# Cyclic Plasticity and Low Cycle Fatigue of an AISI 316L Stainless Steel: Experimental Evaluation of Material Parameters for Durability Design

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Plasticity Models: Theoretical Background

## 3. Experimental Tests

#### 3.1. Material and Testing Setup

^{−3}s

^{−1}. The experiments were stopped before the complete separation of the specimen when the strain exceeded the safety limit imposed to the tensile machine.

#### 3.2. Brief Analysis of the Experimental Material Behaviour

## 4. Plasticity Models: Identification of Material Parameters

#### 4.1. Young’s Modulus and Yield Stress

#### 4.2. Kinematic Hardening Model

#### 4.3. Isotropic Hardening Model

#### 4.4. Model vs. Experiment Comparison

## 5. Low-Cycle Fatigue Curves

#### 5.1. Approximate Strain–Life Curves from Monotonic Tensile Properties

#### 5.2. Statistical Methods and Strain–Life Design Curves

#### 5.2.1. Deterministic Method (“2 sigma” or “3 sigma”)

#### 5.2.2. One-Side Tolerance Interval Method

#### 5.2.3. Prediction Interval Method

#### 5.2.4. Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Polák, J.; Obrtlík, K.; Hájek, M. Cyclic Plasticity in type 316L austenitic stainless steel. Fatigue Fract. Eng. Mater. Struct.
**1994**, 17, 773–782. [Google Scholar] [CrossRef] - Mayama, T.; Sasaki, K.; Kuroda, M. Quantitative evaluations for strain amplitude dependent organization of dislocation structures due to cyclic plasticity in austenitic stainless steel 316L. Acta Mater.
**2008**, 56, 2735–2743. [Google Scholar] [CrossRef] - Pham, M.S.; Solenthaler, C.; Janssens, K.G.F.; Holdsworth, S.R. Dislocation structure evolution and its effects on cyclic deformation response of AISI 316L stainless steel. Mater. Sci. Eng. A
**2011**, 528, 3261–3269. [Google Scholar] [CrossRef] - Pham, M.S.; Holdsworth, S.R.; Janssens, K.G.F.; Mazza, E. Cyclic deformation response of AISI 316L at room temperature: Mechanical behaviour, microstructural evolution, physically-based evolutionary constitutive modelling. Int. J. Plast.
**2013**, 47, 143–164. [Google Scholar] [CrossRef] - Zhou, J.; Sun, Z.; Kanouté, P.; Retraint, D. Experimental analysis and constitutive modelling of cyclic behaviour of 316L steels including hardening/softening and strain range memory effect in LCF regime. Int. J. Plast.
**2018**, 107, 54–78. [Google Scholar] [CrossRef] - Xie, X.-F.; Jiang, W.; Chen, J.; Zhang, X.; Tu, S.-T. Cyclic hardening/softening behavior of 316L stainless steel at elevated temperature including strain-rate and strain-range dependence: Experimental and damage-coupled constitutive modeling. Int. J. Plast.
**2019**, 114, 196–214. [Google Scholar] [CrossRef] - Baudry, G.; Pineau, A. Influence of strain-induced martensitic transformation on the low-cycle fatigue behavior of a stainless steel. Mater. Sci. Eng.
**1977**, 28, 229–242. [Google Scholar] [CrossRef] - Pegues, J.W.; Shao, S.; Shamsaei, N.; Schneider, J.A.; Moser, R.D. Cyclic strain rate effect on martensitic transformation and fatigue behaviour of an austenitic stainless steel. Fatigue Fract. Eng. Mater. Struct.
**2017**, 40, 2080–2091. [Google Scholar] [CrossRef] - Alain, R.; Violan, P.; Mendez, J. Low cycle fatigue behavior in vacuum of a 316L type austenitic stainless steel between 20 and 600 °C part I: Fatigue resistance and cyclic behavior. Mater. Sci. Eng. A
**1997**, 229, 87–94. [Google Scholar] [CrossRef] - Gerland, M.; Alain, R.; Ait Saadi, B.; Mendez, J. Low cycle fatigue behaviour in vacuum of a 316L-type austenitic stainless steel between 20 and 600 °C-Part II: Dislocation structure evolution and correlation with cyclic behaviour. Mater. Sci. Eng. A
**1997**, 229, 68–86. [Google Scholar] [CrossRef] - Runciman, A.; Xu, D.; Pelton, A.R.; Ritchie, R.O. An equivalent strain/Coffin–Manson approach to multiaxial fatigue and life prediction in superelastic Nitinol medical devices. Biomaterials
**2011**, 32, 4987–4993. [Google Scholar] [CrossRef] - Livieri, P.; Salvati, E.; Tovo, R. A non-linear model for the fatigue assessment of notched components under fatigue loadings. Int. J. Fatigue
**2016**, 82, 624–633. [Google Scholar] [CrossRef] - Srnec Novak, J.; De Bona, F.; Benasciutti, D. Benchmarks for Accelerated Cyclic Plasticity Models with Finite Elements. Metals
**2020**, 10, 781. [Google Scholar] [CrossRef] - Lemaitre, J.; Chaboche, J.-L. Mechanics of Solid Materials; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- Chaboche, J.L. Time-independent constitutive theories for cyclic plasticity. Int. J. Plast.
**1986**, 2, 149–188. [Google Scholar] [CrossRef] - Chaboche, J.L. A review of some plasticity and viscoplasticity constitutive theories. Int. J. Plast.
**2008**, 24, 1642–1693. [Google Scholar] [CrossRef] - Voce, E. The relationship between stress and strain for homogeneous deformations. J. Inst. Met.
**1948**, 74, 537–562. [Google Scholar] - Chaboche, J.L.; Van, K.D.; Cordier, G. Modelization of the Strain Memory Effect on the Cyclic Hardening Of 316 Stainless Steel. In Proceedings of the International Association for Structural Mechanics in Reactor Technology, Berlin, Germany, 9–21 August 1979; p. 12. [Google Scholar]
- ASTM E606/E606M-12, Standard Test. Method for Strain-Controlled Fatigue Testing; ASTM International: West Conshohocken, PA, USA, 2012; Available online: www.astm.org (accessed on 11 September 2020).
- Jiang, Y.; Zhang, J. Benchmark experiments and characteristic cyclic plasticity deformation. Int. J. Plast.
**2008**, 24, 1481–1515. [Google Scholar] [CrossRef] - Benasciutti, D.; Srnec Novak, J.; Moro, L.; De Bona, F.; Stanojević, A. Experimental characterisation of a CuAg alloy for thermo-mechanical applications. Part 1: Identifying parameters of non-linear plasticity models. Fatigue Fract. Eng. Mater. Struct.
**2018**, 41, 1364–1377. [Google Scholar] [CrossRef] - Bari, S.; Hassan, T. Anatomy of coupled constitutive models for ratcheting simulation. Int. J. Plast.
**2000**, 16, 381–409. [Google Scholar] [CrossRef] - ASTM E739-10(2015), Standard Practice for Statistical Analysis of Linear or Linearized Stress-Life (S-N) and Strain-Life (ε-N.) Fatigue Data; ASTM International: West Conshohocken, PA, USA, 2015; Available online: www.astm.org (accessed on 11 September 2020).
- 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. [Google Scholar] [CrossRef][Green Version] - 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] - Manson, S.S. Fatigue: A complex subject-Some simple approximations-Both ends of the fatigue spectrum are covered in this lecture. On the one hand, the present state of understanding of the mechanism is reviewed and the complexity of the process observed. On the other hand, some approximations useful in design are outlined and their application illustrated. Exp. Mech.
**1965**, 5, 193–226. [Google Scholar] [CrossRef] - Manson, S.S.; Halford, G.R. Fatigue and Durability of Structural Materials; ASM International: Materials Park, OH, USA, 2006. [Google Scholar]
- Novak, J.S.; Benasciutti, D.; De Bona, F.; Stanojević, A.; De Luca, A.; Raffaglio, Y. Estimation of material parameters in nonlinear hardening plasticity models and strain life curves for CuAg alloy. IOP Conf. Ser.: Mater. Sci. Eng.
**2016**, 119, 012020. [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] - Montgomery, D.C.; Runger, G.C. Applied Statistics and Probability for Engineers. Eur. J. Eng. Educ.
**1994**, 19, 383. [Google Scholar] [CrossRef] - Ralph, I.S.; Fatemi, A.; Stephens, R.R.; Fuchs, H.O. Metal. Fatigue in Engineering, 2nd ed.; Wiley: New York, NY, USA, 2001. [Google Scholar]
- Lee, Y.-L.; Hathaway, R.; Pan, J.; Barkey, M.E. Fatigue Testing and Analysis: Theory and Practice; Butterworth-Heinemann: Oxford, UK, 2004. [Google Scholar]
- Shen, C.L.; Wirsching, P.H.; Cashman, G.T. Design curve to characterize fatigue strength. J. Eng. Mater. Technol. Trans. ASME
**1996**, 118, 535–541. [Google Scholar] [CrossRef] - Owen, D.B. Survey of Properties and Applications of the Noncentral t-Distribution. Technometrics
**1968**, 10, 445–478. [Google Scholar] [CrossRef] - Wirsching, P.H.; Hsieh, S. Linear model in probabilistic fatigue design. ASCE J. Eng. Mech. Div.
**1980**, 106, 1265–1278. [Google Scholar] [CrossRef] - Simo, J.C.; Hughes, T.J.R. Computational Inelasticity; Springer: New York, NY, USA, 1998. [Google Scholar]

**Figure 1.**Experimental behaviour observed: (

**a**) example of three stress-strain cycles in the test with 0.7% strain amplitude; (

**b**) cyclic stress response in all tests carried out with different strain amplitudes.

**Figure 2.**Masing behaviour analysis results: (

**a**) stress–plastic strain hysteresis cycles rigidly translated to the origin (0,0); (

**b**) comparison between the monotonic curve and the cyclic curve passing through the experimental points $\left({\epsilon}_{a},{\sigma}_{a}\right)$.

**Figure 3.**Curve fitting to find kinematic hardening parameters applied to the tensile branch of the 200th cycle of the test at 0.7% strain amplitude.

**Figure 4.**Kinematic hardening parameter ($X$) evaluation: (

**a**) from cyclic curve; (

**b**) from monotonic curve.

**Figure 5.**Example of curve fitting of Equation (9) to find isotropic hardening parameters for the test with 0.5% strain amplitude.

**Figure 6.**Curve fitting for each isotropic model parameter using a 2nd order polynomial function of the strain amplitude: (

**a**) saturated stress ${R}_{\infty ,i}$; (

**b**) speed of stabilisation ${b}_{i}$.

**Figure 7.**Comparison between experimental data and simulation with kinematic hardening parameters calibrated on monotonic curve and isotropic hardening parameters different for each test: (

**a**) stress–strain cycles for test with 0.5% strain amplitude; (

**b**) cyclic stress response.

**Figure 8.**(

**a**) Manson–Coffin strain–life curve from regression analysis, with contributions of elastic, plastic and total strain amplitude; (

**b**) comparison between the Manson–Coffin curve and the Universal Slopes Equation model.

**Figure 9.**(

**a**) Example of “hyperbolic” design curve obtained with the prediction interval method for the elastic strain amplitude; (

**b**) comparison between Manson–Coffin and design curves from different methods.

%C | %Si | %Mn | %P | %S | %N | %Cr | %Mo | %Ni | %Cu | %Co |
---|---|---|---|---|---|---|---|---|---|---|

0.019 | 0.37 | 1.75 | 0.024 | 0.026 | 0.079 | 16.60 | 2.07 | 10.16 | 0.47 | 0.13 |

**Table 2.**Estimated material parameters used for the comparison between experimental data and simulation.

$\mathbf{Strain}\text{}\mathbf{Amplitude},\text{}{\mathit{\epsilon}}_{\mathit{a}}$ | Isotropic Model | Kinematic Model | |||
---|---|---|---|---|---|

${R}_{\infty ,1}$ (MPa) | ${R}_{\infty ,2}$ (MPa) | ${b}_{1}$ | ${b}_{2}$ | ||

0.3% | 13.7 | −58.9 | 90.00 | 0.8841 | ${C}_{1}$ = 189500 MPa ${\gamma}_{1}$ = 2950 ${C}_{2}$ = 33500 MPa ${\gamma}_{2}$ = 350 |

0.4% | 14.8 | −51.8 | 46.26 | 0.7596 | |

0.5% | 25.8 | −49.8 | 29.18 | 0.7792 | |

0.6% | 41.0 | −48.0 | 16.19 | 0.9851 | |

0.7% | 48.6 | −42 | 11.37 | 1.280 |

**Table 3.**Coefficients of the 2nd order polynomials adopted to link the isotropic model parameters for different strain amplitudes.

Parameter | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ |
---|---|---|---|

${R}_{\infty ,1}$ (MPa) | 0.5143106 | 408.5609 | 957143.9 |

${b}_{1}$ | 266.8643 | −77254.43 | 5852143 |

${R}_{\infty ,2}$ (MPa) | −63.42285 | 2054.282 | 128571.8 |

${b}_{2}$ | 1.710213 | −484.2335 | 60392.39 |

**Table 4.**Mean absolute percentage error (MAPE) of the stress peaks for each strain amplitude used for calibration and absolute percentage error (APE) for the stress peak at the end of the softening stage.

$\mathbf{Strain}\text{}\mathbf{Amplitude},\text{}{\mathit{\epsilon}}_{\mathit{a}}$ | Error, MAPE | Error, APE (Last Stress Peak) |
---|---|---|

0.3% | 1.81% | 2.19% |

0.4% | 0.551% | 1.34% |

0.5% | 0.298% | 0.569% |

0.6% | 0.233% | 0.126% |

0.7% | 0.359% | 0.0282% |

**Table 5.**Estimated parameters of “median” and design strain–life curves; ${\epsilon}_{a,d}$ refers to ${\left(2{N}_{f}\right)}_{d}=2\times {10}^{5}$.

Method | $\mathit{K}$ | ${\left(\frac{{\widehat{\mathit{\sigma}}}_{\mathit{f}}^{\prime}}{\mathit{E}}\right)}_{\mathit{d}}$ | ${\widehat{{\mathit{b}}^{\prime}}}_{\mathit{d}}$ | ${\left({\widehat{\mathit{\epsilon}}}_{\mathit{f}}^{\prime}\right)}_{\mathit{d}}$ | ${\widehat{{\mathit{c}}^{\prime}}}_{\mathit{d}}$ | ${\mathit{\epsilon}}_{\mathit{a},\mathit{d}}(\%)$ | $\mathit{e}\%$ |
---|---|---|---|---|---|---|---|

$\mathrm{Regression}\text{}(\alpha =50\%)$ | - | 0.01034 | −0.1748 | 0.05799 | −0.2842 | 0.3031 | - |

$\mathrm{USE}\text{}(\alpha =50\%)$ | - | 0.00636 | −0.12 | 0.90604 | −0.6 | 0.2068 | 31.8% |

$\mathrm{Deterministic}\text{}(\alpha =50\%)$ | 1.645 | 0.00890 | −0.1748 | 0.05138 | −0.2842 | 0.2654 | 12.4% |

$\mathrm{EPI}\text{}(\alpha =5\%$$,\text{}n=8)$ | 2.0187 | 0.00860 | −0.1748 | 0.04999 | −0.2842 | 0.2575 | 15.1% |

1D tolerance interval $(\alpha =5\%$$,\text{}\beta =90\%$$,\text{}n=8)$ | 2.755 | 0.00804 | −0.1748 | 0.04735 | −0.2842 | 0.2427 | 19.9% |

1D tolerance interval Owen $(\alpha =5\%$$,\text{}\beta =90\%$$,\text{}n=8)$ | 2.9864 | 0.00787 | −0.1748 | 0.04655 | −0.2842 | 0.2382 | 21.4% |

${\widehat{S}}_{el}$, ${\widehat{S}}_{pl}$ std. deviation from regression analysis of experimental data |

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**

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.
https://doi.org/10.3390/ma14133588

**AMA Style**

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(13):3588.
https://doi.org/10.3390/ma14133588

**Chicago/Turabian Style**

Pelegatti, Marco, Alex Lanzutti, Enrico Salvati, Jelena Srnec Novak, Francesco De Bona, and Denis Benasciutti.
2021. "Cyclic Plasticity and Low Cycle Fatigue of an AISI 316L Stainless Steel: Experimental Evaluation of Material Parameters for Durability Design" *Materials* 14, no. 13: 3588.
https://doi.org/10.3390/ma14133588