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

Provenance of Source Code

Jeevan Jaidi
Research Scholar (Ph. D),
Mechanical Engineering Department,
Indian Institute of Science,

E-mail: jaidi@mecheng.iisc.ernet.in

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

Added to MAP: May 2003.

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Program to forecast the development of microstructure in the heat-affected zone of low-alloy steel weldments. Given the chemical composition and cooling conditions, it is possible to estimate the fractions of ferrite, pearlite, bainite and martensite) as a function of the austenite grain size and temperature.

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Language: FORTRAN-90
Product form: Source code

Complete program.

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In fusion welding processes, the thermal cycle consists of heating and cooling periods. Within the heat-affected zone (HAZ), the formation of austenite on heating and its decomposition on cooling depends on the position with respect to the fusion boundary.
The method has been described in references 1 and 2, based on original theory in reference 3. Some of the equations in references 1,2 contain significant errors, and hence are reproduced in a corrected form in the presentation that follows. The units of a variety of empirical activation energies were also not stated in the original papers [1-3]; they are presented below in J/mole.

The austenite can, during cooling, transform into ferrite, pearlite, bainite and martensite. The start and finish temperatures of each of the phases are calculated as follows:

Equation x of reference y.

Austenite decomposition is characterised as follows:

Equation x of reference y.

where X is the volume fraction of the transformation product at a given instant of time, G is the ASTM grain size number, and m and p are empirical constants related to the nucleation and growth model.

Austenite is stable at temperatures above the Ae3 line and is unstable below the Ae3 line. As the temperature drops below the Ae3, ferrite begins to form. The austenite-ferrite decomposition rate equation is given by:

Equation x of reference y.

For temperatures belo eutectoid, and depending upon the cooling rate, the untransformed austenite will tend to decompose to pearlite. The austenite-pearlite decomposition rate is given by:

Equation x of reference y.

where D is the diffusion coefficient:

Equation x of reference y.

When the bainite-start temperaure is reached, the pearlite is in this empirical theory, assumed to continue to form bainite at a rate given by:

Equation x of reference y.

where f(X,Ci) is expressed by the following:

Equation x of reference y.

The remaining austenite is thenassumed to form martensite.

In the rate equations, the supercooling is defined with respect to the start-temperature of the phase concerned. The average austenite grain size in micro-meters is calculated numerically using the free grain growth model. This may not be appropriate once transformation has started. The ASTM grain size number is then calcuted from the calculated average grain size multiplied by 1.776 to allow for the conversion of two-dimensional measurements to three dimensions.

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  1. D. F. Watt, L. Coon, M. Bibby, J. Goldak and C. Henwood, 1988, Acta Metall., 36, 3029-3035.
  2. C. Henwood, M. Bibby, J. Goldak and D. Watt, 1988, Acta Metall., 36, 3037-3046.
  3. J. S. Kirkaldy and D. Venugopalan, Phase Transformations in Ferrous Alloys, published by Am. Inst. Min. Engg., Philadelphia, Pa (1984)
  4. ,
  5. O. M. Akselsen, O. Grong, N. Ryum and N. Christensen, 1986, Acta Metall., 34, 1807-1815.
  6. M. F. Ashby and K. E. Easterling, Acta Metallurgica, 30 (1982) 1969-1978.
  7. O. Grong, Metallurgical Modelling of Welding, 2nd edition, published by the Insitute of Materials, London.

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Input parameters

A - real array
Array of coefficients in polynomial equation relating temperature and time.

ASTM_GSN - real
ASTM grain size number.

B - real array
Array of coefficients in polynomial equation relating austenite grain size and time

C - real array
Array containing the concentrations of alloying elements (% wt).

C_ALPHA - real
Carbon concentration in ferrite (% wt).

C_GAMMA - real
Carbon concentration in austenite (% wt).

DT - real
Small time interval (s).

FT - real
FT is the function f

FT1 - real
Derivative of FT.

GVALUE - real
GVALUE is the average austenite grain size in micrometers

R - real
Universal gas constant (J/mol/K).

TIME1 - real
Start time for transformation, (s).

TIME2 - real
Finish time for transformation (s).

TIMEAT - real
Time (s).

TVALUE - real
Absolute temperature.

Undercooling (K).

XFE - real
Equilibrium volume fraction of ferrite.

Output parameters

Ae3 - real
Austenite decomposition start-temperature (oC).

Ps - real
Pearlite-start temperature (oC).

Bs - real
Bainite-start temperature (oC).

Ms - real
Martensite-start temperature (oC).

FERRITE - real
Amount of ferrite.

Amount of pearlite.

BAINITE - real
Amount of bainite.

Amount of martensite.

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Error Indicators

The time steps of the order of 10-4 s are required to integrate the austenite decomposition rate equation.

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No information supplied.

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Further Comments

  1. The peak temperaure of a thermal cycle must be above the temperature at which austenite decomposition starts.
  2. A fourth order polynomial curve fit equation (A(1)*t4+A(2)*t3 +A(3)*t2+A(4)*t+A(5)) is used to describe the nature of temperature vs time during the cooling period.
  3. A second order polynomial curve fit equation (B(1)*t2+B(2)*t+B(3)) is used to describe the nature of average grain size vs time during the cooling period.

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1. Program text

Complete program.

2. Program data

The concentraion (%wt) of alloying elements and the polynomial coefficients for
temperature vs time and average grain size vs time are the input data to be given
in the file haz_input.txt. 

The following sequence is adapted for the concentration of alloying elements.
No.   Element 
C(1)    C
C(2)    Ni
C(3)    Si
C(4)    V
C(5)    Mo
C(6)    W
C(7)    Mn
C(8)    Cr
C(9)    Cu
C(10)   P
C(11)   Al
C(12)   As
C(13)   Ti

Coefficients of polynomial equation for temperature vs time

Coefficients of polynomial equation for average grain size vs time

See file haz_input.txt

3. Program results

See file haz_output.txt

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Auxiliary Routines


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ASTM grain size number, super cooling, heat-affected zone (HAZ), rate equation, austenite decomposition, ferrite, pearlite, bainite and martensite.

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Download source code

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

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