# Micromechanical Modeling of Fatigue Crack Nucleation around Non-Metallic Inclusions in Martensitic High-Strength Steels

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## Abstract

**:**

## 1. Introduction

## 2. Material and Experimental Investigations

#### 2.1. Martensitic Microstructure

#### 2.2. Failure-Relevant Non-Metallic Inclusions

## 3. Modeling Methodology and Constitutive Model

#### 3.1. Modeling Statistical Volume Elements

#### 3.1.1. Modeling Martensitic Statistical Volume Elements without Defects

#### 3.1.2. Modeling Martensitic Statistical Volume Elements with Defects

#### 3.2. Local Crystal Plasticity Model

#### 3.2.1. Constitutive Equations

#### 3.2.2. Calibration of the Crystal Plasticity Model

#### 3.3. Simulation Methodology for Residual Stress Development

#### 3.4. Fatigue Indicator Parameters and Fatigue Crack Nucleation Model

#### 3.5. Strategy of the Numerical Design of Experiment

## 4. Results and Discussion

#### 4.1. Influence of Microstructural Variability

#### 4.2. Effect of Residual Stresses

#### 4.3. Influence of Interface Properties on Fatigue Response

#### 4.4. Influence of Defect Size

#### 4.5. Influence of Aspect Ratio of Voids

#### 4.6. Influence of Manganese Sulfide Aspect Ratio and Orientation

#### 4.7. Fatigue Crack Initiation Lifetime Results

## 5. Conclusions

- Residual stresses around oxidic inclusions
- Microstructural variability around defects
- Varying interface properties between non-metallic inclusions and the surrounding matrix
- Defect size
- Defect shape
- Alignment of manganese sulfides to loading axis

- The consideration of the residual stresses around oxidic inclusions due to previous heat treatments result in an increased fatigue crack nucleation potential, in particular for fatigue loading levels close to the elastic regime.
- The local fatigue crack nucleation responses around defects and inclusions depend strongly on the defect size, defect shape, martensitic block size and the crystallographic properties at the hot spot. Therefore, multiple martensitic microstructural realizations were considered in the numerical study to create a certain kind of statistical basis.
- In the present study, voids with increasing aspect ratios represent the most severe defects under fully reversed strain-controlled loading conditions. In particular, for loading levels within the elastic regime, the different inclusion–matrix interface conditions influence the fatigue crack nucleation lives substantially.
- The micromechanical investigations of the manganese sulfide fatigue crack nucleation behavior exhibit an influence of the aspect ratio (size of manganese sulfide) as well as a significant influence of the alignment of the manganese sulfide to the loading axis. These results correlate qualitatively well with the corresponding experimentally observed fracture behavior.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Martensitic microstructure and corresponding non-metallic inclusions: (

**a**) Schematic visualization of the hierarchical martensitic microstructure, following Kitahara et al. [50]. (

**b**) Inverse pole figure color map with reconstructed prior austenite grain boundaries indicated by white solid lines. (

**c**) Spherical aluminum oxide. (

**d**) Elongated manganese sulfides. The colors in (

**b**) correspond to orientations perpendicular to the observed plane.

**Figure 2.**Statistical evaluation of failure-relevant non-metallic inclusions in SAE 4150: (

**a**) Experimental inclusion data and generalized extreme value distributions for oxidic and sulfidic inclusions with respect to the defect length scale λ. (

**b**) Experimental inclusion data and generalized extreme value distributions for oxidic and sulfidic inclusions with respect to aspect ratio.

**Figure 3.**Finite element model and applied boundary conditions of the reference configuration without defect. Prior austenite grain boundaries are indicated by black solid lines.

**Figure 4.**Schematic visualization of a statistical volume element containing a defect with diameter D exhibiting different possible inclusion–matrix interface configurations. In addition, the two different mesh regions within the SVE are indicated correspondingly.

**Figure 5.**Schematic visualization of the modeling methodology of the manganese sulfides within the martensitic microstructure: (

**a**) Schematic visualization of the martensitic microstructure with the manganese sulfide (Manganese sulfide shown as orange inclusion). (

**b**) Detail of the spatial distribution of tie constraint and contact modeling schemes and (

**c**) detail of the spatial position of the manganese sulfide with respect to loading axis.

**Figure 6.**Schematic representation of the inclusion and defect variants of the micromechanical design of experiment (DOE). Five-level hierarchical structure of the DOE: (1) Inclusion types and interfaces. (2) Inclusion and defect sizes represented by diameters (D) and aspect ratios (AR). (3) Defect geometries expressed by aspect ratios. (4) Microstructural variations ranging from M1–M3 and finally (5) the different applied total strain amplitudes.

**Figure 7.**Effect of local crystallographic variations around voids (AR = 1.0) on fatigue crack initiation behavior for ε

_{a,t}= 0.50%. Radial path plots through the individual hot spot of each SVE. (

**a**) Path plot for void D = 18 μm. (

**b**) Path plot for void D = 36 μm and (

**c**) Path plot for void D = 110 μm. One of the fatigue crack nucleation hot spots of the investigated voids is exemplarily depicted in Figure 10.

**Figure 8.**Residual stress distribution around a perfectly bonded Al

_{2}O

_{3}-inclusions with an aspect ratio of AR = 1. (

**a**) Martensitic microstructure with an embedded inclusion. (

**b**) Radial residual stress distribution. (

**c**) Circumferential residual stress distribution.

**Figure 9.**Residual stresses around oxidic inclusions. (

**a**) Comparison of predicted radial (σ

_{rad}) and circumferential (σ

_{circ}) residual stresses around an inclusion by micromechanical modeling and isotropic material modeling. (

**b**) Effect of superimposed residual stresses on fatigue crack nucleation behavior at different loading levels. The location of one fatigue crack nucleation hot spot of the investigated perfectly bonded inclusions is exemplarily depicted in Figure 10.

**Figure 10.**Contour plots of the von Mises stress in MPa and the dimensionless Fatemi–Socie parameter around the defects with D = 110 μm are shown at the end of fifths fatigue loading cycles in the first and second column, respectively. The bonded inclusion, partly debonded inclusion, cracked inclusion and the void are shown in the first, second, third and fourth row, respectively. The individual fatigue crack nucleation hotspots are indicated with a white arrow in the Fatemi–Socie plot. The contour plot values correspond to the local integration points in the center of each finite element.

**Figure 11.**Comparison of the non-local Fatemi–Socie parameter distribution as a function of the far-field applied total strain amplitude for the different inclusion–matrix interface configurations. (

**a**) Inhomogeneity with diameter D = 18 μm. (

**b**) Inhomogeneity with diameter D = 36 μm. (

**c**) Inhomogeneity with diameter D = 110 μm.

**Figure 12.**Non-local Fatemi–Socie responses of the different inclusion–matrix configurations as a function of the different defect sizes with an aspect ratio of AR = 1.0, for: (

**a**) 0.10% applied total strain amplitude at R

_{ε}= −1 and (

**b**) 0.50% applied total strain amplitude at R

_{ε}= −1. The different colored dashed lines represent the arithmetical average of the three microstructural representatives of each defect type.

**Figure 13.**Schematic visualization of the models to investigate the influence of aspect ratio on fatigue crack initiation. Therefore, voids within the martensitic microstructure are used. These defects exhibit different aspect ratios increasing from left to right from AR = 1.0, 2.0 up to 10.0.

**Figure 14.**Non-local Fatemi–Socie fatigue crack initiation simulation results of voids with a major axis of 110 μm and three different aspect ratios. The aspect ratio is represented as the ratio of the major and minor defect axes. (

**a**) Loading level of ε

_{a,t}= 0.10% and (

**b**) loading level of ε

_{a,t}= 0.50%.

**Figure 15.**Non-local Fatemi–Socie parameter as a function of the manganese sulfide aspect ratio. The colored markers of each aspect ratio represent the different microstructural realizations. The fully reversed strain-controlled loading was perpendicular to the manganese sulfide with: (

**a**) A total strain amplitude of ε

_{a,t}= 0.10% and (

**b**) with a total strain amplitude of ε

_{a,t}= 0.50%.

**Figure 16.**Fatigue crack initiation lifetime results for the martensitic steel with / without varying defects under fully reversed strain-controlled loading. By means of the Fatemi–Socie metric, micromechanical simulations were performed for: (a) Defects with diameters of D = 110 μm. (b) Defects with diameters of D = 36 μm. (c) Defects with diameters of D = 18 μm and (d) for manganese sulfides with varying aspect ratios and orientations to loading axis. Arithmetic average linear fatigue lifetime curves are shown for each defect type in the corresponding color. The abbreviation PD stands for partially debonded and AR for aspect ratio.

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

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

**Table 2.**Derived inclusion properties of oxidic and sulfidic non-metallic inclusions in SAE 4150. Extracted defect length scale $\lambda $, corresponding diameter and aspect ratio values belonging to the 10th, 50th and 90th percentile of the underlying generalized extreme value distributions.

Cumulative Probability | Oxidic Non-Metallic Inclusions | Sulfidic Non-Metallic Inclusions | ||||
---|---|---|---|---|---|---|

P = 10% | P = 50% | P = 90% | P = 10% | P = 50% | P = 90% | |

Defect length scale λ in μm | 16 | 32 | 97 | 16 | 43 | 116 |

Diameter in μm | 18 | 36.2 | 110 | - | - | - |

Aspect ratio [-] | 1.2 | 2.2 | 10 | 10 | 25 | 54 |

**Table 3.**Mechanical properties of the considered inclusion types and the steel matrix [3].

Chemical Composition | Young’s Modulus E | Poisson’s Ratio ν | Coefficient of Thermal Expansion ${\mathit{\alpha}}_{\mathit{T}}$ |
---|---|---|---|

Al_{2}O_{3} | 389 GPa | 0.25 | 8.0 × ${10}^{-6}$ 1/K |

MnS | 69 GPa | 0.30 | 18.1 × ${10}^{-6}$ 1/K |

Steel Matrix | - | - | 12.5 × ${10}^{-6}$ 1/K |

**Table 4.**Set of crystal plasticity parameters for SAE 4150 (39 HRC) at room temperature [49].

Elastic Constants | Flow Rule Parameters | Kinematic Hardening Parameters | |||
---|---|---|---|---|---|

${C}_{11}$ | 253.1 GPa | ${\dot{\gamma}}_{0}$ | 0.001 s${}^{-1}$ | A | 65.506 GPa |

${C}_{22}$ | 132.4 GPa | ${\tau}_{c,0}^{\alpha}$ | 209 MPa | B | 499 |

${C}_{44}$ | 75.8 GPa | m | 100 | M | 8 |

**Table 5.**Ratio of the maximum to the minimum non-local Fatemi–Socie response of three considered microstructural realizations.

Strain Amplitude | D = 110 $\mathsf{\mu}$m | D = 36 $\mathsf{\mu}$m | D = 18 $\mathsf{\mu}$m |
---|---|---|---|

0.10% | 2.35 | 5.10 | 1.98 × ${10}^{5}$ |

0.30% | 1.66 | 1.08 | 1.46 |

0.50% | 1.52 | 1.60 | 2.02 |

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**MDPI and ACS Style**

Schäfer, B.J.; Sonnweber-Ribic, P.; ul Hassan, H.; Hartmaier, A. Micromechanical Modeling of Fatigue Crack Nucleation around Non-Metallic Inclusions in Martensitic High-Strength Steels. *Metals* **2019**, *9*, 1258.
https://doi.org/10.3390/met9121258

**AMA Style**

Schäfer BJ, Sonnweber-Ribic P, ul Hassan H, Hartmaier A. Micromechanical Modeling of Fatigue Crack Nucleation around Non-Metallic Inclusions in Martensitic High-Strength Steels. *Metals*. 2019; 9(12):1258.
https://doi.org/10.3390/met9121258

**Chicago/Turabian Style**

Schäfer, Benjamin Josef, Petra Sonnweber-Ribic, Hamad ul Hassan, and Alexander Hartmaier. 2019. "Micromechanical Modeling of Fatigue Crack Nucleation around Non-Metallic Inclusions in Martensitic High-Strength Steels" *Metals* 9, no. 12: 1258.
https://doi.org/10.3390/met9121258