# Numerical Modeling of Phase Transformations in Dual-Phase Steels Using Level Set and SSRVE Approaches

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of the Phase Transformation Models

## 3. Methodology of Determining SSRVEs

_{i}—coefficient weight, and ε—penalty function value. The penalty function includes checking the shape for intersecting segments.

## 4. Phase Transformation Model

#### 4.1. Level Set Method (LSM)

_{I}—standard interpolation function in the Galerkin method, h—characteristic dimension of the element, u

_{k}—advection velocity, and α—weight parameter of SUPG scheme. The final system of equations takes the form:

**M**and

**K**matrices take the following forms:

_{0}. The solution is delivered in an iterative manner until a steady state is obtained in the entire computational domain of the solution, i.e., the condition is satisfied:

_{p}parameter.

#### 4.2. MLS-DIFF Model

_{0}—mobility constant, Q

_{M}—activation energy, R—gas constant, and T—temperature. The driving force, ΔG, in equation (16) depends on the carbon concentration and takes the form:

_{γα}—equilibrium carbon concentration in austenite, and C

_{γ}—average carbon concentration in austenite.

_{i}

^{c}—value of the i-th level function for a given degree of freedom after correction and φ

_{i}

^{p}—value of the j-th level function before correction. The final grain morphology is obtained using a property of the distance sum function that preserves the position of all the boundaries:

_{i}—value of level set function at a given position.

#### 4.2.1. Austenitic Transformation

_{e}

_{1}. We assume that the functions describing carbon concentration in the austenite at the γ-α boundary (C

_{γα}) and at the γ-cementite boundary (C

_{γβ}) are described by the following relations:

_{e}

_{1}, is the point of intersection of these functions:

_{e}

_{1}, into Function (22). Based on the current simulation time, the temperature is determined, from which the diffusion coefficient and the boundary motion condition are calculated. For simplicity, it is assumed that carbon diffusion occurs only in the austenite, which is dictated by the limited solubility of carbon in ferrite. The boundary motion condition for each grain is that the average carbon concentration at the boundary (c

_{Γ}) exceeds the maximum carbon content in the austenite at that temperature:

#### 4.2.2. Ferritic Transformation

_{v}, to be estimated in the form:

^{13}s

^{−1}), k—Boltzmann constant, T—temperature, R—gas constant, E

_{M}—activation energy of interfacial diffusion, N

_{het}—density of nucleation sites, and $\Delta {G}^{*}$—the nucleation barrier energy that must be overcome to produce an embryo of critical size. The nucleation barrier energy is determined from the relationship:

_{v}—number of carbon atoms per unit volume, ρ—effective grain boundary thickness ($2.5\times {10}^{-10}\mathrm{m}$), d

_{γ}—grain diameter of the parent phase, and v—factor capturing the dimension of nucleation sites (homogeneous nucleation—three, along the grain boundary plane—two, along grain edges—one, and at triple points—0).

_{γα}(T) in the form of the Dirichlet condition. The concentration value at the boundary is calculated from the GS line of the equilibrium diagram (the line below which austenitic transformation begins). The initial condition for the diffusion simulation is a given carbon concentration in the austenite region equal to the average carbon content of the steel. The concentration distribution can be homogeneous or heterogeneous, depending on the assumptions of the simulated process. After solving the diffusion equation, the mass conservation condition (carbon concentration), which is a predictor of the boundary motion, is checked:

## 5. Model Verification

#### 5.1. Laboratory Verification

- Each of the selected temperature cycles was mapped using the MLS-DIFF model. For this purpose, a digital representation of a 100 × 100 μm-base microstructure with an average grain size of 30μm was prepared and simulations were performed on it. The initial microstructure is ferritic–perlitic. Obtained results from the EBSD analysis are presented in Figure 9. It can be seen that in the case of thermal cycle one and two, diffusive transformation products were obtained (with equiaxed ferrite morphologies), while in the case of thermal cycle three and four, acicular ferrite and bainite were produced, respectively.

#### 5.2. Industrial Verification

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

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**Figure 1.**The idea of the statistically similar representative element. (

**a**) Reference microstructure image and (

**b**) periodic SSRVE [16].

**Figure 5.**Finite element. The idea of boundary discretization in the geometric reinitialization method.

**Figure 10.**Final microstructure morphology (ferrite) predicted by the model with carbon concentration distribution in martensite for (

**a**) case one and (

**b**) case two.

**Figure 11.**Kinetics of ferritic transformation according to MLS-DIFF model, comparison to experimental results.

**Figure 13.**(

**a**) Photo of the microstructure of steel B in the cold-rolled condition and (

**b**) image of the ferritic–perlite microstructure after image postprocessing.

**Figure 14.**Examples of the SSRVEs based on the microstructure of B steel with: (

**a**) two, (

**b**) three and (

**c**) four grains.

**Figure 15.**Morphology of the microstructure of steel B and carbon concentration distribution at (

**a**) 870 °C, (

**b**) after cooling to room temperature, and (

**c**) the kinetics of ferritic transformation—comparison of the model with the experimental results.

**Figure 16.**Simulation of phase transformation upon heating using SSRVEs with two (

**a**), three (

**b**), and four (

**c**) grains.

**Figure 17.**Simulation of the phase transformation under cooling using SSRVEs, with three perlite grains (

**a**) and four grains (

**b**). Comparison of transformation kinetics obtained for SSRVEs and the RVE (

**c**).

**Figure 18.**(

**a**) Temperature cycle of continuous annealing; (

**b**) microstructure morphology of steel B at 810 °C and carbon concentration distribution in the austenite grains.

**Figure 19.**Microstructure morphology and carbon concentration distribution predicted by the model at 450 °C after 80 s (

**a**) and at the end of the cooling cycle (

**b**). SEM photography after the same cycle (

**c**) [35].

Steel | C | Al | Cr | Cu | Mn | Mo | Ni | Si | N | P | S |
---|---|---|---|---|---|---|---|---|---|---|---|

A | 0.13 | 0.003 | 0.06 | 0.27 | 0.47 | 0.02 | 0.09 | 0.16 | 0.009 | 0.018 | 0.025 |

B | 0.09 | 0.053 | 0.35 | 0.03 | 1.42 | 0.005 | 0.01 | 0.1 | 0.009 | 0.011 | 0.01 |

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Bzowski, K.; Rauch, Ł.; Pietrzyk, M.; Kwiecień, M.; Muszka, K. Numerical Modeling of Phase Transformations in Dual-Phase Steels Using Level Set and SSRVE Approaches. *Materials* **2021**, *14*, 5363.
https://doi.org/10.3390/ma14185363

**AMA Style**

Bzowski K, Rauch Ł, Pietrzyk M, Kwiecień M, Muszka K. Numerical Modeling of Phase Transformations in Dual-Phase Steels Using Level Set and SSRVE Approaches. *Materials*. 2021; 14(18):5363.
https://doi.org/10.3390/ma14185363

**Chicago/Turabian Style**

Bzowski, Krzysztof, Łukasz Rauch, Maciej Pietrzyk, Marcin Kwiecień, and Krzysztof Muszka. 2021. "Numerical Modeling of Phase Transformations in Dual-Phase Steels Using Level Set and SSRVE Approaches" *Materials* 14, no. 18: 5363.
https://doi.org/10.3390/ma14185363