# Modeling the Local Deformation and Transformation Behavior of Cast X8CrMnNi16-6-6 TRIP Steel and 10% Mg-PSZ Composite Using a Continuum Mechanics-Based Crystal Plasticity Model

^{*}

## Abstract

**:**

## 1. Introduction

^{2}[7]. When these materials with low SFEs are deformed at room temperature, the stabilized Face Centered Cubic (FCC) austenitic phase transforms into the Body-Centered Cubic (BCC) ά-martensitic phase through the highly deformed Hexagonal Closest Packed (HCP) ε-martensitic phase [8,9]. Based on the experimental investigations, researchers have reported properties of the cast X8CrMnNi16-6-6 steel stacking faults [10], the martensitic lath thickness [11], the transformation behavior of the material [12], the nano hardness measurement of different phases [13], the evolution of the hardening coefficient of the material, and critical strains for transformation [9,14,15].

## 2. Micro-Mechanical Modeling

^{2}at room temperature. Therefore, they exhibit TRIP/TWIP phenomena, and with increasing temperature, the SFE increases more than 45 mJ/m

^{2}, which results in dislocation glide only. The model was further developed by Madivala, and he showed the model to be working in the range of SFE 13−118 mJ/m

^{2}[38]. The challenge is that the proposed model comprises of ~50 physical-based and fitting parameters. Although the coefficient values for the high manganese TRIP/TWIP steel model have been proposed and validated with experimental results, the identification and calibration of the fitting parameters for other materials of the same class is a challenge.

## 3. Material Data

_{2}ceramic (d10 = 25 µm, d50 = 35 µm and d90 = 48 µm) with the chemical composition presented in Table 2 was mixed with the steel powder before sintering to manufacture composite samples.

## 4. Experimentation

^{−3}s

^{−1}.

_{2}suspension with 0.02 µm particle size. EBSD scanning was done using a high-magnification field-emission SEM (MIRA 3 XMU, Tescan, Czech Republic), which is equipped with a retractable four-quadrant BSE detector and an EBSD/EDX system of EDAX/TSL (Ametek).

## 5. Numerical Simulation

^{−3}s

^{−1}, which is in the quasi-static strain range. The averaged stress and strain values at each increment obtained from the RVE simulations were compared with the experimental stress–strain (σ–ε) and the strain-hardening (dσ/dε–ε) behavior. The procedure for obtaining optimized constitutive model parameters for dislocation glide, twinning, and martensite transformation is explained elsewhere [38]. The tuned material model is used to run numerical simulations on actual EBSD data of the material. In full phase simulations, the RVEs are simulated in-plane stress conditions. Under this assumption, the three-dimensional crystal model is compelled to undergo two-dimensional deformation. Due to this assumption, quantitative discussions of the results are not possible.

## 6. Results

#### 6.1. Deformation Behavior of Steel Samples

#### 6.1.1. Global Behavior

#### 6.1.2. Local Behavior

_{2})). This martensitic evolution reduces the available mean free path for the dislocation glide, and hence the dislocation pinning occurs near these martensitic islands. This increasing dislocation pinning results in higher dislocation densities, which further reduces the plastic flow of the material in these zones and results in the global hardening of the material. These zones with a high transformed martensitic volume and a high dislocation density also bear the highest amount of stress during deformation. They are the most susceptible zones for void formation [17,18,19,48].

^{15}m

^{−2}in areas of high martensite volume. Von Mises’ true stress evolution during deformation is shown in Figure 5e. The stress is higher in areas of high martensite volume. The study of stress evolution reveals that, as expected, the martensite phase is carrying most of the applied stress when compared with austenitic zones.

#### 6.2. Deformation Behavior of Composite Samples

#### 6.2.1. Global Behavior

^{12}m

^{−2}(assigned initial dislocation density) to 17 × 10

^{13}m

^{−2}up to 20% of the strain, after which the material fails.

#### 6.2.2. Local Behavior

_{3}) at 16% of the global strain that the stress distribution inside the particles is heterogeneous. It is difficult to analyze this heterogeneity at low-magnification simulations. Therefore, a high-magnification simulation model was run to get a closer look at the overall parametric distribution in the microstructure at elevated strains.

^{14}m

^{−2}in areas of high martensite volume.

_{3}), it is observed that the stress evolution in the zirconia particle is higher near the interface. This heterogeneous distribution of stress in zirconia particles is due to three reasons. Firstly, it is due to a variation in the orientation distribution assignment (see Figure 2(a

_{1},b

_{1})). Secondly, it is due to dependence on the neighbor grain orientations. Thirdly, it is due to dependence on particle size.

## 7. Discussion

_{3})). Thus, it appears that the full strain hardening capacity of the microstructure is not exploited during the deformation. In the case of the composite depending upon the ceramic particle distribution, the heterogeneity of plastic strain accommodation differs.

## 8. Conclusions

- The developed material model with identified fitting parameters presented in this research and for the X8CrMnNi16-6-6 TRIP steel matrix and 10% Mg-PSZ composite can accurately predict the deformation behavior of steel matrix and ceramic particles with less than 2% error.
- The individual deformation phenomena of dislocation glide and transformation in the steel matrix for the current model correspond well with in-situ AE analysis data and help in associating the number of AE events with physical attributes.
- There is considerable heterogeneity in dislocation motion and strain evolution in the steel matrix with or without ceramic particles, which results in intense stress partitioning.
- Most of the local strain evolution in the steel matrix is due to dislocation motion, and the strain due to partial dislocation motion or transformation is significantly less, contributing very little to the overall strain in the material.
- It is observed that the zirconia particles take the most stress and undergo significantly less strain during deformation, which leads to high stress/strain partitioning at the ceramic/matrix interface, which is the possible reason for interface decohesion in this material.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Acronyms: | |

SSS | Solid Solution Strength |

SFE | Stacking Fault Energy |

TRIP | Transformation Induced Plasticity |

MMC | Metal Matrix Composite |

Mg-PSZ | Magnesium partially stabilized Zirconia |

EBSD | Electron Back Scatter Diffraction |

XRD | X-ray Diffraction |

AE | Acoustic Emission |

IPF | Inverse Pole Figure |

Symbols: | |

ɛ | True strain |

σ | True stress |

γ | Austenite phase (FCC) |

ρ | Dislocation density |

ΔG | Change in Gibbs free energy |

${\tau}_{eff}^{\alpha}$ | effective resolved shear stress o slip system $\alpha $ |

${\tau}_{sol}$ | Solid Solution Strength |

${b}_{s}$ | Length of Burgers vector for slip |

${v}_{0}$ | Dislocation glide velocity |

${Q}_{s}$ | The activation energy for dislocation slip |

${k}_{b}$ | Boltzmann constant |

$T$ | Temperature, in the current case, is 300 K |

$p$ | Fitting parameter for strain hardening |

$q$ | Fitting parameter for strain hardening |

$d$ | Average grain size |

${\lambda}_{slip}^{\alpha}$ | dislocation pile-up |

${\lambda}_{sliptwin}^{\alpha}$ | twin pile up |

${\lambda}_{sliptrans}^{\alpha}$ | martensite pile up |

${f}_{tw}$ | the total twin volume fraction |

${f}_{tr}$ | the total transformation volume fraction |

${t}_{tw}$ | average twin thickness |

${t}_{tr}$ | average martensite thickness |

${i}_{slip}$ | fitting parameter for slip mean free path |

${i}_{tw}$ | fitting parameter for twin mean free path |

${i}_{tr}$ | fitting parameter for transformation mean free path |

${\xi}_{\alpha \stackrel{\xb4}{\alpha}}$ | Interaction matrix between different slip systems $\alpha $ and $\stackrel{\xb4}{\alpha}$ |

${\xi}_{\beta \stackrel{\xb4}{\beta}}$ | Interaction matrix between different slip systems $\beta $ and $\stackrel{\xb4}{\beta}$ |

${\xi}_{\chi \stackrel{\xb4}{\chi}}$ | Interaction matrix between different slip systems $\chi $ and $\stackrel{\xb4}{\chi}$ |

${f}^{\stackrel{\xb4}{\beta}}$ | the volume fraction of twins for the twin system $\stackrel{\xb4}{\beta}$ |

${f}^{\stackrel{\xb4}{\chi}}$ | the volume fraction of twins for the twin system $\stackrel{\xb4}{\chi}$ |

${\widehat{\tau}}_{tw}$ | the critical stress for twining |

${\widehat{\tau}}_{tr}$ | the critical stress for transformation |

${\tau}^{\beta}$ | resolved shear stress on twin system β |

${\tau}^{\chi}$ | resolved shear stress on transformation system χ |

$A$ | fitting parameter for twin probability |

$B$ | fitting parameter for transformation probability |

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**Figure 1.**Two-dimensional (

**a**) Electron backscatter diffraction (EBSD) maps of X8CrMnNi16-6-6 Transformation-Induced Plasticity (TRIP) steel, which are used for simulations colored with an IPF scheme and an (

**b**) IPF orientation scale.

**Figure 2.**Two-dimensional EBSD maps of X8CrMnNi16-6-6 TRIP steel 10% Mg-PSZ composite, which are used for simulations, (

**a**) medium magnification, and (

**b**) high magnification. The white space in a

_{1}and b

_{1}corresponds to the embedded Zirconia particles shown in a

_{2}and b

_{2}, (

**c**) IPF orientation scale.

**Figure 3.**Numerical simulation results of the tuned model compared with experimental results of cast X8CrMnNi16-6-6 steel, (

**a**) Comparison of experimental and numerical simulation stress—strain and martensite volume evolution, (

**b**) Comparison of experimental and numerical simulation strain hardening curve, (

**c**) Evolution of martensite volume percent, TWIN volume fraction and dislocation density with increasing true strain.

**Figure 4.**Numerical simulation results of the tuned model compared with experimental results of cast X8CrMnNi16-6-6 steel.

**Figure 5.**Simulation results at different levels of applied tensile deformation in principal x-direction (left to right: 5%, 15%, and 30%) for X8CrMnNi16-6-6 TRIP steel. The evolution of the (

**a**) total dislocation density of the microstructure (Σρ

_{disloc.}) calculated by adding edge and dipole dislocation densities, the (

**b**) accumulated strain due to slipping in all planes (ɛ

_{slip}), the (

**c**) total martensitic volume percent (Σφ

_{mar}.), the (

**d**) accumulated strain due to martensitic transformation (ɛ

_{trans.}), (

**e**) VON MISES‘ stress (σ

_{eq}), and (

**f**) VON MISES‘ strain (ɛ

_{eq}).

**Figure 6.**Numerical simulation results of the tuned model compared with experimental results of cast 10% Mg-PSZ composite, (

**a**) Comparison of experimental and numerical simulation stress – strain and martensite volume evolution, (

**b**) Comparison of experimental and numerical simulation strain hardening curve, (

**c**) Evolution of martensite volume percent, TWIN volume fraction and dislocation density with increasing true strain.

**Figure 7.**Simulation results of ceramic particles at different levels of applied tensile deformation in principal x-direction (left to right: 4%, 10%, and 16%) in zirconia particles of X8CrMnNi16-6-6 TRIP steel and 10% Mg-PSZ composite. The evolution of the (

**a**) VON MISES’ stress (σ

_{eq}) and the (

**b**) VON MISES’ strain (ɛ

_{eq}).

**Figure 8.**Simulation results at different levels of applied tensile deformation in principal x-direction (left to right: 4%, 10%, and 16%) in the steel matrix of X8CrMnNi16-6-6 TRIP steel and 10% Mg-PSZ composite. The evolution of the (

**a**) total dislocation density in the microstructure (Σρ

_{disloc.}) calculated by adding edge and dipole dislocation densities, the (

**b**) accumulated strain due to slipping in all planes (ɛ

_{slip}), the (

**c**) total martensitic volume percent (Σφ

_{mar}.), the (

**d**) accumulated strain due to martensitic transformation (ɛtrans.), (

**e**) VON MISES‘ stress (σ

_{eq}), and (

**f**) VON MISES‘ strain (ɛ

_{eq}).

**Figure 9.**Simulation results of the steel matrix at different levels of applied tensile deformation in principal x-direction (left to right: 4%, 10%, and 16%) in the steel matrix of X8CrMnNi16-6-6 TRIP steel and 10% Mg-PSZ composite at high magnification. The evolution of the (

**a**) total dislocation density in the microstructure (Σρ

_{disloc.}) calculated by adding edge and dipole dislocation densities, the (

**b**) accumulated strain due to slipping in all planes (ɛ

_{slip}), the (

**c**) total martensitic volume percent (Σφ

_{mar}.), (

**d**) the accumulated strain due to martensitic transformation (ɛtrans.), (

**e**) VON MISES‘ stress (σ

_{eq}), and (

**f**) VON MISES‘ strain (ɛ

_{eq}).

**Figure 10.**Simulation results of ceramic particles at different levels of applied tensile deformation in principal x-direction (left to right: 4%, 10%, and 16%) in zirconia particles of X8CrMnNi16-6-6 TRIP steel and 10% Mg-PSZ composite at high magnification. The evolution of (

**a**) VON MISES’ stress (σ

_{eq}) and (

**b**) VON MISES’ strain (ɛ

_{eq}).

**Figure 11.**The deformed states of representative volume elements (RVEs) compared with undeformed shapes (grey outline). (

**a**) VON MISES’ stress (σ

_{eq}) states at maximum deformation and (

**b**) VON MISES’ strain (ɛ

_{eq}) states as maximum deformation. (1) is for X8CrMnNi16-6-6 TRIP steel at 30% true strain, (2) is for steel matrix in 10% Mg-PSZ composite at high magnification at 16% true strain, and (3) is for steel matrix in 10% Mg-PSZ composite at high magnification at 16% true strain.

Chemical Composition (wt.%) | Characteristics | ||||||
---|---|---|---|---|---|---|---|

C | Cr | Mn | Ni | Si | Al | N | SFE (mJ/m^{2}) [41] |

0.08 | 16.0 | 6.0 | 6.1 | 1.0 | 0.05 | 0.05 | 9.6−15.6 |

Chemical Composition (wt.%) | ||||||
---|---|---|---|---|---|---|

ZrO_{2} | MgO | Na_{2}O | CaO | TiO_{2} | Fe_{2}O_{3} | SiO_{2} |

0.08 | 16.0 | 6.0 | 6.1 | 1.0 | 0.05 | 0.05 |

**Table 3.**Single-crystal elastic constants for austenite, martensite, and zirconia incorporated in the model during simulations.

Austenite [37] | Martensite [42] | Zirconia [43] | Units |
---|---|---|---|

C_{11} = 175.0 | C_{11} = 242.3 | C_{11} = 191.0 | GPa |

C_{12} = 115.0 | C_{12} = 117.7 | C_{12} = 80.0 | GPa |

C_{44} = 135.0 | C_{13} = 45.0 | C_{44} = 40.0 | GPa |

C_{33} = 315.0 | GPa | ||

C_{44} = 40.5 | GPa |

Symbol | Description | Value | Unit | ||
---|---|---|---|---|---|

martensite transformation parameters | C_{threshhold-trans.} | Adj. parameter controlling trans threshold stress | 0.5 | -- | |

L_{tr} | Width of martensite lath during nucleation | 5.0 × 10^{−8} | m | ||

t_{tr} | Average martensite thickness | 5.0 × 10^{−6} | m | [45] | |

$B$ | ß-exponent in transformation formation probability | 3.0 | -- | ||

${i}_{tr}$ | Adj. parameter controlling trans mean free path | 10 | -- | ||

Twinning formation parameters | $A$ | r-exponent in twin formation probability | 1.0 | -- | |

${i}_{tw}$ | Adj. parameter controlling twin mean free path | 5.0 | -- | ||

C_{threshhold-TWIN.} | Adj. parameter controlling twin threshold stress | 1.3 | -- | ||

${\Gamma}_{sf}$ | Stacking fault energy | 10 | mJ/m² | [7] | |

ΔG | Change in Gibbs free energy | −2.54 × 10^{7} | J/m^{3} | [46] | |

Dislocation glide parameters | ${\tau}_{sol}$ | Solid solution strength | 5 × 10^{7} | Pa | |

${\rho}_{ini}$ | Initial dislocation density | 1.0 × 10^{12} | m^{−2} | [37] | |

${i}_{slip}$ | Adj. parameter controlling dislocation mean free path | 55 | -- | ||

d | Avg. grain size | 50 × 10^{−6} | m | [3] |

**Table 5.**Optimized constitutive model parameters for the X8CrMnNi16-6-6 TRIP steel matrix in 10% Mg-PSZ MMC.

Symbol | Description | Value | Unit | |
---|---|---|---|---|

Dislocation glide parameters | ${\tau}_{sol}$ | Solid solution strength | 8.5 × 10^{7} | Pa |

${i}_{slip}$ | Adj. parameter controlling dislocation mean free path | 350 | -- | |

Martensite transformation parameters | ${i}_{tr}$ | Adj. parameter controlling trans mean free path | 5.0 | -- |

$B$ | ß-exponent in transformation formation probability | 3.0 | -- |

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## Share and Cite

**MDPI and ACS Style**

Qayyum, F.; Guk, S.; Schmidtchen, M.; Kawalla, R.; Prahl, U.
Modeling the Local Deformation and Transformation Behavior of Cast X8CrMnNi16-6-6 TRIP Steel and 10% Mg-PSZ Composite Using a Continuum Mechanics-Based Crystal Plasticity Model. *Crystals* **2020**, *10*, 221.
https://doi.org/10.3390/cryst10030221

**AMA Style**

Qayyum F, Guk S, Schmidtchen M, Kawalla R, Prahl U.
Modeling the Local Deformation and Transformation Behavior of Cast X8CrMnNi16-6-6 TRIP Steel and 10% Mg-PSZ Composite Using a Continuum Mechanics-Based Crystal Plasticity Model. *Crystals*. 2020; 10(3):221.
https://doi.org/10.3390/cryst10030221

**Chicago/Turabian Style**

Qayyum, Faisal, Sergey Guk, Matthias Schmidtchen, Rudolf Kawalla, and Ulrich Prahl.
2020. "Modeling the Local Deformation and Transformation Behavior of Cast X8CrMnNi16-6-6 TRIP Steel and 10% Mg-PSZ Composite Using a Continuum Mechanics-Based Crystal Plasticity Model" *Crystals* 10, no. 3: 221.
https://doi.org/10.3390/cryst10030221