# Crystal-Plasticity-Finite-Element Modeling of the Quasi-Static and Dynamic Response of a Directionally Solidified Nickel-Base Superalloy

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Material Description

#### 2.2. Mechanical Characterisation

**a**) was used in the tests performed at 1.5 ×${10}^{-4}$ s${}^{-1}$ and 500 ${\mathrm{s}}^{-1}$, while the sample (

**b**) was employed in the tests carried out at a strain rate of 150 ${\mathrm{s}}^{-1}$. The latter had a different geometry due to limitations of the set-up used for conducting these particular experiments.

#### 2.2.1. Quasi-Static Tensile Tests

#### 2.2.2. Dynamic Tensile Tests

#### 2.3. Numerical Modelling

#### 2.3.1. Crystal Plasticity Model

#### 2.3.2. Numerical Set-Up

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CP | Crystal plasticity |

CPFEM | Crystal-plasticity-finite-element method |

DS | Directionally solidified |

FCC | Face centred cubic |

SX | Single crystal |

${\mathrm{T}}_{\mathrm{m}}$ | Melting temperature |

VPSC | Visco plastic self consistent |

SHTB | Split Hopkinson tension bar |

fps | Frames per second |

CRSS | Critical-resolved-shear stress |

RVE | Representative volume element |

C3D8 | Eight-node fully-integrated hexahedral element |

Probability density function |

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**Figure 1.**Microstructure of the MAR-M247DS alloy. The images (

**a**,

**b**) show the grains of the cross-section perpendicular to grain-growth direction, while the image (

**c**) is in the parallel direction.

**Figure 2.**Experimental grain-size distribution (blue histogram) and the log-normal fitting (dashed line) of the alloy. The mean size is 5.5 mm. The virtual grain-size distribution generated with Dream3D [29] is also represented.

**Figure 3.**Schematic representation of the axisymmetric samples. Quasi-static and 500 ${\mathrm{s}}^{-1}$ dynamic tests were performed with the sample on the left (

**a**) and the 150 ${\mathrm{s}}^{-1}$ dynamic tests with the sample depicted in (

**b**) .

**Figure 4.**Example of the edge tracing technique used for monitoring the minimum sample diameter during testing. Image 1 (

**a**) corresponds to the frame when the test starts and Image n (

**b**) corresponds to the frame just before to failure.

**Figure 5.**Dimensions of the split Hopkinson tension bar (SHTB) set-up. The strain gauges are in the positions 1, 2 and 3. Image adapted from [33]. For more information about this SHTB apparatus, consult the same citation.

**Figure 6.**Illustration of the cylindrical FE models used to simulate the gauge length of the specimens. Images (

**a**,

**c**) depict models with grains parallel to grain-growth direction (0${}^{\circ}$), while figures (

**b**,

**d**) depict models where the axial load is perpendicular to grain-growth direction (90${}^{\circ}$). Each colour represents a different grain.

**Figure 7.**Stereographic projection of the crystallographic orientation of 1000 grains defined by the Euler-angles normal distribution ${\varphi}_{1}$ = 20 ${}^{\circ}$± 90${}^{\circ}$, $\Phi $ = 0${}^{\circ}$± 4${}^{\circ}$, ${\varphi}_{2}$ = 32${}^{\circ}$± 20${}^{\circ}$.

**Figure 8.**Experimental (lines) and numerical (markers) true stress–strain curves for MAR-M247DS alloy at room temperature under quasi-static (

**a**) and dynamic (

**b**,

**c**) regimes and different loading directions. The green markers in the plot (

**a**) are related to the fitting of the hardening law.

**Figure 9.**Elastic part of the experimental (coloured lines) and numerical (markers) true stress–strain curves for the quasi-static tests.

**Figure 10.**Probability density function of the total plastic shear strain $\Gamma $ depending on the grain Schmid factor for the samples oriented perpendicularly. The data were obtained for three different levels of applied true strain: at the beginning of plastic deformation -0.01-, in the middle of the test -0.045- and in the final stages -0.09-.

**Figure 11.**Histogram to represent, depending on the sample configuration (

**a**) for ${0}^{\circ}$ and (

**b**) for ${90}^{\circ}$, the relative amounts of elements with local von Mises stress ${\overline{\sigma}}_{\mu}$ higher than the global one. The data involve the last frame of all simulations. ${\overline{\sigma}}_{max}$ is the von Mises stress value from the true curves (Figure 8).

**Figure 12.**Contour plot (

**a**,

**c**) of the von Mises stress for the two different specimen configurations (

**b**) = 90${}^{\circ}$ and (

**d**) = 0${}^{\circ}$ at the last frame. Image on the right represents the deformed grain structure. Stresses are in MPa.

Cr | Co | Al | Ti | W | Ta | Mo | C | Hf | Ni |
---|---|---|---|---|---|---|---|---|---|

8.00 | 10.0 | 5.50 | 1.00 | 10.0 | 3.00 | 0.60 | 0.15 | 1.5 | Bal. |

**Table 2.**Parameters of the CP model for MAR-M247. The elastic constants were obtained from [43], while the values that define the expression (10) were fitted from the experimental data in the parallel direction.

${\mathbf{C}}_{11}$ | ${\mathbf{C}}_{12}$ | ${\mathbf{C}}_{44}$ | ${\mathbf{h}}_{0}$ | ${\mathbf{\tau}}_{0}$ | ${\mathbf{\tau}}_{\mathit{s}}$ | m |
---|---|---|---|---|---|---|

258.6 GPa | 167.0 GPa | 125.0 GPa | 1.36${\tau}_{0}$ | ${\tau}_{0}$ | 2.26 ${\tau}_{0}$ | 0.0015 |

${\dot{\gamma}}_{\mathbf{0}}$ | ${\mathbf{q}}_{\mathbf{1}}$ | ${\mathbf{q}}_{\mathbf{2}}$ | ${\mathbf{q}}_{\mathbf{3}}$ | ${\mathbf{q}}_{\mathbf{4}}$ | ${\mathbf{q}}_{\mathbf{5}}$ | ${\mathbf{q}}_{\mathbf{6}}$ |

0.001 s${}^{-1}$ | 1.00 | 1.00 | 5.38 | 0.68 | 1.12 | 0.96 |

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**MDPI and ACS Style**

Sancho, R.; Segurado, J.; Erice, B.; Pérez-Martín, M.-J.; Gálvez, F.
Crystal-Plasticity-Finite-Element Modeling of the Quasi-Static and Dynamic Response of a Directionally Solidified Nickel-Base Superalloy. *Materials* **2020**, *13*, 2990.
https://doi.org/10.3390/ma13132990

**AMA Style**

Sancho R, Segurado J, Erice B, Pérez-Martín M-J, Gálvez F.
Crystal-Plasticity-Finite-Element Modeling of the Quasi-Static and Dynamic Response of a Directionally Solidified Nickel-Base Superalloy. *Materials*. 2020; 13(13):2990.
https://doi.org/10.3390/ma13132990

**Chicago/Turabian Style**

Sancho, Rafael, Javier Segurado, Borja Erice, María-Jesús Pérez-Martín, and Francisco Gálvez.
2020. "Crystal-Plasticity-Finite-Element Modeling of the Quasi-Static and Dynamic Response of a Directionally Solidified Nickel-Base Superalloy" *Materials* 13, no. 13: 2990.
https://doi.org/10.3390/ma13132990