1.1342 | 0.3512 | 0.1234 | −0.325 | 0.309 | 1.0711 | 0.1578 | 0.0249 | −0.919 | 5.292 | 1.1342 | 0.3512 | −0.351 | 0.309 | ||

1.25 | 20 | 0.5466 | 0.2195 | 0.0482 | 0.424 | 0.402 | 0.5029 | 0.1897 | 0.0359 | −0.589 | −0.636 | 0.5466 | 0.2195 | 0.496 | 0.402 | ||

C45 | 1.00 | 20 | 0.7191 | 0.1047 | 0.0109 | −0.364 | 0.146 | 0.7114 | 0.0662 | 0.0044 | −0.577 | −0.448 | 0.7191 | 0.1047 | −0.426 | 0.146 | |

1.25 | 40 | 0.6916 | 0.0957 | 0.0091 | −0.885 | 0.138 | 0.6842 | 0.0668 | 0.0045 | −1.546 | −0.405 | 0.6916 | 0.0957 | −0.956 | 0.138 | ||

1.50 | 20 | 0.7908 | 0.1015 | 0.0103 | 1.189 | 0.128 | 0.7851 | 0.0527 | 0.0028 | 0.771 | −0.502 | 0.7618 | 0.1015 | 1.391 | 0.133 | ||

D16T1 | 1.15 | 20 | 0.7281 | 0.2334 | 0.0545 | −0.021 | 0.320 | 0.6872 | 0.1604 | 0.0257 | −0.939 | −0.985 | 0.7281 | 0.2334 | −0.025 | 0.321 | |

${A}_{2}$ | 15Cr2MoVa | 1.00 | 20 | 1.8269 | 1.1217 | 1.2583 | 1.319 | 0.614 | 1.5389 | 0.2668 | 0.0712 | −0.137 | 1.425 | 1.8203 | 1.1217 | 1.543 | 0.616 |

1.12 | 40 | 1.2532 | 0.4659 | 0.2121 | 1.163 | 0.367 | 1.1816 | 0.1483 | 0.0220 | 0.305 | 2.047 | 1.2086 | 0.4606 | 1.256 | 0.381 | ||

1.25 | 20 | 0.9176 | 0.2111 | 0.0445 | 0.028 | 0.230 | 0.8931 | 0.4064 | 0.0113 | −0.598 | −2.167 | 0.9176 | 0.2111 | 0.033 | 0.230 | ||

C45 | 1.00 | 20 | 1.0524 | 0.2196 | 0.0482 | −0.521 | 0.209 | 1.0263 | 0.1061 | 0.0113 | −1.534 | 9412 | 1.0524 | 0.2196 | −0.609 | 0.209 | |

1.25 | 40 | 0.9211 | 0.1155 | 0.0133 | −0.954 | 0.125 | 0.9131 | 0.0596 | 0.0035 | −1.402 | −1.511 | 0.9210 | 0.1155 | −1.030 | 0.125 | ||

1.50 | 20 | 0.8987 | 0.0975 | 0.0095 | 1.115 | 0.108 | 0.8940 | 0.0451 | 0.0020 | 0.709 | −0.926 | 0.8560 | 0.0974 | 1.304 | 0.114 | ||

${A}_{2f}$ | 15Cr2MoVa | 1.00 | 20 | 1.3976 | 0.4440 | 0.1972 | −0.067 | 0.318 | 1.3186 | 0.1629 | 0.0265 | −1.159 | 1.357 | 1.3976 | 0.4440 | −0.078 | 0.318 |

1.12 | 40 | 1.1358 | 0.3519 | 0.1239 | −0.325 | 0.309 | 1.0725 | 0.1579 | 0.0249 | −0.920 | 5.192 | 1.1358 | 0.3519 | −0.351 | 0.309 | ||

1.25 | 20 | 0.5674 | 0.2302 | 0.0530 | 0.464 | 0.406 | 0.5216 | 0.1901 | 0.0361 | −0.555 | −0.673 | 0.5674 | 0.2302 | 0.542 | 0.406 | ||

C45 | 1.00 | 20 | 0.7287 | 0.1109 | 0.0123 | −0.188 | 0.152 | 0.7203 | 0.0684 | 0.0047 | −0.423 | −0.481 | 0.7287 | 0.1109 | −0.220 | 0.152 | |

1.25 | 40 | 0.6995 | 0.0954 | 0.0091 | −0.912 | 0.136 | 0.6922 | 0.0658 | 0.0043 | −1.583 | −0.412 | 0.6995 | 0.0954 | −0.985 | 0.136 | ||

1.50 | 20 | 0.8069 | 0.0999 | 0.0099 | 1.333 | 0.124 | 0.8016 | 0.0506 | 0.0026 | 0.886 | −0.527 | 0.7587 | 0.0999 | 1.559 | 0.132 | ||

$\beta \xb7{10}^{-3}$ | 15Cr2MoVa | 1.00 | 20 | 0.2355 | 0.1915 | 0.0367 | 2.012 | 0.813 | 0.1840 | 0.3098 | 0.0959 | 0.121 | −0.421 | 0.2209 | 0.1915 | 2.354 | 0.866 |

1.12 | 40 | 1.4325 | 1.2270 | 1.5056 | 1.679 | 0.856 | 0.9942 | 0.4037 | 0.1629 | −0.389 | −0.134 | 1.4325 | 1.2270 | 1.813 | 0.856 | ||

1.25 | 20 | 6.8502 | 3.2973 | 10.872 | 0.341 | 0.481 | 5.9727 | 0.2540 | 0.0645 | −0.853 | 0.327 | 6.8502 | 3.2973 | 0.399 | 0.481 | ||

$c\xb7{10}^{-4}$ | 15Cr2MoVa | 1.00 | 20 | 5.7665 | 3.5335 | 12.485 | 0.889 | 0.613 | 4.8351 | 0.2679 | 0.0718 | 0.010 | 0.391 | 5.7665 | 3.5335 | 1.039 | 0.613 |

1.12 | 40 | 8.4257 | 5.8851 | 34.634 | 1.081 | 0.698 | 6.5179 | 0.3335 | 0.1112 | −0.345 | 0.409 | 8.4257 | 5.8851 | 1.167 | 0.698 | ||

1.25 | 20 | 14.730 | 7.1011 | 50.426 | 0.344 | 0.482 | 13.010 | 0.2309 | 0.0533 | 0.343 | 0.207 | 14.730 | 7.1011 | 0.402 | 0.482 | ||

${\overline{S}}_{T}$ | 15Cr2MoVa | 80 | 1.2500 | 0.492 | 0.0024 | −0.493 | 0.039 | 1.2491 | 0.0169 | 0.0003 | −0.369 | 0.175 | 1.2500 | 0.0492 | −0.577 | 0.039 | |

C45 | 80 | 1.0500 | 0.0196 | 0.0004 | 0.185 | 0.018 | 1.0578 | 0.0079 | 0.000 | 0.104 | 0.242 | 1.0500 | 0.0196 | 0.217 | 0.018 | ||

D16T1 | 20 | 1.6540 | 0.1034 | 0.0107 | 0.534 | 0.062 | 1.6509 | 0.0267 | 0.0007 | 0.443 | 0.122 | 1.6540 | 0.1034 | 0.625 | 0.064 | ||

$\alpha $ | D16T1 | 1.15 | 20 | 0.5595 | 0.0949 | 0.0090 | 0.147 | 0.169 | 0.5518 | 0.0746 | 0.0056 | −0.168 | −0.289 | 0.5595 | 0.172 | 0.172 | 0.169 |

Material | ${\overline{\mathit{\sigma}}}_{0}$ | p | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{1\mathit{f}}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{2\mathit{f}}$ | $\mathit{\beta}\mathit{\xb7}{10}^{-3}$ | $\mathit{c}\mathit{\xb7}{10}^{-4}$ | ${\overline{\mathit{S}}}_{\mathit{T}}$ | α | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{x}}_{\mathit{p}}^{\mathit{U}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{L}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{U}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{L}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{U}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{L}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{U}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{L}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{U}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{L}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{U}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{L}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{U}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{L}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{U}}$ | ${\mathit{x}}_{\mathit{p}}^{\mathit{L}}$ | |||

15Cr2MoVa | 1.00 | 0.01 | 0.1497 | −1.4552 | 0.6080 | 0.0889 | −0.1288 | −1.4356 | 0.6233 | 0.1060 | 0.1252 | 0.0968 | 3.6242 | 2.9893 | 1.1486 | 1.1223 | - | - |

0.50 | 2.4702 | 1.1647 | 1.6444 | 1.1254 | 2.4803 | 1.1735 | 1.6562 | 1.1389 | 0.2092 | 0.1618 | 5.3238 | 4.3912 | 1.2631 | 1.2368 | - | - | ||

0.99 | 5.0902 | 3.7847 | 2.6807 | 2.1617 | 5.0894 | 3.7826 | 2.6890 | 2.1717 | 0.3498 | 0.2705 | 7.8204 | 6.4503 | 1.3775 | 1.3512 | - | - | ||

1.12 | 0.01 | 0.3589 | 0.0082 | 0.4536 | 0.1807 | 0.3505 | −0.0115 | 0.4538 | 0.1803 | 0.4806 | 0.3591 | 3.9686 | 3.2527 | 1.1486 | 1.1223 | - | - | |

0.50 | 1.4091 | 1.0583 | 1.2706 | 0.9977 | 1.4342 | 1.0722 | 1.2725 | 0.9991 | 1.1503 | 0.8594 | 7.1994 | 5.9007 | 1.2631 | 1.2368 | - | - | ||

0.99 | 2.4598 | 2.1085 | 2.0876 | 1.8147 | 2.5179 | 2.1559 | 2.0911 | 1.8227 | 2.7523 | 2.0569 | 13.060 | 10.704 | 1.3775 | 1.3512 | - | - | ||

1.25 | 0.01 | 0.5173 | 0.2673 | 0.1638 | −0.0919 | 0.5495 | 0.3035 | 0.1659 | −0.1022 | 4.6094 | 3.8771 | 10.509 | 9.1092 | 1.1486 | 1.1223 | - | - | |

0.50 | 1.0116 | 0.7665 | 0.6744 | 0.4187 | 1.0406 | 0.7945 | 0.7015 | 0.4333 | 6.5124 | 5.4777 | 13.981 | 12.118 | 1.2631 | 1.2368 | - | - | ||

0.99 | 1.5156 | 1.3906 | 1.1852 | 0.9294 | 1.5317 | 1.2857 | 1.2369 | 0.9687 | 9.1981 | 7.7367 | 18.600 | 16.122 | 1.3775 | 1.3512 | - | - | ||

C45 | 1.00 | 0.01 | 0.6639 | 0.4075 | 0.5365 | 0.4145 | 0.6694 | 0.4135 | 0.5353 | 0.4061 | - | - | - | - | 1.0095 | 0.9990 | - | - |

0.50 | 1.1759 | 0.9195 | 0.7801 | 0.6581 | 1.1803 | 0.9244 | 0.7933 | 0.6641 | - | - | - | - | 1.0552 | 1.0447 | - | - | ||

0.99 | 1.6879 | 1.4315 | 1.0236 | 0.9016 | 1.6912 | 1.4353 | 1.0513 | 0.9221 | - | - | - | - | 1.1008 | 1.0903 | - | - | ||

1.25 | 0.01 | 0.6908 | 0.6016 | 0.5061 | 0.4317 | 0.6974 | 0.6076 | 0.5146 | 0.4404 | - | - | - | - | 1.0095 | 0.9990 | - | - | |

0.50 | 0.9581 | 0.8688 | 0.7288 | 0.6544 | 0.9659 | 0.8762 | 0.7366 | 0.6624 | - | - | - | - | 1.0552 | 1.0447 | - | - | ||

0.99 | 1.2253 | 1.1361 | 0.9515 | 0.8771 | 1.2346 | 1.1448 | 0.9586 | 0.8844 | - | - | - | - | 1.1008 | 1.0903 | - | - | ||

1.50 | 0.01 | 0.7166 | 0.6032 | 0.6138 | 0.4956 | 0.7287 | 0.6151 | 0.6325 | 0.5160 | - | - | - | - | 1.0095 | 0.9990 | - | - | |

0.50 | 0.9432 | 0.8298 | 0.8499 | 0.7317 | 0.9555 | 0.8419 | 0.8651 | 0.7486 | - | - | - | - | 1.0552 | 1.0447 | - | - | ||

0.99 | 1.1698 | 1.0564 | 1.0860 | 0.9678 | 1.1822 | 1.0686 | 1.0977 | 0.9812 | - | - | - | - | 1.1008 | 1.0903 | - | - | ||

D16T1 | 1.15 | 0.01 | 0.2234 | −0.0342 | 0.3211 | 0.0492 | - | - | - | - | - | - | - | - | 1.4737 | 1.3533 | 0.3941 | 0.2835 |

0.50 | 0.7380 | 0.4803 | 0.8640 | 0.5921 | - | - | - | - | - | - | - | - | 1.7142 | 1.5938 | 0.6148 | 0.5042 | ||

0.99 | 1.2525 | 0.9948 | 1.4068 | 1.1349 | - | - | - | - | - | - | - | - | 1.9546 | 1.8342 | 0.8355 | 0.7249 |

## References

- Makhutov, N.A.; Gadenin, M.M. Integrated Assessment of the Durability, Resources, Survivability, and Safety of Machinery Loaded under Complex Conditions. J. Mach. Manuf. Reliab.
**2020**, 49, 292–300. [Google Scholar] [CrossRef] - Makhutov, N.A.; Panov, A.N.; Yudina, O.N. The development of models of risk assessment complex transport systems. In IOP Conference Series: Materials Science and Engineering, Proceedings of the V International Scientific Conference, Survivability and Structural Material Science (SSMS 2020), Moscow, Russia, 27–29 October 2020; Institute of Physics Publishing (IOP): Moscow, Russia; Volume 1023, Available online: https://scholar.google.lt/scholar?hl=lt&as_sdt=0%2C5&q=The+development+of+models+of+risk+assessment+complex+transport+systems&btnG= (accessed on 16 November 2021).
- Sekhar, A.P.; Nandy, S.; Bakkar, M.A.; Ray, K.K.; Das, D. Low cycle fatigue response of differently aged AA6063 alloy: Statistical analysis and microstructural evolution. Materialia
**2021**, 20, 101219. [Google Scholar] [CrossRef] - Jiang, Z.; Han, Z.; Li, M. A probabilistic model for low-cycle fatigue crack initiation under variable load cycles. Int. J. Fatigue
**2022**, 155, 106528. [Google Scholar] [CrossRef] - Makhutov, N.A.; Zatsarinny, V.V.; Reznikov, D.O. Fatigue prediction on the basis of analysis of probabilistic mechanical properties. AIP Conf. Proc.
**2020**, 2315, 040025. [Google Scholar] [CrossRef] - Tomaszewski, T.; Strzelecki, P.; Mazurkiewicz, A.; Musiał, J. Probabilistic estimation of fatigue strength for axial and bending loading in high-cycle fatigue. Materials
**2020**, 13, 1148. [Google Scholar] [CrossRef][Green Version] - Li, X.-K.; Chen, S.; Zhu, S.-P.; Ding Liao, D.; Gao, J.-W. Probabilistic fatigue life prediction of notched components using strain energy density approach. Eng. Fail. Anal.
**2021**, 124, 105375. [Google Scholar] [CrossRef] - Pelegatti, M.; Lanzutti, A.; Salvati, E.; Srnec Novak, J.; De Bona, F.; Benasciutti, D. Cyclic Plasticity and Low Cycle Fatigue of an AISI 316L Stainless Steel: Experimental Evaluation of Material Parameters for Durability Design. Materials
**2021**, 14, 3588. [Google Scholar] [CrossRef] - Strzelecki, P. Determination of fatigue life for low probability of failure for different stress levels using 3-parameter Weibull distribution. Int. J. Fatigue
**2021**, 145, 106080. [Google Scholar] [CrossRef] - Correia, J.; Apetre, N.; Arcari, A.; Jesus, A.D.; Muñiz-Calvente, M.; Calçada, R.; Berto, F.; Fernández-Canteli, A. Generalized probabilistic model allowing for various fatigue damage variables. Int. J. Fatigue
**2017**, 100, 187–194. [Google Scholar] [CrossRef] - Castillo, E.; Fernández-Canteli, A.; Pinto, H.; López-Aenlle, M. A general regression model for statistical analysis of strain–life fatigue data. Mater. Lett.
**2008**, 62, 3639–3642. [Google Scholar] [CrossRef] - 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][Green Version] - Xiang, Y.; Liu, L. Application of inverse first-order reliability method for probabilistic fatigue life prediction. Probabilistic Eng. Mech.
**2011**, 26, 148–156. [Google Scholar] [CrossRef] - Xu, L.; Yu, X.; Hui, L.; Zhou, S. Fatigue life prediction of aviation aluminium alloy based on quantitative pre-corrosion damage analysis. Trans. Nonferrous Met. Soc. China
**2017**, 27, 1353–1362. [Google Scholar] [CrossRef] - 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] - Gu, Z.; Ma, X. A feasible method for the estimation of the interval bounds based on limited strain-life fatigue data. Int. J. Fatigue
**2018**, 116, 172–179. [Google Scholar] [CrossRef] - Sakin, R.; Ay, I. Statistical analysis of bending fatigue life data using Weibull distribution in glass-fiber reinforced polyester composites. Mater. Des.
**2008**, 29, 1170–1181. [Google Scholar] [CrossRef] - Haidyrah, A.S.; Newkirk, J.W.; Castaño, C.H. Weibull statistical analysis of Krouse type bending fatigue of nuclear materials. J. Nucl. Mater.
**2016**, 470, 244–250. [Google Scholar] [CrossRef] - Liu, C.L.; Lu, Z.Z.; Xu, Y.L.; Yue, Z.F. Reliability analysis for low cycle fatigue life of the aeronautical engine turbine disc structure under random environment. Mater. Sci. Eng. A
**2005**, 395, 218–225. [Google Scholar] [CrossRef] - Khelif, R.; Chateauneuf, A.; Chaoui, K. Statistical analysis of HDPE fatigue lifetime. Meccanica
**2008**, 43, 567–576. [Google Scholar] [CrossRef] - Zhu, S.-P.; Huang, H.-Z.; Smith, R.; Ontiveros, V.; He, L.-P.; Modarres, M. Bayesian framework for probabilistic low cycle fatigue life prediction and uncertainty modeling of aircraft turbine disk alloys. Probabilistic Eng. Mech.
**2013**, 34, 114–122. [Google Scholar] [CrossRef] - Zhao, Y.X.; Wang, J.N.; Gao, Q. Statistical evolution of small fatigue crack in 1Cr18Ni9Ti weld metal. Theor. Appl. Fract. Mech.
**1999**, 32, 55–64. [Google Scholar] [CrossRef] - Sun, Q.; Dui, H.-N.; Fan, X.-L. A statistically consistent fatigue damage model based on Miner’s rule. Int. J. Fatigue
**2014**, 69, 16–21. [Google Scholar] [CrossRef] - Daunys, M.; Šniuolis, R. Statistical evaluation of low cycle loading curves parameters for structural materials by mechanical characteristics. Nucl. Eng. Des.
**2006**, 236, 13. [Google Scholar] [CrossRef] - Daunys, M.; Šniuolis, R.; Stulpinaite, A. Evaluation of cyclic instability by mechanical properties for structural materials. Mechanics
**2012**, 18, 280–284. [Google Scholar] [CrossRef][Green Version] - Raslavičius, L.; Bazaras, Ž.; Lukoševičius, V.; Vilkauskas, A.; Česnavičius, R. Statistical investigation of the weld joint efficiencies in the repaired WWER pressure vessel. Int. J. Press. Vessel. Pip.
**2021**, 189, 1–9. [Google Scholar] [CrossRef] - Bazaras, Z. Analysis of probabilistic low cycle fatigue design curves at strain cycling. Indian J. Eng. Mater. Sci.
**2005**, 12, 411–418. Available online: https://scholar.google.lt/scholar?hl=lt&as_sdt=0%2C5&q=Bazaras%2C+Z.+2005.+Analysis+of+probabilistic+low+cycle+fatigue+design+curves+at+strain+cycling%2C+Indian+Journal+of+Engineering+%26+Material+Sciences+2%3A+411-418.&btnG= (accessed on 16 November 2021). - Daunys, M.; Bazaras, Z.; Timofeev, B.T. Low cycle fatigue of materials in nuclear industry. Mechanics
**2008**, 73, 12–17. Available online: https://scholar.google.lt/scholar?hl=lt&as_sdt=0%2C5&q=28.%09Daunys%2C+M.%3B+Bazaras%2C+Z.%3B+Timofeev%2C+B.+T.+Low+cycle+fatigue+of+materials+in+nuclear+industry&btnG= (accessed on 16 November 2021). - GOST 25502-79 Standard.; Strength Analysis and Testing in Machine Building. Methods of Metals Mechanical Testing. Methods of Fatigue Testing. Standardinform: Moscow, Russia, 1993.
- GOST 22015–76 Standard.; Quality of Product. Regulation and Statistical Quality Evaluation of Metal Materials and Products on Speed-torque Charac-teristics. Standardinform: Moscow, Russia, 2010.
- Daunys, M. Cycle Strength and Durability of Structures; Technologija: Kaunas, Lithuania, 2005. (In Lithuanian) [Google Scholar]
- Serensen, S.V.; Shneiderovich, R.M. Deformations and rupture criteria under low-cycles fatigue. Exp. Mech.
**1966**, 6, 587–592. [Google Scholar] [CrossRef] - Daunys, M.; Česnavičius, R. Low cycle stress strain curves and fatigue under tension—Compression and torsion. Mechanics
**2009**, 6, 5–11. Available online: https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Low+cycle+stress+strain+curves+%20and+fatigue+under+tension-compression+and+torsion&btnG= (accessed on 16 November 2021). - Bazaras, Ž.; Lukoševičius, V.; Vilkauskas, A.; Česnavičius, R. Probability Assessment of the Mechanical and Low-Cycle Properties of Structural Steels and Aluminium. Metals
**2021**, 11, 918. [Google Scholar] [CrossRef] - Stepnov, M.N. Statistical Methods for Computation of the Results of Mechanical Experiments, Mechanical Engineering; Mashinostroeniya: Moscow, Russia, 1985; pp. 87–89. (In Russian) [Google Scholar]
- Coffin, L.F., Jr. A study of the effects of cyclic thermal stresses on a ductile metal. Trans. Metall. Soc. ASME
**1954**, 76, 931–950. [Google Scholar] - Manson, S.S. Fatigue: A complex subject—Some simple approximations. Exp. Mech.
**1965**, 7, 193–225. Available online: https://link.springer.com/article/10.1007%2FBF02321056 (accessed on 20 November 2021). [CrossRef]

**Figure 1.**Samples of circular cross section for low-cycle tension-compression fatigue experiments (Units in mm).

**Figure 3.**Hysteresis loop widths dependence of softening materials on the number of loading semi-cycles (1—${\overline{e}}_{0}=9.6;$ 2—${\overline{e}}_{0}=5.2;$ 3—${\overline{e}}_{0}=2.4;$ 4—${\overline{e}}_{0}=1.4;$ 5—${\overline{e}}_{0}=1.2$ ).

**Figure 4.**Histograms of low cyclic load diagram characteristics of steel 15Cr2MoVa (1, 4, 7), steel 45 (2, 5, 8)—loading level ${\overline{\sigma}}_{0}=1.00$ and aluminum alloy (3, 6, 9)—loading level ${\overline{\sigma}}_{0}=1.12$ (

**1**–

**3**—A

_{1};

**4**–

**6**—A

_{1f};

**7**–

**9**—${\overline{S}}_{T})$.

**Figure 5.**Coefficients of variation of cyclic properties (x-axis—normal and Weibull; y-axis—log-normal); (

**a**) $1\u2014{A}_{1}$, 2$\u2014{A}_{1f}$, 3$\u2014{A}_{2}$; (

**b**) 4$\u2014\beta $, 5$\u2014c$.

**Figure 6.**Log-normal distribution curves of the cyclic properties: (

**a**,

**b**)—steel 15Cr2MoVa ($1\u2014{\overline{\sigma}}_{0}=1.25,2\u2014{\overline{\sigma}}_{0}=1.12,3\u2014{\overline{\sigma}}_{0}=1.00$); (

**c**,

**d**)—steel C45 ($1\u2014{\overline{\sigma}}_{0}=1.50,2\u2014{\overline{\sigma}}_{0}=1.12,3\u2014{\overline{\sigma}}_{0}=1.00$ ); (

**e**)—aluminum alloy D16T1 (${\overline{\sigma}}_{0}=1.12)$ (

**f**)—normalized stress of cyclic proportionality limit: 1—steel 15Cr2MoVa, 2—steel C45, 3—aluminum alloy D16T.

**Figure 7.**Dependence of relative value K under the log-normal distribution law on mean square deviation s and coefficient of variation V: (

**a**,

**b**) $1\u2014{A}_{1},2\u2014{A}_{1f},3\u2014{A}_{2},4\u2014{A}_{2f},5\u2014{\overline{S}}_{T},6\u2014\propto $; (

**c**,

**d**) $1\u2014\beta ,2\u2014c$.

**Figure 8.**Comparison of the experimental (dashed lines) and theoretical (straight lines) curves under loading with controlled stress (1–7 analytical probability 1–99%; 8–14 experimental probability 1–99%): (

**a**)—steel 15Cr2MoVa (Equation (11)); (

**b**)—steel C45.

Material | C | Si | Mn | Cr | Ni | Mo | V | S | P | Mg | Cu | Al |
---|---|---|---|---|---|---|---|---|---|---|---|---|

% | ||||||||||||

15Cr2MoVA (GOST 5632-2014) | 0.18 | 0.27 | 0.43 | 2.7 | 0.17 | 0.67 | 0.30 | 0.019 | 0.013 | - | - | - |

C45 (GOST 1050-2013) | 0.46 | 0.28 | 0.63 | 0.18 | 0.22 | - | - | 0.038 | 0.035 | - | - | - |

D16T1 (GOST 4784-97) | - | - | 0.70 | - | - | - | - | - | - | 1.6 | 4.5 | 9.32 |

Material | e_{pr} | σ_{pr} | σ_{0.2} | σ_{u} | S_{k} | ψ |
---|---|---|---|---|---|---|

% | MPa | % | ||||

15Cr2MoVA (GOST 5632-2014) | 0.200 | 280 | 400 | 580 | 1560 | 80 |

C45 (GOST 1050-2013) | 0.260 | 340 | 340 | 800 | 1150 | 39 |

D16T1 (GOST 4784-97) | 0.600 | 290 | 350 | 680 | 780 | 14 |

**Table 3.**Goodness-of-Fit estimation of cyclic properties that the result samples have a log-normal distribution using the Smirnov compatibility criterion ${\omega}^{2}$.

Cyclic Properties | 15Cr2MoVa | C45 | D16T1 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\overline{\mathit{\sigma}}}_{0}=1.00$ | ${\overline{\mathit{\sigma}}}_{0}=1.12$ | ${\overline{\mathit{\sigma}}}_{0}=1.25$ | ${\overline{\mathit{\sigma}}}_{0}=1.00$ | ${\overline{\mathit{\sigma}}}_{0}=1.12$ | ${\overline{\mathit{\sigma}}}_{0}=1.50$ | ${\overline{\mathit{\sigma}}}_{0}=1.15$ | ||||||||

${\mathit{\omega}}^{2}$ | ${\mathit{a}}_{1}$ | ${\mathit{\omega}}^{2}$ | ${\mathit{a}}_{1}$ | ${\mathit{\omega}}^{2}$ | ${\mathit{a}}_{1}$ | ${\mathit{\omega}}^{2}$ | ${\mathit{a}}_{1}$ | ${\mathit{\omega}}^{2}$ | ${\mathit{a}}_{1}$ | ${\mathit{\omega}}^{2}$ | ${\mathit{a}}_{1}$ | ${\mathit{\omega}}^{2}$ | ${\mathit{a}}_{1}$ | |

${A}_{1}$ | 0.5937 | 0.333 | 0.9375 | 0.598 | 0.8452 | 0.540 | 0.8936 | 0.573 | 0.735 | 0.458 | 0.9783 | 0.627 | 0.6753 | 0.398 |

${A}_{1f}$ | 0.6356 | 0.371 | 0.8477 | 0.540 | 0.9658 | 0.621 | 0.7364 | 0.458 | 0.8456 | 0.540 | 0.9967 | 0.637 | 0.7986 | 0.504 |

${A}_{2}$ | 0.6953 | 0.424 | 0.7569 | 0.474 | 0.6973 | 0.424 | 0.7970 | 0.504 | 0.8478 | 0.547 | 0.8951 | 0.573 | - | - |

${A}_{2f}$ | 0.7354 | 0.458 | 0.8757 | 0.567 | 0.7072 | 0.441 | 0.9651 | 0.598 | 0.9996 | 0.643 | 1.0651 | 0.668 | - | - |

$\beta \xb7{10}^{-3}$ | 0.4652 | 0.202 | 0.6759 | 0.407 | 0.5671 | 0.313 | - | - | - | - | - | - | - | - |

$c\xb7{10}^{-4}$ | 1.1456 | 0.643 | 0.9954 | 0.637 | 1.2345 | 0.740 | - | - | - | - | - | - | - | - |

${\overline{S}}_{T}$ | 0.9983 | 0.637 | 0.8751 | 0.560 | 1.0784 | 0.677 | 0.7954 | 0.504 | 0.9672 | 0.615 | 0.8756 | 0.560 | 0.8971 | 0.580 |

$\propto $ | - | - | - | - | - | - | - | - | - | - | - | - | 0.6978 | 0.424 |

Parameters | Probability, % | ||||||
---|---|---|---|---|---|---|---|

1 | 10 | 30 | 50 | 70 | 90 | 99 | |

15Cr2MoVa | |||||||

${A}_{1}$ | 0.23 | 0.87 | 1.32 | 1.60 | 1.94 | 2.40 | 3.04 |

${A}_{2}$ | 0.30 | 0.94 | 1.34 | 1.64 | 2.00 | 2.43 | 3.06 |

${\overline{S}}_{T}$ | 1.15 | 1.20 | 1.25 | 1.28 | 1.30 | 1.35 | 1.40 |

${m}_{0}$ | 0.15 | 0.17 | 0.19 | 0.21 | 0.23 | 0.26 | 0.30 |

$c\xb7{10}^{-4}$ | 1.1 | 3.0 | 4.9 | 7.4 | 11.0 | 21.0 | 50.0 |

C 45 | |||||||

${A}_{1}$ | 0.60 | 0.76 | 087 | 0.95 | 1.04 | 1.15 | 1.29 |

${A}_{2}$ | 0.61 | 0.80 | 0.89 | 0.96 | 1.05 | 1.16 | 1.30 |

${\overline{S}}_{T}$ | 0.92 | 1.02 | 1.07 | 1.11 | 1.15 | 1.21 | 1.29 |

${m}_{0}$ | 0.14 | 0.16 | 0.18 | 0.19 | 0.20 | 0.22 | 0.25 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bazaras, Ž.; Lukoševičius, V.
Statistical Estimation of Resistance to Cyclic Deformation of Structural Steels and Aluminum Alloy. *Metals* **2022**, *12*, 47.
https://doi.org/10.3390/met12010047

**AMA Style**

Bazaras Ž, Lukoševičius V.
Statistical Estimation of Resistance to Cyclic Deformation of Structural Steels and Aluminum Alloy. *Metals*. 2022; 12(1):47.
https://doi.org/10.3390/met12010047

**Chicago/Turabian Style**

Bazaras, Žilvinas, and Vaidas Lukoševičius.
2022. "Statistical Estimation of Resistance to Cyclic Deformation of Structural Steels and Aluminum Alloy" *Metals* 12, no. 1: 47.
https://doi.org/10.3390/met12010047