# Probabilistic Modeling of Slip System-Based Shear Stresses and Fatigue Behavior of Coarse-Grained Ni-Base Superalloy Considering Local Grain Anisotropy and Grain Orientation

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## Abstract

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## 1. Introduction

- Uniaxial stress states in each grain with global uniaxial load
- No influence of the deformation behavior of the surrounding grains
- Only Young’s modulus and Schmid factor in direction of a uniaxial stress were considered
- Homogeneous resolved shear stress field at slip system within the grain.

^{®}2017; considering the elastic anisotropy of the crystals. These simulations allow approximating the local mechanical properties in a coarse-grained polycrystalline, uniaxially loaded LCF specimen where the local grain orientations and their interaction lead to various multiaxial stress states. The polycrystalline Finite Element Analysis (FEA) simulations were carried out for the case of randomly and for preferentially oriented grains. These preconditions led to distinct distributions of local resolved shear stresses and therefore Schmid factors. Deterministic formulae are used to derive statistical distributions for the LCF crack initiation life of both cases. In order to verify the differences seen in the simulation results, LCF test data of two René80 batches (isothermal 850 °C) was generated in strain-controlled experiments. One batch has had coarse grains with random orientation, while the other batch has had smaller grains with preferential orientation. Hence, polycrystalline FEA simulations according to this grain orientation were carried out. The thereof derived life distribution was shifted to a higher median compared to the case of random grain orientation. Using the large LCF test data set of Seibel [29] (coarse, randomly oriented grains) for calibration of the Schmid factor based crack initiation life model, it was possible to predict the observed crack initiation lives of the René80 batch with preferential grain orientation.

## 2. Materials and Methods

#### 2.1. Material

#### 2.2. Experimental Isothermal LCF Testing

^{2}and to compare the results to [29].

#### 2.3. FEA Models for Polycrystalline Microstructure Modelling

^{®}(Dessault systemes, Vélizy-Villacoublay, France). In order to account for anisotropic stiffness of René80, a global, anisotropic, linear-elastic material law was defined in ABAQUS

^{®}.

^{®}transfers the globally defined material law by means of tensor rotation with $\mathit{U}$ into the local coordinate system of the individual grains. With this procedure, the grains interact according to their orientation dependent stiffness. However, grain boundary interactions are not explicitly modeled by physics-based considerations. In order to have high comparability to the LCF experiments, all FEA simulations were modeled with given displacements using a material model for $T=850$°C. Since the latter was linear-elastic and a stable cyclic behavior for low total strain amplitudes was observed [43], only one load case was evaluated. All nodes of the cylinder top face were displaced by 0.045 mm which is equal to a total strain of 0.25%. The nodes at the bottom face were fixed but allowed transverse contraction.

#### 2.4. Derivation of the Schmid Factor Distribution

#### 2.5. Calibration of the Probabilistic LCF Fatigue Model and the Cyclic Material Strength Model

## 3. Results

#### 3.1. Microscopic Material Examinations and Orientation Distributions

#### 3.2. Results of the Isothermal LCF Tests at 850 °C

#### 3.3. Results of the Finite Element Simulation

#### 3.4. Results of the Schmid-Factor Distribution Calculations

#### 3.5. Procedure of the LCF Life Calibration and Prediction

#### Prediction of the Wöhler Curve from a Weibull Distribution

#### Prediction of the Wöhler Curve from the Single Grain Schmid Factor Distribution

#### Prediction of the Wöhler Curve from the Modified Schmid factor Distribution

## 4. Discussion

#### 4.1. Influence of the Grain Orientation Distribution on the Mechanical Properties

#### 4.2. Influence of Grain Orientation Distribution on the Fatigue Behavior

#### 4.3. Comparison of Fit and Prediction Quality of Probabilistic Models

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Specimen geometry. Only the cylindrical gauge section is considered in Finite Element Analysis (FEA) simulations.

**Figure 3.**(

**a**) Scanning electron microscope (SEM) image of the γ,γ′microstructure; (

**b**) light microscope image of cross section shows dendritic grain growth.

**Figure 4.**Light microscopy image of the vertical cross section of a $20\mathrm{mm}$ bar of René80. Coarse grains with random orientation solidified in the center of the gauge section but also fine, dendritically solidified grains are also visible in the edge area. The right-hand side shows the section that remained after specimen machining.

**Figure 5.**Light microscopy image of the vertical cross section of a $12\mathrm{mm}$ bar of René80. A preferential orientation of the grains was likely caused by the temperature gradients in the mold during solidification.

**Figure 6.**Stabilized cyclic stress response data and respective Ramberg-Osgood calibration curve for coarse grain batch and fine grain batch.

**Figure 7.**Stress and strain distribution in loading direction for a specimen with 49 grains and random orientation at 0.25% total strain. An anisotropic elasticity model of IN 738 LC at 850 °C was applied.

**Figure 8.**Stress and strain distribution in loading direction of a free cut grain from the specimen model with 49 grains at 0.25% total strain and 850 °C.

**Figure 9.**Frequency distribution of the parameter κ to evaluate the stress state on every simulated node of both specimen models.

**Figure 10.**Stress and strain distribution in loading direction for the specimen model with 500 grains and directed orientation distribution at 0.25% total strain. An anisotropic elasticity model of IN 738 LC at 850 °C was applied.

**Figure 11.**Distribution of maximum resulting shear stress in the <111>{110} slip systems at total strain of 0.25%. Coarse grain morphology and random orientation to the left and fine grain morphology with preferential orientation ($\vartheta =25\xb0$) to the right.

**Figure 12.**Distribution of maximum resolved shear stress at free cut grains in the [111] (110) slip systems for globally applied strain of 0.25%.

**Figure 13.**Comparison of the Schmid factor distribution density functions for a single crystal ${f}_{SF}\left(\tilde{m}\right)$ and a polycrystal ${f}_{SF}\left({\tilde{m}}_{mod}\right)$ (from FEA). The random Euler angle $\vartheta $ (coarse grain batch) refers to isotropically and $\vartheta =25\xb0$ (fine grain batch) refers to preferentially distributed grain orientations. A uniaxial stress was applied to all models. Vertical lines indicate the median values.

**Figure 17.**Orientation distribution functions and resulting local stiffness distributions (z-direction). (

**a**) Isotropically distributed grain orientation: Resulting local stiffness distribution from a single crystal Monte-Carlo simulation (${E}_{sg}$, cyan) and from polycrystalline FEA results (${E}_{FEA}$, violet). (

**b**) Preferential grain orientations: Resulting local stiffness distribution from a single crystal Monte-Carlo simulation (${E}_{sg}$, gray) and from polycrystalline FEA results (${E}_{FEA}$, green).

**Figure 18.**Stress-life plot of test data and fit and prediction curves for the modified Schmid factor distribution model.

**Table 1.**Chemical composition of René80 and IN 738 LC (taken from Hermann, W. (2014) [26]) in wt.%.

Element | Ni | Cr | Co | Ti | Mo | W | Al | C | B | Zr | Ta | Nb | Fe |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

René80 | Bal. | 14.04 | 9.48 | 5.08 | 4.03 | 4.02 | 2.93 | 0.17 | 0.015 | 0.011 | - | - | - |

IN 738 LC | Bal. | 16 | 8.3 | 8.7 | 3.4 | 1.8 | 2.7 | 3.4 | 0.11 | - | 1.9 | 0.9 | 0.1 |

Distribution | Modeling Approach |
---|---|

${F}_{SF}\left(\tilde{m}\right)$ | Monte-Carlo sampling of statistically distributed orientations of a single crystal, i.e., single grain. Maximum normalized resolved shear stresses calculated at global uniaxial stress state. Explicit consideration of elastic stiffness anisotropy. |

${F}_{SF}\left({\tilde{m}}_{mod}\right)$ | Monte-Carlo sampling of statistically distributed orientations of single crystals in polycrystalline FEA simulations. Maximum normalized resolved shear stresses calculated at all FEA nodes from local (multiaxial) stress states. Explicit consideration of elastic stiffness anisotropy. |

**Table 3.**Comparison of the determined global Young’s moduli from polycrystalline Finite Element Analysis (FEA).

Value | Random Orientation $\mathit{\vartheta}=\mathit{r}\mathit{a}\mathit{n}\mathit{d}.$ | Preferential Orientation $\mathit{\vartheta}=25\xb0$ | Shift |
---|---|---|---|

${E}_{global}$ | 160 GPa | 142 GPa | $-11\%$ |

Standard deviation | ±1.5 GPa | ±0.3 GPa | - |

**Table 4.**Comparison of the median values of the Schmid factor distributions ${f}_{SF}\left(\tilde{m}\right)$ (geometric approach) and ${f}_{SF}\left({\tilde{m}}_{mod}\right)$ (from FEA).

Schmid Factor | Random Orientation $\mathit{\vartheta}=\mathit{r}\mathit{a}\mathit{n}\mathit{d}\mathit{o}\mathit{m}$ | Preferential Orientation $\mathit{\vartheta}=25\xb0$ | Shift |
---|---|---|---|

${\tilde{m}}^{50\%}$ | $0.512$ | $0.493$ | $-4\%$ |

${\tilde{m}}_{mod}^{50\%}$ | $0.468$ | $0.431$ | $-8\%$ |

**Table 5.**Comparison of crack initiation life distribution densities for the applied probabilistic crack initiation model approaches.

Model Nr. | Distribution Approach | Distribution Density Visualization |
---|---|---|

1 | Weibull distributed life ${f}_{N}\left(n|\eta ,m\right)$ | |

2 | Life distribution from ${\tau}_{RSS}$ (Schmid factors) in single grain, ${f}_{N}\left(n\left(\tilde{m}\right)\right)$ | |

3 | Life distribution from ${\tau}_{RSS}$ (Schmid factors) in polycrystalline FEA, ${f}_{N}\left(n\left({\tilde{m}}_{mod}\right)\right)$ |

Model Combination: Scale Model + Life Distribution | Neg. Log-Likelihood per Data Point at Calibration | Neg. Log-Likelihood per Data Point at Prediction |
---|---|---|

CMB + Weibull distribution | 10.7 | 8.7 |

CMB + Schmid factor-corrected life distribution | 0.87 | 12.38 |

CMB + modified Schmid factor-corrected life distribution | 0.57 | 0.61 |

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**MDPI and ACS Style**

Engel, B.; Mäde, L.; Lion, P.; Moch, N.; Gottschalk, H.; Beck, T.
Probabilistic Modeling of Slip System-Based Shear Stresses and Fatigue Behavior of Coarse-Grained Ni-Base Superalloy Considering Local Grain Anisotropy and Grain Orientation. *Metals* **2019**, *9*, 813.
https://doi.org/10.3390/met9080813

**AMA Style**

Engel B, Mäde L, Lion P, Moch N, Gottschalk H, Beck T.
Probabilistic Modeling of Slip System-Based Shear Stresses and Fatigue Behavior of Coarse-Grained Ni-Base Superalloy Considering Local Grain Anisotropy and Grain Orientation. *Metals*. 2019; 9(8):813.
https://doi.org/10.3390/met9080813

**Chicago/Turabian Style**

Engel, Benedikt, Lucas Mäde, Philipp Lion, Nadine Moch, Hanno Gottschalk, and Tilmann Beck.
2019. "Probabilistic Modeling of Slip System-Based Shear Stresses and Fatigue Behavior of Coarse-Grained Ni-Base Superalloy Considering Local Grain Anisotropy and Grain Orientation" *Metals* 9, no. 8: 813.
https://doi.org/10.3390/met9080813