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MAP_STEEL_NUCSOLVE finds the heterogeneous nucleation rate required to form a given volume fraction of allotriomorphic ferrite, during continuous cooling from austenite, for a specified length of time. It uses subroutine MAP_STEEL_AVOLF to calculate the volume fraction of allotriomorphic ferrite as a function of nucleation rate [1]. The bisection method [2] is used to find the solution.

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References

H.K.D.H. Bhadeshia, L-E Svensson, and B. Gretoft, 1987, Proc. Conf. Welding Metallurgy and Structural Steels, ed. J.Y. Koo, TMS AIME, Warrendale, Penn., 517-530.
W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, 1986, Numerical Recipes, Cambridge University Press, Cambridge, U.K.
H.K.D.H. Bhadeshia, 1982, Metal Science, 16, 156-165.

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Parameters

Input parameters

V - real: V is the required volume fraction of allotriomorphic ferrite.
SV - real: SV is the austenite grain boundary surface area per unit volume (m^-1).
BPRE - real: BPRE is the previous value of the function MAP_STEEL_NUCSOLVE for the allotriomorphic ferrite boundary nucleation rate per unit area (in m^-2s^-1), and is used as an initial value for the nucleation rate in the calculations. When the function is called for the first time, this value should be set to 1.0 on entry.
ALP - real: ALP is the one-dimensional parabolic thickening rate constant perpendicular to the grain boundary for ferrite allotriomorphs (ms^-0.5).
ETA - real: ETA is the ratio of the one-dimensional parabolic lengthening rate constant parallel to the grain boundary, to the 1-d parabolic thickening rate constant perpendicular to the grain boundary (ALP). It is usually defined as 3.
XBAR - real: XBAR is the mole fraction of carbon in the bulk austenite.
XAGA - real: XAGA is the mole fraction of carbon in the ferrite at the alpha/gamma interface.
XGAG - real: XGAG is the mole fraction of carbon in the ferrite at the alpha/alpha+gamma boundary.
TIME - real: TIME is the incubation time for ferrite formation, allowing for the effects of austenite grain size as well as niobium and boron in solid solution (seconds).

Output parameters

AREA - real: AREA is the fraction of austenite grain boundary covered by ferrite.
ANS - real: ANS is the value of the integral of the extended area over all space, corrected for extended volume effects.
ZETA - real: ZETA is the volume fraction of ferrite formed, divided by the equilibrium volume fraction of ferrite.
IFAIL - integer: IFAIL = 0 on exit if a solution was found. IFAIL is set to 1 or 2 if no solution is found (see Error Indicators below)
MAP_STEEL_NUCSOLVE - real: MAP_STEEL_NUCSOLVE is the nucleation rate required to form an allotriomorphic ferrite volume fraction, V, in time, TIME.

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Error Indicators

IFAIL=0 on successful exit of the subroutine.

IFAIL=1 if the number of iterations in the subroutine exceeds 40; MAP_STEEL_NUCSOLVE is set to the last estimated value for the nucleation rate.

IFAIL=2 if the number of iterations exceeds 40 when attempting to bracket the solution; MAP_STEEL_NUCSOLVE remains unchanged on exit.

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Accuracy

A solution for the nucleation rate is accepted if the predicted volume fraction satisfies the criterion:

Delta_V/V < 0.01

where V is the target volume fraction, and Delta_V is the error in its determination.

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Further Comments

None.

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Example

1. Program text

      IMPLICIT NONE
      DOUBLE PRECISION ALP,AREA,ANS,AREA,BPRE,ETA,RATE,SV,TIME
      DOUBLE PRECISION V,XAGA,XBAR,XGAG,ZETA
      DOUBLE PRECISION MAP_STEEL_NUCSOLVE
      INTEGER IFAIL
C
      WRITE (*,*)
      WRITE (*,*) 'Input SV, ALP, ETA, TIME'
      READ  (*,*) SV,ALP,ETA,TIME
      WRITE (*,*) 'Input XBAR, XAGA, XGAG'
      READ  (*,*) XBAR,XAGA,XGAG
      WRITE (*,*) 'Input V: required volume fraction:'
      READ  (*,*) V
      BPRE = 1D0
      RATE = MAP_STEEL_NUCSOLVE
     &     (AREA,ANS,V,SV,BPRE,ALP,ETA,ZETA,XBAR,XAGA,XGAG,TIME,IFAIL)
      IF ( IFAIL.NE.0 ) THEN
         WRITE(*,*) 'No solution found after 40 iterations'
      ELSE
      WRITE (*,1) V, RATE
      WRITE (*,2) ANS, ZETA, AREA
      ENDIF
      STOP
    1 FORMAT( // 'Required volume fraction = ',D12.3 /
     &           'Nucleation rate needed   = ',D12.3,' 1/m(2)s')
    2 FORMAT( //'Value of the integral, ANS                  = ',
     &   D12.5/'Normalised volume fraction of ferrite, ZETA = ',D12.5/
     &   'Fraction of austenite GB covered by ferrite = ',D12.5)
      END

2. Program data

 Input SV, ALP, ETA, TIME
  10000.0  3E-6 3.0   20
 Input XBAR, XAGA, XGAG
       0.02 0.0002 0.06
 Input V: required volume fraction: 
 0.01

3. Program results

Required volume fraction =    0.100D-01
Nucleation rate needed   =    0.944D+06 1/m(2)s


Value of the integral, ANS                  =  0.37609D-01
Normalised volume fraction of ferrite, ZETA =  0.14974D-01
Fraction of austenite GB covered by ferrite =  0.46894D-01

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Language:	FORTRAN
Product form:	Source code

Materials Algorithms Project
Program Library

Function MAP_STEEL_NUCSOLVE

Provenance of Source Code

Purpose

Specification

Description

References

Parameters

Input parameters

Output parameters

Error Indicators

Accuracy

Further Comments

Example

1. Program text

2. Program data

3. Program results

Auxiliary Routines

Keywords

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Input parameters

Output parameters

1. Program text

2. Program data

3. Program results

Materials Algorithms Project
Program Library