# Micromechanical Modelling of the Cyclic Deformation Behavior of Martensitic SAE 4150—A Comparison of Different Kinematic Hardening Models

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## Abstract

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## 1. Introduction

## 2. Material and Experiments

#### 2.1. Microstructure Characterization

#### 2.2. Mechanical Cyclic Properties

## 3. Modelling Methodology and Constitutive Calibration Strategy

#### 3.1. Generation of Representative Volume Elements

#### 3.2. Constitutive Framework

#### 3.2.1. Crystal Plasticity Model

#### 3.2.2. Phenomenological Hardening Models

#### 3.3. Parametrization of the Crystal Plasticity Model

#### Calibration Strategy

- ${\mathsf{\Gamma}}_{el}=[{C}_{11},{C}_{12},{C}_{44}]$ a set of elastic constants of the crystal
- ${\mathsf{\Gamma}}_{flow}=[{\dot{\gamma}}_{0},m,{\tau}_{c,0}^{\alpha}]$ a set related to the flow rule
- ${\mathsf{\Gamma}}_{iso}=[{h}_{0},{\tau}_{s},a]$ a set related to the isotropic hardening
- ${\mathsf{\Gamma}}_{kin}=[{A}_{i},{B}_{i},{M}_{i}]$ a set related to the KH

## 4. Results and Discussion

#### 4.1. Calibration of the Constitutive Models

#### 4.2. Cyclic Deformation Behavior at R${}_{\epsilon}$ = −1—KH Model Comparison

#### 4.3. Cyclic Deformation Behavior at R${}_{\epsilon}$ = 0—KH Model Comparison

## 5. Conclusions

- A comparison of experimental EBSD data and calculated OR indicates that the Nishiyama– Wassermann orientation relationship is well suited for representing the martensitic variants.
- A microstructure model for lath martensite incorporating the Nishiyama–Wassermann orientation relationship is proposed using a multiscale Voronoi tessellation technique.
- A multi-objective calibration procedure for the determination of the cyclic crystal plasticity constitutive parameters based on experimental LCF data is proposed. Thereby, a simplified numerical efficient RVE is used for the combined sensitivity analysis and the subsequent genetic algorithm. The simplified RVE shows macroscopically an almost equivalent stress-strain behavior than the most complex proposed martensitic RVE with deviations smaller than 1.0%.
- For fully reversed loadings (R${}_{\epsilon}$ = −1), the OW model shows an excellent agreement to experimental data, in particular for small applied total strain amplitudes. The agreement of the AF and Chaboche model to the experimental data are adequate, but with increasing errors. Consequently, the OW model is superior to the AF and Chaboche model for all applied total strain amplitudes, except for ${\epsilon}_{a,t}$ = 0.90% where the Chaboche model agrees better to experimental data than the OW model.
- Simulations of the KH model dependent mean stress relaxation behavior for (R${}_{\epsilon}$ = 0) indicate that the AF model is not able to capture mean stress effects. The mean stress relaxation predicted by the Chaboche model shows a non-consistent characteristic due to high mean stresses for large applied strain amplitudes and small mean stresses for small applied strain amplitudes. The OW model shows a good agreement to the experimental stress-strain hystereses at R${}_{\epsilon}$ = 0. Furthermore, the predicted mean stress relaxation characteristic by the OW model agrees well with the experimental characteristic with small deviations.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Experimental analysis of the SAE 4150 microstructure: (

**a**) EBSD image quality map (

**b**) EBSD map in inverse pole figure color code.

**Figure 2.**(001) pole figures of experimental martensitic orientations within a single prior austenite grain, corresponding simulated martensitic orientations and the underlying prior austenite grain orientation: (

**a**) with simulated Kurdjumov–Sachs OR (

**b**) with simulated Nishiyama–Wassermann OR.

**Figure 4.**Experimental low cycle fatigue test results: (

**a**) Evolution of maximum and minimum stresses versus normalized lifetime at R${}_{\epsilon}$ = −1 for different strain amplitudes and (

**b**) cyclic stress-strain behavior at R${}_{\epsilon}$ = −1, in cyclic stable material conditions (normalized lifetime = 0.5).

**Figure 5.**Experimental mean stress evolutions of SAE 4150 during cycling at different constant total strain amplitudes at R${}_{\epsilon}$ = 0.

**Figure 6.**Two-dimensional schematic procedure of the hierarchical modelling approach for lath martensitic microstructures of low carbon-alloyed steels. (

**a**) Generation of prior austenite grains. (

**b**) Partition of each prior austenite grain into packets. (

**c**) Generation of blocks parallel to the corresponding habit plane within individual packets.

**Figure 7.**Three-dimensional synthetic martensitic microstructure: (

**a**) Representative volume element with 125,000 elements (

**b**) two-dimensional representation of a transformed PAG with an initial orientation of (001)[100] (

**c**) (100) pole figure of Figure 7b.

**Figure 9.**(100), (110) and (111) pole figures of the EBSD data from Figure 1b.

**Figure 10.**Flow chart of the calibration procedure for the crystal plasticity parameters using a genetic algorithm.

**Figure 11.**Experimental and simulated stress-strain hysteresis loops of SAE 4150 at cyclic stable material conditions, for three different total strain amplitudes, (

**a**) ${\epsilon}_{a,t}$ = 0.35%, (

**b**) ${\epsilon}_{a,t}$ = 0.60% and (

**c**) ${\epsilon}_{a,t}$ = 0.90%.

**Figure 12.**Influence of the exponent M of the Ohno–Wang model, (

**a**) on the effective cyclic properties at zero mean stress; (

**b**) on the mean stress relaxation behavior at ${\epsilon}_{a,t}$ = 0.50% at R${}_{\epsilon}$ = 0.

**Figure 13.**Plastic strain energy density characteristics for three different kinematic hardening models and experimental data of SAE 4150 at R${}_{\epsilon}$ = −1; (

**a**) with logarithmic y-axis (

**b**) with linear y-axis.

**Figure 14.**Comparison of the mean stress relaxation behavior at R${}_{\epsilon}$ = 0 for three different total strain amplitudes ${\epsilon}_{a,t}$ = 0.30%, ${\epsilon}_{a,t}$ = 0.40% and ${\epsilon}_{a,t}$ = 0.50% between the micromechanical model prediction and the experimental results; (

**a**) for the AF KH model (

**b**) for the CH KH model and (

**c**) for the OW KH model.

**Figure 15.**Plastic strain energy density characteristics at R${}_{\epsilon}$ = 0, for the three different KH models in cycle stable conditions and the corresponding experimental data of SAE 4150 at half lifetime.

**Figure 16.**Effective cyclic properties at R${}_{\epsilon}$ = 0, simulation and experimental results. The applied total strain amplitudes are (

**a**) ${\epsilon}_{a,t}$ = 0.30%, (

**b**) ${\epsilon}_{a,t}$ = 0.40% and (

**c**) ${\epsilon}_{a,t}$ = 0.50%.

Material | C | Si | Mn | P | S | Cr | Mo |
---|---|---|---|---|---|---|---|

SAE 4150 | 0.52 | 0.26 | 0.74 | 0.014 | 0.008 | 1.31 | 0.18 |

Orientation Relationship | Plane Parallel | Direction Parallel |
---|---|---|

Kurdjumov–Sachs | ${\left(111\right)}_{\gamma}$//${\left(110\right)}_{{\alpha}^{\prime}}$ | ${\left[110\right]}_{\gamma}//{\left[111\right]}_{{\alpha}^{\prime}}$ |

Nishiyama–Wassermann | ${\left(111\right)}_{\gamma}$//${\left(110\right)}_{{\alpha}^{\prime}}$ | ${\left[112\right]}_{\gamma}//{\left[110\right]}_{{\alpha}^{\prime}}$ |

${\mathbf{C}}_{11}$ | ${\mathbf{C}}_{12}$ | ${\mathbf{C}}_{44}$ |
---|---|---|

253.1 GPa | 132.4 GPa | 75.8 GPa |

**Table 4.**Optimized parameter sets for different kinematic hardening models for quenched and tempered SAE4150 with 39 HRC.

Armstrong Frederick (AF) | Chaboche (CH) | Ohno–Wang (OW) | |
---|---|---|---|

${\tau}_{c,0}^{\alpha}$ | 187 MPa | 155 MPa | 209 MPa |

${A}_{1}$ | 79,397 MPa | 254,385 MPa | 65,506 MPa |

${B}_{1}$ | 491 | 2636 | 499 |

${A}_{2}$ | - | 90,000 MPa | - |

${B}_{2}$ | - | 8547 | - |

${A}_{3}$ | - | 20,303 MPa | - |

${B}_{3}$ | - | 0 | - |

M | - | - | 8 |

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**MDPI and ACS Style**

Schäfer, B.J.; Song, X.; Sonnweber-Ribic, P.; ul Hassan, H.; Hartmaier, A.
Micromechanical Modelling of the Cyclic Deformation Behavior of Martensitic SAE 4150—A Comparison of Different Kinematic Hardening Models. *Metals* **2019**, *9*, 368.
https://doi.org/10.3390/met9030368

**AMA Style**

Schäfer BJ, Song X, Sonnweber-Ribic P, ul Hassan H, Hartmaier A.
Micromechanical Modelling of the Cyclic Deformation Behavior of Martensitic SAE 4150—A Comparison of Different Kinematic Hardening Models. *Metals*. 2019; 9(3):368.
https://doi.org/10.3390/met9030368

**Chicago/Turabian Style**

Schäfer, Benjamin J., Xiaochen Song, Petra Sonnweber-Ribic, Hamad ul Hassan, and Alexander Hartmaier.
2019. "Micromechanical Modelling of the Cyclic Deformation Behavior of Martensitic SAE 4150—A Comparison of Different Kinematic Hardening Models" *Metals* 9, no. 3: 368.
https://doi.org/10.3390/met9030368