# Identifying Structure–Property Relationships Through DREAM.3D Representative Volume Elements and DAMASK Crystal Plasticity Simulations: An Integrated Computational Materials Engineering Approach

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## Abstract

Predicting, understanding, and controlling the mechanical behavior is the most important task when designing structural materials. Modern alloy systems—in which multiple deformation mechanisms, phases, and defects are introduced to overcome the inverse strength–ductility relationship—give raise to multiple possibilities for modifying the deformation behavior, rendering traditional, exclusively experimentally-based alloy development workflows inappropriate. For fast and efficient alloy design, it is therefore desirable to predict the mechanical performance of candidate alloys by simulation studies to replace time- and resource-consuming mechanical tests. Simulation tools suitable for this task need to correctly predict the mechanical behavior in dependence of alloy composition, microstructure, texture, phase fractions, and processing history. Here, an integrated computational materials engineering approach based on the open source software packages DREAM.3D and DAMASK (Düsseldorf Advanced Materials Simulation Kit) that enables such virtual material development is presented. More specific, our approach consists of the following three steps: (1) acquire statistical quantities that describe a microstructure, (2) build a representative volume element based on these quantities employing DREAM.3D, and (3) evaluate the representative volume using a predictive crystal plasticity material model provided by DAMASK. Exemplarily, these steps are here conducted for a high-manganese steel.

## Keywords

Representative Volume Element Stack Fault Energy Crystal Plasticity Trip Steel TWIP Steel## Introduction

Controlling the mechanical behavior is *the* key task when developing materials for structural applications. Replacing mechanical tests by simulation studies to evaluate the mechanical performance of candidate alloys is highly desirable as it enables a significant reduction of resource allocation in the alloy design process. However, in order to get reliable results, the simulation tool needs to correctly predict the mechanical behavior in dependence of alloy composition, microstructure, and texture.

- 1.
Acquire

*statistical*quantities that describe a microstructure. - 2.
Build a

*representative*volume element (RVE) based on these quantities using DREAM.3D. - 3.
Evaluate the RVE using a

*predictive*crystal plasticity material model implemented in the DAMASK framework.

In the proof-of-concept study presented here, statistical microstructural quantities are retrieved from an existing and experimentally well-characterized material. This allows for a comparison of calculated results with experimental data to evaluate the capabilities of the approach. However, as outlined below, using a predictive crystal plasticity model and synthetic microstructure generation raises the opportunity to use the procedure to also investigate *new* materials with the aim of forecasting suitable microstructures for different loading states.

The study is organized as follows. First, to give the reader a background of the investigated model alloy, a concise synopsis on HMnS is provided. Next, we explain how the microstructure features obtained from the experimental characterization are translated into appropriate statistical quantities. Then, it is discussed how these microstructure measures are subsequently used for the generation of representative microstructures using DREAM.3D. After that, we explain the simulation details of the crystal plasticity model implemented into the DAMASK software package. Finally, after presentation of the simulation results, we draw conclusions and provide an outlook on how to further improve and apply the methodology.

## Model Material

The concept of HMnS is based on stabilizing the face-centered cubic (fcc) austenite phase. This is usually accomplished by adding a high amount of manganese (15–30 wt.%). Small proportions of carbon (0.05–1.00 wt.%), aluminium (0.0–3.0 wt.%) and/or silicon (0.0–3.0 wt.%) can be added for tuning the stacking fault energy and the oxidation layers.9 HMnS are considered to be part of the second generation of advanced high-strength steels (AHSS). The active deformation mechanism(s) in HMnS mainly depend(s) on the stacking fault energy (SFE), which is normally below \(20 \hbox { mJ}\,\hbox {m}^{-2}\) for TRIP steels and in the range between \(20 \hbox { mJ}\,\hbox {m}^{-2}\) and \(40 \hbox { mJ}\,\hbox {m}^{-2}\) for TWIP steels. In HMnS with SFE values above \(45 \hbox { mJ}\,\hbox {m}^{-2}\), dislocation glide dominates plastic deformation, whereas TRIP and TWIP effects are suppressed.10,11 The selected TWIP steel has a composition of 22.5Mn-1.2Al-0.3C wt.%. Its SFE was determined by a subregular solution thermodynamic model12 as approximately \(25 \hbox { mJ}\,\hbox {m}^{-2}\). Hence, deformation twinning is expected to be the only active deformation mechanism besides dislocation glide. Since the material has been extensively investigated in previous works,13, 14, 15, 16, 17, 18 details on the production and post-processing procedure15,19 are not repeated here.

## Integrated Computational Materials Engineering Procedure

### Experimental Characterization

From the initial material in hot-rolled condition, six different microstructural states have been produced by imposing different (thermo-) mechanical treatments. First, the material was cold-rolled to 30%, 40%, and 50% thickness reduction. From these three states, recrystallized samples have been produced by subsequent annealing at \(700^{\circ }\) for 15 min after 30% reduction and for 10 min after 40% and 50% reductions.

Details on sample preparation for the following electron backscatter diffraction (EBSD) and x-ray diffraction analyses used for the material characterization can be found in an already published study.15

The crystallographic texture was characterized by means of x-ray pole figure measurements. Three incomplete (\(0 ^{\circ }\;\hbox {to}\; 85 ^{\circ }\)) pole figures—\(\{111\}, \{200\}\), and \(\{220\}\)—were acquired at the mid-thickness layer of the sheet on a Bruker D8 Advance diffractometer, equipped with a HI-STAR area detector, operating at 30 kV and 25 mA using filtered iron radiation and polycapillary focusing optics. The orientation distribution functions (ODFs) were calculated in the MATLAB-based MTEX package.20,21 \(\varphi _2=45 ^{\circ }\) sections of the ODF of the six investigated states are shown in Fig. 2. The corresponding legend of the ideal texture components is given elsewhere.22 With increasing rolling degree, the texture transformed gradually from Cu-type to Brass-type, as indicated by the decreasing \(\{112\}\langle 111\rangle \) Cu texture component and the more pronounced \(\mathrm {\alpha }\)-fiber (\(\langle 110\rangle \,\mathrm {\parallel }\,\)ND) with a spread towards the \(\{552\}\langle 115\rangle \) CuT texture component. After recrystallization, the rolling texture was retained but significantly weakened in intensity as a result of the oriented nucleation and the formation of recrystallization twins.23,24

### Representative Volume Element Generation Using DREAM.3D

*d*) along RD, ND, and transverse direction (TD) are similar (\(0.8< d_\mathrm {ND}/d_\mathrm {RD}< d_\mathrm {TD}/d_\mathrm {RD} < 1.0\)). The average grain diameters for each state were then determined by calculating the mean lineal intercept in 2D and extrapolating to 3D by the relationship \(d_\mathrm {3D} = 1.5 d_\mathrm {2D}\). The crystallographic texture in each state was used directly from the calculated ODFs. The microstructure was simplified in such a way that a single orientation was assigned to each grain and no internal orientation gradients exist for either cold-rolled nor cold-rolled and recrystallized states. It is important to note that the constructed volume elements are microstructurally representative, and multiple instantiations will produce volumes with similar microstructural characteristics. However, the volumes can be termed properly representative only if multiple instantiations yield similar simulated properties (Fig. 3; Table I).

Grain morphology properties of the generated representative volume elements

State | Size/\(\upmu \hbox {m}^{3}\) | Grain count | Shape |
---|---|---|---|

30% CR | 640.0 | 4086 | Ellipsoidal |

30% CR + RX | 128.0 | 3959 | Spherical |

40% CR | 640.0 | 4041 | Ellipsoidal |

40% CR + RX | 64.0 | 4585 | Spherical |

50% CR | 640.0 | 4044 | Ellipsoidal |

50% CR + RX | 51.2 | 4664 | Spherical |

### Crystal Plasticity Simulation Using DAMASK

Values of the parameters and their symbols used in the crystal plasticity model8 for all simulations

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

| | |

(b) Dislocation glide | |||
---|---|---|---|

Parameter | Symbol | Value | Unit |

Burgers vector magnitude | | 2.56 × 10 | m |

Activation energy slip | | 1.5 × 10 | J |

Activation energy climb | | 2.0 × 10 | J |

Activation volume climb | | 1.7 × 10 | m |

Obstacle profile top | | 1.0 | |

Obstacle profile bottom | | 1.0 | |

Mean free path in multiples of ... | |||

...dislocation spacing | | 45.0 | |

...twin spacing | | 1.0 | |

Self-diffusion prefactor | | 4.0 × 10 | m |

Minimum dipole spacing | | 8.96 × 10 | m |

Velocity prefactor | | 1.0 × 10 | ms |

Solution strengthening | | 2.0 × 10 | Pa |

(c) Twinning | |||
---|---|---|---|

Burgers vector | | 1.47 × 10 | m |

Nucleus width | | 2.56 × 10 | m |

Avg. twin thickness | | 5.0 × 10 | m |

Activation volume cross-slip | | 5.0 × 10 | m |

Profile width exponent | | 13.96 | |

SFE | | 25.0 | mJm |

Values of the parameters (grain size: *d*; edge dislocation density \(\rho _\mathrm {e}^0\); dipole dislocation density \(\rho _\mathrm {d}^0\); twinned volume fraction \(f_\mathrm {t}^0\)) used in the crystal plasticity model8 adjusted for specific material states

State | d/\(\upmu \hbox {m}\) | \(\rho _\mathrm {e}^0/\hbox {m}^{-2}\) | \(\rho _\mathrm {d}^0/\hbox {m}^{-2}\) | \(f_\mathrm {t}^0\) |
---|---|---|---|---|

30% CR | 28.35 | 4.5 × 10 | 8.1 × 10 | 0.0033 |

30% CR + RX | 10.04 | 1.0 × 10 | 1.0 | 0.0 |

40% CR | 24.29 | 7.0 × 10 | 1.4 × 10 | 0.0039 |

40% CR + RX | 4.78 | 1.0 × 10 | 1.0 | 0.0 |

50% CR | 20.25 | 9.2 × 10 | 1.7 × 10 | 0.0041 |

50% CR + RX | 3.80 | 1.0 × 10 | 1.0 | 0.0 |

The same material parameters as used in a previous study15 have been applied with the exception of the grain size *d* and initial values for the edge dislocation density \(\rho _\mathrm {e}\), the dipole dislocation density \(\rho _\mathrm {d}\), and the twinned volume fraction \(f_\mathrm {t}\).

For the cold-rolled and recrystallized samples, the average grain size diameter is determined from the RVE characteristics given in Table I: first, the average grain volume is computed as the RVE size divided by number of grains, followed by the calculation of the diameter of a sphere with this volume.

For the cold-rolled states, ellipsoidal grain shapes are assumed which result from the rolling of the initially spherical grains with diameter \(d=27\,\upmu \hbox {m}\). The shortest axis of the ellipsoid (along ND) \(d_\mathrm {ND} = 27\,\upmu \hbox {m} \times 3/2 \times \varepsilon \) with \(\varepsilon \in \{0.3,0.4,0.5\}\) is taken as the limiting size that determines the mean free path for dislocation glide.

The initial values of \(\rho _e^0\), \(\rho _d^0\), and \(f_\mathrm {t}^0\) are derived from the results of a plane strain compression simulation15 by calculating the average over all 12 slip and 12 twin systems at the respective deformation level, i.e. neglecting any partitioning to specific systems.

Simulations are performed using a spectral method34 coupled with DAMASK.2,35,36 The RVEs are subjected to uniaxial tension at a loading rate of \(1\times 10^{-3} \hbox {s}^{-1}\). The microstructures representing the cold-rolled states are loaded along all three directions (RD, ND, TD) to investigate the anisotropy introduced by the preceding deformation. As no such anisotropy is expected for their recrystallized counterparts, owing to the weak crystallographic textures37 and the absence of grain shape effects, those are loaded only along RD. The final strain levels have been adjusted to the experimentally obtained values for the cold-rolled states and are set to a true strain of \(\varepsilon =0.16\) for simulation in cold-rolled and recrystallized conditions.

## Simulation Results Obtained by DAMASK

^{1}, a good qualitative agreement in the elastic–plastic deformation regimes, i.e. before damage is expected to set in, can be observed. Also, the evolution of the twin volume fraction \(f_\mathrm {t}\) (not shown here) follows the experimental observations: essentially, there is no twinning activity at the simulated deformation levels15

^{2}. The yield stress values for the cold-rolled and recrystallized state are, however, lower by approximately 180 MPa and the hardening rate is underestimated.

## Conclusion and outlook

The presented ICME approach can assist, accelerate, and guide the design of new alloys and suitable processing pathways. Based on statistical descriptions of microstructures, RVEs are created on the basis of which the material’s performance is evaluated. The use of DREAM.3D enables microstructure RVE generation at high fidelity, incorporating statistical features which can be evaluated using the spectral solver and physical-based crystal plasticity laws for specific materials8,41,42 available in DAMASK. Such a strictly microstructure-oriented ICME approach enables the drastic reduction in the number of experiments typically required in alloy, microstructure, and process development.

To further strengthen the link between measured microstructural features and generated RVEs, additional statistical quantities can be taken into account. Features that might have an influence on the mechanical behavior could be the in-grain orientation distribution (especially for the cold-rolled states), the grain-to-grain misorientation in the form of a misorientation distribution function,43 and the exact grain shape. While incorporating rules to construct microstructures based on these additional pieces of information is a challenging task by itself, for a comparison with experimental results, data acquisition efforts will also increase. For both crystallographic features, the experimental characterization would need to be expanded to obtain not only global orientation information but also spatially resolved and neighborhood-sensitive quantities^{3}. Similarly, measurement complexity for a more exact grain shape determination requires statistically relevant 3D information, e.g., from EBSD measurements conducted on mutually orthogonal surfaces.

The employed crystal plasticity model has revealed its predictive capabilities for different chemical compositions and various temperatures.8 Most of the parameters used in this model have a physical meaning and can be obtained from small-scale simulations. However, two weak points in the model can be identified that are deemed to impede a quantitative agreement with experimental results. First, the grain size is only considered as a fixed average value to limit the mean free path for dislocation slip and does not affect the local hardening based on dislocation density gradients. Out of the various approaches of taking gradients into account, the physically most sound approaches are based on the flux of dislocations.42,44 While, on the one hand, the associated computational costs prohibit the use of such constitutive models for large-scale simulations as presented here, on the other hand the effect on macroscopic properties is probably not that pronounced.45 Secondly, solid solution strengthening is also not implemented in a physics-based way and its temperature dependence is neglected in the current approach.8

Another area of current research where progress will increase the quality of the results is the coupling of damage and crystal plasticity models. As damage is the dominating mechanism for the highly cold-rolled states, their behavior cannot be captured correctly within the current simulation framework. Nevertheless, current approaches of implementing damage models into DAMASK46,47 will allow the tackling of this challenge in the near future.

## Footnotes

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society. MD acknowledges the funding of the *TCMPrecipSteel* project in the framework of the SPP 1713 *Strong coupling of thermo-chemical and thermo-mechanical states in applied materials* by the Deutsche Forschungsgemeinschaft (DFG). CH, DAM, FR, and DR acknowledge gratefully the financial support of the DFG within the SFB 761 *Steel - ab initio: Quantum mechanics guided design of new Fe based materials*.

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