# Combining Single- and Poly-Crystalline Measurements for Identification of Crystal Plasticity Parameters: Application to Austenitic Stainless Steel

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Constitutive Model

## 3. Two-Scale Calibration Strategy

- The hardening parameters are identified on a grain scale by fitting the calculated tensile responses to corresponding experimental stress-strain curves. Simultaneous fitting is performed against single and polycrystalline tensile data.
- Before being used in the calibration, the length scale effects due to grain boundary strengthening need to be subtracted from the measured polycrystalline data. The Hall–Petch relation is used to obtain the adjusted (infinite-grain-size) macroscopic tensile stress (${\sigma}_{0}\left(\u03f5\right)$ in Equation (1)).
- The RVE model of the polycrystalline microstructure is identified and used in the identification procedure. To allow for fast simulations during calibration iterations, a small representative polycrystalline model is determined, containing a comparable number of finite elements as single crystalline FE models.
- The automated optimization procedure for hardening parameters is introduced where parameters are constrained to take only discrete values with a prescribed precision (step size).

#### 3.1. Subtracting Grain Boundary Strengthening Effects from Raw Polycrystalline Data

#### 3.2. Representative Volume Element

#### 3.3. Single-Crystal Models

#### 3.4. Automated Optimization of Model Parameters Defined on Discrete Range

## 4. Results of Model Calibration

#### 4.1. Introduction of Length Scale into Crystal Plasticity

## 5. Discussion

#### 5.1. Comparison with Simpler Hardening Models

#### 5.2. Local Fields

## 6. Conclusions

- A relatively small value of ${\gamma}_{0}^{I}\sim 0.02$ and a high value of ${f}_{0}\sim 0.67$ in the Bassani and Wu hardening law indicate a short Stage I region and hard activation of cross slip during Stage II hardening. A low value of $q\sim 0.2$ also suggests that self-hardening dominates over latent hardening.
- High similarities between two-stage and three-stage Bassani and Wu hardening models indicate a very slow Stage II to Stage III transition with slip.
- Grain boundary strengthening effects become relatively weak when average grain size $\langle D\rangle \gtrsim 90$ $\mathsf{\mu}$m, thus providing negligible additional hardening in the tensile response.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Typical ${\tau}^{\alpha}-{\gamma}^{\alpha}$ curve of a pure FCC single crystal loaded in a uniaxial tension with an initial orientation for single slip. (

**b**) Polycrystalline tensile data of 316L stainless steel at room temperature for various average grain sizes $\langle D\rangle $. Data were extracted from [31]. The red line denotes the extrapolated tensile curve ${\sigma}_{0}\left(\u03f5\right)$ using the Hall–Petch relation.

**Figure 2.**Examples of polycrystalline aggregate models with realistic gauge geometry: (

**a**) ${N}_{g}=64$, ${N}_{eg}=8$, (

**b**) ${N}_{g}=64$, ${N}_{eg}=64$, and (

**c**) ${N}_{g}=512$, ${N}_{eg}=64$. (

**d**) Single crystal model with ${N}_{eg}=48$. The colors denote different grains with common crystallographic orientation. Grains are meshed by linear hexahedral elements (C3D8) in polycrystalline models and by quadratic hexahedral elements with reduced integration (C3D20R) in a single crystal model. Arrows in (

**a**,

**d**) denote the boundary conditions used in tensile simulations.

**Figure 3.**Calculated tensile responses of some of the aggregate models from Table 1 with (

**a**) fixed ${N}_{g}=64$ and increasing ${N}_{eg}$ and (

**b**) fixed ${N}_{eq}=1$ and increasing ${N}_{g}$. The two arrows depict the values of strains for which a convergence analysis was performed.

**Figure 4.**Convergence analysis of tensile stress ${\sigma}_{33}$ calculated at (

**a**,

**b**) ${\u03f5}_{33}=0.1$ and (

**c**,

**d**) ${\u03f5}_{33}=0.2$ as a function of the number of grains ${N}_{g}$ and the number of elements per grain ${N}_{eg}$. Dashed lines represent a fit with a power law; Equation (9). Extrapolated stress value ${\sigma}_{33}^{*}$ is marked by a horizontal dashed line. The representative volume element (RVE) model is marked by a red square.

**Figure 5.**Results of the two-scale calibration showing a comparison between calculated tensile curves of the crystal plasticity (CP) model with the Bassani and Wu hardening law (BW3) hardening model and corresponding measurements on 316L stainless steel [30,31]. The calculated lines are fitted to the experimental data using four different fitting domains: (

**a**) ${e}_{max}=0.1$, (

**b**) ${e}_{max}=0.2$, (

**c**) ${e}_{max}=0.3$ and (

**d**) ${e}_{max}=0.4$. The corresponding hardening parameters are shown in Table 4.

**Figure 6.**(

**a**) Results of the calibration of the modified BW3 hardening model, Equations (14) and (15), calculated with newly identified parameters ${k}_{0}$ and ${k}_{1}$ (shown in legend in units of MPa$\sqrt{\mathsf{\mu}\mathrm{m}}$) and previously identified set ${P}_{4}$ from Table 4. A comparison is shown with tensile measurements on 316L stainless steel at room temperature [31]. (

**b**) Same as (a), but using Equations (18) and (19) for the modified BW3 hardening model and experimental Hall–Petch slope $K(\u03f5,\langle D\rangle )$ (shown in the legend in units of MPa$\sqrt{\mathsf{\mu}\mathrm{m}}$) from [31].

**Figure 7.**Results of the two-scale calibration showing a comparison between calculated tensile curves and corresponding measurements on 316L stainless steel [30,31]. The calculated lines are fitted to the experimental data on a fixed strain domain ${e}_{max}=0.2$ using four different strategies: (

**a**) BW2 hardening model, (

**b**) PAN hardening model, (

**c**) BW3 hardening model, but with raw polycrystalline data with $\langle D\rangle =3.1$ $\mathsf{\mu}$m, and (

**d**) BW3 hardening model, but with raw polycrystalline data with $\langle D\rangle =86.7$ $\mathsf{\mu}$m. The corresponding hardening parameters are shown in Table 4 and Table 5.

**Figure 8.**(

**a**) Voronoi aggregate model with 1000 grains denoted by different colors. Grains are meshed by quadratic tetrahedral elements C3D10. (

**b**) Tensile curves calculated with a Voronoi model using best-fit parameters ${P}_{2}$, ${P}_{6}$ and ${P}_{10}$ from Table 4. In the inset, also the results of the RVE model are shown for comparison.

**Figure 10.**(

**a**) Difference of the von Mises stress fields $\Delta {\sigma}_{mis}$ between the BW3 and BW2 models and between the BW3 and PAN models from Figure 9. (

**b**) Distributions of $\Delta {\sigma}_{mis}$ calculated from all integration points of the Voronoi model at $e=0.2$.

**Table 1.**A list of aggregate models used in this study. ${N}_{g}$ stands for the number of grains, ${N}_{eg}$ for the number of elements per grain and ${N}_{e}$ for the number of total elements.

i | ${\mathit{N}}_{\mathit{g}}$ | ${\mathit{N}}_{\mathbf{eg}}$ | ${\mathit{N}}_{\mathit{e}}$ |
---|---|---|---|

1 | $2\times 4\times 8=64$ | ${1}^{3}=1$ | 64 |

2 | 64 | ${2}^{3}=8$ | 512 |

3 | 64 | ${4}^{3}=64$ | 4096 |

4 | 64 | ${8}^{3}=512$ | 32,768 |

5 | 64 | ${16}^{3}=4096$ | 262,144 |

6 | $4\times 8\times 16=512$ | 1 | 512 |

7 | 512 | 8 | 4096 |

8 | 512 | 64 | 32,768 |

9 | 512 | 512 | 262,144 |

10 | $8\times 16\times 32=4096$ | 1 | 4096 |

11 | 4096 | 8 | 32,768 |

12 | 4096 | 64 | 262,144 |

13 | $16\times 32\times 64$ = 32,768 | 1 | 32,768 |

14 | 32,768 | 8 | 262,144 |

15 | $32\times 64\times 128$ = 262,144 | 1 | 262,144 |

${\mathit{\u03f5}}_{33}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | ${\mathit{\sigma}}_{33}^{*}$ |
---|---|---|---|---|---|

$0.1$ | 126 MPa | $0.244$ | $56.1$ MPa | $0.230$ | 359 MPa |

$0.2$ | 177 MPa | $0.260$ | $94.6$ MPa | $0.184$ | 461 MPa |

**Table 3.**A list of fitting parameters with corresponding minimum and maximum values used in the calibration procedure.

${\mathit{p}}_{\mathit{i}}$ | ${\mathit{p}}_{\mathit{i},\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{p}}_{\mathit{i},\mathit{m}\mathit{a}\mathit{x}}$ |
---|---|---|

${\tau}_{0}$ | 50 MPa | 300 MPa |

${\tau}_{s}$ | 50 MPa | 3000 MPa |

${h}_{0}$ | 0 MPa | 1000 MPa |

${h}_{s}^{I}$ | 0 MPa | 1000 MPa |

${h}_{s}^{III}$ | 0 MPa | 1000 MPa |

${\gamma}_{0}$ | 0 | 1 |

${\gamma}_{0}^{III}$ | 0 | 1 |

${f}_{0}$ | 0 | 1 |

q | 0 | $1.4$ |

**Table 4.**Hardening parameters for 316L stainless steel obtained from the two-scale calibration procedure. Different hardening models were considered in the calibration: BW3 (Bassani and Wu [29] with Stages I, II, III), BW2 (Bassani and Wu [28] with Stages I, II) and PAN (Peirce, Asaro and Needleman [40] with Stage I). Parameters ${h}_{i}$, ${\tau}_{i}$ and $\sqrt{{\chi}^{2}}$ are shown in units of MPa.

Model | Set | ${\mathit{e}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{\tau}}_{0}$ | ${\mathit{\tau}}_{\mathit{s}}$ | ${\mathit{h}}_{0}$ | ${\mathit{h}}_{\mathit{s}}^{\mathit{I}}$ | ${\mathit{h}}_{\mathit{s}}^{\mathit{III}}$ | ${\mathit{\gamma}}_{0}^{\mathit{I}}$ | ${\mathit{\gamma}}_{0}^{\mathit{III}}$ | ${\mathit{f}}_{0}$ | q | $\sqrt{{\mathit{\chi}}^{2}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

BW3 | ${P}_{1}$ | $0.1$ | 83.1 | 88.3 | 214 | 159 | 111 | 0.034 | 0.186 | 0.328 | 0.836 | 6.70 |

${P}_{2}$ | $0.2$ | 82.8 | 96.0 | 394 | 71 | 86 | 0.018 | 0.104 | 0.674 | 0.175 | 12.6 | |

${P}_{3}$ | $0.3$ | 82.8 | 96.0 | 410 | 98 | 86 | 0.019 | 0.158 | 0.674 | 0.172 | 14.6 | |

${P}_{4}$ | $0.4$ | 82.8 | 96.0 | 459 | 125 | 89 | 0.020 | 0.186 | 0.647 | 0.172 | 20.7 | |

BW2 | ${P}_{5}$ | $0.1$ | 83.1 | 84.9 | 210 | 150 | 0.026 | 0.321 | 0.836 | 7.20 | ||

${P}_{6}$ | $0.2$ | 82.8 | 97.9 | 386 | 70 | 0.018 | 0.674 | 0.175 | 12.9 | |||

${P}_{7}$ | $0.3$ | 84.5 | 96.0 | 418 | 98 | 0.019 | 0.674 | 0.172 | 16.3 | |||

${P}_{8}$ | $0.4$ | 82.0 | 93.9 | 418 | 91 | 0.016 | 0.659 | 0.172 | 21.6 | |||

PAN | ${P}_{9}$ | $0.1$ | 78.6 | 2754 | 224 | 1.51 | 9.90 | |||||

${P}_{10}$ | $0.2$ | 78.6 | 200 | 215 | 1.75 | 27.5 | ||||||

${P}_{11}$ | $0.3$ | 78.6 | 200 | 215 | 1.75 | 29.7 | ||||||

${P}_{12}$ | $0.4$ | 78.6 | 204 | 228 | 1.68 | 35.4 |

**Table 5.**Hardening parameters for the BW3 model [29] for 316L stainless steel obtained from the two-scale calibration procedure when using raw polycrystalline data with different $\langle D\rangle $ and fixed ${e}_{max}=0.2$. Parameters ${h}_{i}$, ${\tau}_{i}$ and $\sqrt{{\chi}^{2}}$ are shown in units of MPa and $\langle D\rangle $ in $\mathsf{\mu}$m.

Model | Set | $\langle \mathit{D}\rangle $ | ${\mathit{\tau}}_{0}$ | ${\mathit{\tau}}_{\mathit{s}}$ | ${\mathit{h}}_{0}$ | ${\mathit{h}}_{\mathit{s}}^{\mathit{I}}$ | ${\mathit{h}}_{\mathit{s}}^{\mathit{III}}$ | ${\mathit{\gamma}}_{0}^{\mathit{I}}$ | ${\mathit{\gamma}}_{0}^{\mathit{III}}$ | ${\mathit{f}}_{0}$ | q | $\sqrt{{\mathit{\chi}}^{2}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

BW3 | ${P}_{13}$ | $3.1$ | 91.1 | 104 | 292 | 0 | 99 | 0.001 | 0.150 | 0.930 | 0.060 | 60.3 |

${P}_{14}$ | $86.7$ | 82.8 | 96.0 | 378 | 93 | 87 | 0.016 | 0.203 | 0.674 | 0.175 | 16.1 | |

${P}_{2}$ | ∞ | 82.8 | 96.0 | 394 | 71 | 86 | 0.018 | 0.104 | 0.674 | 0.175 | 12.6 |

**Table 6.**Hardening parameters for 316L stainless steel taken from the literature. Note that only (raw) polycrystalline data were used in the identification procedure. Parameters ${h}_{i}$ and ${\tau}_{i}$ are shown in units of MPa.

Model | ${\mathit{\tau}}_{0}$ | ${\mathit{\tau}}_{\mathit{s}}$ | ${\mathit{h}}_{0}$ | ${\mathit{h}}_{\mathit{s}}^{\mathit{I}}$ | ${\mathit{h}}_{\mathit{s}}^{\mathit{III}}$ | ${\mathit{\gamma}}_{0}^{\mathit{I}}$ | ${\mathit{\gamma}}_{0}^{\mathit{III}}$ | ${\mathit{f}}_{0}$ | q | Source |
---|---|---|---|---|---|---|---|---|---|---|

BW2 | 150 | 75 | 75 | 30 | 0 | 1 | 1 | [6,15] | ||

PAN | 150 | 75 | 75 | 1 | [14,45] | |||||

PAN | 90 | 175 | 675 | 1 | [32] |

© 2017 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/).

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**MDPI and ACS Style**

El Shawish, S.; Cizelj, L. Combining Single- and Poly-Crystalline Measurements for Identification of Crystal Plasticity Parameters: Application to Austenitic Stainless Steel. *Crystals* **2017**, *7*, 181.
https://doi.org/10.3390/cryst7060181

**AMA Style**

El Shawish S, Cizelj L. Combining Single- and Poly-Crystalline Measurements for Identification of Crystal Plasticity Parameters: Application to Austenitic Stainless Steel. *Crystals*. 2017; 7(6):181.
https://doi.org/10.3390/cryst7060181

**Chicago/Turabian Style**

El Shawish, Samir, and Leon Cizelj. 2017. "Combining Single- and Poly-Crystalline Measurements for Identification of Crystal Plasticity Parameters: Application to Austenitic Stainless Steel" *Crystals* 7, no. 6: 181.
https://doi.org/10.3390/cryst7060181