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The PyRolL JMAK-Recrystallization Plugin

This plugin provides a set of JMAK-type microstructure evolution equations for static recrystallization, dynamic recrystallization, metadynamic recrystallization and grain growth under constant process conditions (time, temperature strain , strain rate).
Four sample material data sets are included in the plugin for C45, S355J2, 54SiCr6 and C20.

Model Equations

The JMAK model was originally founded and named after Johnson and Mehl 1, Avrami 2 3 4 and Kolmogorov 5. The following equations do not represent the basic theory, but are adapted versions for the four recrystallization stages (dynamic, metadynamic, static, grain growth) published elsewhere in literature (given at respective position). All of them need a set of empirical parameters which characterize the behavior of a material with a certain chemical composition under defined conditions. Note, that although the parameters are named equally for all mechanisms, they have distinct values for each mechanism. A few sample sets of these parameters are included in the package, additional can be found in literature or measured and determined by the user.

The following equations try to catch up as many of the various forms found in literature as possible, to be able to use existing coefficient sets. They cannot be found in literature in these exact forms, but are a merge of existing ones. Also, it was tried to name the parameters for the different mechanisms as coherent as possible. The use of the Zener-Holomon-parameter 6 is avoided here, but individual Arrhenius-terms are introduced to allow usage of distinct activation energies in each equation. The factors of $10^6$ are introduced with the grain sizes, since commonly the grain size is given in $\mathrm{\mu m}$, but PyRolL uses the meter as plain SI-unit everywhere and validity of existing coefficient sets shall be maintained. Minus signs in the equations are avoided, as the sign shall be caught in the value of the parameter to avoid confusion.

Definitions

The following table defines the mathematical symbols used in the equations.

Symbol Meaning
$X$ recrystallized fraction
$D$ mean grain size
index $\mathrm{in}$ incoming/start value
index $\mathrm{out}$ outgoing/end value
$\varphi$ equivalent strain
$a_i$, $b_i$, $c_i$, $d_i$ material dependent empirical parameters
$Q_i$ activation energy
$Z$ Zener-Holomon-Parameter
$T$ temperature
$t$ time
RX recrystallization
DRX dynamic recrystallization
SRX static recrystallization
MRX metadynamic recrystallization
GG grain growth

Recrystallization

The following equations describing the recrystallization kinetics using a JMAK-type approach, are a merge of common forms found in literature. It was tried to make them as general as possible, to be able to use most coefficient sets published. The approach uses a critical time for start of recrystallization $t_\mathrm{cr}$ and a reference time $t_ \mathrm{ref}$. $t_\mathrm{ref}$ is often taken as $t_{0.5}$, the time of half recrystallization. In this case, the factor $k = \ln \frac12$. For dynamic recrystallization the time is substituted with the equivalent strain $\varphi$ under the assumption of constant strain rate. Except from that, the equations are equivalent for dynamic, static and metadynamic recrystallization. For static and metadynamic recrystallization the strain rates equal those of the latest forming step.

The newly recrystallized fraction is given as:

$$ X = 1 - \exp \left[ k \left( \frac{t - t_\mathrm{c}}{t_\mathrm{ref} - t_\mathrm{c}} \right)^n \right] $$

The critical time of the recrystallization start (incubation time) is given as (sometimes assumed as just zero):

$$ t_\mathrm{c} = a_1 \cdot \varphi_\mathrm{in}^{a_2} \cdot \dot\varphi^{a_3} \cdot (D_\mathrm{in} \cdot 10^6)^{a_4} \cdot \exp \left[ \frac{Q_a}{RT} \right] $$

The reference time is given as:

$$ t_\mathrm{ref} = b_1 \cdot \varphi_\mathrm{in}^{b_2} \cdot \dot\varphi^{b_3} \cdot (D_\mathrm{in} \cdot 10^6)^{b_4} \cdot \exp \left[ \frac{Q_b}{RT} \right] $$

The mean diameter of freshly recrystallized grains is given as:

$$ D = c_1 \cdot \varphi_\mathrm{in}^{c_2} \cdot \dot\varphi^{c_3} \cdot (D_\mathrm{in} \cdot 10^6)^{c_4} \cdot \exp \left[ \frac{Q_d}{RT} \right] \cdot 10^{-6} $$

The mean grain size at the output of a unit is given by a law of mixture as:

$$ D_\mathrm{out} = D_\mathrm{in} + (D - D_\mathrm{in}) X $$

The recrystallized fraction at the output is given by a law of mixture as:

$$ X_\mathrm{out} = X_\mathrm{in} + (1 - X_\mathrm{in}) X $$

Grain Growth

Grain growth kinetics modelled as a root law rather than sigmoidal was originally given by Sellars 7 as:

$$ D_\mathrm{out} = \sqrt[d_1]{(D_\mathrm{in} \cdot 10^6)^{d_1} + d_2 t \exp \left[ \frac{Q}{RT} \right] } $$

Usage

To use the sample datasets provided, it is sufficient to provide the respective key in Profile.material. For own coefficient sets, give the Profile.jmak_dynamic_recrystallization_parameters, Profile.jmak_metadynamic_recrystallization_parameters, Profile.jmak_static_recrystallization_parameters, Profile.jmak_grain_growth_parameters hooks, whose values must be an instance of the JMAKRecrystallizationParameters resp. JMAKGrainGrowthParameters class provided with this package. If one of these hooks does not provide a value for the used material, the respective mechanisms is disabled. Especially missing parameters for metadynamic recrystallization will cause a fallback to static recrystallization, even if the conditions for metadynamic where met. For example:

import pyroll.core as pr
import pyroll.jmak_recrystallization as prj

in_profile = pr.Profile.round(
    ...,
    jmak_dynamic_recrystallization_parameters=prj.JMAKRecrystallizationParameters(
        k=-1.4952,
        n=1.7347,
        a1=1.2338e-3 * 0.79,
        a3=0.1971,
        a4=0.3007,
        qa=258435.17 * 0.1971,
        b1=6.6839e-4,
        b3=0.2265,
        b4=0.4506,
        qb=258435.17 * 0.2265,
        c1=1072.98,
        c3=-0.1629,
        qc=258435.17 * -0.1629,
    ),
    jmak_static_recrystallization_parameters=prj.JMAKRecrystallizationParameters(
        n=1.505,
        b1=3.7704e-8,
        b2=-1.1988,
        b3=-1.003,
        b4=-0.1886,
        qb=163457.62,
        c1=0.1953,
        c2=-0.7016,
        c3=-0.0101,
        c4=1.2052,
        qc=6841.34,
    ),
    jmak_metadynamic_recrystallization_parameters=prj.JMAKRecrystallizationParameters(
        n=2.038,
        b1=6.9235e-2,
        b3=-0.9245,
        qb=248617.4 - 258435.17 * 0.9245,
        c1=840.57,
        c3=-0.1629,
        qc=258435.17 * -0.1629,
    ),
    jmak_grain_growth_parameters=prj.JMAKGrainGrowthParameters(
        d1=6.0,
        d2=1.9144e8,
        qd=-30000.0,
    )
    ...
)

Most remarkable hooks for the user defined by this plugin are the following:

Host Name Meaning Range
Profile recrystallized_fraction portion of the microstructure that is considered as recrystallized (without deformation experienced) 0 to 1
Profile recrystallization_state verbal classification of the recrystallization state "full", "partial" or "none"
Unit recrystallized_fraction portion of the microstructure that is recrystallized within this unit 0 to 1
Unit recrystallized_grain_size grain size of the newly created grains in this unit positive float (meters)

The strain value of out profiles in each unit are lowered by the recrystallized_fraction. The grain_size hook is calculated by the weighted mean of incoming grain size and recrystallized_grain_size.

Implementation Notes

In roll passes, there is always the dynamic recrystallization mechanism in operation. The type of recrystallization mechanism happening in a transport is selected by the value of the newly introduced Unit.recrystallization_mechanism hook, which is determined for transports as follows:

Value Condition Mechanisms
"none" if the recrystallization_state of the in profile is "full" only grain growth
"metadynamic" if the recrystallization_state of the in profile is "partial" metadynamic recrystallization and grain growth
"static" otherwise, especially if the recrystallization_state of the in profile is "none" static recrystallization and grain growth

For roll passes it is either "dynamic" or "none", depending on available parameters and if the critical strain is reached.

The value of the Profile.recrystallization_state hook is determined as follows using a threshold value that is set in the pyroll.jmak_recrystallization.Config class:

Value Condition
"none" if the recrystallized_fraction" of the profile is smaller than THRESHOLD
"full" if the recrystallized_fraction" of the profile is larger than 1 - THRESHOLD
"partial" otherwise

These string keys are selected in the other hook implementations to select there appropriateness for the current unit and with that choosing the equation set to use.

Footnotes

  1. W. A. Johnson and R. F. Mehl, “Reaction Kinetics in Processes of Nucleation and Growth,” Trans. Am. Inst. Min. Metall. Eng., vol. 135, pp. 416–458, 1939. ↩

  2. M. Avrami, “Kinetics of Phase Change. I General Theory,” The Journal of Chemical Physics, vol. 7, no. 12, pp. 1103–1112, Dec. 1939, doi: 10.1063/1.1750380. ↩

  3. M. Avrami, “Kinetics of Phase Change. II Transformation-Time Relations for Random Distribution of Nuclei,” The Journal of Chemical Physics, vol. 8, no. 2, pp. 212–224, Feb. 1940, doi: 10.1063/1.1750631. ↩

  4. M. Avrami, “Granulation, Phase Change, and Microstructure Kinetics of Phase Change. III,” The Journal of Chemical Physics, vol. 9, no. 2, pp. 177–184, Feb. 1941, doi: 10.1063/1.1750872. ↩

  5. A. Kolmogorov, “К статистической теории кристаллизации металлов,” Известия академии наук СССР, vol. 1, no. 3, pp. 355–359, 1937. ↩

  6. C. Zener and H. C. Holomon, "Effect of Strain Rate Upon Plastic Flow of Steel", Journal of Applied Physics, vol. 15, no.1, pp. 22-32, Jan. 1944, doi: 10.1063/1.1707363 ↩

  7. C. M. Sellars and J. A. Whiteman, “Recrystallization and grain growth in hot rolling,” Metal Science, vol. 13, no. 3–4, pp. 187–194, Mar. 1979, doi: 10.1179/msc.1979.13.3-4.187. ↩