# Combined Effects of Texture and Grain Size Distribution on the Tensile Behavior of α-Titanium

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Preparation, Microstructural and Mechanical Characterization of the Specimens

^{−1}, which corresponds to an initial strain rate of $7.5\xb7{10}^{-4}$ s

^{−1}. The extension direction was parallel either to the last rolling direction or to the last transverse direction. Throughout the paper, the nomenclature to name the specimens is the following: the first letter corresponds to the last rolling direction (R means rolling along the previous rolling direction and T along the previous transverse direction) whereas the second letter refers to the direction of tension (R means tension along the last rolling direction and T along the last transverse direction) and the number to the mean grain size in microns. The names of the seven selected specimens used in this study and their conditions of preparation are reported in Table 1. Figure 1 exhibits examples of experimental tensile curves obtained for three specimens of RR type (i.e., rolling and tension along the previous rolling direction) with different mean grain sizes.

#### 2.2. Micromechanical Modeling Including Grain Size Effects

#### 2.2.1. Micro-Macro Scale Transition

#### 2.2.2. Single Crystal Constitutive Laws

#### 2.2.3. Model Parameters

## 3. Results and Discussion

- the 0.2% yield stress: ${\sigma}_{e}$,
- the maximal engineering stress: ${\sigma}_{max}$,
- the ${\sigma}_{max}$ corresponding engineering strain: ${\epsilon}^{*}$,
- the hardening amplitude: $\Delta \sigma ={\sigma}_{max}-{\sigma}_{e}$.

#### 3.1. Yield Stress

#### 3.2. Local Mechanical Fields

#### 3.3. Slip Activity

#### 3.4. Hardening Behavior

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Ti | titanium |

cp | commercially pure |

RD | rolling direction |

TD | transverse direction |

EBSD | electron backscatter diffraction |

SEM | scanning electron microscopy |

FEG | field emission gun |

EVPSC | elasto-visco-plastic self-consistent |

CRSS | critical resolved shear stress |

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**Figure 1.**Engineering stress versus engineering plastic strain for three specimens of RR type with different mean grain sizes.

**Figure 2.**Probability distributions of grain sizes for specimens RR 2.8 and RR 21.3. In the bottom part, the distributions are normalized by the mean grain size ${D}_{m}$ and the maximum likelihood estimate for a log-normal distribution computed thanks to the MATLAB software (R2015a, MathWorks, Natick, Massachusetts, US) being superimposed.

**Figure 3.**Pole figures for the specimens RR 2.8 (

**a**,

**b**) and RR 11.7 (

**c**,

**d**) plotted thanks to the ATEX software [25]. x represents the last rolling direction and y the transverse direction.

**Figure 4.**Tensile curves (engineering stress and strain) for model predictions and experimental measurements for the six specimens deformed along the last rolling direction.

**Figure 5.**Tensile curves (engineering stress and strain) for model predictions and experimental measurements for the specimen deformed along the last transverse direction.

**Figure 6.**Comparisons between experimental and calculated values of ${\sigma}_{e}$, ${\sigma}_{max}$, $\Delta \sigma $ and ${\epsilon}^{*}$ (see text for definitions).

**Figure 7.**Comparisons of experimental and calculated values of the yield stress ${\sigma}_{e}$ with respect to the mean grain size ${D}_{m}$ for the specimens of RR type. Linear fit estimates obtained by MATLAB between ${\sigma}_{e}$ and $1/\sqrt{{D}_{m}}$ are shown in green (Model) and black (Exp.) dotted lines.

**Figure 8.**Distribution of predicted tensile stress–strain relationships for each grain at 1% (magenta), 5% (red), 10% (blue) and 15% (green) of macroscopic strain for specimens RR 2.8, RR 4.8, RR 9.8 and RR 21.3. The corresponding model tensile curve is superimposed on each plot (black thick line).

**Figure 9.**Relative activities of slip families ($\alpha $) predicted by the model for specimen RR 9.8 (

**left**) and RT 9.8 (

**right**).

**Figure 11.**Model predictions for the evolution of the total forest (${\rho}_{f}$) and mobile (${\rho}_{m}$) dislocation densities for the specimens of RR type. Densities are cumulated over the 30 slip systems and averaged over the grains’ population.

Specimen | Cold Rolling Reduction | Annealing Conditions | Number of Grains in the Data Set |
---|---|---|---|

RR 2.8 | 75% | 500 °C - 40 mn | 6254 |

TR 2.8 | 75% | 500 °C - 40 mn | 7328 |

RR 4.8 | 75% | 650 °C - 1 h | 8793 |

RR 9.8 | 75% | 730 °C - 2 h | 3262 |

RT 9.8 | 75% | 730 °C - 2 h | 3262 |

RR 11.7 | 75% | 740 °C - 2 h | 4075 |

RR 21.3 | 30% | 840 °C - 4 h | 1273 |

**Table 2.**Model parameters that are specific to slip families (Prismatic: P, Basal: B, Pyramidal <a>: ${\Pi}_{1}^{<a>}$, 1st order Pyramidal <c+a>: ${\Pi}_{1}^{<c+a>}$, 2nd order Pyramidal <c+a>: ${\Pi}_{2}^{<c+a>}$).

P | ${\mathsf{\Pi}}_{1}^{<\mathit{a}>}$ | B | ${\mathsf{\Pi}}_{1}^{<\mathit{c}+\mathit{a}>}$ | ${\mathsf{\Pi}}_{2}^{<\mathit{c}+\mathit{a}>}$ | |
---|---|---|---|---|---|

b (Å) | 2.95 | 2.95 | 2.95 | 5.53 | 5.53 |

$\mu $ (GPa) | 35.0 | 37.1 | 46.5 | 47.7 | 49.2 |

${\mathsf{\tau}}_{0}$ (MPa) | 50 | 90 | 120 | 75 | 150 |

${C}_{1}$ (${m}^{-1}$) | 80 | 15 | 15 | 15 | 15 |

n | 65 | 32 | 32 | 32 | 32 |

**Table 3.**Model parameters that are not specific to slip families. ${C}_{ij}$ are the stiffness constants. ${a}_{coli}$ denotes the interaction coefficient related to collinear interactions, i.e., reactions between dislocations with parallel Burgers vectors gliding in different planes, whereas ${a}_{\ne coli}$ refers to non-collinear interactions (see details in [19]).

$\mathit{c}/\mathit{a}$ | ${\mathit{C}}_{\mathbf{11}}$ (GPa) | ${\mathit{C}}_{\mathbf{33}}$ (GPa) | ${\mathit{C}}_{\mathbf{44}}$ (GPa) | ${\mathit{C}}_{\mathbf{12}}$ (GPa) | ${\mathit{C}}_{\mathbf{13}}$ (GPa) | ${\mathit{k}}_{\mathit{H}\mathit{P}}$ (MPa.m^{0.5}) | |
---|---|---|---|---|---|---|---|

1.587 | 160 | 181 | 46.5 | 90 | 66 | 0.1 | |

${\mathit{v}}_{\mathbf{0}}$ (ms^{−1}) | ${\mathit{\rho}}_{{\mathit{m}}_{\mathbf{0}}}$ (m^{−2}) | ${\mathit{\rho}}_{{\mathit{f}}_{\mathbf{0}}}$ (m^{−2}) | ${\mathit{C}}_{\mathbf{2}}$ | $\mathit{K}$ | ${\mathit{a}}_{\mathit{coli}}$ | ${\mathit{a}}_{\ne \mathit{coli}}$ | ${\mathit{k}}_{\mathit{c}}$ |

$3\times {10}^{-5}$ | $1\times {10}^{10}$ | $2\times {10}^{12}$ | 75 | 80 | 0.7 | 0.1 | 10 |

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**MDPI and ACS Style**

Richeton, T.; Wagner, F.; Chen, C.; Toth, L.S.
Combined Effects of Texture and Grain Size Distribution on the Tensile Behavior of *α*-Titanium. *Materials* **2018**, *11*, 1088.
https://doi.org/10.3390/ma11071088

**AMA Style**

Richeton T, Wagner F, Chen C, Toth LS.
Combined Effects of Texture and Grain Size Distribution on the Tensile Behavior of *α*-Titanium. *Materials*. 2018; 11(7):1088.
https://doi.org/10.3390/ma11071088

**Chicago/Turabian Style**

Richeton, Thiebaud, Francis Wagner, Cai Chen, and Laszlo S. Toth.
2018. "Combined Effects of Texture and Grain Size Distribution on the Tensile Behavior of *α*-Titanium" *Materials* 11, no. 7: 1088.
https://doi.org/10.3390/ma11071088