Materials Algorithms Project
Program Library

Subroutine MAP_STEEL_SIM_TRANS

1. Provenance of code.
2. Purpose of code.
3. Specification.
4. Description of subroutine's operation.
5. References.
6. Parameter descriptions.
7. Error indicators.
8. Accuracy estimate.
9. Any additional information.
10. Example of code
11. Auxiliary subroutines required.
12. Keywords.
13. Download source code.
14. Links.

Provenance of Source Code

S.J. Jones^* and H.K.D.H. Bhadeshia,
Phase Transformations Group,
Department of Materials Science and Metallurgy,
University of Cambridge,
Cambridge, U.K.

^*Dr S.J. Jones is now with Ford Motor Co. (UK) Ltd.

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Purpose

To calculate the volume fraction of simultaneously nucleating phases. Parabolic and linear growth from both random and grain boundary nuclei is considered.

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Specification

 SUBROUTINE MAP_STEEL_SIM_TRANS(I2,TIME,PRETIME,DTIME,PARA1D,
& PARA3D,DIFFUSY,GRWID,AR,ARWID,VIINCA,VIINCWID,VIA,VIWID,
& EQVOLFRAC,OBOUND,PERCENTRANS,VTOTTRANS,VAREAL,
& VWIDREAL,VAREALINC,VWIDREALINC,IFLAG,SATTIME)

 IMPLICIT NONE

 DOUBLE PRECISION
& QA(1001),PREQA(1001),OREA(3000),PREOREA(3000),QWID(1001),
& PREQWID(1001),OREWID(3000),PREOREWID(3000),Y(3000),
& NOPARTINCA(1001),NOPARTINCWID(1001),NOPARTGBA(1001),
& NOPARTGBWID(1001),TIMEOFNUCL(1001)

 DOUBLE PRECISION TIME,PRETIME,DTIME,PARA1D,PARA3D,DIFFUSY,
& GRWID,AR,ARWID,VIINCA,VIINCWID,VIA,VIWID,EQVOLFRAC,OBOUND,
& PERCENTRANS,VTOTTRANS,VAREAL,VWIDREAL,VAREALINC,VWIDREALINC,
& SATTIME,PI,VOLTOT,YEND,DYSMA,DYMED,DYLAR,SUMINCA,
& SUMINCWID,TIMESNINC,NEWCONINCA,NEWCONINCWID,
& DVEXTAINC,DVEXTWIDINC,TIMESN,
& MAXHGTA,MAXHGTWID,MAXHGT,ALLA,ALLWID,DOEXA,
& DOEXWID,DOREA,DOREWID,FRACOBUT,PORA,PORWID,PREVAEXT,
& VAEXT,PREVWIDEXT,VWIDEXT,DVAEXT,DVWIDEXT,FRACUT,
& DVAREALINC,DVAREAL,
& VTOTAREAL,PREVTFERR,GROWCONINCA,QAINCRE,
& GROWCONINCWID,QWIDINCRE,DVWIDREALINC,DVWIDREAL,VTOTWIDREAL

 INTEGER I2,IFLAG,INOSMA,INOMED,INOLAR,IYPLANE,I20,
& I610,I642,I810,I10,I400,I500

 PARAMETER (PI=3.141592654)

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Description

The volume of any number of simultaneously nucleating phases in a system is calculated using a new numerical approach which is based on the methods of Johnson, Mehl and Avrami. Four possibilities are allowed :-

1. Parabolic reconstructive growth of spheres from random nuclei.
2. Parabolic reconstructive growth of discs from boundary nuclei.
3. Linear growth of displacive plates from random nuclei.
4. Linear growth of displacive plates from boundary nuclei.

The model may be modified to allow any number of phases to nucleate at any number of sites.

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References

1. J.W. Christian, "The Theory of Phase Transformations in Metals and Alloys", Part 1, 2nd ed., Permagon, Oxford (1965).
2. J.W. Cahn, "The Kinetics of Grain Boundary Nucleated Reactions", Acta. Metall., 4, 449-459, (1956).
3. R.C. Reed & H.K.D.H. Bhadeshia, Mat. Sci. Tech., 8, 421-435, (1992).
4. S.J. Jones, PhD. Thesis, University of Cambridge, (1996)

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Parameters

Input parameters

I2 - integer

I2 represents the time interval.

TIME - real

TIME is the elapsed time from the start of the transformation to the end of the current time interval (s).

PRETIME - real

PRETIME is the elapsed time from the start of the transformation to the end of the previous time interval (s).

DTIME - real

DTIME is the current time interval (s).

PARA1D - real

PARA1D is the natural 1-dimensional parabolic thickening parameter for the reconstructive transformation product.

PARA3D - real

PARA3D is the natural 3-dimensional parabolic thickening parameter for the reconstructive transformation product.

DIFFUSY - real

DIFFUSY is the diffusivity in the parent phase of the atomic species which determines the parabolic growth rate of the reconstructive product (m² s^-1).

GRWID - real

GRWID is the lengthening rate of the displacive plates (m s^-1).

AR - real

AR is the aspect ratio of reconstructive discs which are boundary nucleated..

ARWID - real

ARWID is the aspect ratio of displacive plates.

VIINCA - real

VIINCA is the steady state nucleation rate per unit volume for reconstructive products which nucleate randomly throughout the parent phase (m^-3 s^-1).

VIINCWID - real

VIINCWID is the steady state nucleation rate per unit volume for displacive products which nucleate randomly throughout the parent phase (m^-3 s^-1).

VIA - real

VIA is the steady state nucleation rate per unit area of grain boundary for reconstructive products (m^-2 s^-1).

VIWID - real

VIWID is the steady state nucleation rate per unit area of grain boundary for displacive products (m^-2 s^-1).

EQVOLFRAC - real

EQVOLFRAC is the total equilibrium volume fraction of the product phase.

OBOUND - real

OBOUND is the grain boundary area per unit volume in the parent phase (m^-1).

Output parameters

PERCENTRANS - real

PERCENTRANS represents the ratio of the total volume fraction of all transformation products to the total equilibrium volume fraction, expressed as a percentage.

VTOTTRANS - real

VTOTTRANS is the total volume fraction of all reconstructive and displacive products.

VAREALINC - real

VAREALINC is the real volume fraction of randomly nucleated reconstructive product.

VWIDREALINC - real

VWIDREALINC is the real volume fraction of randomly nucleated displacive product.

VAREAL - real

VAREAL is the real volume fraction of boundary nucleated reconstructive product.

VWIDREAL - real

VWIDREAL is the real volume fraction of boundary nucleated displacive product.

IFLAG - integer

IFLAG is set to zero at the start of the transformation, it is set to unity when 99% of the grain boundary of the parent phase is saturated.

SATTIME - real

SATTIME is the time when 99 % of the grain boundary of the parent phase is saturated (s).

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Error Indicators

If MAXHGT exceeds YEND then an error is reported and the program execution is stopped.

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Accuracy

Numerical accuracy is dependent on the size of the time interval compared to the total time required for complete transformation. Exact solutions for a single nucleating phase indicate that when the nucleation and growth conditions remain invariant and the time interval used is 1% of the total transformation time, then an error of less than 1% is generated. This error may be reduced by using a smaller time interval. However, the computation time becomes enormous if the total number of time intervals is large. A variable time interval should therefore be used to allow both the transformation start and finish to be accurately represented. As the transformation proceeds the time interval should be gradually increased. For brevity this is not shown in the example program.

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Further Comments

The numerical nature of the approach taken permits many simplifications to be removed so that the nucleation and growth conditions may vary according to the time, temperature and the degree of transformation. This permits the modelling of solute enrichmenment/depletion, anisothermal transformation conditions and the use of a variable time interval. The model could easily be extended to deal with nucleation on linear defects such as grain edges and dislocations.

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Example

1. Program text

       PROGRAM MAP_STEEL_SIM_TRANS_EX
       IMPLICIT NONE
       DOUBLE PRECISION DTIME,PARA1D,PARA3D,DIFFUSY,GRWID,AR
       DOUBLE PRECISION ARWID,VIWID,EQVOLFRAC,OBOUND,VIINCA
       DOUBLE PRECISION VIINCWID,VIA
C
       DOUBLE PRECISION TIME,PRETIME,PERCENTRANS,VTOTTRANS,
      & VAREAL,VWIDREAL,VAREALINC,VWIDREALINC,SATTIME
C
       INTEGER I100,IFLAG
C
       READ(*,*) DTIME,PARA1D,PARA3D,DIFFUSY,GRWID,AR,ARWID,
      & VIINCA,VIINCWID,VIA,VIWID,EQVOLFRAC,OBOUND
       TIME=0.0
       DO 100 I100=1,10,1
        PRETIME=TIME
        TIME=DTIME*I100
        CALL MAP_STEEL_SIM_TRANS(I100,TIME,PRETIME,DTIME,PARA1D,PARA3D,
      & DIFFUSY,GRWID,AR,ARWID,VIINCA,VIINCWID,VIA,VIWID,
      & EQVOLFRAC,OBOUND,PERCENTRANS,VTOTTRANS,VAREAL,
      & VWIDREAL,VAREALINC,VWIDREALINC,IFLAG,SATTIME)
        WRITE(*,200) TIME,VTOTTRANS,VAREAL,VWIDREAL,
      & VAREALINC,VWIDREALINC
        IF (PERCENTRANS.GE.100.0) GOTO 300
100    CONTINUE
200    FORMAT(E10.3,1X,E10.3,1X,E10.3,1X,E10.3,1X,E10.3,1X,E10.3)
300    STOP
       END

2. Program data

5.0  2.5  3.0  1.0E-13  1.0E-6  3.0  0.02  4.0E13  4.0E13  1.0E9  1.0E9  0.85  4.0E4

3. Program results

0.500E+01  0.714E-01   0.624E-01   0.500E-03  0.800E-02   0.500E-03
0.100E+02  0.192E+00   0.127E+00   0.363E-01  0.266E-01   0.236E-02
0.150E+02  0.262E+00   0.134E+00   0.628E-01  0.564E-01   0.882E-02
0.200E+02  0.351E+00   0.134E+00   0.933E-01  0.991E-01   0.247E-01
0.250E+02  0.472E+00   0.134E+00   0.132E+00  0.152E+00   0.542E-01
0.300E+02  0.620E+00   0.134E+00   0.179E+00  0.209E+00   0.980E-01
0.350E+02  0.773E+00   0.134E+00   0.228E+00  0.262E+00   0.150E+00
0.400E+02  0.899E+00   0.134E+00   0.267E+00  0.300E+00   0.198E+00

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Auxiliary Routines

None.

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Keywords

simultaneous, phase, transformation, nucleation, growth, random, boundary, reconstructive, displacive, numerical, anisothermal, depletion, enrichment

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Download

Download source code

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