2.1 Recrystallization
The evolution of the polycrystalline microstructure after deformation is expressed in terms of the nucleation and growth kinetics of recrystallized grains. The formation of recrystallization nuclei occurs on the junctions of high-angle grain boundaries (HAGB) and low-angle grain boundaries (LAGB) in the sense of the Bailey–Hirsch mechanism.[
17] For instance, the experimental results of Reference
18 confirm the occurrence of recrystallization nucleation at grain boundaries in low-alloyed steel. Consequently, the nucleation rate,
\( \dot{N}_{\text{rx}} \), is written as
$$ \dot{N}_{{{\text{rx}}}} = \left\{ {\begin{array}{ll} {C_{{{\text{rx}}}} \left( {\frac{\pi }{6}\delta ^{2} D} \right)^{{ - 1}} exp(\frac{{ - Q_{{{\text{rx}}}} }}{{RT}})(1 - X_{{{\text{rx}}}} )} & {,\delta \ge \delta _{{{\text{crit}}}} } \\ 0 & {,\delta < \delta _{{{\text{crit}}}} } \\ \end{array} } \right., $$
(1)
where
δ is the subgrain diameter,
D is the mean unrecrystallized grain diameter,
C
rx is a calibration coefficient,
Q
rx is an activation energy similar in value to that for substitutional self-diffusion along grain boundaries,
X
rx is the recrystallized fraction, and
R is the universal gas constant. The parent austenite grain is assumed to be of spherical geometry. The criterion for nucleation is determined by the ratio between the surface energy of a subgrain,
\( \gamma_{\text{LB}} \), and the driving force for recrystallization,
\( P_{\text{D}} \), which is provided by the excess of deformation-induced dislocations. The corresponding relation[
19] reads as
$$ \delta_{\text{crit}} = \frac{{3\gamma_{\text{LB}} }}{{P_{\text{D}} }} = \frac{{3\gamma_{\text{LB}} }}{{0.5Gb^{2} \rho_{{}} }} . $$
(2)
The energy contribution of dislocations is calculated
via the shear modulus,
G, the burgers vector,
b, and the excess dislocation density
\( \rho \). Once the nucleus exceeds a critical size, its further growth rate,
\( \dot{D}_{\text{rx}} \), is expressed in terms of an effective HABG mobility,
\( M_{\text{eff,HB}} \), and the driving force as
$$ \dot{D}_{\text{rx}} = M_{\text{eff,HB}} P_{\text{D}} (1 - X_{\text{rx}} ) . $$
(3)
In the course of recrystallization, the overall growth velocity of recrystallizing grains is assumed to decrease as a consequence of decreasingly available unrecrystallized volume. The evolution of the recrystallized fraction, which represents the ratio between the velocity of recrystallized volume gain,
\( \dot{V}_{\text{rx}} \), and total volume,
\( V_{\text{tot}} \), is expressed as superposition of a term related to the nucleation of newly recrystallized grains and growth of existing ones as
$$ \dot{X}_{\text{rx}} = \frac{\pi }{6}\left( {\dot{N}_{\text{rx}} D_{\text{rx}}^{3} + 3N_{\text{rx}} \dot{D}_{\text{rx}} D_{\text{rx}}^{2} } \right) = \frac{{\dot{V}_{\text{rx}} }}{{V_{\text{tot}} }} . $$
(4)
The evolution of the dislocation density is described by means of an extended Kocks–Mecking model considering the processes of dislocation generation as well as dynamic and static recovery. In this context, we closely follow the approach introduced by Sherstnev
et al.[
20] describing the rate of the total dislocation density evolution as
$$ \dot{\rho } = \frac{{M\sqrt {\rho_{{}} } }}{Ab}\dot{\phi } - 2B\frac{{d_{\text{ann}} }}{b}\rho_{{}} M\dot{\phi } - 2CD_{\text{Dis}} \frac{{Gb^{3} }}{{k_{B} T}}(\rho_{{}} - \rho_{\text{RS}} ) , $$
(5)
with the Taylor factor,
M, the critical dislocation annihilation distance,
d
ann, the substitutional self-diffusion coefficient at dislocations,
D
Dis, the strain rate
\( \dot{\phi } \), and material parameters
A,
B, and
C. However, in contrast to the original Sherstnev
et al. model, where the authors assume that the driving force for static recovery is given by the difference of actual and equilibrium dislocation density, we introduce a limiting degree of static recovery, here, given by the amount of geometrically necessary dislocations,
\( \rho_{\text{RS}} \), for maintaining the subgrain microstructure. In the Read–Shockley model,[
21] which is adopted here, the mean subgrain misorientation angle,
\( \theta_{\text{mean}} \), and the mean subgrain size,
δ, in a periodic network in the grain boundary plane, define the required dislocation density,
\( \rho_{\text{RS}} \), as
$$ \rho_{\text{RS}} = \frac{{\tan \left( {\theta_{\text{mean}} } \right)}}{b\delta } . $$
(6)
The deformation-induced subgrain size is assumed to be correlated with the dislocation density by means of the principle of similitude.[
22,
23] This mainly empirical relation delivers a cell/subgrain size, which is directly linked to the dislocation density evolution during deformation with
$$ \delta = \frac{K}{\sqrt \rho } , $$
(7)
where
K is a material parameter. After deformation, and before the onset of recrystallization, subgrain coarsening takes place. The mean growth rate of subgrains is expressed in terms of an effective LAGB mobility,
M
eff,LB, and a driving force provided by curvature
$$ \dot{\delta } = M_{\text{eff,LB}} \frac{{3\gamma_{\text{LB}} }}{\delta } , $$
(8)
with
\( M_{\text{eff,LB}} \) as an effective LAGB mobility.
The numerical integration of the presented microstructure evolution model is performed by the thermokinetic software tool MatCalc, which utilizes an automatic and adaptive time-step control.
2.2 Precipitates and Solute Atoms
Micro-alloying elements in steel can have two effects on recrystallization kinetics: Zener pinning by carbo-nitride particles and solute drag by solid solution atoms.[
7] For the effect of precipitates, the Zener pressure,
\( P_{\text{Z}} \), can be expressed[
10] as
$$ P_{\text{Z}} = \frac{{3f\gamma_{\text{HB}} }}{{2r_{{}} }} , $$
(9)
with
\( f_{{}} \) being the precipitated phase fraction and
\( r_{{}} \) being the mean precipitate radius. Since the MatCalc precipitation kinetics framework offers detailed information on the size distribution of precipitates also, in the simulations, a size class-based formulation of the Zener pressure is used as introduced by Rath and Kozeschnik[
24] in a recent treatment of coupled precipitation and grain growth. To account for different precipitate types,
i, and size classes,
k, we use the following expression which reads
$$ P_{\text{Z}} (k,i) = \frac{3}{2}\gamma_{\text{HB}} \sum\limits_{i} {\sum\limits_{k} {\frac{{f_{k,i} }}{{r_{k,i} }}} } . $$
(10)
To describe the impact of precipitation on recrystallization, we assume that the precipitates, which potentially pin the boundaries, are interconnected along high-velocity diffusion paths,
i.e., the grain boundaries. Due to the fast diffusion kinetics along the boundaries, the precipitates are subject to significantly accelerated coarsening. When the number density of precipitates pinning the boundary decreases due to coarsening, the Zener pressure decreases and the grain boundary becomes locally released. The free grain boundary then continues to move further into the deformed microstructure until it encounters a new front of pinning precipitates, where the local coarsening procedure repeats. On average, the grain boundary can thus continuously move through the material even if the Zener pressure determined by the initial precipitate distribution exceeds the driving pressure for recrystallization. This issue is discussed in detail in Reference
25.
In support of this concept, Yazawa
et al.[
26] and Jones and Ralph[
27] experimentally observed this special precipitate coarsening behavior in the presence of recrystallization. The precipitates in front of the moving boundary and behind had significantly different average size and number density. To mimic this behavior in our model, we include the Zener pressure into the mobility term instead of reducing the available driving force by the Zener pressure to obtain an effective driving force. The resulting mobility taking into account the particle pinning effect reads as
$$ M_{{{\text{prec}}}} = \left\{ {\begin{array}{*{20}l} {\left( {\frac{{P_{{\text{D}}} - P_{{\text{Z}}} }}{{P_{{\text{D}}} }}} \right)M_{{{\text{free}}}} + \left( {1 - \frac{{P_{{\text{D}}} - P_{{\text{Z}}} }}{{P_{{\text{D}}} }}} \right)M_{{{\text{pinned}}}} } \hfill & {,P_{{\text{D}}} > P_{{\text{Z}}} } \hfill \\ {M_{{{\text{pinned}}}} } \hfill & {,P_{{\text{D}}} \le P_{{\text{Z}}} } \hfill \\ \end{array} ,} \right. $$
(11)
where
\( M_{\text{prec}} \) is the effective mobility of the grain boundary in the presence of precipitates,
\( M_{\text{free}} \) is the free mobility without any dragging and retarding influences of particles and/or solute atoms, and
\( M_{\text{pinned}} \) is the limiting (non-zero) mobility, which is adopted by the grain boundary when the Zener pressure exceeds the driving pressure for recrystallization.
In the present model, the impact of solute drag is modeled on the basis of the work of Cahn.[
3] Accordingly, the dragging effect of solute atoms, which are segregated into the grain boundary, is incorporated into the mobility term with
$$ M_{{{\text{SD}}}} = \frac{1}{{\alpha C_{{{\text{GB}}}} }}, $$
(12)
where
\( M_{\text{SD}} \) is the mobility of the grain boundary in the presence of solute drag,
\( C_{\text{GB}} \) is the grain boundary concentration of the solute drag element, and
\( \alpha \)is an inverse mobility. The latter determines the temperature dependency of the solute drag effect
via the grain boundary/atom interaction energy,
\( E_{\text{B}} \), given as
$$ \alpha = \frac{{\omega \left( {RT} \right)^{2} }}{{E_{\text{B}} D_{\text{CB}} V_{\text{M}} }}\left( {\sinh \left( {\frac{{E_{\text{B}} }}{RT}} \right) - \left( {\frac{{E_{\text{B}} }}{RT}} \right)} \right) , $$
(13)
where
\( \omega \) is the grain boundary width,
\( V_{\text{M}} \) is the molar volume of the matrix phase, and
\( D_{\text{CB}} \) is the cross-boundary diffusion coefficient of the solute drag element. For convenience, in the present analysis, the grain boundary concentration is assumed to be identical to that of the matrix without any regard of additional element segregation into the boundary.
The integral effective mobility is finally evaluated as
$$ M_{\text{eff,HB}} = \left( {\frac{1}{{M_{\text{prec}} }} + \frac{1}{{M_{\text{SD}} }}} \right)^{ - 1} , $$
(14)
which is in accordance with Cahn´s original suggestion of combining the solute drag mobility with the free mobility.
2.4 Model Parameters
Apart from the parameters
C
rx and
Q
rx, Eq. [
1], which determine the nucleation rate of recrystallizing grains, a major input quantity into the recrystallization simulations is the effective mobility of the recrystallization front,
i.e., the grain boundary mobility. This quantity (Eq. [
14]) is basically determined by three partial mobilities: (i)
\( M_{\text{free}} , \) (ii)
\( M_{\text{pinned}}, \) and (iii)
\( M{}_{\text{SD}} \), which are discussed in more detail next.
(i)
The free mobility is parameterized in accordance to the suggestion of Turnbull[
29] as
$$ M_{\text{free}} = \eta_{\text{free}} \cdot M_{\text{TB}} = \eta_{\text{free}} \cdot \frac{{\omega D_{\text{GB}} V_{\text{m}} }}{{b^{2} RT}} , $$
(15)
where
\( \eta_{\text{free}} \) is a linear pre-factor,
\( M_{\text{TB}} \) is the Turnbull mobility,
\( \omega \) is the grain boundary width, and
\( D_{\text{GB}} \) is the substitutional self-diffusion coefficient along grain boundaries. The latter is adopted from a recent independent assessment of Stechauner and Kozeschnik,[
30] providing the essential information on the temperature dependence of the free mobility, which thus becomes a fixed quantity in our treatment instead of being an unknown fitting parameter. The absolute value of this quantity is adjusted such that it is in accordance to the mobility suggestion for low-alloyed austenite reported in Reference
31. A pre-factor of
\( \eta_{\text{free}} \) = 1.5 pct is chosen in the present work. A grain boundary width of
\( \omega \) = 1 nm is adopted from Reference
32.
(ii)
The pinned mobility concept, as utilized in the present work, is based on the assumption of local precipitate coarsening along grain boundaries. This concept has been introduced recently in Reference
25 and it was briefly described earlier in Section
II–B. In an analysis of grain boundary precipitate coarsening, Kirchner[
33] showed that coarsening at grain boundaries should obey a temperature dependence determined by the grain boundary diffusion coefficient. We thus conclude that the temperature dependence of the Turnbull mobility is also determining the local coarsening kinetics. Therefore, we adopt this concept for the pinned mobility and express it as a fraction of the Turnbull mobility with
$$ M_{\text{pinned}} = \eta_{\text{pinned}} \cdot M_{\text{free}} = \eta_{\text{pinned}} \cdot \eta_{\text{free}} \cdot M_{\text{TB}} , $$
(16)
with a dimensionless pre-factor,
\( \eta_{\text{pinned}} \). In the present work, its value is set to 3 pct.
(iii)
The empirical studies by Andrade
et al.[
4] show that the solute drag effect of V during recrystallization is considerably smaller than that of Ti or Nb, however, it is supposed to be still conceivable at lower temperatures. Unfortunately, Andrade
et al. do not report absolute values for the binding energy of V to the grain boundary within the framework of the Cahn model.[
3] We assume that the trapping energy of V to the austenite grain boundaries is of the order of 2.5 kJ/mol, because this value delivers good agreement with experimental evidence.
The driving pressure for recrystallization is mainly determined by the amount of excess defects (dislocations) that are introduced into the material during deformation. The dislocation density evolution is, in turn, determined by the material parameters
A,
B, and
C (Eq. [
5]) and, in the present work, adjusted to the flow curve data of Hernandez
et al.[
34] utilizing the Taylor forest hardening law. For the deformation conditions reported there and used here, the computed dislocation densities reach maximum values below 8 × 10
14 m
−2. The parameters used in the present study are summarized in Table
II. In a recent contribution,[
25] the basic functionality of the elaborated model has been demonstrated with similar input parameters. Both, the nucleation and growth behavior, showed reasonable agreement with experiment.
Table II
Input Parameters for Recrystallization Simulation
D
Dis
| 4.5 × 10−5 exp(185000/RT) | m2/s | |
D
GB
| 5.5 × 10−5 exp(145000/RT) | m2/s | |
D
CB
| 2D
B
| m2/s | |
Q
rx
| 145 | kJ/mol | |
C
rx
| 1.5 × 106
| — | This work |
γ
HB
| 1.3111 − 0.0005T
| J/m2
| |
γ
LB
| 0.5 γ
HB
| J/m2
| This work |
A, B, C
| 50; 5; 5 × 10−5
| — | This work |
K
| A | — | This work |
θmean
| 3 | degree | This work |
EB,V
| 2.5 | kJ/mol | This work |
ω
| 1 × 10−9
| m | |
η
free,HB
| 1.5 × 10−2
| — | |
η
pinned,HB
| 3 × 10−2
| — | This work |