# Transformation of 2D RVE Local Stress and Strain Distributions to 3D Observations in Full Phase Crystal Plasticity Simulations of Dual-Phase Steels

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulation Model

## 3. Method

## 4. Results

#### 4.1. Local Stress/Strain Distribution in 2D and 3D

#### 4.2. Step by Step for Transformation from 2D to 3D

#### 4.3. Convergence and Statistical Analysis

#### 4.4. Derivation of 3D Stress and Strain Distribution from the 2D Result

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Constants ${a}_{m}$, ${b}_{m}$, and ${R}_{m}^{2}$ (m = 1, 2, 3, 4) for the martensite phase of linear equation with stress condition by the proposed iterative method.

Stress | |||||||||
---|---|---|---|---|---|---|---|---|---|

Analysis Parameter | Total Grain | Strain Level | Volume Fraction | ||||||

8400 | 1900 | 770 | 5% | 15% | 25% | 0.1 | 0.15 | 0.2 | |

Case | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

RVE Iterative Const. | A | B | C | D | D | D | D | E | F |

${a}_{1}$ | −0.86 | −0.78 | −0.88 | −0.7375 | −0.7938 | −0.82 | −0.7938 | −0.725 | −0.766 |

${b}_{1}$ | 1875 | 1614 | 1746 | 1526 | 2002 | 2222 | 2002 | 2092 | 1956 |

${R}_{1}^{2}$ | 0.31 | 0.23 | 0.36 | 0.14 | 0.24 | 0.34 | 0.24 | 0.21 | 0.17 |

${a}_{2}$ | 0.67 | 0.76 | 0.63 | 0.84 | 0.74 | 0.64 | 0.74 | 0.77 | 0.81 |

${b}_{2}$ | 6.02 | 2.87 | −1.003 | 28.96 | 14.86 | 6.64 | 14.86 | 20.6 | −25.19 |

${R}_{2}^{2}$ | 0.67 | 0.76 | 0.63 | 0.84 | 0.74 | 0.64 | 0.74 | 0.79 | 0.81 |

${a}_{3}$ | 0.32 | 0.23 | 0.36 | 0.15 | 0.25 | 0.35 | 0.25 | 0.23 | 0.18 |

${b}_{3}$ | 4.67 | 1.05 | −1.00 | 5.69 | 6.49 | 6.37 | 6.49 | 6.55 | −5.52 |

${R}_{3}^{2}$ | 0.31 | 0.23 | 0.36 | 0.14 | 0.25 | 0.35 | 0.25 | 0.23 | 0.17 |

${a}_{4}$ | 0.68 | 0.76 | 0.63 | 0.86 | 0.75 | 0.65 | 0.75 | 0.77 | 0.82 |

${b}_{4}$ | 5.68 | 1.35 | −1.00 | 8.39 | 8.43 | 7.14 | 8.43 | 6.24 | −6.69 |

${R}_{4}^{2}$ | 0.68 | 0.76 | 0.63 | 0.86 | 0.75 | 0.65 | 0.75 | 0.77 | 0.82 |

**Table A2.**Constants ${a}_{m}$, ${b}_{m}$, and ${R}_{m}^{2}$ (m = 1, 2, 3, 4) for the martensite phase of linear equation with strain condition by the proposed iterative method.

Strain | |||||||||
---|---|---|---|---|---|---|---|---|---|

Analysis Parameter | Total Grain | Strain Level | Volume Fraction | ||||||

8400 | 1900 | 770 | 5% | 15% | 25% | 0.1 | 0.15 | 0.2 | |

Case | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

RVE Iterative Const. | A | B | C | D | D | D | D | E | F |

${a}_{1}$ | −0.31 | 0.09 | 0.5354 | 0.26 | 0.23 | 0.18 | 0.23 | 0.22 | −0.03 |

${b}_{1}$ | 2.19 | −0.023 | −1.3028 | −0.30 | −0.93 | −1.38 | -0.93 | −0.3 | 0.60 |

${R}_{1}^{2}$ | 0.005 | 0.0007 | 0.015 | 0.003 | 0.002 | 0.002 | 0.002 | 0.006 | 0.001 |

${a}_{2}$ | 0.99 | 0.99 | 0.98 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 |

${b}_{2}$ | 0.17 | −0.12 | −0.4026 | 0.028 | 0.104 | 0.16 | 0.104 | −0.32 | −0.53 |

${R}_{2}^{2}$ | 0.99 | 0.99 | 0.98 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 |

${a}_{3}$ | 0.005 | 0.0008 | 0.0164 | 0.003 | 0.003 | 0.002 | 0.003 | 0.006 | 0.001 |

${b}_{3}$ | 0.0009 | −0.00008 | −0.0058 | 0 | 0 | 0 | 0 | 0 | 0 |

${R}_{3}^{2}$ | 0.005 | 0.0007 | 0.0156 | 0.003 | 0.003 | 0.002 | 0.003 | 0.006 | 0.001 |

${a}_{4}$ | 0.99 | 0.99 | 0.9850 | 0.996 | 0.997 | 0.998 | 0.997 | 0.993 | 0.99 |

${b}_{4}$ | 0.001 | −0.00015 | −0.0091 | 0 | 0 | 0 | 0 | 0 | 0 |

${R}_{4}^{2}$ | 0.99 | 0.99 | 0.98 | 0.996 | 0.997 | 0.998 | 0.998 | 0.993 | 0.99 |

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**Figure 1.**The schematic diagram represents the flow chart of the current work. The upper half (yellow background) indicates the flow chart of crystal simulation, and the bottom half (blue background) represents numerical analysis. The blue arrow indicates the flow of data. The red arrow represents tensile loading, and the yellow arrow separates the ferrite (F) and martensite (M) phases.

**Figure 2.**(

**a**) The resulting RVE-C initial microstructure colored according to the inverse pole figure (IPF). (

**b**) All RVE models (RVE-A to RVE-F) were sliced into geometries with 1-layer (h = 0.01) RVE in terms of texture style.

**Figure 3.**Local von Mises stress and strain distribution for ferrite phase from case 1 to case 9. Note that case 1 to case 3 indicate different total grain numbers; case 4 to case 6 indicate different strain levels; case 7 to case 9 indicate different volume fractions. The extreme left and right column indicates the 2D and 3D local stress and strain distribution differences, respectively.

**Figure 4.**Scatter diagram showing the numerical tendency of stress in (

**a**) case 1 (

**b**) case 3 and strain in (

**c**) case 1 (

**d**) case 3 by the proposed iterative method for ferrite phase at 25% global true strain and 0.1 volume fraction.

**Figure 5.**Convergence analysis for the ferrite phase at 25% global true strain. (

**a**) Stress and (

**b**) strain by the proposed iterative method with different iterative steps.

**Figure 6.**Probability distribution function of different (

**a**) stress and (

**b**) strain distributions between 3D result and iteratively revised 2D result for the ferrite phase of case 1 with different iterative steps.

**Figure 7.**Local von Mises stress and strain distribution for the ferrite phase of case 1 with different iterative steps. The extreme left and right column indicate the 2D and 3D local stress and strain distribution differences, respectively.

**Table 1.**Grain size distribution, number of total grain, volume fraction, and strain level for the microstructure [37].

Case | RVE | Ferrite Grains | Martensite Grains | Total Grain | Strain Level [%] | Volume Fraction | ||||
---|---|---|---|---|---|---|---|---|---|---|

Min. | Max. | Avg. | Min. | Max. | Avg. | |||||

1 | A | 5.1 | 7.6 | 6.35 | 3.5 | 5.7 | 4.6 | 8400 | 25 | 0.1 |

2 | B | 8.5 | 12.7 | 10.6 | 5.4 | 8.8 | 7.1 | 1900 | 25 | 0.1 |

3 | C | 11.2 | 16.8 | 14.0 | 9.8 | 15.8 | 12.8 | 700 | 25 | 0.1 |

4 | D | 8.5 | 12.7 | 10.6 | 5.4 | 8.8 | 7.1 | 1900 | 5 | 0.1 |

5 | D | 8.5 | 12.7 | 10.6 | 5.4 | 8.8 | 7.1 | 1900 | 15 | 0.1 |

6 | D | 8.5 | 12.7 | 10.6 | 5.4 | 8.8 | 7.1 | 1900 | 25 | 0.1 |

7 | D | 8.5 | 12.7 | 10.6 | 5.4 | 8.8 | 7.1 | 1900 | 15 | 0.1 |

8 | E | 8.5 | 12.7 | 10.6 | 5.4 | 8.8 | 7.1 | 1900 | 15 | 0.15 |

9 | F | 8.5 | 12.7 | 10.6 | 5.4 | 8.8 | 7.1 | 1900 | 15 | 0.2 |

**Table 2.**Mechanical properties of multiphase (ferrite and martensite) were adopted from [37] for simulation modeling.

Parameter | Symbol | Ferrite | Martensite | Unit |
---|---|---|---|---|

First elastic stiffness constant with normal strain | C_{11} | 233.3 | 417.4 | GPa |

Second elastic stiffness constant with normal strain | C_{12} | 135.5 | 242.4 | GPa |

First elastic stiffness constant with shear strain | C_{44} | 128.0 | 211.1 | GPa |

Initial shear resistance on [111] | S_{0} [111] | 95 | 406 | MPa |

Saturation shear resistance on [111] | S_{∞} [111] | 222 | 873 | MPa |

Initial shear resistance on [112] | S_{0} [112] | 96 | 457 | MPa |

Saturation shear resistance on [112] | S_{∞} [112] | 412 | 971 | MPa |

Slip hardening parameter | h_{0} | 1.0 | 563 | GPa |

Interaction hardening parameter | h_{α},_{β} | 1.0 | 1.0 | - |

Stress exponent | $n$ | 20 | 20 | - |

Curve fitting parameter | $w$ | 2.0 | 2.0 | - |

**Table 3.**Constants ${a}_{m}$, ${b}_{m}$, and ${R}_{m}^{2}$ (m = 1, 2, 3, 4) for the ferrite phase of linear equation with stress condition by the proposed iterative method.

Stress | |||||||||
---|---|---|---|---|---|---|---|---|---|

Analysis Parameter | Total Grain | Stress/Strain Level, % | Volume Fraction, % | ||||||

8400 | 1900 | 770 | 5 | 15 | 25 | 0.1 | 0.15 | 0.2 | |

Case | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

RVE Iterative Const. | A | B | C | D | D | D | D | E | F |

${a}_{1}$ | −0.76 | −0.81 | −0.75 | −0.63 | −0.74 | −0.78 | −0.74 | −0.66 | −0.66 |

${b}_{1}$ | 543.73 | 572.10 | 572.15 | 357.41 | 470.71 | 579.71 | 470.71 | 444.5 | 435.8 |

${R}_{1}^{2}$ | 0.35 | 0.37 | 0.34 | 0.31 | 0.38 | 0.41 | 0.38 | 0.33 | 0.31 |

${a}_{2}$ | 0.64 | 0.62 | 0.65 | 0.68 | 0.61 | 0.59 | 0.61 | 0.64 | 0.66 |

${b}_{2}$ | −8.62 | −10.73 | −5.70 | −11.48 | −9.46 | −7.43 | −9.46 | −13.85 | −12.25 |

${R}_{2}^{2}$ | 0.64 | 0.64 | 0.68 | 0.69 | 0.62 | 0.60 | 0.62 | 0.69 | 0.72 |

${a}_{3}$ | 0.35 | 0.36 | 0.32 | 0.31 | 0.37 | 0.40 | 0.37 | 0.30 | 0.27 |

${b}_{3}$ | −1.41 | −2.97 | −3.13 | −1.38 | -2.44 | −3.03 | −2.44 | −4.95 | −4.28 |

${R}_{3}^{2}$ | 0.35 | 0.36 | 0.32 | 0.31 | 0.37 | 0.40 | 0.37 | 0.30 | 0.27 |

${a}_{4}$ | 0.65 | 0.64 | 0.68 | 0.69 | 0.62 | 0.60 | 0.62 | 0.70 | 0.72 |

${b}_{4}$ | −0.18 | −0.54 | −1.03 | −0.253 | −0.36 | −0.42 | −0.36 | −2.2 | −2.4 |

${R}_{4}^{2}$ | 0.65 | 0.64 | 0.68 | 0.69 | 0.62 | 0.59 | 0.62 | 0.70 | 0.72 |

**Table 4.**Constants ${a}_{m}$, ${b}_{m}$, and ${R}_{m}^{2}$ (m = 1, 2, 3, 4) for the ferrite phase of linear equation with strain condition by the proposed iterative method.

Strain | |||||||||
---|---|---|---|---|---|---|---|---|---|

Analysis Parameter | Total Grain | Strain Level, % | Volume Fraction, % | ||||||

8400 | 1900 | 770 | 5 | 15 | 25 | 0.1 | 0.15 | 0.2 | |

Case | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

RVE Iterative Const. | A | B | C | D | D | D | D | E | F |

${a}_{1}$ | −0.78 | −0.85 | −0.82 | -0.72 | −0.73 | −0.75 | −0.73 | −0.74 | −0.73 |

${b}_{1}$ | 22.09 | 25.30 | 25.16 | 4.57 | 13.95 | 23.85 | 13.95 | 14.91 | 14.94 |

${R}_{1}^{2}$ | 0.50 | 0.57 | 0.49 | 0.49 | 0.49 | 0.49 | 0.49 | 0.51 | 0.48 |

${a}_{2}$ | 0.50 | 0.44 | 0.50 | 0.5 | 0.50 | 0.49 | 0.50 | 0.48 | 0.51 |

${b}_{2}$ | 0.01 | 0.51 | −0.31 | −0.05 | −1.87 | −0.38 | −1.87 | 0.21 | 0.16 |

${R}_{2}^{2}$ | 0.50 | 0.44 | 0.51 | 0.50 | 0.50 | 0.49 | 0.50 | 0.49 | 0.53 |

${a}_{3}$ | 0.50 | 0.56 | 0.49 | 0.49 | 0.49 | 0.50 | 0.49 | 0.50 | 0.47 |

${b}_{3}$ | −0.06 | −0.11 | −0.30 | −0.02 | −0.08 | −0.16 | −0.08 | −0.21 | −0.21 |

${R}_{3}^{2}$ | 0.50 | 0.56 | 0.48 | 0.49 | 0.49 | 0.50 | 0.49 | 0.50 | 0.47 |

${a}_{4}$ | 0.50 | 0.43 | 0.51 | 0.50 | 0.50 | 0.49 | 0.50 | 0.49 | 0.53 |

${b}_{4}$ | −0.005 | −0.01 | −0.04 | −0.02 | −0.007 | −0.01 | −0.007 | −0.04 | −0.04 |

${R}_{4}^{2}$ | 0.50 | 0.43 | 0.51 | 0.50 | 0.50 | 0.49 | 0.50 | 0.49 | 0.52 |

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## Share and Cite

**MDPI and ACS Style**

Tseng, S.; Qayyum, F.; Guk, S.; Chao, C.; Prahl, U.
Transformation of 2D RVE Local Stress and Strain Distributions to 3D Observations in Full Phase Crystal Plasticity Simulations of Dual-Phase Steels. *Crystals* **2022**, *12*, 955.
https://doi.org/10.3390/cryst12070955

**AMA Style**

Tseng S, Qayyum F, Guk S, Chao C, Prahl U.
Transformation of 2D RVE Local Stress and Strain Distributions to 3D Observations in Full Phase Crystal Plasticity Simulations of Dual-Phase Steels. *Crystals*. 2022; 12(7):955.
https://doi.org/10.3390/cryst12070955

**Chicago/Turabian Style**

Tseng, Shaochen, Faisal Qayyum, Sergey Guk, Chingkong Chao, and Ulrich Prahl.
2022. "Transformation of 2D RVE Local Stress and Strain Distributions to 3D Observations in Full Phase Crystal Plasticity Simulations of Dual-Phase Steels" *Crystals* 12, no. 7: 955.
https://doi.org/10.3390/cryst12070955