# Influence of the Elastoplastic Strain on Fatigue Durability Determined with the Use of the Spectral Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{a}) is determined based on the spectral density of stress from the linear elastic analysis, which has to be used. This treatment gave good results for proportional loads and materials corresponding to the Masing rule [16] (the Masing material model is often used in solid mechanics to describe materials with stable elastoplastic properties). Using Masing’s rule in relation to fatigue tests, one can understand the special properties of a material, which are manifested stably under cyclic loads. Such a material, with the same load parameters (force or bending moment), draws the same path on a graph (ε-σ) for each subsequent load cycle. Hysteresis loops snap during a fatigue test under constant amplitude fatigue loading. Therefore, we needed to find corresponding stress amplitudes for the elastic and elastoplastic values, which corresponded to the Neuber hyperbola for the intersection points with the elastoplastic model and cyclic deformation curve. These new amplitudes were then used in the process of new probability density estimations with the correction for the elastoplastic stress values.

_{m}, yield strength R

_{e}, Young modulus E, and Poisson ratio υ, are presented in Table 1 for the average of 6 specimens. A stretching diagram is shown in Figure 4.

_{a}—elastic stress amplitude (superscript e) or elastoplastic stress (superscript e−p), and E—Young’s modulus.

## 3. Results

_{i}moments obtained from the power spectral densities for i = 0,…,4, K

_{1}, K

_{2}, K

_{3}, K

_{4}, and Z are model coefficients described in detail elsewhere [23,24].

^{+}—is the expected number of peaks per unit of time, N

_{f}—is the number of cycles, and p(σ

_{a})—stress amplitude probability distribution.

## 4. Discussion

## 5. Conclusions and Observations

- An algorithm for taking into account the elastoplastic strain in the process of fatigue life assessment with the use of spectral method is presented.
- In order to use the proposed correction, we need to use the Neuber hyperbola to obtain the values corresponding to the elastic stress amplitudes obtained during the probability density calculation and their corresponding elastoplastic stress amplitudes.
- Correction due to elastoplastic stress is successfully applied to determine fatigue life in combination with all four distribution models.
- In the case of no correction, in terms of elastoplastic strain, we obtain overestimated fatigue calculation results.
- Comparison of experimental and computational durability with correction shows that calculations are within a safe scatter band of 3.
- All models used to calculate the probability density function enable obtaining results in the desired scatter band, and the computation results are on the safe side, as they do not overestimate the experimental results.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Banvillet, A.; Łagoda, T.; Macha, E.; Niesłony, A.; Palin-Luc, T.; Vittori, J.-F. Fatigue life under non-Gaussian random loading from various models. Int. J. Fatigue
**2004**, 26, 349–363. [Google Scholar] [CrossRef] [Green Version] - Benasciutti, D.; Tovo, R. Comparison of spectral methods for fatigue analysis of broad-band Gaussian random processes. Probabilistic Eng. Mech.
**2006**, 21, 287–299. [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] - Böhm, M.; Nieslony, A. Strain-based multiaxial fatigue life evaluation using spectral method. In Proceedings of the 3rd International Conference on Material and Component Performance under Variable Amplitude Loading, VAL 2015, Prague, Czech Republic, 23–26 March 2015; Papuga, J., Ruzicka, M., Eds.; Elsevier Science BV: Amsterdam, The Netherlands, 2015; Volume 101, pp. 52–60. [Google Scholar]
- Łagoda, T.; Macha, E.; Niesłony, A. Fatigue life calculation by means of the cycle counting and spectral methods under multiaxial random loading. Fatigue Fract. Eng. Mater. Struct.
**2005**, 28, 409–420. [Google Scholar] [CrossRef] - Nieslony, A.; Macha, E. Spectral Method in Multiaxial Random Fatigue. In Lecture Notes in Applied and Computational Mechanics; Springer: Berlin/Heidelberg, Germany, 2007; ISBN 978-3-540-73822-0. [Google Scholar]
- Benzing, J.T.; Liu, Y.; Zhang, X.; Luecke, W.E.; Ponge, D.; Dutta, A.; Oskay, C.; Raabe, D.; Wittig, J.E. Experimental and numerical study of mechanical properties of multi-phase medium-Mn TWIP-TRIP steel: Influences of strain rate and phase constituents. Acta Mater.
**2019**, 177, 250–265. [Google Scholar] [CrossRef] - Das, A.; Tarafder, S.; Sivaprasad, S.; Chakrabarti, D. Influence of microstructure and strain rate on the strain partitioning behaviour of dual phase steels. Mater. Sci. Eng. A
**2019**, 754, 348–360. [Google Scholar] [CrossRef] - Khosravani, M.R.; Anders, D.; Weinberg, K. Influence of strain rate on fracture behavior of sandwich composite T-joints. Eur. J. Mech.-A/Solids
**2019**, 78, 103821. [Google Scholar] [CrossRef] - Rognon, H.; Da Silva Botelho, T.; Tawfiq, I.; Galtier, A.; Bennebach, M. Modeling of Plasticity in Spectral Methods for Fatigue Damage Estimation of Narrowband Random Vibrations. In Proceedings of the Volume 1: 23rd Biennial Conference on Mechanical Vibration and Noise, Parts A and B, Washington, DC, USA, 28–31 August 2011; ASMEDC: Washington, DC, USA, 2011; pp. 771–779. [Google Scholar]
- Pitoiset, X.; Preumont, A. Spectral methods for multiaxial random fatigue analysis of metallic structures. Int. J. Fatigue
**2000**, 22, 541–550. [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, Ljubljana, Slovenia, 7–10 September 2015; pp. 743–752. [Google Scholar]
- Neuber, H. Kerbspannungslehre: Theorie der Spannungskonzentration Genaue Berechnung der Festigkeit, 4th ed.; Klassiker der Technik; Springer: Berlin/Heidelberg, Germany, 2001; ISBN 978-3-540-67657-7. [Google Scholar]
- Palmieri, M.; Česnik, M.; Slavič, J.; Cianetti, F.; Boltežar, M. Non-Gaussianity and non-stationarity in vibration fatigue. Int. J. Fatigue
**2017**, 97, 9–19. [Google Scholar] [CrossRef] - Braccesi, C.; Cianetti, F.; Lori, G.; Pioli, D. The frequency domain approach in virtual fatigue estimation of non-linear systems: The problem of non-Gaussian states of stress. Int. J. Fatigue
**2009**, 31, 766–775. [Google Scholar] [CrossRef] - Chiang, D.-Y. The generalized Masing models for deteriorating hysteresis and cyclic plasticity. Appl. Math. Model.
**1999**, 23, 847–863. [Google Scholar] [CrossRef] - Knop, M.; Jones, R.; Molent, L.; Wang, C. On the Glinka and Neuber methods for calculating notch tip strains under cyclic load spectra. Int. J. Fatigue
**2000**, 22, 743–755. [Google Scholar] [CrossRef] - Böhm, M.; Kowalski, M.; Nieslony, A. Multiaxial Fatigue Test Stand Concept—Stand and Control Design. In Mechatronics: Ideas for Industrial Applications; Awrejcewicz, J., Szewczyk, R., Trojnacki, M., Kaliczynska, M., Eds.; Springer International Publishing AG: Cham, Switzerland, 2015; Volume 317, pp. 437–445. ISBN 978-3-319-10990-9. [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]
- Gadamchetty, G.; Pandey, A.; Gawture, M. On Practical Implementation of the Ramberg-Osgood Model for FE Simulation. SAE Int. J. Mater. Manf.
**2016**, 9, 200–205. [Google Scholar] [CrossRef] - Kamaya, M. Ramberg–Osgood type stress–train curve estimation using yield and ultimate strengths for failure assessments. Int. J. Press. Vessel. Pip.
**2016**, 137, 1–12. [Google Scholar] [CrossRef] - Dirlik, T. Application of Computers in Fatigue Analysis. Ph.D. Thesis, University of Warwick, Coventry, UK, 1985. [Google Scholar]
- Nieslony, A.; Böhm, M. Mean stress effect correction in frequency-domain methods for fatigue life assessment. In Proceedings of the 3rd International Conference on Material and Component Performance under Variable Amplitude Loading, VAL 2015, Prague, Czech Republic, 23–26 March 2015; Papuga, J., Ruzicka, M., Eds.; Elsevier Science BV: Amsterdam, The Netherlands, 2015; Volume 101, pp. 347–354. [Google Scholar]
- Niesłony, A.; Böhm, M.; Łagoda, T.; Cianetti, F. The use of spectral method for fatigue life assessment for non-gaussian random loads. Acta Mech. Autom.
**2016**, 10, 100–103. [Google Scholar] [CrossRef] [Green Version] - Zhao, W.; Baker, M. On the probability density function of rainflow stress range for stationary Gaussian processes. Int. J. Fatigue
**1992**, 14, 121–135. [Google Scholar] [CrossRef] - Lalanne, C. Mechanical Vibration and Shock Analysis, Fatigue Damage; John Wiley & Sons: Hoboken, NJ, USA, 2013; ISBN 978-1-118-61893-6. [Google Scholar]
- 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] - Łagoda, T.; Macha, E. Energy approach to fatigue under combined cyclic bending with torsion of smooth and notched specimens. Mater. Sci.
**1998**, 34, 630–639. [Google Scholar] [CrossRef] - MSC. Fatigue User’s Guide. Available online: https://kupdf.net/download/msc-fatigue-user-39-s-guide_589dd9c76454a7865ab1e8d6_pdf (accessed on 18 December 2019).

**Figure 1.**Calculation algorithm for determining fatigue life using the spectral method and stress correction due to elastoplastic deformation.

**Figure 4.**The average tensile curve for 0H18N9 steel for 6 specimens (solid line blue color) with a standard deviation of 5% (dashed lines red color).

**Figure 6.**Test stand for tests with uniaxial and biaxial load. The image on the right shows a magnification of the position in which the specimen is clamped inside the stand.

**Figure 7.**Calculations: Maximum stress (

**a**), maximum deformation (

**b**), and maximum plastic deformation for specimen (

**c**).

**Figure 8.**A fragment of the deformation process registered with the strain gauges (

**a**), distribution of deformation amplitudes (

**b**), and its characteristics due to matrices counted by the rainflow method (

**c**).

**Figure 10.**Neuber hyperbola together with the linear elastoplastic model (Hooke’s law) and deformation curve used in the derivation of the model.

**Figure 11.**Obtained shapes of a wide range of probability distribution models after transformation due to elastoplastic deformations.

**Figure 12.**Comparison of experimental and computational durability calculated with the use of four probability density distributions (Dirlik, Benasciutti–Tovo, Lalanne and Zhao–Baker).

**Figure 13.**Comparison of experimental and computational durability calculated with the use of four probability density distributions (Dirlik, Benasciutti–Tovo, Lalanne, and Zhao–Baker), but no elastoplastic correction.

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

750 | 515 | 200 | 0.29 |

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**MDPI and ACS Style**

Böhm, M.; Kowalski, M.; Niesłony, A.
Influence of the Elastoplastic Strain on Fatigue Durability Determined with the Use of the Spectral Method. *Materials* **2020**, *13*, 423.
https://doi.org/10.3390/ma13020423

**AMA Style**

Böhm M, Kowalski M, Niesłony A.
Influence of the Elastoplastic Strain on Fatigue Durability Determined with the Use of the Spectral Method. *Materials*. 2020; 13(2):423.
https://doi.org/10.3390/ma13020423

**Chicago/Turabian Style**

Böhm, Michał, Mateusz Kowalski, and Adam Niesłony.
2020. "Influence of the Elastoplastic Strain on Fatigue Durability Determined with the Use of the Spectral Method" *Materials* 13, no. 2: 423.
https://doi.org/10.3390/ma13020423