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

  • S. N. S. Mortazavi
  • S. SerajzadehEmail author
Original Paper
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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

An elastic–plastic finite element model has been utilized for the determination of strain and stress fields under symmetrical and asymmetrical plate rolling operations while the temperature rise and its effect on mechanical behavior were ignored. In the modeling, the principle of minimum potential work was employed as follows:
$$ \int\limits_{V} {\rho_{\text{mat}} \ddot{u}\delta v{\text{d}}V} + \int\limits_{V} {\sigma \delta D{\text{d}}V} - \int\limits_{S} {q\delta v{\text{d}}S} = 0, $$
(1)
where \( \sigma \) is the Cauchy stress tensor, 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):
$$ M\ddot{U} + KU = F, $$
(2)
where 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:
$$ u^{i + 1} = u^{i} + \Delta t^{{i + \frac{1}{2}}} \dot{u}^{{i + \frac{1}{2}}} , $$
(3)
where the indices “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.
Figure 1 shows the employed meshing and deformation geometry in asymmetrical rolling with reduction of 40%, rolling speed of 40 rpm, the friction coefficient at the upper and the lower sides were taken as 0.3 and 0.1, respectively. Finally, the distributions of effective strain and stress were computed by means of the interpolation techniques (Belytschko et al. 2000) and the stored energy was then defined based on the predicted local flow stress of the rolled plate. Accordingly, the flow stresses at the centroid of the elements were computed and used to calculate the local dislocation density, i.e., \( \rho_{\text{dis}} \), according to the following equation (Dieter 1986):
$$ \rho_{\text{dis}} = \left( {\frac{\sigma }{\alpha Gb}} \right)^{2} , $$
(4)
where G denotes shear modulus, b is the magnitude of Burgers vector, \( \alpha \) is the dislocation interaction term, and \( \sigma \) is the predicted effective stress.
Fig. 1

Illustration of the employed meshing system and its distortion during asymmetrical rolling with reduction of 40% and rolling speed of 40 rpm

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.

$$ \frac{\partial }{\partial y}\left( {k\frac{\partial T}{\partial y}} \right) + \frac{\partial }{\partial z}\left( {k\frac{\partial T}{\partial z}} \right) = \rho_{\text{mat}} c\frac{\partial T}{\partial t}, $$
(5)
where 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.
$$ -\, k\frac{\partial T}{\partial n} = h_{\text{eff}} (T - T_{\text{a}} ), $$
(6)
where n is the normal direction to the surface boundary, Ta is the surrounding temperature, and heff 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:
$$ \begin{aligned}&\int {N^{T} \left( {k\frac{\partial T}{\partial n}} \right){\text{d}}c}\\ &\quad - \int {\left[ {\frac{{\partial N^{T} }}{\partial y}k\frac{\partial T}{\partial y} + \frac{{\partial N^{T} }}{\partial z}k\frac{\partial T}{\partial z} + N^{T} \rho c\frac{\partial T}{\partial t}} \right]} {\text{d}}A = 0,\end{aligned} $$
(7)
where 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:
$$ Ka^{\text{e}} + C\frac{{{\text{d}}a^{\text{e}} }}{{{\text{d}}t}} = f, $$
(8)
where ae 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).
$$ \dot{n} = c_{0} \Delta E\exp \left( { - \frac{{Q_{\text{N}} }}{RT}} \right), $$
(9)
where \( \dot{n} \) is the nucleation rate per width unit. \( \Delta E \) denotes the difference in energy levels of the deformed and the recrystallized states and QN 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.
$$ \Delta E = 0.5VGb^{2} \Delta \rho_{\text{dis}} + \Delta (A_{\varGamma } \gamma ). $$
(10)
In this equation, 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).
$$ \gamma = \gamma_{\text{m}} \frac{\Delta \theta }{{\Delta \theta_{\text{m}} }}\left( {1 - \ln \frac{\Delta \theta }{{\Delta \theta_{\text{m}} }}} \right), $$
(11)
where \( \gamma_{\text{m}} \) is the high-angle grain boundary energy. \( \Delta \theta \) and \( \Delta \theta_{\text{m}} \) are boundary misorientation and misorientation of high-angle boundaries which are calculated by the following equation (Shabaniverki and Serajzadeh 2016; Lan et al. 2006).
$$ \Delta \theta = 2\pi \frac{{\left| {S_{j} - S_{i} } \right|}}{Q} \, . $$
(12)
Here, Si and Sj 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.
$$ P_{{\rm Nucleation}}^{{\rm K}} = \dot{n}^{\text{K}} \lambda^{3} \Delta t, $$
(13)
where \( \lambda \) is the length of cells and \( \Delta t \) is the time step in cellular automata model calculated as
$$ \Delta t = \frac{\lambda }{{v_{\hbox{max} } }}. $$
(14)

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.

$$ v = M_{0} \exp \left( { - \frac{Q}{RT}} \right)[\kappa \gamma + \Delta E], $$
(15)
where M0 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 0.5Tmp 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.
Fig. 2

a The employed algorithm in the thermal–microstructural simulation, b the steps used in the CA simulation

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.

After initial annealing, the specimens were symmetrically and asymmetrically rolled with different reductions of 30 and 40% using work-rolls having diameter of 150 mm and different frictional conditions listed in Table 1. To determine friction factor at the roll/metal interface, symmetrical rolling experiment was carried out and the maximum reduction in thickness, \( \Delta h_{\hbox{max} } \), was measured and then the following equation was used to estimate the mean friction coefficient at the roll/metal interface (Dieter 1986).
$$ \mu = \left( {\frac{{\Delta h_{\hbox{max} } }}{R}} \right)^{1/2} , $$
(16)
where 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:
Table 1

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

$$ X_{\text{rex}} = \frac{{H_{\text{t}} - H_{0} }}{{H_{\text{Ann}} - H_{0} }}, $$
(17)
where Ht and HAnn denote harnesses of the deformed and the recrystallized steel, respectively, and Ht is the hardness of partially recrystallized steel. Also, the mean grain size of fully recrystallized samples was defined using Clemex pro software.
To record the temperature variations during annealing processing, a K-type thermocouple connected to a data-logger was inserted in some samples and temperature histories were recorded with the rate of 1 Hz. For instance, Fig. 3 shows the achieved time–temperature diagrams in the furnace having the temperatures of 550 and 700 °C.
Fig. 3

The experimental and predicted time–temperature diagrams, a at 550 °C, b at 700 °C

Regarding the recoded heating diagrams and the results of thermal modeling, the effective convection coefficients at 550, 600, 650 and 700 °C were determined and then, the following equation was derived using least square interpolation method.
$$ h_{\text{eff}} = -\, 0.001T^{2} + 1.45T - 435\;({\text{W}}/{\text{m}}^{2} \,^\circ {\text{C}}). $$
(18)

4 Results and discussion

Figure 4 presents the predicted effective strain distribution along thickness direction for the rolling conditions listed in Table 1. Figure 5 also shows the distribution of effective plastic strain in samples A, B and C where these samples were rolled under the same reduction of 40% but different frictional conditions. It can be found that the effective strain increases as the sum of the friction coefficients increases, e.g., the sum of the friction coefficients for the sample A is taken as 0.2 while this increases to 0.6 in sample C. In other words, the values of friction factor at both sides can significantly change the distribution of plastic strain and as a result, a considerable inhomogeneity in stored energy could be made in asymmetrically rolled samples. For instance, in sample B, the effective strain at the oil-lubricated side is about 0.62, however, on the other side which was dried by acetone, the effective strain reaches 0.75. This phenomenon can result in microstructural inhomogeneity after annealing treatments.
Fig. 4

Predicted effective strain distribution after rolling

Fig. 5

Predicted effective strain field during rolling, a sample A, b sample B, c sample C

In the next stage, the annealing treatments of deformed steel were carried out under non-isothermal conditions. As noted, stored strain energy and temperature are two important factors that can considerably affect the rate of static recrystallization. The stored strain energy was predicted by the mechanical model and then it was employed as the initial condition in the microstructural simulation while the temperature history was defined by the thermal analysis simultaneously. It is worth noting that different positions of the rolled plate were considered in the CA model and the stored energy in each location was defined based on the results of mechanical modeling, in other words, the stored energy within each CA domain was assumed to be uniform but it changes in different locations regarding the predicted strain field after rolling. Furthermore, the softening data of sample A were employed for determination of material constants. As a matter of fact, this sample was used to calibrate the parameters in the microstructural simulation while the other samples were utilized to validate the model. In this regard, the experimental data including the rate of recrystallization and the microstructural images for different annealing times at 600 °C for sample A were employed to define the material constants. In this regard, the activation energies for nucleation and growth were defined as 140 and 155 kJ/mole, respectively. Table 2 illustrates the material parameters used in simulation of recrystallization kinetics.
Table 2

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

G

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

\( \gamma \)

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

b

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

\( D_{0} \)

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

Figure 6 shows the generated microstructure for the undeformed steel and the rolled plates with reductions of 30 and 40%, i.e., samples B and D. Note that Fig. 6b, c displays the grain structure at the surface regions of rolled steel subjected to equivalent plastic strains of 0.52 and 0.75, respectively. As the reduction increases the grains are more elongated and the grain aspect ratio is increasing as well. As explained earlier, the CA model as a probabilistic analysis needs to determine proper positions as the nucleation sites. In ferritic low-carbon steels, the most probable positions might be the triple junctions of grain boundaries because of the high stored energy in these regions and the most second probable location could be the boundaries of the adjacent grains. In the third place, nucleation can be occurred within the grains for the case of heavily deformed samples, i.e., homogenous nucleation. Accordingly, the proposed model recognizes the triple points and the grain boundaries as the most suitable locations for initiation of new phase; however, for the case of high-reduction rolling particularly in asymmetrical rolling process, the homogenous nucleation might be operative either. Figure 7 shows the initial stage of nucleation at the surface region of samples C and D. Figure 8a presents recrystallization progress at surface of specimen D for different annealing temperatures achieved from the experiments. As seen, by increasing temperature, the rate of static recrystallization increases and also, recrystallization kinetics becomes considerably slow for annealing at 600 °C. Also, Fig. 8b–d shows the microstructures of the surface of annealed plate after 300 s. As it was mentioned, the recrystallization rate for the rolled steel at 600 °C becomes very slow and, therefore, the annealed microstructures should contain both equiaxed and elongated grains. This microstructural pattern can be observed in Fig. 8 where a combination of equiaxed and elongated grains can be detected after annealing at 600 °C. However, at annealing temperatures of 650 and 700 °C, the recrystallization has been completed and a fine-grain microstructure. Figure 9 compares the experimental and the predicted recrystallization rates and the final microstructures at the surface of sample D. The real and predicted mean grain sizes in this case were determined as 36.5 and 33.4 µm, respectively. A reasonable consistency can be observed between the two set of data. The progress of recrystallization at the surface of sample D at 700 °C is also given in Fig. 10. As seen, in the early stages of annealing the higher rate of nucleation occurs at the grain boundaries; however, for the longer recrystallization periods the nucleation rate decreases significantly and instead the growth of the existing grains dominates the recrystallization progress. Moreover, the free-strain grains cover the domain completely in Fig. 10d after 300 s which has an appropriate agreement with the experiments.
Fig. 6

The generated microstructure, a the initial microstructure, b cold rolled at reduction of 30%, surface region, c cold rolled at reduction of 40%, surface region

Fig. 7

Suitable locations for initiation of new phase, a sample C, surface region, b sample D, surface region

Fig. 8

a Experimental recrystallization progress at different annealing temperatures at surface of sample D, b microstructure of sample D after 300 s at 600 °C, c microstructure sample D after 300 s at 650 °C, d microstructure of sample D after 300 s at 700 °C

Fig. 9

a Comparing the experimental and predicted recrystallization kinetics for sample D at different annealing temperatures, b the real microstructure after annealing at 700 °C for 300 s, c predicted microstructure after annealing at 700 for 300 s

Fig. 10

The microstructural changes during annealing at the surface of sample D at 700 °C, a after 90 s, b after 150 s, c after 170 s, d after 300 s

Initial straining is an important factor that can affect recrystallization behavior of the steel. Figure 11 shows the recrystallization progress for specimen B at upper surface with effective strain of 0.75 for annealing at 700 °C. By comparing Figs. 9 and 11, it can found that the rate of recrystallization increases for higher initial strains, as a result, the condition of rolling could affect the recrystallization rate, i.e., lower rate of recrystallization and more homogenous microstructure are expected after annealing for the case of symmetrical rolling. Figure 12 presents the recrystallization rate and microstructure for different positions of sample B with total reduction of 40% in which the effective strain decreases from 0.75 at the upper surface to 0.62 at the lower side. The grain size at upper and lower sides were predicted about 15.1 and 22.4 µm, respectively, while these were measured as 13.5 and 22.3 µm from the real samples. It shows that the side in contact with the cleaned work-roll has a faster recrystallization rate because of the higher stored strain energy compared to the other side in contact with the lubricated work-roll. Furthermore, the model can predict the grain size and microstructure properly. It may be noted that the grain size for the case of asymmetrically rolled sample is noticeably lesser than that of symmetrically rolled sample. This means that the role of initial stored energy plays a significant role on the final macrostructures owing to high-density nucleation at the early stage of recrystallization.
Fig. 11

Comparing the experimental and predicted recrystallization kinetics at upper surface of sample B and annealing temperature of 700 °C

Fig. 12

a Experimental and predicted recrystallization kinetics for sample B at 700 °C, b comparing microstructure after annealing at 700 °C for 150 s, upper surface, c comparing microstructures after annealing at 700 °C for 150 s, lower surface

Figure 13 presents the results of annealing under both isothermal and non-isothermal conditions for specimen B at the center of the rolled plate and furnace temperature of 700 °C. There is a significant difference in softening behaviors. Accordingly, the kinetics of recrystallization under isothermal condition has been noticeably faster than that under non-isothermal condition owing to higher rate of nucleation in early stages of softening in isothermal annealing. On the other hand, it takes considerable time to complete the softening process for the case of non-isothermal annealing operation. This phenomenon shows the importance of nucleation rate on the overall rate of softening and thus, the recrystallization kinetics can be controlled by the applying a proper thermal cycle for accelerating or lowering the rate of softening.
Fig. 13

a Comparing the predicted recrystallization kinetics of specimen B at the center region under isothermal and non-isothermal heat treatments, b microstructure of sample B at the central region after isothermal annealing, c microstructure of specimen B at the central region after non-isothermal annealing

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.

References

  1. Belytschko T, Liu WK, Moran B (2000) Nonlinear finite element methods for continua and structures. Wiley, ChichesterzbMATHGoogle Scholar
  2. Davies CHJ, Hong L (1999) The cellular automaton simulation of static recrystallization in cold-rolled AA1050. Scr Mater 40:1145–1150CrossRefGoogle Scholar
  3. Dieter GE (1986) Mechanical metallurgy, 3rd edn. McGraw Hill, New YorkGoogle Scholar
  4. Humphreys FJ, Hatherly M (2006) Recrystallization and related annealing phenomena. Elsevier Science, OxfordGoogle Scholar
  5. Janssens K, Frans G (2007) Computational materials engineering: an introduction to microstructure evolution. Elsevier, San DiegoGoogle Scholar
  6. Kremeyer K (1998) Cellular automata investigations of binary solidification. J Comput Phys 142:243–262CrossRefzbMATHGoogle Scholar
  7. Kugler G, Turk R (2006) Study of the influence of initial microstructure topology on the kinetics of static recrystallization using a cellular automata model. Comput Mater Sci 37:284–291CrossRefGoogle Scholar
  8. Lan YJ, Li DZ, Li YY (2006) A mesoscale cellular automaton model for curvature driven grain growth. Metall Mater Trans B 37B:119–129CrossRefGoogle Scholar
  9. Lin YC, Liu YX, Chen MS, Huang MH, Ma X, Long ZL (2016) Study of static recrystallization behavior in hot deformed Ni-based superalloy using cellular automaton model. Mater Des 99:107–114CrossRefGoogle Scholar
  10. Marx V, Reher FR, Gottstein G (1999) Simulation of primary recrystallization using a modified three-dimensional cellular automaton. Acta Mater 47:1219–1230CrossRefGoogle Scholar
  11. Porter DA, Easterling KE (1981) Phase transformations in metals and alloys, 1st edn. Van Nostrand Reinhold, BerkshireGoogle Scholar
  12. Raabe D (2000) Yield surface simulation for partially recrystallized aluminum polycrystals on the basis of spatially discrete data. Comput Mater Sci 19:13–26CrossRefGoogle Scholar
  13. Raabe D, Hantcherli L (2005) 2D cellular automaton simulation of the recrystallization texture of an IF sheet steel under consideration of Zener pinning. Comput Mater Sci 34:299–313CrossRefGoogle Scholar
  14. Richelsen AB (1997) Elastic–plastic analysis of the stress and strain distributions in asymmetric rolling. Int J Mech Sci 39:1190–1211CrossRefzbMATHGoogle Scholar
  15. Schafer C, Pomana G, Mohles V, Gottstein G, Engler O, Hirsch J (2010) Recrystallization modeling of AA8XXX alloys with cellular automata considering recovering kinetics. Adv Eng Mater 12:131–140CrossRefGoogle Scholar
  16. Seyed Salehi M, Serajzadeh S (2012) Simulation of static recrystallization in non-isothermal annealing using a coupled cellular automata and finite element model. Comput Mater Sci 53:145–152CrossRefGoogle Scholar
  17. Shabaniverki S, Serajzadeh S (2016) Simulation of softening kinetics and microstructural events in aluminum alloy subjected to single and multi-pass rolling operations. Appl Math Model 40:7571–7582MathSciNetCrossRefGoogle Scholar
  18. Stasa FL (1985) Applied finite element analysis for engineers. CBS Publishing, TokyoGoogle Scholar
  19. Svyetlichnyy DS (2012) Simulation of microstructure evolution during shape rolling with the use of frontal cellular automata. ISIJ Int 52:559–568CrossRefGoogle Scholar
  20. Xiao N, Zheng C, Li D, Li Y (2008) A simulation of dynamic recrystallization by coupling a cellular automaton method with a topology deformation technique. Comput Mater Sci 41:366–374CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Materials Science and EngineeringSharif University of TechnologyTehranIran

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