# Application of the S-N Curve Mean Stress Correction Model in Terms of Fatigue Life Estimation for Random Torsional Loading for Selected Aluminum Alloys

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## Abstract

**:**

## 1. Introduction

## 2. Materials

## 3. Methods

_{max}and minimum τ

_{min}value of the stress during the cyclic tests:

_{m}and stress amplitude τ

_{a}:

_{t}is the torsional moment, and W

_{o}is the section modulus for the torsion calculated with the use of the outer diameter D and inner diameter d:

_{af}), as well as any other fatigue strength amplitude. The model works on the basis of the interpolation of the values between two S-N curves. In the case where we compensate for the R values between −1 and 0, we will obtain an interpolated value for every compensated amplitude. In the case of this study, it will be for R = −1 (τ

_{af}) and R = 0 (τ

_{afR}= 0). It can be presented in the iterated form of Equation (7):

_{a}= τ

_{m}= τ

_{afR}= 0, the model takes the form of Equation (10):

_{b}) or torsional moment (M

_{t}). The maximum allowable value of the bending moment is 80 Nm, and the value of the static moment is 60 Nm. The maximum frequency that can be obtained during the bending or torsional test for the stand is 29 Hz. Taking the limitations of the stand into account, a simple static finite element method (FEM) calculation has been performed in order to assess what maximum stress values can be obtained during the experimental tests for the dimensions and mechanical properties of the specimen. The shear stress distribution along the specimen is presented in Figure 3a. The view of the cross section in the middle of the specimen is presented in Figure 3b. The computations have been performed with the use of the Siemens FEMAP v.11 program and the NX NASTRAN solver. The force that was simulating the leverage work (84.5 N) was simulated with the use of an independent node placed at a distance of 200 mm to the specimen position’s center in the holder, at the point of the lever where it connected to the rotating disks, and this node was connected to the surface of the specimen with rigid elements. As one can note, our test stand imposes the torsion with the use of a lever, and due to this a parasitic shear stress remains present, which results from bending caused by the lever. The influence is negligible, but as one can notice this is the cause for such a distribution.

_{af}, the fatigue strength N

_{f}, as well as the transformed stress amplitude resulting from the mean stress effect τ

_{aiT}:

_{obs}is the observation time taken from the rainflow algorithm.

## 4. Results

_{af}corresponding to the value of the fatigue strength N

_{f}have been calculated and are presented for all the tested materials in Table 7. We can note that the slope values of PA4 and PA6 are close to each other, which can be explained by the fact that they have similar mechanical properties, such as the ultimate strength R

_{m}or yield strength R

_{e}.

## 5. Discussion

## 6. Conclusions and Observations

- The new design of the test samples has served its purpose, as it allowed us to obtain a pure shear distribution inside the sample.
- The S-N curve Niesłony−Böhm (N-B) mean stress compensation model can also be applied to torsional loading conditions.
- The literature results for the missing fatigue strength amplitude for R = 0 used with the Niesłony−Böhm model have improved the results substantially in comparison to fatigue strengths obtained with the SWT model.
- The calculation results obtained for the generated narrowband loading signal have allowed us to perform calculations for three new S-N curves in the case of no mean stress correction for the ratio R = −1, and two mean stress effect compensations for R = 0 with different approaches to obtain the fatigue strength for the N−B model.
- It can be noted that the narrowband curves for R = 0 were within the scatter band of the cyclic results for R = −1, within or below the 20% area, whereas the calculation results for the narrowband curves for R = −1 were only in the scatter band of the PA6 alloy.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Wang, Q.; Zhang, W.; Jiang, S. Fatigue life prediction based on crack closure and equivalent initial flaw size. Materials
**2015**, 8, 7145–7160. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhang, J.; Li, W.; Dai, H.; Liu, N.; Lin, J. Study on the elastic–plastic correlation of low-cycle fatigue for variable asymmetric loadings. Materials
**2020**, 13, 2451. [Google Scholar] [CrossRef] [PubMed] - Xing, Z.; Wang, Z.; Wang, H.; Shan, D. Bending fatigue behaviors analysis and fatigue life prediction of 20Cr2Ni4 gear steel with different stress concentrations near non-metallic inclusions. Materials
**2019**, 12, 3443. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhang, J.; Fu, X.; Lin, J.; Liu, Z.; Liu, N.; Wu, B. Study on damage accumulation and life prediction with loads below fatigue limit based on a modified nonlinear model. Materials
**2018**, 11, 2298. [Google Scholar] [CrossRef] [Green Version] - Benasciutti, D.; Tovo, R. Frequenzbasierte analyse zufalliger ermudungsbelastungen: Modelle, hypothesen, praxis. Mater. Werkst.
**2018**, 49, 345–367. [Google Scholar] [CrossRef] - Yu, Z.-Y.; Zhu, S.-P.; Liu, Q.; Liu, Y. Multiaxial fatigue damage parameter and life prediction without any additional material constants. Materials
**2017**, 10, 923. [Google Scholar] [CrossRef] [Green Version] - Li, Y.; Hu, W.; Sun, Y.; Wang, Z.; Mosleh, A. A life prediction model of multilayered PTH based on fatigue mechanism. Materials
**2017**, 10, 382. [Google Scholar] [CrossRef] [Green Version] - Xue, L.; Shang, D.G.; Li, L.J. Online fatigue damage evaluation method based on real-time cycle counting under multiaxial variable amplitude loading. IOP Conf. Ser. Mater. Sci. Eng.
**2020**, 784, 012014. [Google Scholar] [CrossRef] - Zhu, S.-P.; Yue, P.; Yu, Z.-Y.; Wang, Q. A combined high and low cycle fatigue model for life prediction of turbine blades. Materials
**2017**, 10, 698. [Google Scholar] [CrossRef] [Green Version] - Niesłony, A.; Böhm, M. Universal method for applying the mean-stress effect correction in stochastic fatigue-damage accumulation. MPC
**2016**, 5, 352–363. [Google Scholar] [CrossRef] - Benasciutti, D.; Tovo, R. Spectral methods for lifetime prediction under wide-band stationary random processes. Int. J. Fatigue
**2005**, 27, 867–877. [Google Scholar] [CrossRef] - Benasciutti, D. An analytical approach to measure the accuracy of various definitions of the “equivalent von Mises stress” in vibration multiaxial fatigue. In Proceedings of the International Conference on Engineering Vibration (ICoEV 2015), Ljubljana, Slovenia, 7–10 September 2015; pp. 743–752. [Google Scholar]
- Dirlik, T. Application of Computers in Fatigue Analysis. Ph.D. Thesis, University of Warwick, Coventry, UK, 1985. [Google Scholar]
- Rozumek, D.; Faszynka, S. Surface cracks growth in aluminum alloy AW-2017A-T4 under combined loadings. Eng. Fract. Mech.
**2020**, 226, 106896. [Google Scholar] [CrossRef] - Kluger, K. Fatigue life estimation for 2017A-T4 and 6082-T6 aluminium alloys subjected to bending-torsion with mean stress. Int. J. Fatigue
**2015**, 80, 22–29. [Google Scholar] [CrossRef] - Szusta, J.; Seweryn, A. Experimental study of the low-cycle fatigue life under multiaxial loading of aluminum alloy EN AW-2024-T3 at elevated temperatures. Int. J. Fatigue
**2017**, 96, 28–42. [Google Scholar] [CrossRef] - Peč, M.; Zapletal, J.; Šebek, F.; Petruška, J. Low-cycle fatigue, fractography and life assessment of EN AW 2024-T351 under various loadings. Exp. Tech.
**2019**, 43, 41–56. [Google Scholar] [CrossRef] - Gadolina, I.; Zaynetdinov, R. Advantages of the rain-flow method at the post-processing stage in comparison with the spectral approach. IOP Conf. Ser. Mater. Sci. Eng.
**2019**, 481, 012005. [Google Scholar] [CrossRef] - Böhm, M.; Kowalski, M. Fatigue life assessment algorithm modification in terms of taking into account the effect of overloads in the frequency domain. AIP Conf. Proc.
**2018**, 2028, 020003. [Google Scholar] [CrossRef] - Niesłony, A.; Böhm, M. Mean stress effect correction using constant stress ratio S–N curves. Int. J. Fatigue
**2013**, 52, 49–56. [Google Scholar] [CrossRef] - Boller, C.; Seeger, T. Materials Data for Cyclic Loading; Elsevier: Amsterdam, The Netherlands, 1987; ISBN 978-0-444-42873-8. [Google Scholar]
- Goodman, J. Mechanics Applied to Engineering; Longmans, Green & Company: Harlow, UK, 1899. [Google Scholar]
- Gerber, W.Z. Bestimmung der zulässigen Spannungen in Eisen-Konstruktionen (Calculation of the allowable stresses in iron structures). Z. Bayer Arch. Ing. Ver.
**1874**, 6, 101–110. [Google Scholar] - Soderberg, C. Factor of safety and working stress. Trans. ASME
**1939**, 52, 13–28. [Google Scholar] - Morrow, J. Fatigue properties of metals. In Section 3.2, Fatigue Design Handbook; Society of Automotive Engineers: Warrendale, PA, USA, 1968; Volume AE-4. [Google Scholar]
- Zhu, S.-P.; Lei, Q.; Huang, H.-Z.; Yang, Y.-J.; Peng, W. Mean stress effect correction in strain energy-based fatigue life prediction of metals. Int. J. Damage Mech.
**2016**. [Google Scholar] [CrossRef] - Tomčala, J.; Papuga, J.; Horák, D.; Hapla, V.; Pecha, M.; Čermák, M. Steps to increase practical applicability of PragTic software. Adv. Eng. Softw.
**2019**, 129, 57–68. [Google Scholar] [CrossRef] - Gates, N.R.; Fatemi, A. Fatigue Life of 2024-T3 Aluminum under Variable Amplitude Multiaxial Loadings: Experimental Results and Predictions. Procedia Eng.
**2015**, 101, 159–168. [Google Scholar] [CrossRef] [Green Version] - Wu, Z.-R.; Hu, X.-T.; Song, Y.-D. Multiaxial fatigue life prediction for titanium alloy TC4 under proportional and nonproportional loading. Int. J. Fatigue
**2014**, 59, 170–175. [Google Scholar] [CrossRef] - Niezgodziński, M.E. Wzory, Wykresy i Tablice Wytrzymałościowe; Wydawnictwa Naukowo-Techniczne: Warsaw, Poland, 2007; ISBN 978-83-204-3380-7. [Google Scholar]
- Smith, K.; Watson, P.; Topper, T. A stress strain function for the fatigue of metals. J. Mater. ASTM
**1970**, 5, 767–778. [Google Scholar] - Papuga, J.; Fojtík, F. Multiaxial fatigue strength of common structural steel and the response of some estimation methods. Int. J. Fatigue
**2017**, 104, 27–42. [Google Scholar] [CrossRef] - Endo, T. Damage evaluation of metals for random on varying loading-three aspects of rain flow method. Mech. Behav. Mater.
**1974**, 1, 374. [Google Scholar] - ASTM International. ASTM E1049-85 (2011) Practices for Cycle Counting in Fatigue Analysis; E08 Committee; ASTM International: West Conshohocken, PA, USA, 2011. [Google Scholar]
- Palmgren, A. Die lebensdauer von kugellagern. Z. Ver. Dtsch. Ing
**1924**, 14, 339–341. [Google Scholar] - Nieslony, A.; Böhm, M. Determination of fatigue life on the basis of experimental fatigue diagrams under constant amplitude load with mean stress. In Fatigue Failure and Fracture Mechanics; Skibicki, D., Ed.; Trans Tech Publications Ltd: Stafa-Zurich, Switzerland, 2012; Volume 726, pp. 33–38. [Google Scholar]

**Figure 2.**The fatigue test stand MZGS-100 for the bending and torsional tests, where: 1—bed, 2—rotational head with a holder, 3—specimen, 4—holder, 5—lever (effective length = 0.2 m), 6—motor, 7—rotating disk with mounted unbalanced mass, 8—flat springs, 9—spring actuator.

**Figure 3.**(

**a**) Shear stress distribution (MPa) along the specimen for the maximum possible torsional moment (

**b**) and the cross-section view of the sample at the spot with the maximum value.

**Figure 4.**Section of the generated narrowband stress signal with a dominating frequency of 5 Hz generated in order to calculate the lifetime via the rainflow method for R = 0.

**Figure 5.**S-N curve for the experimental results for PA4 for R = −1, with the marked results for other tested stress asymmetry ratios.

**Figure 6.**S-N curve for the experimental results for PA6 for R = −1, with the marked results for other tested stress asymmetry ratios.

**Figure 7.**S-N curve for the experimental results for PA7 for R = −1, with the marked results for other tested stress asymmetry ratios.

**Figure 8.**Calculation results obtained for the generated narrowband signal for the cases of R = −1 and R = 0, together with the cyclic experimental results for R = −1 for PA4.

**Figure 9.**Calculation results obtained for the generated narrowband signal for the cases of R = −1 and R = 0, together with the cyclic experimental results for R = −1 for PA6.

**Figure 10.**Calculation results obtained for the generated narrowband signal for the cases of R = −1 and R = 0, together with the cyclic experimental results for R = −1 for PA7.

Standard | ||||
---|---|---|---|---|

PN | EN | WNR | ISO | DIN |

PA4 | AW-6082-T6 | 3.2315 | AlSi1MgMn | AlMgSi1 |

PA6 | AW-2017A-T4 | 3.1325 | AlCu4MgSi(A) | AlCuMg1 |

PA7 | AW-2024-T3 | 3.1354 | AlCu4Mg1 | AlCuMg2 |

R_{m} MPa | R_{e} MPa | E MPa | υ | |
---|---|---|---|---|

PA4 | 344 ± 3 | 322 ± 2 | 69,000 | 0.33 |

PA6 | 330 ± 2 | 312 ± 2 | 72,000 | 0.33 |

PA7 | 497 | 359 | 73,200 | 0.33 |

Fe | Si | Zn | Ti | Mg | Mn | Cu | Cr | Other | Al | |
---|---|---|---|---|---|---|---|---|---|---|

PA4 | 0.50 | 1.30 | 0.20 | 0.10 | 1.20 | 1 | 0.10 | 0.25 | 0.15 | balanced |

PA6 | 0.70 | 0.80 | 0.25 | 0.25 | 1 | 1 | 4.50 | 0.10 | 0.15 | balanced |

PA7 | 0.5 | 0.5 | 0.25 | 0.15 | 1.2 | 0.3 | 3.8 | 0.1 | 0.15 | balanced |

Specimen No. | τ_{m} | τ_{a} | τ_{max} | N_{f} |
---|---|---|---|---|

PA4-9 | 0 | 88 | 90 | 1,321,873 |

PA4-14 | 0 | 91 | 90 | 1,848,426 |

PA4-1 | 0 | 105 | 93 | 221,164 |

PA4-8 | 0 | 104 | 104 | 457,156 |

PA4-16 | 0 | 105 | 103 | 411,196 |

PA4-2 | 0 | 121 | 119 | 174,696 |

PA4-4 | 0 | 121 | 118 | 96,357 |

PA4-3 | 0 | 137 | 131 | 21,409 |

PA4-5 | 17 | 106 | 123 | 456,183 |

PA4-10 | 19 | 106 | 125 | 230,671 |

PA4-12 | 18 | 105 | 123 | 374,000 |

PA4-6 | 35 | 88 | 122 | 1,700,497 |

PA4-11 | 35 | 86 | 121 | 2,667,863 |

PA4-13 | 37 | 89 | 126 | 2,048,000 |

PA4-25 | 36 | 108 | 144 | 547,886 |

PA4-26 | 35 | 105 | 140 | 464,294 |

PA4-23 | 50 | 94 | 144 | 1,382,063 |

PA4-24 | 49 | 97 | 146 | 1,905,095 |

Specimen No. | τ_{m} | τ_{a} | τ_{max} | N_{f} |
---|---|---|---|---|

PA6-12 | 0 | 70 | 71 | 10,567,970 |

PA6-13 | 0 | 69 | 70 | 11,021,452 |

PA6-9 | 0 | 79 | 81 | 4,714,524 |

PA6-4 | 0 | 87 | 84 | 5,825,588 |

PA6-5 | 0 | 88 | 87 | 2,793,997 |

PA6-6 | 0 | 90 | 88 | 2,700,912 |

PA6-10 | 0 | 103 | 105 | 1,104,340 |

PA6-11 | 0 | 104 | 102 | 582,611 |

PA6-7 | 0 | 113 | 111 | 227,991 |

PA6-8 | 0 | 110 | 108 | 329,501 |

PA6-1 | 0 | 125 | 119 | 65,574 |

PA6-2 | 0 | 123 | 113 | 48,248 |

PA6-3 | 0 | 124 | 114 | 64,380 |

PA6-15 | 18 | 104 | 122 | 441,360 |

PA6-16 | 17 | 107 | 124 | 864,694 |

PA6-17 | 35 | 90 | 125 | 1,710,746 |

PA6-18 | 37 | 91 | 127 | 2,273,590 |

PA6-20 | 32 | 104 | 135 | 381,996 |

PA6-21 | 29 | 104 | 133 | 117,770 |

PA6-22 | 43 | 89 | 132 | 842,576 |

PA6-23 | 46 | 90 | 136 | 981,805 |

Specimen No. | τ_{m} | τ_{a} | τ_{max} | N_{f} |
---|---|---|---|---|

PA7-13 | 0 | 60 | 58 | 11,035,860 |

PA7-14 | 0 | 61 | 62 | 9,047,500 |

PA7-10 | 0 | 71 | 72 | 3,119,563 |

PA7-11 | 0 | 73 | 70 | 4,091,781 |

PA7-8 | 0 | 88 | 91 | 2,514,519 |

PA7-9 | 0 | 90 | 91 | 1,410,229 |

PA7-4 | 0 | 105 | 105 | 993,766 |

PA7-5 | 0 | 107 | 107 | 576,193 |

PA7-7 | 0 | 104 | 103 | 827,959 |

PA7-2 | 0 | 122 | 120 | 581,857 |

PA7-6 | 0 | 126 | 125 | 332,553 |

PA7-1 | 0 | 142 | 138 | 150,606 |

PA7-3 | 0 | 146 | 145 | 113,728 |

PA7-16 | 20 | 125 | 146 | 675,295 |

PA7-17 | 20 | 125 | 146 | 511,227 |

PA7-15 | 35 | 106 | 141 | 1,013,935 |

PA7-18 | 34 | 105 | 139 | 684,876 |

PA7-19 | 34 | 107 | 141 | 614,049 |

PA7-20 | 34 | 120 | 154 | 359,856 |

PA7-21 | 35 | 124 | 159 | 75,166 |

PA7-22 | 58 | 107 | 166 | 459,271 |

PA7-23 | 58 | 109 | 167 | 855,168 |

m | τ_{af} MPa | N_{f} Cycle | |
---|---|---|---|

PA4 | 9.25 | 90 | 1,848,426 |

PA6 | 9.14 | 70 | 11,021,452 |

PA7 | 4.69 | 60 | 11,035,860 |

**Table 8.**Missing fatigue strength values for R = 0 obtained for the SWT model and from literature data.

Material | SWT | Lit. Data |
---|---|---|

τ_{afR} = 0 MPa | τ_{afR} = 0 MPa | |

PA4 | 63.64 | 76.5 |

PA6 | 49.50 | 61.25 |

PA7 | 42.43 | 49.8 |

**Table 9.**Ratio between the calculated and literature fatigue strengths for the presented materials and the R = −1 fatigue strength.

Material | SWT | Lit. Data |
---|---|---|

τ_{afR=0}/τ_{af} | τ_{afR=0}/τ_{af} | |

PA4 | 0.707 | 0.85 |

PA6 | 0.707 | 0.875 |

PA7 | 0.707 | 0.83 |

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**MDPI and ACS Style**

Böhm, M.; Kluger, K.; Pochwała, S.; Kupina, M.
Application of the S-N Curve Mean Stress Correction Model in Terms of Fatigue Life Estimation for Random Torsional Loading for Selected Aluminum Alloys. *Materials* **2020**, *13*, 2985.
https://doi.org/10.3390/ma13132985

**AMA Style**

Böhm M, Kluger K, Pochwała S, Kupina M.
Application of the S-N Curve Mean Stress Correction Model in Terms of Fatigue Life Estimation for Random Torsional Loading for Selected Aluminum Alloys. *Materials*. 2020; 13(13):2985.
https://doi.org/10.3390/ma13132985

**Chicago/Turabian Style**

Böhm, Michał, Krzysztof Kluger, Sławomir Pochwała, and Mariusz Kupina.
2020. "Application of the S-N Curve Mean Stress Correction Model in Terms of Fatigue Life Estimation for Random Torsional Loading for Selected Aluminum Alloys" *Materials* 13, no. 13: 2985.
https://doi.org/10.3390/ma13132985