# Inverse Method to Determine Fatigue Properties of Materials by Combining Cyclic Indentation and Numerical Simulation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material and Experiment

#### 2.1. Materials Specifications

#### 2.2. Indentation and Fatigue Testing

#### 2.3. Numerical Models

#### 2.4. Material Model

_{i}represents the kinematic hardening moduli. The change in the yield surface of the combined hardening evolution for monotonic tension and in the stress space is graphically presented in Figure 4 [27].

## 3. Inverse Parameter Identification

^{−5}) is met or the maximum allowed iterations are reached. The yield stress and Young’s modulus are kept constant at 1060 MPa and 204 GPa, respectively, based on monotonic stress–strain experimental data. It is known that the yield strength cannot be uniquely determined only based on force–displacement curves with sharp indenters. Hence, this material parameter must be assumed as known and can be determined by other methods, e.g., tensile tests or other inverse methods based on indentation (e.g., see [1,2,3,4,5,6,7,8,9,10,11]).

_{sim}= ΔF

_{exp}.

## 4. Results and Discussion

#### 4.1. Method Development

^{−5}.

^{−5}; nevertheless, the value of ΔF from the simulation is lower than the experimental ΔF, which has a direct impact on the uniaxial stress–strain hysteresis prediction. The inclusion of the ΔF into “objective function 2” leads to a better prediction of the uniaxial stress–strain hysteresis, which is shown by the dotted blue line hysteresis in Figure 7.

#### 4.2. Validation

^{−4}; Figure 9a) and 100 N force amplitude (NMSE = 1.6 × 10

^{−4}; Figure 9b).

#### 4.3. Transferability of the Method

#### 4.3.1. Transferability to Higher Force Amplitude (75 N)

^{−5}) between the experimental and the simulated curves for both force–displacement and uniaxial stress–strain hysteresis. The parameters obtained after the simulation are reported in Table 2 and not much different from the parameters obtained for the 50 N force amplitude.

#### 4.3.2. Transferability to Higher Hardness (47 HRC)

#### 4.3.3. Transferability for Other Material (Cu)

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Huber, N.; Tsakmakis, C. Determination of constitutive properties from spherical indentation data using neural networks. Part I: The case of pure kinematic hardening in plasticity laws. J. Mech. Phys. Solids
**1999**, 47, 1569–1588. [Google Scholar] [CrossRef] - Huber, N.; Tsakmakis, C. Determination of constitutive properties from spherical indentation data using neural networks. Part II: Plasticity with nonlinear isotropic and kinematic hardening. J. Mech. Phys. Solids
**1999**, 47, 1589–1607. [Google Scholar] [CrossRef] - Wymysłowski, A.; Dowhań, Ł. Application of nanoindentation technique for investigation of elasto-plastic properties of the selected thin film materials. Microelectron. Reliab.
**2013**, 53, 443–451. [Google Scholar] [CrossRef] - Oliver, W.C.; Pharr, G.M. Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology. J. Mater. Res.
**2004**, 19, 3–20. [Google Scholar] [CrossRef] - Peirce, D.; Asaro, R.J.; Needleman, A. Material rate dependence and localized deformation in crystalline solids. Acta Metall.
**1983**, 31, 1951–1976. [Google Scholar] [CrossRef] - Hyung-Yil, L. Ball Indenter Utilizing Fea Solutions for Property Evaluation. WO2003010515A1, 2002. Available online: https://patents.google.com/patent/WO2003010515A1 (accessed on 10 July 2020).
- Suresch, A.; Alcala, S.; Giannakopoulos, J. Depth Sensing Indentation and Methodology for Mechanical Property Measurements. WO1997039333A2, 1996. Available online: https://patents.google.com/patent/WO1997039333A2 (accessed on 10 July 2020).
- Suresh, T.A.; Dao, S.; Chollacoop, M.; Van, N.; Venkatesh, K.V. Systems and Methods for Estimation and Analysis of Mechanical Property Data. WO2002073162A2, 2002. Available online: https://patents.google.com/patent/WO2002073162A2 (accessed on 10 July 2020).
- Fontanari, V.; Beghini, M.; Bertini, L. Method and Apparatus for Determining Mechanical Features of a Material with Comparison to Reference Database. WO2006013450A2, 2004. Available online: https://patents.google.com/patent/WO2006013450A2/en (accessed on July 10 2020).
- Schmaling, B.; Hartmaier, A. Method for Testing Material, particularly for Hardness Testing, Involves Producing Impression in to Be Tested Material in Experimental Manner with Test Body with Known Geometry and with Known Test Load. DE102011115519A1, 2011. Available online: https://patents.google.com/patent/DE102011115519A1/de (accessed on 10 July 2020).
- Broitman, E. Indentation Hardness Measurements at Macro-, Micro-, and Nanoscale: A Critical Overview. Tribol. Lett.
**2017**, 65, 23. [Google Scholar] [CrossRef] [Green Version] - Strzelecki, P. Analytical Method for Determining Fatigue Properties of Materials and Construction Elements in High Cycle Life; Uniwersytet Technologiczno-Przyrodniczy w Bydgoszczy: Bydgoszcz, Poland, 2014. [Google Scholar]
- Murakami, Y. Effects of small defects and nonmetallic inclusions on the fatigue strength of metals. JMSE Int. J.
**1989**, 32, 167–180. [Google Scholar] [CrossRef] [Green Version] - Bandara, C.S.; Siriwardane, S.C.; Dissanayake, U.I.; Dissanayake, R. Developing a full range S-N curve and estimating cumulative fatigue damage of steel elements. Comput. Mater. Sci.
**2015**, 96, 96–101. [Google Scholar] [CrossRef] - Bandara, C.S.; Siriwardane, S.C.; Dissanayake, U.I.; Dissanayake, R. Full range S-N curves for fatigue life evaluation of steels using hardness measurements. Int. J. Fatigue
**2016**, 82, 325–331. [Google Scholar] [CrossRef] - Strzelecki, P.; Tomaszewski, T. Analytical models of the S-N curve based on the hardness of the material. Procedia Struct. Integr.
**2017**, 5, 832–839. [Google Scholar] [CrossRef] - Lyamkin, V.; Starke, P.; Boller, C. Cyclic indentation as an alternative to classic fatigue evaluation. In Proceedings of the 7th International Symposium on Aircraft Materialsno, Compiegne, France, 24–26 April 2018. [Google Scholar]
- Faisal, N.H.; Prathuru, A.K.; Goel, S.; Ahmed, R.; Droubi, M.G.; Beake, B.D.; Fu, Y.Q. Cyclic Nanoindentation and Nano-Impact Fatigue Mechanisms of Functionally Graded TiN/TiNi Film. Shape Mem. Superelasticity
**2017**, 3, 149–167. [Google Scholar] [CrossRef] [Green Version] - Haghshenas, M.; Klassen, R.J.; Liu, S.F. Depth-sensing cyclic nanoindentation of tantalum. Int. J. Refract. Met. Hard Mater.
**2017**, 66, 144–149. [Google Scholar] [CrossRef] - Prakash, R.V. Evaluation of fatigue damage in materials using indentation testing and infrared thermography. Trans. Indian Inst. Met.
**2010**, 63, 173–179. [Google Scholar] [CrossRef] - Prakash, R.V. Study of Fatigue Properties of Materials through Cyclic Automated Ball Indentation and Cyclic Small Punch Test Methods. Key Eng. Mater.
**2017**, 734, 273–284. [Google Scholar] [CrossRef] - Xu, B.X.; Yue, Z.F.; Chen, X. Numerical investigation of indentation fatigue on polycrystalline copper. J. Mater. Res.
**2009**, 24, 1007–1015. [Google Scholar] [CrossRef] - Schäfer, B.; Song, X.; Sonnweber-Ribic, P.; Hassan, H.U.; Hartmaier, A. Micromechanical Modelling of the Cyclic Deformation Behavior of Martensitic SAE 4150—A Comparison of Different Kinematic Hardening Models. Metals (Basel)
**2019**, 9, 368. [Google Scholar] [CrossRef] [Green Version] - Kramer, H.S.; Starke, P.; Klein, M.; Eifler, D. Cyclic hardness test PHYBALCHT - Short-time procedure to evaluate fatigue properties of metallic materials. Int. J. Fatigue
**2014**, 63, 78–84. [Google Scholar] [CrossRef] - DIN EN ISO 6507-2. Metallic Materials—Vickers Hardness Test—Part 2: Verification and Calibration of Testing Machines; NSAI: Dublin, Ireland, 2005. [Google Scholar]
- Mises, R.V. Mechanik der festen Körper im plastisch- deformablen Zustand. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse
**1913**, 1913, 582–592. [Google Scholar] - Srnec Nova, J.; 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, 12020. [Google Scholar] [CrossRef] - Chaboche, J.L.L. Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int. J. Plast.
**1989**, 5, 247–302. [Google Scholar] [CrossRef] - Lemaitre, J.; Chaboche, J.-L. Mechanics of Solid Materials; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- Frederick, C.O.; Armstrong, P.J. A mathematical representation of the multiaxial Bauschinger effect. Mater. High Temp.
**2007**, 24, 1–26. [Google Scholar] [CrossRef] - Sajjad, H.M.; Hanke, S.; Güler, S.; ul Hassan, H.; Fischer, A.; Hartmaier, A. Modelling cyclic behaviour of martensitic steel with J2 plasticity and crystal plasticity. Materials
**2019**, 12, 1767. [Google Scholar] [CrossRef] [PubMed] [Green Version] - About LS-OPT—DYNAmore GmbH. Available online: https://www.dynamore.de/de/produkte/opt/ls-opt (accessed on 26 April 2020).
- Chaparro, B.M.; Thuillier, S.; Menezes, L.F.; Manach, P.Y.; Fernandes, J.V. Material parameters identification: Gradient-based, genetic and hybrid optimization algorithms. Comput. Mater. Sci.
**2008**, 44, 339–346. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Experimental results for Cu and 50CrMo4: (

**a**) indentation cycle at 50 N for 50CrMo4; (

**b**) stress–strain from fatigue experiments of 50CrMo4; (

**c**) indentation cycle at 10 N for Cu (

**d**) Stress–strain from fatigue experiments of Cu.

**Figure 2.**Force–displacement curve from the cyclic indentation curve. Components of the cycle are shown in different colors for clarity.

**Figure 3.**Details of the numerical model used. (

**a**) Specimen is held fixed from the bottom and the indenter is placed in the center of the specimen that moves in and out during loading and unloading, respectively. (

**b**) The fine meshing is performed at the center of the specimen.

**Figure 4.**Graphical depiction of the combined hardening growth in the (

**a**) stress space shown by the yield surface and (

**b**) under monotonic tension presented as a stress–strain diagram, redrawn from [27] under the CC-BY license.

**Figure 5.**Optimization loop used to identify the material parameters by fitting the experimental curve with the simulation curve.

**Figure 6.**(

**a**) Complete cycle of the force–displacement curve from indentation, with normalized mean square error (NMSE) = 2.0 × 10

^{−5}. (

**b**) Predicted uniaxial stress–strain hysteresis with a plastic work error of 2.5%.

**Figure 7.**Effect of ΔF on the uniaxial stress–strain hysteresis prediction. The solid blue force-displacement (FD) loop displays the fitting of the FD loop to the blue experimental FD loop by using objective function 1, while the dotted solid stress–strain hysteresis is the prediction of stress–strain hysteresis. Similarly, the dotted blue FD loop shows the fitting of the FD loop by using objective function 2, while the dotted blue stress–strain hysteresis represents the prediction of the stress–strain hysteresis.

**Figure 9.**Validation of the method at higher force amplitudes: (

**a**) predicted force–displacement at 75 N; (

**b**) predicted force–displacement at 100 N.

**Figure 10.**(

**a**) Predicted uniaxial stress–strain hysteresis for the 10th cycle. (

**b**) Stress amplitude over the number of cycles for the first 10 cycles.

**Figure 11.**Transferability of method: (

**a**) indentation of 38 HRC at 75 N; (

**b**) prediction of stress–strain hysteresis of 38 HRC.

**Figure 12.**Transferability of method: (

**a**) simulated indentation force–displacement at 50 N for 47 HRC; (

**b**) prediction of uniaxial stress–strain hysteresis of 47 HRC.

**Figure 13.**Transferability of method: (

**a**) simulated indentation force–displacement of Cu at 10 N; (

**b**) prediction of uniaxial stress–strain hysteresis.

**Table 1.**Identified material parameters for 50CrMo4 (38 HRC) after fitting of force–displacement at 50 N.

Symbol | Value |
---|---|

C_{1} (MPa) | 262,197 |

γ_{1} | 373 |

C_{2} (MPa) | 4714 |

γ_{2} | 0.25 |

Q (MPa) | −575 |

b | 262 |

**Table 2.**Identified material parameters for 50CrMo4 (38 HRC) after fitting of force–displacement at 75 N force amplitude.

Symbol | Value |
---|---|

C_{1} (MPa) | 257,503 |

γ_{1} | 354 |

C_{2} (MPa) | 3663 |

γ_{2} | 0.2837 |

Q (MPa) | −611 |

b | 163 |

**Table 3.**Identified material parameters for 50CrMo4 (47 HRC) of force–displacement at 50 N force amplitude.

Symbol | Value |
---|---|

C_{1} (MPa) | 337,885 |

γ_{1} | 374 |

C_{2} (MPa) | 6681 |

γ_{2} | 2.3 |

Q (MPa) | −724 |

b | 273 |

**Table 4.**Identified material parameters of the force–displacement loop for Cu at 10 N force amplitude.

Symbol. | Value |
---|---|

C_{1} (MPa) | 154,790 |

γ_{1} | 2,257 |

C_{2} (MPa) | 11,586 |

γ_{2} | 82 |

Q (MPa) | −12 |

b | 47 |

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**MDPI and ACS Style**

Sajjad, H.M.; ul Hassan, H.; Kuntz, M.; Schäfer, B.J.; Sonnweber-Ribic, P.; Hartmaier, A.
Inverse Method to Determine Fatigue Properties of Materials by Combining Cyclic Indentation and Numerical Simulation. *Materials* **2020**, *13*, 3126.
https://doi.org/10.3390/ma13143126

**AMA Style**

Sajjad HM, ul Hassan H, Kuntz M, Schäfer BJ, Sonnweber-Ribic P, Hartmaier A.
Inverse Method to Determine Fatigue Properties of Materials by Combining Cyclic Indentation and Numerical Simulation. *Materials*. 2020; 13(14):3126.
https://doi.org/10.3390/ma13143126

**Chicago/Turabian Style**

Sajjad, Hafiz Muhammad, Hamad ul Hassan, Matthias Kuntz, Benjamin J. Schäfer, Petra Sonnweber-Ribic, and Alexander Hartmaier.
2020. "Inverse Method to Determine Fatigue Properties of Materials by Combining Cyclic Indentation and Numerical Simulation" *Materials* 13, no. 14: 3126.
https://doi.org/10.3390/ma13143126