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  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

C.Capdevila-Montes and C.Garcia de Andres,
Departamento de Metalurgia Fisica,
Centro Nacional de Investigaciones Metalurgicas- Consejo Superior de Investigaciones Cientificas,
Avenida Gregorio del Amo 8,
28040 Madrid Spain

H.K.D.H. Bhadeshia,
Phase Transformations Group,
Department of Materials Science and Metallurgy,
University of Cambridge,
Cambridge, U.K.

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Uses the the Gilmour et al approximation but with an analytical treatment (for alpha formation under NPLE mechanism) of the soft impingement problem in Fe-C-X alloys.

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Complete Program.

Product form:Source code

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The following calculations deal with the one-dimensional growth of planar grain boundary allotriomorphs from opposite side of grain.

This process may be considered in two stages: the first involves parabolic growth from both sides according to the earlier exact calculations; during the second stage of inmpingment, the driving force due to the carbon activity gradient is gradually reduced and the transformation rate correspondingly decreases to a value consistent with a uniform and equal carbon chemical potential in both phases [1].

The concentration of carbon in the middle of the austenite grain size is calculated by means of balancing the amount of carbon enrichment of the austenite against the carbon depletion of the ferrite [2]. At fixed time and temeprature, it is possible to study the volume fraction of pearlite formed. The pearlite appears when the carbon concentration in remaining austenite enriches the carbon concentration of the extrapolated tie-line austenite/austenite + cementite at this temperature [3], and when the moving interface is slower than maximum rate to allow nucleation in the interface [4].

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  1. J.B Gilmour, G.R. Purdy and J.S. Kirkaldy, Met. Trans, 3, 3213, (1972).
  2. H.K.D.H. Bhadeshia, Progress in Materials Science, 29, 321, (1985).
  3. M. Takahashi, PhD. Thesis, University of Cambridge, (1992).
  4. H.I. Aaronson, M.R. Plitchta, G.W. Franti and K.C. Russell, Met. Trans., 9A, 363, (1978).
  5. H.K.D.H. Bhadeshia, Metal Science, 16, 159, March 1982.
  6. K.C. Russell, Acta Metall., 17, 1123, (1969).
  7. D.E. Porter and K.E. Easterling, Phase Transformation in Metals and Alloys, Van Nostrand Reinhold, Wokingham, England, (1981).
  8. O.Grong, Metallurgical Modelling of Welding, ed. H.K.D.H. Bhadeshia, Institute of Materials, London, (1995).
  9. L.Kaufman and S.V. Radcliffe, Decomposition of austenite by diffusional process, ed. by V.F. Zackay and H.I. Aaronson, Interscience Publishers, New York, NY, 313, (1962).

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Input parameters

CARBON - real
Overall concentration of carbon (in weight percent).

SI - real
Overall concentration of silicon (in weight percent).

MN - real
Overall concentration of manganese (in weight percent).

NI - real
Overall concentration of nickel (in weight percent).

MO - real
Overall concentration of molybdenum (in weight percent).

CR - real
Overall concentration of chromium (in weight percent).

V - real
Overall concentration of vanadium (in weight percent).

CO - real
Overall concentration of cobalt (in weight percent).

CU - real
Overall concentration of copper (in weight percent).

AL - real
Overall concentration of aluminium (in weight percent).

WO - real
Overall concentration of tungsten (in weight percent).

LINEAL - real
Prior austenite grain size in microns.

CTEMP - real
Isothermal heat treatment temperature (in celsius).

TIME - real
Time at CTEMP temperature (in seconds).

TIME2 - real
Time required to study the carbon profile ahead allotriomorphic ferrite of half-thickness HALF at temperature T (in seconds).

Output parameters

All output is to STDOUT.

TIME - real
Time at CTEMP temperature (in seconds).

Z-IMP - real
Half-thickness of ferrite under impingment (in centimeters).

Z-PARA - real
Half-thickness of ferrite under parabolic growth (in centimeters).

CL - real
Concentration of carbon in the middle of austenite grain (in weight percent).

VOL_FRAC - real
Volume fraction of ferrite formed.

Velocity of the moving alpha/gamma interface (in meters per second).

C - real
Carbon profile ahead the interface (in weight percent).

Position ahead the alpha/gamma interface at CTEMP and TIME2 (in centimeters).

Volume fraction of pearlite estimated.

VTOT - real
Total volume fraction of ferrite and pearlite.

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Error Indicators


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If holding time at temperature close to Ae3 is higher than 10000 seconds, a problem with the convergence of data is obtained.

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Further Comments

The nucleation time is incorporated according to [5-6], the effect of different alloying elements is taken into account.

The interface energy for nucleation of cementite in a moving interface is calculated according to [7-9].

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1. Program text

       Complete program.

2. Program data


Key :
Chemical composition C,Si,Mn,Ni,Mo,Cr,V,Co,Cu,Al,W
Prior austenite grain size (in microns)
Temperature (in centigrade)
Time at this temperature (in seconds)
Time required to study the carbon profile ahead interface

3. Program results

            ****Chemical composition****

  Carbon=0.3700               Silicon=0.5600
  Manganese=1.4500            Nickel=0.0000
  Molibdenon=0.0250           Cromium=0.0400


     Isothermal temperature= 640.0
     Holding time=1200.00
     Austenite grain size ,in microns= 70.00
     TIMESTEP in calculations=  10.00

     C-C interaction energy in austenite  in J/mol=8394.418
     C-C interaction energy in ferrite  in J/mol=   48570.000

               ~~~~~~~MOLE FRACTION~~~~~~~~

     Mole fraction of C in austenite=0.01680
     Concentration of carbon in alfa/gama interface into ferrite=0.00074
     Concentration of carbon in alfa/gama interface into austenite=0.04039958

               ~~~~~WEIGHT PERCENTAGE~~~~~~

     Carbon concentration of carbon at the bulk=0.37000
     Concentration of carbon in alpha/gamma interface into austenite=0.90232597
     Concentration of carbon in alpha/gamma interface into ferrite=0.01596

               <<<<<< CRITICAL TEMPERATURES >>>>>>

                  Ae3 temperature (C)= 726.85
                  Ae1 temperature (C)= 656.85


     C diffusion coefficient into austenite(cm**/s)=   0.12011E-08
     Parabolic-thickening rate constant (3-D) in cm/s**(1/2) =   0.10603E-03
     Parabolic-thickening rate constant (1-D) in cm/s**(1/2) =   0.48376E-04

     Half-thickness of the layer of ferrite  in cm. NO INMPINGMENT =   0.16758E-02

     Impingement began at   0.146436E-02
     Maximum half-thickness in Fe-C system =  0.251849E-02
     Maximum half-thickness in this system =  0.206482E-02

     Time for nucleation =     25.71 seconds

 Time(s)    Z-imp.(cm)    Z-para.(cm)   CL (%wt)    Vol.Frac  Migration Rate(m/s)


                     ********Parabolic Growth********

       0.00   0.0000E+00   0.0000E+00   0.3700E+00   0.0000E+00   0.0000E+00
      10.00   0.0000E+00   0.0000E+00   0.3700E+00   0.0000E+00   0.0000E+00
      20.00   0.0000E+00   0.0000E+00   0.3700E+00   0.0000E+00   0.0000E+00
      30.00   0.1002E-03   0.1002E-03   0.3700E+00   0.2864E-01   0.1167E-06
      40.00   0.1829E-03   0.1829E-03   0.3700E+00   0.5226E-01   0.6398E-07
      50.00   0.2384E-03   0.2384E-03   0.3700E+00   0.6813E-01   0.4907E-07
      60.00   0.2833E-03   0.2833E-03   0.3700E+00   0.8094E-01   0.4130E-07
      70.00   0.3220E-03   0.3220E-03   0.3700E+00   0.9199E-01   0.3634E-07
      80.00   0.3565E-03   0.3565E-03   0.3700E+00   0.1018E+00   0.3283E-07
      90.00   0.3879E-03   0.3879E-03   0.3700E+00   0.1108E+00   0.3017E-07
     100.00   0.4170E-03   0.4170E-03   0.3700E+00   0.1191E+00   0.2806E-07
     110.00   0.4441E-03   0.4441E-03   0.3700E+00   0.1269E+00   0.2634E-07
     120.00   0.4698E-03   0.4698E-03   0.3700E+00   0.1342E+00   0.2491E-07
     130.00   0.4940E-03   0.4940E-03   0.3700E+00   0.1412E+00   0.2368E-07
     140.00   0.5172E-03   0.5172E-03   0.3700E+00   0.1478E+00   0.2262E-07
     150.00   0.5393E-03   0.5393E-03   0.3700E+00   0.1541E+00   0.2170E-07
     160.00   0.5606E-03   0.5606E-03   0.3700E+00   0.1602E+00   0.2087E-07
     170.00   0.5811E-03   0.5811E-03   0.3700E+00   0.1660E+00   0.2014E-07
     180.00   0.6009E-03   0.6009E-03   0.3700E+00   0.1717E+00   0.1947E-07
     190.00   0.6201E-03   0.6201E-03   0.3700E+00   0.1772E+00   0.1887E-07
     200.00   0.6387E-03   0.6387E-03   0.3700E+00   0.1825E+00   0.1832E-07
     210.00   0.6567E-03   0.6567E-03   0.3700E+00   0.1876E+00   0.1782E-07
     220.00   0.6743E-03   0.6743E-03   0.3700E+00   0.1927E+00   0.1735E-07
     230.00   0.6914E-03   0.6914E-03   0.3700E+00   0.1976E+00   0.1692E-07
     240.00   0.7082E-03   0.7082E-03   0.3700E+00   0.2023E+00   0.1652E-07
     250.00   0.7245E-03   0.7245E-03   0.3700E+00   0.2070E+00   0.1615E-07
     260.00   0.7405E-03   0.7405E-03   0.3700E+00   0.2116E+00   0.1580E-07
     270.00   0.7561E-03   0.7561E-03   0.3700E+00   0.2160E+00   0.1548E-07
     280.00   0.7714E-03   0.7714E-03   0.3700E+00   0.2204E+00   0.1517E-07
     290.00   0.7864E-03   0.7864E-03   0.3700E+00   0.2247E+00   0.1488E-07
     300.00   0.8012E-03   0.8012E-03   0.3700E+00   0.2289E+00   0.1460E-07
     310.00   0.8157E-03   0.8157E-03   0.3700E+00   0.2330E+00   0.1435E-07
     320.00   0.8299E-03   0.8299E-03   0.3700E+00   0.2371E+00   0.1410E-07
     330.00   0.8439E-03   0.8439E-03   0.3700E+00   0.2411E+00   0.1387E-07
     340.00   0.8576E-03   0.8576E-03   0.3700E+00   0.2450E+00   0.1364E-07
     350.00   0.8712E-03   0.8712E-03   0.3700E+00   0.2489E+00   0.1343E-07
     360.00   0.8845E-03   0.8845E-03   0.3700E+00   0.2527E+00   0.1323E-07
     370.00   0.8976E-03   0.8976E-03   0.3700E+00   0.2565E+00   0.1304E-07
     380.00   0.9106E-03   0.9106E-03   0.3700E+00   0.2602E+00   0.1285E-07
     390.00   0.9233E-03   0.9233E-03   0.3700E+00   0.2638E+00   0.1267E-07
     400.00   0.9359E-03   0.9359E-03   0.3700E+00   0.2674E+00   0.1250E-07
     410.00   0.9483E-03   0.9483E-03   0.3700E+00   0.2710E+00   0.1234E-07
     420.00   0.9606E-03   0.9606E-03   0.3700E+00   0.2745E+00   0.1218E-07
     430.00   0.9727E-03   0.9727E-03   0.3700E+00   0.2779E+00   0.1203E-07
     440.00   0.9846E-03   0.9846E-03   0.3700E+00   0.2813E+00   0.1188E-07
     450.00   0.9965E-03   0.9965E-03   0.3700E+00   0.2847E+00   0.1174E-07
     460.00   0.1008E-02   0.1008E-02   0.3700E+00   0.2880E+00   0.1161E-07
     470.00   0.1020E-02   0.1020E-02   0.3700E+00   0.2913E+00   0.1148E-07
     480.00   0.1031E-02   0.1031E-02   0.3700E+00   0.2946E+00   0.1135E-07
     490.00   0.1042E-02   0.1042E-02   0.3700E+00   0.2978E+00   0.1123E-07
     500.00   0.1054E-02   0.1054E-02   0.3700E+00   0.3010E+00   0.1111E-07
     510.00   0.1065E-02   0.1065E-02   0.3700E+00   0.3042E+00   0.1099E-07
     520.00   0.1076E-02   0.1076E-02   0.3700E+00   0.3073E+00   0.1088E-07
     530.00   0.1086E-02   0.1086E-02   0.3700E+00   0.3104E+00   0.1077E-07
     540.00   0.1097E-02   0.1097E-02   0.3700E+00   0.3134E+00   0.1067E-07
     550.00   0.1108E-02   0.1108E-02   0.3700E+00   0.3165E+00   0.1056E-07
     560.00   0.1118E-02   0.1118E-02   0.3700E+00   0.3195E+00   0.1046E-07
     570.00   0.1129E-02   0.1129E-02   0.3700E+00   0.3225E+00   0.1037E-07
     580.00   0.1139E-02   0.1139E-02   0.3700E+00   0.3254E+00   0.1027E-07
     590.00   0.1149E-02   0.1149E-02   0.3700E+00   0.3283E+00   0.1018E-07
     600.00   0.1159E-02   0.1159E-02   0.3700E+00   0.3312E+00   0.1009E-07
     610.00   0.1169E-02   0.1169E-02   0.3700E+00   0.3341E+00   0.1001E-07
     620.00   0.1179E-02   0.1179E-02   0.3700E+00   0.3369E+00   0.9922E-08
     630.00   0.1189E-02   0.1189E-02   0.3700E+00   0.3398E+00   0.9839E-08
     640.00   0.1199E-02   0.1199E-02   0.3700E+00   0.3426E+00   0.9759E-08
     650.00   0.1209E-02   0.1209E-02   0.3700E+00   0.3453E+00   0.9681E-08
     660.00   0.1218E-02   0.1218E-02   0.3700E+00   0.3481E+00   0.9604E-08
     670.00   0.1228E-02   0.1228E-02   0.3700E+00   0.3508E+00   0.9529E-08
     680.00   0.1237E-02   0.1237E-02   0.3700E+00   0.3535E+00   0.9456E-08
     690.00   0.1247E-02   0.1247E-02   0.3700E+00   0.3562E+00   0.9385E-08
     700.00   0.1256E-02   0.1256E-02   0.3700E+00   0.3589E+00   0.9315E-08
     710.00   0.1265E-02   0.1265E-02   0.3700E+00   0.3616E+00   0.9246E-08
     720.00   0.1275E-02   0.1275E-02   0.3700E+00   0.3642E+00   0.9180E-08
     730.00   0.1284E-02   0.1284E-02   0.3700E+00   0.3668E+00   0.9114E-08
     740.00   0.1293E-02   0.1293E-02   0.3700E+00   0.3694E+00   0.9050E-08
     750.00   0.1302E-02   0.1302E-02   0.3700E+00   0.3720E+00   0.8988E-08
     760.00   0.1311E-02   0.1311E-02   0.3700E+00   0.3745E+00   0.8926E-08
     770.00   0.1320E-02   0.1320E-02   0.3700E+00   0.3771E+00   0.8866E-08
     780.00   0.1329E-02   0.1329E-02   0.3700E+00   0.3796E+00   0.8807E-08
     790.00   0.1337E-02   0.1337E-02   0.3700E+00   0.3821E+00   0.8749E-08
     800.00   0.1346E-02   0.1346E-02   0.3700E+00   0.3846E+00   0.8692E-08
     810.00   0.1355E-02   0.1355E-02   0.3700E+00   0.3871E+00   0.8637E-08
     820.00   0.1363E-02   0.1363E-02   0.3700E+00   0.3895E+00   0.8582E-08
     830.00   0.1372E-02   0.1372E-02   0.3700E+00   0.3920E+00   0.8529E-08
     840.00   0.1380E-02   0.1380E-02   0.3700E+00   0.3944E+00   0.8476E-08
     850.00   0.1389E-02   0.1389E-02   0.3700E+00   0.3968E+00   0.8425E-08
     860.00   0.1397E-02   0.1397E-02   0.3700E+00   0.3992E+00   0.8374E-08
     870.00   0.1406E-02   0.1406E-02   0.3700E+00   0.4016E+00   0.8324E-08
     880.00   0.1414E-02   0.1414E-02   0.3700E+00   0.4040E+00   0.8275E-08
     890.00   0.1422E-02   0.1422E-02   0.3700E+00   0.4063E+00   0.8227E-08
     900.00   0.1430E-02   0.1430E-02   0.3700E+00   0.4087E+00   0.8180E-08
     910.00   0.1439E-02   0.1439E-02   0.3700E+00   0.4110E+00   0.8134E-08
     920.00   0.1447E-02   0.1447E-02   0.3700E+00   0.4133E+00   0.8088E-08
     930.00   0.1455E-02   0.1455E-02   0.3700E+00   0.4156E+00   0.8043E-08
     940.00   0.1463E-02   0.1463E-02   0.3700E+00   0.4179E+00   0.7999E-08

                     ********Impingement begins********

     950.00   0.1463E-02   0.1471E-02   0.3700E+00   0.4179E+00   0.4983E-08
     960.00   0.1469E-02   0.1479E-02   0.3727E+00   0.4196E+00   0.4963E-08
     970.00   0.1474E-02   0.1487E-02   0.3764E+00   0.4213E+00   0.4943E-08
     980.00   0.1480E-02   0.1494E-02   0.3801E+00   0.4230E+00   0.4922E-08
     990.00   0.1486E-02   0.1502E-02   0.3838E+00   0.4246E+00   0.4901E-08
    1000.00   0.1492E-02   0.1510E-02   0.3875E+00   0.4263E+00   0.4880E-08
    1010.00   0.1498E-02   0.1518E-02   0.3913E+00   0.4280E+00   0.4859E-08
    1020.00   0.1504E-02   0.1525E-02   0.3951E+00   0.4296E+00   0.4837E-08
    1030.00   0.1510E-02   0.1533E-02   0.3989E+00   0.4313E+00   0.4815E-08
    1040.00   0.1515E-02   0.1541E-02   0.4027E+00   0.4330E+00   0.4792E-08
    1050.00   0.1521E-02   0.1548E-02   0.4065E+00   0.4346E+00   0.4769E-08
    1060.00   0.1527E-02   0.1556E-02   0.4104E+00   0.4363E+00   0.4746E-08
    1070.00   0.1533E-02   0.1563E-02   0.4143E+00   0.4379E+00   0.4723E-08
    1080.00   0.1539E-02   0.1571E-02   0.4182E+00   0.4396E+00   0.4699E-08
    1090.00   0.1544E-02   0.1578E-02   0.4221E+00   0.4413E+00   0.4675E-08
    1100.00   0.1550E-02   0.1586E-02   0.4260E+00   0.4429E+00   0.4650E-08
    1110.00   0.1556E-02   0.1593E-02   0.4300E+00   0.4446E+00   0.4625E-08
    1120.00   0.1562E-02   0.1600E-02   0.4339E+00   0.4462E+00   0.4600E-08
    1130.00   0.1568E-02   0.1608E-02   0.4379E+00   0.4479E+00   0.4574E-08
    1140.00   0.1573E-02   0.1615E-02   0.4419E+00   0.4495E+00   0.4548E-08
    1150.00   0.1579E-02   0.1622E-02   0.4460E+00   0.4512E+00   0.4522E-08
    1160.00   0.1585E-02   0.1629E-02   0.4500E+00   0.4528E+00   0.4495E-08
    1170.00   0.1591E-02   0.1636E-02   0.4541E+00   0.4544E+00   0.4468E-08
    1180.00   0.1596E-02   0.1644E-02   0.4582E+00   0.4561E+00   0.4441E-08
    1190.00   0.1602E-02   0.1651E-02   0.4623E+00   0.4577E+00   0.4413E-08
    1200.00   0.1608E-02   0.1658E-02   0.4664E+00   0.4594E+00   0.4385E-08

~~~~~~~~~~~~~~~~~Carbon Profile ahead interface~~~~~~~~~~~~~~~~~~~~~~~~~~

     Number of particles per unit area =  0.2810D+11
     DALPHA =  0.9399D-05 m 
     DGAMMA =   0.7000D-04 m

     Isothermal temperature= 640.0
     Holding time at this temperature=  40.00
     Carbon concen.(%wt) at gamma/theta interf.  in austenite=   0.58919E+00
     Position of interface at this temperature=   0.18290E-03
     Migration rate of alpha/gamma boundary (m/s)=  0.6398E-07
     Highest rate of alpha/gamma to allow nucleation  (m/s)=  0.5740E-07
     Volume fraction of allotromorphic ferrite=   0.52257E-01

    Position(cm)      C(%wt)       Vpearlite        Vtot


               No impingement takes place

    0.18290E-03     0.9023E+00     0.0000E+00     0.5226E-01
    0.28290E-03     0.6835E+00     0.0000E+00     0.5226E-01
    0.38290E-03     0.4647E+00     0.0000E+00     0.5226E-01
    0.48290E-03     0.3700E+00     0.0000E+00     0.5226E-01
    0.58290E-03     0.3700E+00     0.0000E+00     0.5226E-01
    0.68290E-03     0.3700E+00     0.0000E+00     0.5226E-01
    0.78290E-03     0.3700E+00     0.0000E+00     0.5226E-01
    0.88290E-03     0.3700E+00     0.0000E+00     0.5226E-01
    0.98290E-03     0.3700E+00     0.0000E+00     0.5226E-01
    0.10829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.11829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.12829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.13829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.14829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.15829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.16829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.17829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.18829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.19829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.20829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.21829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.22829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.23829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.24829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.25829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.26829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.27829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.28829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.29829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.30829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.31829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.32829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.33829E-02     0.3700E+00     0.0000E+00     0.5226E-01
    0.34829E-02     0.3700E+00     0.0000E+00     0.5226E-01

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Auxiliary Routines

Subroutines :-

Functions :-


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ferrite growth kinetics, allotriomorphic ferrite, soft impingement

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Download source code

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