# Quantifying the Contribution of Crystallographic Texture and Grain Morphology on the Elastic and Plastic Anisotropy of bcc Steel

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## Abstract

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## 1. Introduction

## 2. Material: Composition, Processing and Characterization

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_{3}particles. Subsequent EBSD measurements were carried out on all three samples with a Tescan Mira3 feg scanning electron microscope at a distance of 170 $\mathsf{\mu}$$\mathrm{m}$ from the flange top surface. The size of the characterized area was adapted to the microstructure so that the maps contained at least 2000 grains and about 2.5 million measurement points to ensure an accurate representation of texture data and grain morphologies at the same time. The three obtained microstructure maps are shown in Figure 2 with color code assigned according to the inverse pole figure (IPF) in the respective sample surface normal direction. It can be seen that the material exhibits a microstructure with highly elongated, “pancake shaped” grains (see Figure 2) with average grain dimensions of $0.2$ $\mathsf{\mu}$$\mathrm{m}$ in ND, $0.8$ $\mathsf{\mu}$$\mathrm{m}$ in TD and $1.4$ $\mathsf{\mu}$$\mathrm{m}$ in RD. The apparent grain aspect ratios in the cross sectional measurements are therefore about 6.9 in the RD-section, 4.0 in the TD-section and 1.7 in the ND-section. The microstructure features a strong bcc-rolling texture including a distinct α-fiber (〈110〉 ‖ RD) with a dominant rotated cube orientation ({001}〈110〉 (the {001} crystal planes are parallel to the sheet plane (ND) and the 〈110〉 crystallographic directions are parallel to the rolling direction (RD).) having maximum intensity of about 20 times random and a typical γ-fiber (〈111〉 ‖ ND). The ${\phi}_{2}=45{}^{\circ}$-section of the orientation distribution function (ODF) of the texture data from the TD-section is shown in Figure 3.

## 3. Simulation Setup

#### 3.1. Microstructure Representation

**I a****Direct takeover 2D:**These 2D full-field models are based on a direct takeover of the measured crystallographic orientation on each of the $1601\times 1600$ = 2,561,600 points (see Figure 2).**I b****Random orientation assignment 2D:**By randomly shuffling the measured crystallographic orientations among the points, a second set of $1601\times 1600$ resolved 2D microstructures has been created.**I c****Random orientation assignment 3D:**The random distribution of almost all (Less than 2% of the discrete crystallographic orientations had to be discarded when distributing them on an equi-gridded cube (${136}^{3}<1601\times 1600<{137}^{3}$).) measured orientations on a 3D grid with $136\times 136\times 136$ = 2,515,456 points gives a third set of microstructure variants.

**I d****2D****Voronoi****tessellation:**A regular grid of $2024\times 2024$ = 4,096,576 pixel is divided into 1000 grains with a periodic Voronoi tessellation. Each grain gets a homogeneous initial orientation assigned.**I e****3D****Voronoi****tessellation:**Similarly, a $160\times 160\times $ 160 = 4,096,000 voxel grid is divided into 1000 equiaxed grains with a periodic Voronoi tessellation. The resulting microstructure for the RD-section is shown in Figure 4b.

**II a****3D microstructure without grain information:**This TCCP-FEM model is conceptually a combination of variant I c (Random orientation assignment 3D) and I e (3D Voronoi tessellation): 1000 orientations are assigned to the points of a $10\times 10\times 10$ grid.**II b****3D microstructure with globular grains:**The same geometric representation as for variant I e (3D Voronoi tessellation) is used but the 1000 orientations represent the texture of all three measurements. To investigate the influence of the grain shape separately from the influence of the strong crystallographic texture present in the probed material, a variant of this microstructure is created in which 1000 randomly sampled orientations are assigned to the grains.**II c****3D microstructure with elongated grains:**To generate elongated grains, a standard Voronoi tesselation of 1000 seed points is performed on a $160\times 160\times (160\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}8)$ grid from which only every eights plane along the last direction is used. The resulting grain structure with a grain aspect ratio of 8:8:1 (RD:TD:ND) and initial homogeneous orientation per grain is shown in Figure 4c. To investigate the influence of the grain shape separately from the influence of the strong crystallographic texture present in the probed material, a variant of this microstructure is created in which 1000 randomly sampled orientations are assigned to the grains.

#### 3.2. Constitutive Model for Crystal Plasticity

#### 3.3. Numerical Solver and Boundary Conditions

## 4. Results

#### 4.1. Average Behaviour

- The yield stress calculated for the individual sections with the analytic approach depends slightly on the data set, it differs by 30 $\mathrm{M}$$\mathrm{Pa}$ (i.e., 3.4%) for the yield stress in TD direction ${\sigma}_{\mathrm{y},\mathrm{TD}}$, see Table 3a.
- The various microstructure models used for the individual data (I a to I e) predict differences of up to 38 $\mathrm{M}$$\mathrm{Pa}$ (${\sigma}_{\mathrm{y},\mathrm{TD}}$ calculated from ND-section data), see Table 3a.
- The yield stress in RD, ${\sigma}_{\mathrm{y},\mathrm{RD}}$, predicted by all simulations is lower than the value obtained from the analytic expression.
- Sampling 1000 orientations from the combined texture results in an increase of the predicted yield stress by 4 $\mathrm{M}$$\mathrm{Pa}$–12 $\mathrm{M}$$\mathrm{Pa}$ when employing the analytic approach, see Table 3b.
- Employing the simpler models (II a: 3D microstructure without grain information and II b: 3D microstructure with globular grains) lowers ${\sigma}_{\mathrm{y},\mathrm{TD}}$ and ${\sigma}_{\mathrm{y},\mathrm{ND}}$ and increases ${\sigma}_{\mathrm{y},\mathrm{RD}}$ in comparison to model II c (3D microstructure with elongated grains) which has the most realistic grain geometry, see Table 3b.

#### 4.2. Micro-Mechanical Behaviour

## 5. Discussion

## 6. Conclusions

- The grain morphology only has a minor impact on anisotropic elastic and plastic properties, with differences of less than 3% between microstructure based and solely texture based numerical models.
- Statistically sufficient orientation measurements are more important than grain morphology. Even measuring 2000 grains does not ensure obtaining a representative orientation data.
- The HybridIa method enables a significant reduction of the orientation data that is required to accurately represent the texture.
- The simple analytic approach based on the geometric mean is suitable for estimating anisotropic elastic properties, since it yields very similar results as more complex numerical simulations.
- The underlying isostrain assumption of the Taylor model renders it an unsuitable choice for materials consisting of non-equiaxed grains with very strong anistropic behaviour.

## 7. Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Upper half of the double-Y-profile produced by linear flow splitting with marked positions of the tensile samples (

**left**) and their geometry (

**right**).

**Figure 2.**Microstructure maps in three mutually perpendicular directions of the material after linear flow splitting. Crystallographic orientation is given in terms of the inverse pole figure parallel to the measurement direction. Note the lower magnification of the normal direction (ND)-section in comparison to the rolling direction (RD)- and transverse direction (TD)-section. (

**a**) ND-section. (

**b**) RD-section. (

**c**) TD-section.

**Figure 3.**${\phi}_{2}$-section of the orientation distribution function (ODF) calculated from the TD-section using a harmonic series expansion approach. ${\phi}_{1}$, $\mathsf{\Phi}$ and ${\phi}_{2}$ are the Bunge–Euler angles.

**Figure 4.**Microstructural models created from the measured crystallographic orientation. ND is aligned with the vertical direction, morphologically there is no difference between RD and TD for all three models. (

**a**) Microstructure I c: Point-wise random orientation distribution, exemplarily shown for the ND-section. The legend is shown in Figure 2a. (

**b**) Microstructure I e: 1000 globular grains with homogeneous crystallographic orientation, exemplarily shown for the RD-section. The legend is shown in Figure 2b. (

**c**) Microstructure II c: 1000 elongated grains with homogeneous crystallographic orientation. The legend is shown in Figure 2c.

**Figure 5.**YOUNG’s modulus in dependence of loading direction. Left: ND-section, Center: RD-section, Right: TD-section. (

**a**) Results from simulations and the geometric mean calculation using the data of the individual measurements. The range between highest and lowest simulation result from all five microstructure variants (model

**I a**to

**I e**) is indicated by the background color (

**b**) Results from the simulations and the geometric mean calculations using the combined texture data.

**Figure 6.**Experimental results of the tensile tests from the samples cut from the flange material. (

**a**) Engineering stress–strain curves. (

**b**) Flow curves the with 0.05% proof stress indicated. φ: Plastic deformation.

**Figure 7.**Yield point in dependence of loading direction. Left: ND-section, Center: RD-section, Right: TD-section. Results from the combined simulations obtained from the individual measurements and from the geometric mean calculations.

**Figure 8.**Lankford coefficient in dependence of the loading direction in the ND section. Results for the 3D models using the combined texture information (II a–c) are shown.

**Figure 9.**Stress (top row) and strain (bottom row) in loading direction for the TD-section (direct takeover, microstructure representation I a). The left image shows loading along ND (vertical direction), the right image loading along RD (horizontal direction) and the central loading aligned at $\mathsf{\Theta}=45{}^{\circ}$ in-between ND and RD. A logarithmic mapping from value to color is employed for stress and strain.

**Figure 10.**Distribution of the stress–strain correlation (“heat map”) in models created from all crystallographic orientations measured in the RD-section for loading along TD and ND using a kernel density estimation. Note: Modeling the response by an isostrain assumption would result in a vertical line, the isostress assumption would result in an horizontal line. (

**a**) Direct takeover 2D (I a), loading along TD. (

**b**) Direct takeover 2D (I a), loading along ND. (

**c**) Random orientation assignment 2D (I b), loading along TD. (

**d**) Random orientation assignment 2D (I b), loading along ND.

**Figure 11.**Distribution of the stress–strain correlation (“heat map”) in models created from the measurement in the RD-section and in a model created from the combination of all three measurements for loading along ND using a kernel density estimation. Note: Modeling the response by an isostrain assumption would result in a vertical line, the isostress assumption would result in an horizontal line. (

**a**) Random orientation assignment 3D (I c), created from the RD-section. (

**b**) 2D VORONOI tessellation (I d), created from the RD-section. (

**c**) 3D VORONOI tessellation (I e), created from the RD-section. (

**d**) 3D model with elongated grains (II c).

(a) | ||
---|---|---|

Property | Value | Unit |

${C}_{11}$ | 230 | $\mathrm{G}$$\mathrm{Pa}$ |

${C}_{12}$ | 134 | $\mathrm{G}$$\mathrm{Pa}$ |

${C}_{44}$ | 116 | $\mathrm{G}$$\mathrm{Pa}$ |

(b) | ||
---|---|---|

Property | Value | Unit |

${\dot{\gamma}}_{0}$ | 1.0 | $\mathrm{m}$$\mathrm{m}$/$\mathrm{s}$ |

${\tau}_{0,\left\{110\right\}}$ | 354 | $\mathrm{M}$$\mathrm{Pa}$ |

${\tau}_{\infty ,\left\{110\right\}}$ | 837 | $\mathrm{M}$$\mathrm{Pa}$ |

${\tau}_{0,\left\{112\right\}}$ | 361 | $\mathrm{M}$$\mathrm{Pa}$ |

${\tau}_{\infty ,\left\{112\right\}}$ | 1538 | $\mathrm{M}$$\mathrm{Pa}$ |

${h}_{0}$ | 1.0 | $\mathrm{G}$$\mathrm{Pa}$ |

Coplanar ${h}_{\alpha \beta}$ | 1.0 | |

Non-coplanar ${h}_{\alpha \beta}$ | 1.4 | |

n | 20.0 | |

a | 2.0 |

(a) | |||
---|---|---|---|

ND-Section | RD-Section | TD-Section | |

${E}_{\mathrm{ND}}/\mathrm{G}\mathrm{Pa}$ | - | ${205}_{202}^{206}$ | ${194}_{191}^{195}$ |

${E}_{\mathrm{RD}}/\mathrm{G}\mathrm{Pa}$ | ${217}_{219}^{220}$ | - | ${215}_{214}^{217}$ |

${E}_{\mathrm{TD}}/\mathrm{G}\mathrm{Pa}$ | ${233}_{234}^{237}$ | ${231}_{231}^{235}$ | - |

(b) | |||||
---|---|---|---|---|---|

Geometric Mean | Simulation | ||||

All Orientations | 1000 Orientations | II a | II b | II c | |

${E}_{\mathrm{ND}}/\mathrm{G}\mathrm{Pa}$ | 198 | 198 | 199 | 198 | 196 |

${E}_{\mathrm{RD}}/\mathrm{G}\mathrm{Pa}$ | 215 | 215 | 216 | 216 | 215 |

${E}_{\mathrm{TD}}/\mathrm{G}\mathrm{Pa}$ | 233 | 234 | 235 | 234 | 236 |

(a) | |||
---|---|---|---|

ND-Section | RD-Section | TD-Section | |

${\sigma}_{\mathrm{y},\mathrm{ND}}/\mathrm{M}\mathrm{Pa}$ | - | ${786}_{876}^{902}$ | ${769}_{832}^{862}$ |

${\sigma}_{\mathrm{y},\mathrm{RD}}/\mathrm{M}\mathrm{Pa}$ | ${889}_{844}^{873}$ | - | ${873}_{818}^{837}$ |

${\sigma}_{\mathrm{y},\mathrm{TD}}/\mathrm{M}\mathrm{Pa}$ | ${904}_{873}^{911}$ | ${874}_{885}^{898}$ | - |

(b) | |||||
---|---|---|---|---|---|

Geometric Mean | Simulation | ||||

All Orientations | 1000 Orientations | II a | II b | II c | |

${\sigma}_{\mathrm{y},\mathrm{ND}}/\mathrm{M}\mathrm{Pa}$ | 778 | 782 | 857 | 853 | 877 |

${\sigma}_{\mathrm{y},\mathrm{RD}}/\mathrm{M}\mathrm{Pa}$ | 874 | 883 | 842 | 837 | 825 |

${\sigma}_{\mathrm{y},\mathrm{TD}}/\mathrm{M}\mathrm{Pa}$ | 890 | 902 | 888 | 885 | 890 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Diehl, M.; Niehuesbernd, J.; Bruder, E.
Quantifying the Contribution of Crystallographic Texture and Grain Morphology on the Elastic and Plastic Anisotropy of bcc Steel. *Metals* **2019**, *9*, 1252.
https://doi.org/10.3390/met9121252

**AMA Style**

Diehl M, Niehuesbernd J, Bruder E.
Quantifying the Contribution of Crystallographic Texture and Grain Morphology on the Elastic and Plastic Anisotropy of bcc Steel. *Metals*. 2019; 9(12):1252.
https://doi.org/10.3390/met9121252

**Chicago/Turabian Style**

Diehl, Martin, Jörn Niehuesbernd, and Enrico Bruder.
2019. "Quantifying the Contribution of Crystallographic Texture and Grain Morphology on the Elastic and Plastic Anisotropy of bcc Steel" *Metals* 9, no. 12: 1252.
https://doi.org/10.3390/met9121252