# Simulation of non-isothermal recrystallization kinetics in cold-rolled steel

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## Abstract

In this work, a model has been developed to determine thermal and microstructural events during non-isothermal annealing of rolled carbon steels. In the first place, the process of cold rolling under both symmetrical and asymmetrical conditions was mathematically modeled employing an elastic–plastic finite element formulation to define the distribution of plastic strain and internal stored energy. In the next step, two-dimensional model based on cellular automata was generated to assess softening kinetics in annealing treatment. At the same time, a thermal model based on Galerkin-finite element analysis was coupled with the microstructural model to consider temperature variations during heat treatment. The impact of different parameters such as heating rate, annealing temperature, and initial microstructures were all taken into account. To validate the employed algorithm, the predictions were compared with the experimental results and a reasonable agreement was found. Accordingly, the simulation results can be employed for designing a proper mechanical–thermal treatment to achieve the desired microstructure as well as mechanical properties under practical processing conditions.

## Keywords

Static recrystallization Finite element method Cellular automata Non-isothermal heat treatment Carbon steels## 1 Introduction

Heat treatment of cold-deformed metals has been a significant route to achieve the desired microstructures and mechanical properties during which different types of softening mechanisms might be operative such as static recovery and static recrystallization. Static recrystallization (SRX) is associated with nucleation and growth of strain-free grains within the deformed structure while both nucleation and growth processes are considerably affected by the level of stored energy as well as annealing temperature and its distribution (Porter and Easterling 1981). Therefore, various studies were conducted to determine microstructural changes in annealing treatments under different working conditions. For instance; Marx et al. (1999) used a three-dimensional model based on cellular automata for predicting the rate of static recrystallization after cold working. Raabe (2000) developed a cellular automata-crystal plasticity model to define the deformation behavior of partially recrystallized aluminum alloys. Lin et al. (2016) employed a probabilistic cellular automaton (CA) model to predict the rate of isothermal static recrystallization in Ni-based alloys. Raabe and Hantcherli (2005) employed two-dimensional cellular automata modeling to evaluate recrystallization texture under isothermal conditions in heavily-deformed IF-steels. Svyetlichnyy (2012) developed a three-dimensional CA model to predict microstructural changes after hot shape rolling of steels. Salehi and Serajzadeh (2012) used a two-dimensional CA finite element model to simulate microstructural changes during static recrystallization within ferritic steels. Moreover, further works based on the cellular automata can be noted dealing with modeling of softening kinetics and microstructural changes during annealing processing of different alloy systems (Kugler and Turk 2006; Shabaniverki and Serajzadeh 2016; Davies and Hong 1999; Schafer et al. 2010).

Regarding the published works, the impact of temperature variations and its distribution and/or the influence of initial strain field have been ignored or widely simplified; however, in practice recrystallization treatments mainly take place under non-isothermal conditions within a non-uniformly deformed specimen, i.e., rolled samples. In this work, static recrystallization kinetics is predicted within the cold-rolled plate under non-isothermal conditions. For doing so, an elastic–plastic finite element analysis is first performed for determination of distribution stored energy after cold rolling operations and then the results of the modeling are considered as the input data for the microstructural–thermal model. Both symmetrical and asymmetrical rolling processes are considered for producing a non-uniform strain field prior to annealing treatment. In the next stage, a two-dimensional probabilistic cellular automata model coupled with a finite element analysis is developed to predict the progress of static recrystallization under non-isothermal heat treatments. To validate the employed algorithm, cold-rolled steels are subjected to annealing treatment and then, the produced microstructures are examined and compared with the simulation results.

## 2 Modeling

*D*is the rate of deformation tensor,

*v*is the velocity vector, \( \rho_{\text{mat}} \) is the material density, and

*q*represents the surface traction tensor due to friction at roll/metal contact region. The above minimization can be managed by means of finite element formulation for an elastic–plastic material obeying Prandtl–Reuss constitutive equations under plain strain conditions, i.e., two-dimensional rolling condition (Richelsen 1997). The finite element formulation together with the Abaqus/Explicit solver was used to solve the deformation problem. In this regard, the above equation may be rewritten in matrix form utilizing a proper interpolation function as below (Belytschko et al. 2000):

*M*is the mass matrix,

*K*is the stiffness matrix relating to the strain energy,

*F*is vector of external forces, i.e., natural boundary conditions, and \( U \) denotes the nodal displacement vector. In the next stage, the above system of differential equations was managed using central difference scheme where the displacement at the new time step was estimated as follows:

*i*+ 1” and “

*i*” denote the deformation steps and \( \Delta t \) is the employed time step. Note that the work-rolls were taken as rigid bodies and thus, the displacement increment along work-roll radius at the roll/metal interface was assumed to be zero and the Coulomb friction model was used to compute the surface traction. Furthermore, a velocity-dependent stress model was utilized to consider the impact of changing the direction of frictional stress at neutral position. Moreover, in the initial step of modeling, the plate was fed into the roll-gap by applying velocity boundary condition at the end of the plate and after roll-bite was done the velocity boundary condition was disabled. In the modeling, quadrilateral elements were utilized for discretization of the deforming plate in which the numbers of elements along thickness and length directions were taken as 6 and 420, respectively and the run-time duration was about 120 min on Core i7-3.0 GHz processor. Note that a mesh sensitivity analysis was first performed to determine the optimum number of elements using the rolling force as the convergence criterion.

*G*denotes shear modulus,

*b*is the magnitude of Burgers vector, \( \alpha \) is the dislocation interaction term, and \( \sigma \) is the predicted effective stress.

After cold rolling, the heating stage was applied to initiate the softening operation. Thus, it needs to predict the heating rate and temperature variations in different positions of the steel subjected to heat treatment. The governing heat conduction equation in Lagrangian framework can be described as below assuming that the heat conduction along longitudinal direction can be ignored because of high length/width and length/thickness ratios of the rolled plate as well as uniformity of boundary condition in this direction.

*T*is temperature,

*y*and

*z*show width and thickness directions, respectively,

*k*is the coefficient of heat conduction, and

*c*is the specific heat. The initial temperature was taken at room temperature, i.e., 25 °C and the following boundary condition was defined on the boundaries of the working domain where an effective heat convection factor was used to include both radiation and convection mechanisms.

*n*is the normal direction to the surface boundary,

*T*

_{a}is the surrounding temperature, and

*h*

_{eff}represents the effective heat transfer coefficient that was determined based on the experimental time–temperature diagrams recorded for different heating conditions. The heat conduction problem was solved employing Galerkin-finite element approach (Stasa 1985) using the Gauss–Green theorem to reveal the natural boundary conditions as:

*c*and

*A*are the boundary and the area of working domain, respectively.

*N*is the matrix of the shape function for a scalar-variable (Stasa 1985). Afterward, the temperature distribution over each element was estimated as \( T^{\text{e}} = Na^{\text{e}} \). Considering this approximate solution and using Eq. 7, the governing partial differential equation for each element can be converted into a system of first-order differential equations as follows:

*a*

^{e}is the nodal temperature vector, and

*K*and

*C*are the conduction and the capacitance matrices, respectively. The vector

*f*is due to the natural boundary conditions, i.e., the first term in Eq. 7. Finally, the central difference approach was used to solve the resulting system of equations and the temperature distribution under continuous cooling conditions can be computed in successive time steps. At the same time, a two-dimensional cellular automaton model was coupled with the thermal analysis to assess the kinetics of static recrystallization. It is worth noting that the effect of static recovery has been ignored owing to the relatively low stalking fault of low-carbon steels (Humphreys and Hatherly 2006). In the CA model, the working domain containing 500 × 500 rectangular cells was generated. It should be noted that misorientation of adjacent grains can affect the grain boundary energy and to include this phenomenon, a random number ranging between 1 and 50 was assigned to each grain for describing its orientation. As a matter of fact, 50 different crystallographic originations were included in the microstructural model. Furthermore, in the CA domain, the cell size was taken 0.5 μm and the periodic boundary conditions were applied on both directions. The initial grain structure was generated by applying random nuclei, normal grain growth rules and the modified Moore-neighborhood definition (Janssens and Frans 2007). In the next step, the grain structure was elongated by a mapping matrix based on a pure-stretch deformation gradient matrix (Xiao et al. 2008). The modeling of the static recrystallization was then performed by coupling the governing physical rules and the probabilistic cellular automata algorithm. The nucleation stage was assumed to be performed randomly assuming the following temperature-dependent equation (Janssens and Frans 2007).

*Q*

_{N}is the activation energy for nucleation process. It should be mentioned that the parameter, \( \Delta E \), was computed based on the initial stored energy, and the grain boundary energy, which is presented as Eq. 10.

*V*,

*G*and

*b*are the volume of new nuclei, shear modulus and Burger vector, respectively. \( \Delta \rho_{\text{dis}} \) and \( \Delta ( {\text{A}}_{\varGamma } \gamma ) { } \) denote the change in dislocation density and surface energy of nuclei during recrystallization. The stored energy was estimated by the results of mechanical modeling and employing Eq. 4 while the Read–Shockley model was used to estimate the grain boundary energy (Xiao et al. 2008).

*S*

_{i}and

*S*

_{j}are the orientation numbers and

*Q*in the maximum amount of orientation number between all cells in domain. To generate distribution of nuclei over the working domain according to the physical rules, the domain was first scanned and the triple junctions and high-energy grain boundaries were identified in which the nucleation of new grains was associated with the larger reduction in stored energy and accordingly, larger probability was given to these locations. Then, in each CA time step, the nucleation rate was determined while these positions were given a higher chance to be formed into recrystallized nuclei by applying probabilistic CA algorithm. The formation chance into recrystallized nuclei is measured as precise values of \( P_{{\rm Nucleation}}^{{\rm K}} \) as following equation.

In this equation \( v_{\hbox{max} } \) is the highest boundary velocity during the growth process among all the boundaries in each step.

After generation of stable nuclei, they start to grow into the initial matrix. The velocity of the recrystallized grain boundary may be affected by various factors including temperature, the curvature of the moving boundaries, the level of initial stored energy, and the mobility of grain boundaries. In this regard, the following equation was employed to calculate the velocity of moving boundaries.

*M*

_{0}is the material constant,

*R*is universal gas constant,

*T*is the recrystallization temperature,

*Q*is activation energy for growth.\( \kappa \) and \( \gamma \) are the curvature and the grain boundary energy that were calculated by kink template method (Kremeyer 1998) and Read–Shockley equations, respectively. It is worth noting that during non-isothermal annealing, static recrystallization may occur during heating stage when the temperature of steel reaches 0.5

*T*

_{mp}or above. Accordingly, this phenomenon is considered in the simulation by coupling heat conduction–microstructural models in heating stage where the temperature is updated after each CA time step regarding the results of the thermal model. According to the above-mentioned equations and algorithm, a code in MATLAB 7.6 was generated for performing thermal–microstructural model while it needs about 75 min to achieve a complete solution of thermal problem and microstructural simulation using a Core i7-3.0 GHz processor. Note that the cold rolling simulation was first made using Abaqus/Explicit and the distribution of stored energy after rolling, e.g., the stored energy at surface and center of the rolled plate, was then defined according to the results of mechanical modeling. Figure 2 shows the employed algorithm and steps utilized in the thermal–microstructural simulation and the CA model.

## 3 Experimental procedures

Carbon steel with the chemical composition of 0.037%C, 0.194%Mn, 0.02%Si, 0.007%P, and 0.004%S (in wt) was examined. The initial thickness of as-received plate was 3 mm. In the first stage, the samples were cut with the dimensions of \( 110 \times 40 \times 3 \) (in mm) and then they were annealed at 900 °C for 25 min and air-cooled to eliminate the influence of pervious deformation processing. The tensile testing was carried out on as-annealed sample for determining the flow stress behavior of the examined steel and then, the achieved results were employed as the input data in modeling of cold rolling. The tensile testing was conducted according to ASTM-E8 and constant crosshead speed of 2 mm/min.

*R*is the work-roll radius. As a result, the friction coefficients have been calculated about 0.3 when the work-rolls were cleaned by acetone and 0.1 for the oil-lubricated condition. Non-isothermal annealing treatments were then carried out on the rolled plates at different temperatures in the rage of 550–700 °C where static recrystallization occurs within ferrite, i.e., the annealing temperature is below the temperature of ferrite–austenite transformation. After that, the samples were cooled rapidly to prevent further softening during cooling stage. The microstructural observation and hardness testing were conducted on the annealed plates to evaluate the produced microstructures and mechanical properties. Optical metallography was carried using mechanical polishing followed by chemical etching in Nital 2%; also, the hardness measurements were made by Vickers micro-hardness and then, recrystallized fraction was defined by the following equation:

The conditions used in the rolling experiments

Sample | Upper lubrication | Lower lubrication | Reduction (%) | Rolling speed (rpm) |
---|---|---|---|---|

A | Oil | Oil | 40 | 40 |

B | Acetone | Oil | 40 | 40 |

C | Acetone | Acetone | 40 | 40 |

D | Acetone | Acetone | 30 | 40 |

*H*

_{t}and

*H*

_{Ann}denote harnesses of the deformed and the recrystallized steel, respectively, and

*H*

_{t}is the hardness of partially recrystallized steel. Also, the mean grain size of fully recrystallized samples was defined using Clemex pro software.

## 4 Results and discussion

The data used in the microstructural model (Seyed Salehi and Serajzadeh 2012)

| \( 6.92 \times 10^{10} \left( {1 - \frac{{1.31\left( {T - 300} \right)}}{1810}} \right) \) (GPa) |

\( \gamma \) | 0.56 \( ({\text{J}}\;{\text{m}}^{ - 2} ) \). |

| \( 2.48 \times 10^{ - 10} \) (m) |

\( D_{0} \) | \( 5.4 \times 10^{ - 8} ({\text{m}}^{2} {\text{s}}^{ - 1} ) \) |

## 5 Conclusions

In this study, a thermal–microstructural model was employed to predict recrystallization kinetics as well as final microstructures after annealing of cold-rolled low-carbon steels. The process of cold rolling including both asymmetrical and symmetrical rolling layouts were first simulated using Abaqus/Explicit and the distribution of stored strain energy was determined and used in the microstructural model as the input data; then, the cellular automata associated with thermal finite element analysis was utilized to predict the kinetics of recrystallization and resulting microstructures during subsequent annealing treatment. The effects of different process parameters including the level of stored energy and its distribution as well as the heating rate during annealing are considered in the simulation. Rolling experiments and non-isothermal annealing treatments were conducted on low-carbon steel and the rate of static recrystallization and microstructure of the annealed steel were determined and compared with the predictions. It was found that there is a good consistency between the experimental observations and the predicted results. According to the modeling results, the activation energies for nucleation and growth were computed about 140 and 155 kJ/mole, respectively. It was found that the rate of nucleation was strongly affected with the imposed heating rate while the highest nucleation rate as well as the recrystallization rate was achieved under isothermal condition.

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