# Numerical Investigation into the Influence of Grain Orientation Distribution on the Local and Global Elastic-Plastic Behaviour of Polycrystalline Nickel-Based Superalloy INC-738 LC

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Material

#### 2.2. Material Modelling

_{11}, C

_{12}and C

_{44}, multiplied by the elastic strain ε according to Equation (1).

#### 2.3. Material Parameters

^{−1}. Figure 1 shows the experimental stress-strain data as well as the corresponding fitted elastic plastic material model.

#### 2.4. Simulations

^{2}cross section and 18 mm in length. 49 grains were generated using NEPER, which leads to an average grain size of 3.25 mm in diameter. Figure 2a represents the simulated specimen, within the visualization tool of NEPER.

## 3. Results

**x**for each curve whereby it was defined as the point at which the simulated elastic stress (based on the Hooke’s law) differs by more than 5% from the simulated elastic-plastic results.

- For the [100] orientation distribution the lowest Young’s moduli of 91.19 GPa can be found. The yield point is reached at 622 MPa with an elastic strain of 0.7%.
- For the soft orientation distribution, which slightly differs from the [100] orientation distribution (see Figure 3), the Young’s moduli is marginal higher with 97.58 GPa. However, yield is distinct lower and reached at 568 MPa and a related elastic strain of 0.6%.
- Highest Young’s moduli is reached for the [111] orientation distribution with 250.45 GPa and yield at 973 MPa with an elastic strain of 0.4%.
- The stiff orientation distribution reaches yield first, i.e., shows lowest elastic strain compared to all other investigated orientation distributions. Besides a quite high Young’s modulus of 220.37 GPa, yield is reached at 0.3% with a yield stress of 655 MPa.
- The [110] orientation distribution as well as the random orientation distribution show same elastic strains at the yield point with 0.35%. But due to slightly higher Young’s modulus for the [110] orientation distribution with 182.67 GPa, the yield stress is also slightly higher with 625 MPa.
- In order to consider the influence of random orientation distributions and the resulting scatter, six simulations were statistically evaluated, where the average stress-strain curve is shown in Figure 4. For the random orientation distribution, the Young’s modulus is 163.8 ± 4.8 GPa and yield stress is reached at 561.63 ± 15.17 MPa.
- The mix orientation distribution shows a Young’s modulus of 136.37 GPa and yields at 590 MPa and 0.45% total strain.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Experimental stress-strain data for IN-738 LC at 850 °C [24] and corresponding fitted elastic plastic polycrystalline model.

**Figure 2.**(

**a**) Polycrystalline model with 49 grains generated by NEPER and (

**b**) meshed equivalent for finite element analysis.

**Figure 3.**(

**a**) Inverse pole figure of chosen grain orientation distributions and (

**b**) $E\xb7m$ values at 850 °C.

**Figure 4.**Simulated stress-strain curves for different grain orientation distributions for an INC-738 LC at 850 °C.

**Figure 5.**(

**a**) Elastic strain S33 in loading direction and (

**b**) equivalent plastic strain for a randomly orientated specimen.

**Figure 6.**Distribution of equivalent plastic strain under (

**a**) 0.2%, (

**b**) 0.4%, (

**c**) 0.6% and (

**d**) 0.8% total strain for the randomly orientated grain distribution.

**Figure 7.**(

**a**) Elastic strain S33 in loading direction and (

**b**) equivalent plastic strain, for the soft orientated specimen.

**Figure 8.**(

**a**) Strain S33 and (

**b**) equivalent plastic strain in loading direction for the stiff orientated specimen.

**Figure 9.**Elastic strain S33 in loading direction (

**a**) and equivalent plastic strain (

**b**) for the mix orientated specimen.

**Figure 10.**Inverse pole plot of the Schmid factor (

**a**), inverse pole plot of the Young’s moduli of IN-738 LC at 850 °C (

**b**) and the chosen orientation distributions.

**Figure 11.**Distribution of random orientations in the inverse $E\xb7m$ plot (

**a**) and its histogram (

**b**).

**Figure 12.**Equivalent total strain for (

**a**) 0.4% global applied total strain and (

**b**) 1% global applied total strain.

**Table 1.**Composition of IN-738LC [16].

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

Wt.% | Bal. | 16 | 8.58 | 3.42 | 3.36 | 2.67 | 1.81 | 1.9 | 0.105 | 0.9 |

${\mathbf{C}}_{11}\left(\mathrm{GPa}\right)$ | ${\mathbf{C}}_{12}\left(\mathrm{GPa}\right)$ | ${\mathbf{C}}_{44}\left(\mathrm{GPa}\right)$ |
---|---|---|

225.83 | 161.45 | 98.79 |

$\dot{{\mathit{\gamma}}_{0}}\left({\mathit{s}}^{-1}\right)$ | $\mathit{m}$ | ${\mathit{h}}_{0}\left(\mathrm{MPa}\right)$ | ${\mathit{g}}_{0}\left(\mathit{M}\mathit{P}\mathit{a}\right)$ | ${\mathit{n}}^{\prime}$ | ${\mathit{g}}_{1}\left(\mathit{M}\mathit{P}\mathit{a}\right)$ | $\dot{{\mathit{\gamma}}_{\mathit{s}}}\left({\mathit{s}}^{-1}\right)$ | ${\mathit{m}}^{\prime}$ |
---|---|---|---|---|---|---|---|

1.0 | 0.05 | 200 | 210 | 1 | 330 | 5 × 10^{10} | 5 × 10^{−3} |

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**MDPI and ACS Style**

Engel, B.; Huth, M.; Hyde, C.
Numerical Investigation into the Influence of Grain Orientation Distribution on the Local and Global Elastic-Plastic Behaviour of Polycrystalline Nickel-Based Superalloy INC-738 LC. *Crystals* **2022**, *12*, 100.
https://doi.org/10.3390/cryst12010100

**AMA Style**

Engel B, Huth M, Hyde C.
Numerical Investigation into the Influence of Grain Orientation Distribution on the Local and Global Elastic-Plastic Behaviour of Polycrystalline Nickel-Based Superalloy INC-738 LC. *Crystals*. 2022; 12(1):100.
https://doi.org/10.3390/cryst12010100

**Chicago/Turabian Style**

Engel, Benedikt, Mark Huth, and Christopher Hyde.
2022. "Numerical Investigation into the Influence of Grain Orientation Distribution on the Local and Global Elastic-Plastic Behaviour of Polycrystalline Nickel-Based Superalloy INC-738 LC" *Crystals* 12, no. 1: 100.
https://doi.org/10.3390/cryst12010100