# A Comparative Study of Deterministic and Stochastic Models of Microstructure Evolution during Multi-Step Hot Deformation of Steels

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{e}

_{3}). The task of the hot forming model is to supply the input data (histograms of the dislocation density and the grain size) for the further simulation of the phase transformations.

## 2. Constitutive Model Based on Internal Variables

**p**: σ

_{p}= f(

**p**). The weak point of these models is that they do not account for the history of the process. Whenever the temperature or the strain rate change rapidly, the calculated flow stress immediately moves to a new equation of state and is a function of the new values of the external variables. On the other hand, it was observed experimentally that the material’s response is delayed [38]. This delay is more significant for pure metals, but it was also observed for alloys (steels) [39]. These observations led to the development of the models based on the internal variables (IVM). In the IVM, the output is a function of time t, with some process parameters

**p**and internal variables grouped in the vector

**q**: σ

_{p}= f(t,

**p**,

**q**). Since the internal variables represent the state of the material, the IVMs account for the delay of the change in this state due to kinetics of the changes of the microstructure. Dislocation density is the main internal variable. Below is a brief introduction and physical background for modelling deformation, based on the dislocation theory.

#### 2.1. Deterministic Model

_{1}= 1/(bl), b—length of the Burgers vector, l = ${a}_{1}{Z}^{-{a}_{2}}$—mean free path for the dislocations, Z—Zener-Hollomon parameter, A

_{2}= a

_{3}exp[−a

_{4}/(RT)], R—gas constant, T—temperature in K, a

_{4}—activation energy for self-diffusion, a

_{7}—strain rate sensitivity of the dynamic recovery, and a

_{1}, a

_{2}, and a

_{3}—other coefficients.

_{cr}—time at which recrystallization begins, as a consequence of reaching critical dislocation density ρ

_{cr}and, A

_{3}= a

_{5}exp[−a

_{6}/(RT)], a

_{6}—activation energy for recrystallization, a

_{5}—coefficient, and ${1}_{\left({t}_{cr},+\infty \right)}\left(t\right)$—indicator function of $({t}_{cr},+\infty )$. The indicator function represents the delay in response to the change in processing conditions as it switches on the last part of the equation when t > t

_{cr}. Consequently, the Equation (2) is a delayed differential equation (DDE) with respect to time. Besides the numerical solution for this equation, a detailed theoretical analysis of this equation in the case when $\dot{\epsilon}=1$ was performed in [51]. Classical approaches to the ordinary differential equations (ODE), presented in the literature e.g., [52,53], require some regularity of right-hand side function, usually Lipschitz condition, which is not satisfied by (2). Therefore, Ref. [51] also contains rigorous analysis of the error for the considered equations. It is also interesting that while [54] shows that simple DDEs in the form of (2) cannot be easily used in the modelling of biological or physical phenomena, a natural application to materials science is discussed in [51]. Specifically, it appears that admissible solutions of (2) for the case when $\dot{\epsilon}=1$ exist are bounded and, furthermore, when ${a}_{8}\in \left\{0,1\right\},$it is possible to derive the exact formulas for ${t}_{cr}$, which is the time when the recrystallization occurs (Equation (15) in [49]):

_{p}accounting for softening due to recrystallization and recovery. The flow stress is proportional to the square root of the dislocation density:

_{6}—coefficient, depending on the material, and G—shear modulus.

#### 2.2. Case Study—Deterministic Internal Variable Model

**σ**, $\dot{\epsilon}$—stress and strain rate tensors, respectively, ${\dot{\epsilon}}_{i}$—effective strain rate, and σ

_{p}—the flow stress provided by (6). An effect of this approach is briefly presented below.

_{0}equals 10

^{4}m

^{−2}. The entry temperature is equal to 1060 °C. The results were determined using the Euler method with time-dependent coefficients ${A}_{1}\left(t\right),{A}_{2}\left(t\right),{A}_{3}\left(t\right)$. Analysis of these results shows that they correctly reproduce the effect of distinct temperature and strain rate histories for the two considered areas. One can see in Figure 2 that, in the center of the strip, while the temperature increases (Figure 1) due to deformation heating, the dislocation density decreases as an effect of the dynamic recrystallization. Additionally, the dislocation density in the surface increases during the temperature decrease because of heat transfer to the cool roll. Consequently, the critical dislocation density for DRX is higher and reached later. In the central part, the strain rate decreases monotonically by cause of the monotonic deformation of this part. The results from the Euler method presented in Figure 2 replicate proper material behavior in these conditions of the deformation. For a detailed description, see [51].

## 3. Stochastic Model for Hot Forming

_{cr}, which is an artificial parameter. The reason for this situation is that in a real material, the onset of the recrystallization may occur in a different time in various parts (various material points) and this process is highly stochastic in nature. The model based on Equation (2) describes the process on average; therefore, it cannot completely and adequately reproduce the stochastic behavior of the material, providing only partial insight into the process. Beyond this, the critical time t

_{cr}is not a physical quantity. It is important to realize that using this time in the model only allows use to average the material response to deformation, which is a weak point of the model (2). A stochastic approach based on Equation (2) is a method to avoid the artificial critical time and build the model, which is closer to the reality.

#### 3.1. State-of-the-Art in Stochastic Approach to Modelling Recrystallization

#### 3.2. Mathematical Background

#### 3.3. Main Equations of the Model

_{cr}with a nucleation probability. To complete this task, authors of [51] first presented the Equation (2) in a finite difference form:

_{i}—the time of ith iteration and Δt—a time step.

_{i}). This way, the evolution of the dislocation density is no longer a deterministic function of time, becoming stochastic:

_{1}and A

_{2}which, similarly to (1), represent hardening and recovery in the material (see [51] for more details). At the same time, we removed coefficient A

_{3}connected with critical time in (2); its role is taken by ξ(t

_{i}). Its distribution is defined by the following formulas:

_{i})). The second is the impingement factor (1 − X(t

_{i})), which takes into account the fact that the migrating boundary encounters ever-more regions which have already been recrystallized. Since in the Poissonian model of the nuclea-tion the recrystallized volume fraction X(t

_{i}) is not known for individual Monte Carlo points, it was substituted by the probability

**P**(ξ(t

_{i}

_{−1}) = 0). Thus, in the first approach the migrating boundary area related to the grain size is expressed by the following equation:

**P**(ξ(t

_{−}

_{1}) = 0) = 0. The authors of [57] set it to q = 0.1 and also define ξ(t

_{0}) = 0.

_{i}) according to rules in (13). Initial values ρ(t

_{0}) are drawn from the Gauss distribution, with the expected value selected as 10

^{4}m

^{−2}for start time t

_{0}= 0. The results of these computations were then aggregated into histograms, one for each time step t

_{i}.

_{i}

_{−1}) = n

_{X}/n

_{MC}, n

_{X}—number of points, which have recrystallized before the time t

_{i}, and n

_{MC}—number of Monte Carlo points.

**a**. The coefficients were identified using an inverse solution for the experimental data, some of which were in the form of histograms describing the grain size.

#### 3.4. Identification

**a**= {a

_{1}, …, a

_{16}}. The general algorithm of identification is well known [37] and beyond this survey. It is sufficient to say that we have to perform an optimization task, with vector

**a**becoming input variable of objective function [31]. The authors in [35] proposed this function to be:

_{c}(

**a**)—histogram calculated on the basis of several model simulations of, H

_{m}—histogram measured in the experiment, and d—a ranking function comparing two histograms.

_{m}). As its counterpart, model flow stress (σ

_{c}) was obtained by the application of (6). This way, an extended version of (15) was proposed as an objective function:

_{σ}, w

_{D}—weighted coefficients.

_{ci}(

**a**), σ

_{mi}) between measured and calculated average dislocation density in the i-th experiment was measured as the mean square root error (MSRE):

#### 3.5. Numerical Tests

_{b}), Monte Carlo points (n

_{MC}), and time steps (n

_{t}) were performed and the following optimal values were selected: n

_{b}= 10 and n

_{t}= 20,000. In order to test the impact of a time step on a solution, the cumulative probability of dislocation density reduction after the given time was computed in [35]. Roughly speaking, it is the probability that ξ(t

_{i}) (Equation (10)) obtained the value ξ(t

_{i}) = 1 at least once during the process. While probabilities in Equation (10) depend on length of time step, the results of the numerical tests in [35] showed cumulative probabilities that looked similar despite different time steps. The optimal number of time steps was obtained by adapting these steps to the temperature variations so that the time step does not exceed 0.1 s and the temperature change in the one-time step does not exceed 0.1 °C. The optimal parameters were selected by searching for the balance between the accuracy (repeatability of distributions, convergence) and the computing time. Five various metrics were analyzed in [35] as a measure of the distance between histograms and, as a result, the Bhattacharyya function [65] was selected as the best performing. Its utility was independent of the optimization method (two were tested: Particle Swarm Optimization (PSO) and Nelder–Mead Simplex Method).

## 4. Case Studies—Stochastic Model

#### 4.1. Varying Strain Rate Test

^{−1}and 1 s

^{−1}was at the strain of 0.4 and the second change between 1 s

^{−1}and 10 s

^{−1}was at the strain of 0.6. Symmetrically, simulations were performed for the strain rate decreasing from 10 s

^{−1}to 1 s

^{−1}and to 0.1 s

^{−1}. The average austenite grain size after pre-heating was 35 μm and the sample temperature was 1150 °C, which meant that, in the slow test at the strain of 0.4, the dynamic recrystallization (DRX) had already begun. The Finite Element (FE) program was applied to calculate current local temperatures and strain rates in the sample, accounting for the inhomogeneity of deformation. Our in-house FE program [69] was used and it was possible to solve the stochastic model in each Gauss integration point in the FE mesh using calculated temperatures and strain rates as inputs. The results for the center of the sample are presented below. Changes in the flow stress in the varying strain rate tests are shown in Figure 3. A delay in the material response is well seen in this figure. Histograms of the dislocation density after these tests are shown in Figure 3b and histograms of the grain size are in Figure 3c. The histograms’ full bars represent dislocation density and grain size after the constant strain rate tests. The bars with a pattern represent dislocation density and the grain size after the varying strain rate tests.

^{−1}). The recrystallized volume fraction is above 60%. However, after the strain of 0.4 the strain rate is increased and the effect of the DRX is negligible. In contrast, when the strain is decreased during the test to 0.1 s

^{−1}, the flow stress (dislocation density) decreases below the value predicted for the constant strain rate of 0.1 s

^{−1}test. This outcome is due to the fact that, during the fast part of the test, energy is accumulated in the material (dislocation density increases). This energy accelerates the DRX during the final slow part of the test. This observation is confirmed by Figure 3d, where changes of the dislocation density right after the strain rate change are shown.

^{−1}.

#### 4.2. Industrial Hot Strip Rolling Process

_{e}

_{3}).

## 5. Future Applications of the Stochastic Model

#### 5.1. Laminar Cooling of Hot Rolled Strips

#### 5.2. Cooling of Rods in the Stelmor System

## 6. Summary

- Capability to predict histograms of different microstructural features instead of their average values is the main advantage of the model. It reflects real-world situations more adequately than models using averaged values of dislocation density or grain size.
- All considered models are classified as mean-field models that do not needing explicit representation of the microstructure. In consequence, their computing costs are low.

- Due to the stochastic nature of Equation (9), the repeatability of histograms depends on the number of points. However, it can be kept at a reasonably low level depending on the design of the experiment. In considered cases, we observed that 20,000 simulations with 10 bins allowed us to reduce differences between generated histograms to the level of 3%.
- In the Inverse Analysis, error on target to computed histogram was decreased to 6%, which is a reasonable score bearing in mind that comparison of the two histograms at exactly the same parameters can result in 3% difference.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Calculated distribution of strain (

**a**), strain rate (

**b**) and temperature (

**c**) in roll gap for rolling of DP steel [51].

**Figure 2.**Calculated variations of dislocation density along the two flow lines in roll gap during rolling of DP steel.

**Figure 3.**Changes of flow stress during various tests (

**a**), histograms of dislocation density (

**b**), and grain size (

**c**) during constant and varying strain rate tests. Changes of dislocation density directly after change in strain rate (

**d**).

**Figure 4.**Calculated histograms of dislocation density (

**a**,

**c**) and austenite grain size (

**b**,

**d**) in final two passes of hot strip rolling according to rolling schedule V1 (

**a**,

**b**) and V2 (

**c**,

**d**).

**Figure 5.**Calculated histograms of ferrite volume fraction (

**a**) and ferrite grain size (

**b**) after laminar cooling of steel strip.

**Figure 6.**Calculated histograms of ferrite volume fraction (

**a**) and ferrite grain size (

**b**) after cooling of steel flat rods.

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## Share and Cite

**MDPI and ACS Style**

Oprocha, P.; Czyżewska, N.; Klimczak, K.; Kusiak, J.; Morkisz, P.; Pietrzyk, M.; Potorski, P.; Szeliga, D.
A Comparative Study of Deterministic and Stochastic Models of Microstructure Evolution during Multi-Step Hot Deformation of Steels. *Materials* **2023**, *16*, 3316.
https://doi.org/10.3390/ma16093316

**AMA Style**

Oprocha P, Czyżewska N, Klimczak K, Kusiak J, Morkisz P, Pietrzyk M, Potorski P, Szeliga D.
A Comparative Study of Deterministic and Stochastic Models of Microstructure Evolution during Multi-Step Hot Deformation of Steels. *Materials*. 2023; 16(9):3316.
https://doi.org/10.3390/ma16093316

**Chicago/Turabian Style**

Oprocha, Piotr, Natalia Czyżewska, Konrad Klimczak, Jan Kusiak, Paweł Morkisz, Maciej Pietrzyk, Paweł Potorski, and Danuta Szeliga.
2023. "A Comparative Study of Deterministic and Stochastic Models of Microstructure Evolution during Multi-Step Hot Deformation of Steels" *Materials* 16, no. 9: 3316.
https://doi.org/10.3390/ma16093316