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N. Chester,

Phase Transformations Group,

Department of Materials Science and Metallurgy,

University of Cambridge,

Cambridge, U.K.

E-mail: Naomi_Chester@technology.britishsteel.co.uk

Added to MAP: November 1999

To calculate the time required to form a given volume fraction of bainitic ferrite.

Language: | FORTRAN |

Product form: | Source code |

DOUBLE PRECISION FUNCTION & MAP_STEEL_NEWF2 (TK,GM0,PSI,THETA,ND,CD,XB,U) DOUBLE PRECISION TK,GM0,PSI,THETA,ND,CD(4),XB,U

This program finds the time required for a given volume fraction of bainitic ferrite to form during an isothermal transformation in a steel alloy. Reference 1 describes the theory and gives an expression for the overall transformation rate of bainite (equation 28). However the final solution for the time required which is given in reference 1 contains an error. A correct analytical solution was obtained using equation 410 in reference 2, giving the result expressed by equation 4.35 in reference 4. This latter equation contains 4 fitting parameters; values for these parameters were obtained by fitting the equation to experimental data values for low alloy steels [4]. These values must be passed to the function from the calling routine in the array CD:

CD(2) = 0.14750D+03

CD(3) = 0.20980D+05

CD(4) = 0.30327D+02

Calculations using equation 4.35 [4] were found to give correct predictions only for a temperature of 405 deg. C, which was attributed to a change in plate volume with transformation temperature. This program incorporates a temperature dependent expression for the plate width u_{w} [5] given by:

subject to a minimum value of 0.05 microns and with temperature T in Centigrade.

- G.I. Rees and H.K.D.H. Bhadeshia, 1992,
*Materials Science and Technology*,**8**, 985-993. - K. Rektorys, 1969,
*Survey of Applicable Mathematics*, ILIFFE Books Ltd, London, p543. - H.K.D.H. Bhadeshia,
*J. Phys. (Orsay)*,**43**, (C4), 443-448. - N.A. Chester, 1997, PhD Thesis, University of Cambridge,
*Mathematical Modelling of Microstructural Development in Continuously Annealed High-Strength Steels*. - L.C. Chang, 1995, PhD Thesis, University of Cambridge,
*Bainite Transformation and Novel Bainitic Rail Steels*. - N.A. Chester and H.K.D.H. Bhadeshia, 1997,
*IV European Symposium on Martensitic Transformation (Esomat '97) Conference Proceedings*, Enschede, The Netherlands.

**TK**- real- TK is the temperature at which bainite is to form (Kelvin).
**GM0**- real- GMO is the initial value of the maximum nucleation free energy change (Jmol
^{-1}). **PSI**- real- PSI is the normalised volume fraction of bainitic ferrite,
*i.e.*the actual volume fraction divided by the maximum volume fraction. **THETA**- real- THETA is the maximum volume fraction of bainite, taken to be that at which
the carbon concentration of the austenite reaches the T
_{0}' line. **ND**- real- ND is the grain size of the parental austenite (microns).
**CD**- real array of dimension 4- CD must contain the four fitting parameters:
- CD(1) = 0.57919D-06
- CD(2) = 0.14750D+03
- CD(3) = 0.20980D+05
- CD(4) = 0.30327D+02

**XB**- real- XB is the mole fraction of carbon in the alloy.

**MAP_STEEL_NEWF2**- real- MAP_STEEL_NEWF2 is the time taken to form the given volume fraction of bainite (seconds).
**U**- real- U is the volume of the bainite plates used in the calculations (mm
^{3}).

None.

No information supplied.

None.

IMPLICIT NONE DOUBLE PRECISION TK,GM0,PSI,THETA,ND,CD(4),XB,NEWFX,U DOUBLE PRECISION MAP_STEEL_NEWF2 C WRITE(*,*) 'Input T(K), GM0, norm. vol. fraction:' READ (*,*) TK, GM0, PSI WRITE(*,*) 'Max. vol. fraction, grain size, C mole fraction:' READ (*,*) THETA, ND, XB CD(1) = 0.57919D-06 CD(2) = 0.14750D+03 CD(3) = 0.20980D+05 CD(4) = 0.30327D+02 NEWFX = MAP_STEEL_NEWF2(TK,GM0,PSI,THETA,ND,CD,XB,U) U = U * 1D9 WRITE (*,1) NEWFX,U STOP 1 FORMAT(' Time for bainite formation = ',F7.2,' seconds'/ & ' Plate volume used in calculations = ',F7.2,' cubic microns') END

Input T(K), GM0, norm. vol. fraction: 653.0 -1523.03 0.2015 Max. vol. fraction, grain size, C mole fraction: 0.397 16.0 0.0172

Time for bainite formation = 93.24 seconds Plate volume used in calculations = 14.12 cubic microns

**Utility Functions**:

MAP_UTIL_FUNCT

MAP_UTIL_FACT (Called by MAP_UTIL_FUNCT)

bainite, kinetics, volume fraction, bainitic ferrite, time

**
MAP originated from a joint project of the National Physical Laboratory and the University of Cambridge.
**