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Subroutine MAP_STEEL_AN2

  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

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

Added to MAP: July 1999.

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To calculate the volume fractions of the microstructures formed in low-alloy steel weld deposits during cooling. Also given are values for the allotriomorphic ferrite half-thickness and the time available for unrestricted Widmanstätten ferrite growth.

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Product form:Source code



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The model, used here to calculate the volume fractions of the microstructures formed during cooling in weld deposits, is described in reference 1. The austenite is assumed to form columnar grains which are hexagonal in cross-section with sides of length L/2. The allotriomorphic ferrite grows inwards from the grain boundaries to form a layer of thickness q (half-thickness). The subroutine requires, as input, values for the parabolic thickening rate constant of allotriomorphic ferrite as a function of temperature. (Suitable values can be obtained using subroutine MAP_STEEL_RATE2.) These values are fitted to a 3rd order polynomial in temperature. The equation is re-expressed as a function of time by substituting into it an expression for temperature using equation (1b) of reference 2:

dT/dt = (C1/(Qo.eta)).(T - Ti)C2 ,
dT/dt is the cooling rate,
Qo is the heat input (Jm-1) (Qo= current*voltage/speed),
eta is the arc weld efficiency,
Ti is the interpass temperature and
C1 and C2 are the heat flow constants.

(Further details and values for C1 and C2 can be obtained by referring to subroutine MAP_STEEL_COOLCU.) The half thickness of allotriomorphic ferrite, q, obtained by continuous cooling from the temperature at which ferrite formation starts to the temperature at which it ceases and Widmanstätten ferrite starts, can then be calculated from equation 5 of reference 1:

Equation 5 of reference 1.

where the integration is made over time t as the weld cools from the allotriomorphic ferrite start temperature at t=0 to the stop temperature at tl, and al is the parabolic thickening rate constant as a function of temperature.

The volume fraction, Va, of allotriomorphic ferrite is obtained using equations 6(a) and 6(b) of reference 1:

va = 8.q.tan(30°){L-2.q.tan(30°)} / L2
Va = 2.04 va + 0.035.

The second equation is an empirical correction to the calculated volume fraction. The half thickness of ferrite is then recalculated from the volume fraction using equation 6(c) of reference 1:

q' = {L - (L2 - VaL2)0.5} / 4tan(30°)

If the flag JI3 is set to 5, however, no empirical corrections are made and the volume fraction of allotriomorphic ferrite is read into the subroutine as an input.

The volume fraction of Widmanstätten ferrite formed, Vw, is calculated from equation 8 of reference 1:

Vw = C4.Vmax.(L-4q'.tan(30°)).t22 / L2

where Vmax is the lengthening rate of a Widmanstätten ferrite plate, t2 is the time available for Widmanstätten ferrite to grow without interference from the formation of acicular ferrite and C4 is a factor which depends only on the concentration of boron in solution. The time t2 is set equal to the time required for a Widmanstätten ferrite plate to grow unhindered across an austenite grain, t3, given by equation 11 of reference 1:

t3 = (L.sin(60°) - 2q') / Vmax ,

unless it is greater than a critical time, tc, after which the formation of acicular ferrite interferes with the growth of Widmanstätten ferrite. In this case the time t2 is set equal to tc, which is taken to be 2.11 seconds.

The factor C4 is defined as 6.8717 unless the flag JBOR=1. In this case the effect of soluble boron in the austenite of concentration xB (in parts per million) is taken into account and C4 is calculated using the equation:

C4 = 6.8717 - xB * FBORW

FBORW defines the strength of the interaction and is an input to the subroutine. A value of 5.0 for FBORW is generally used [3].

The volume fraction of acicular ferrite formed, Vaf, is taken to be equal to the volume fraction of the remaining untransformed austenite:

Vaf = 1 - Va - Vw .

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  1. H.K.D.H. Bhadeshia, L.-E. Svensson and Gretoft, 1985, Acta metall., 33, 1271-1283.
  2. L.-E. Svensson, B. Gretoft and H.K.D.H. Bhadeshia, 1986, Scandinavian Journal of Metallurgy, 15, 97-103.
  3. H.K.D.H. Bhadeshia. Unpublished.

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Input parameters

X - real array of dimension 6
See below.

Y - real array of dimension 6

X and Y contain values for the parabolic thickening rate constant for allotriomorphic ferrite as a function of temperature. X contains the temperature values (°C) and Y the corresponding parabolic thickening rate constants (cms-0.5).

HIGHT - real
HIGHT is the allotriomorphic ferrite start temperature (°C).

LOWT - real
LOWT is the allotriomorphic ferrite stop temperature (°C).

L - real
L is twice the hexagon side length of the austenite grains (microns).

VMAX - real
VMAX is the lengthening rate of the Widmanstätten ferrite plates (microns/second).

C1 - real
C1 is the heat flow constant C1 described above.

C2 - real
C2 is the heat flow constant C2 described above.

TINT - real
TINT is the interpass or preheat temperature (°C).

CURR - real
CURR is the welding current (Amps).

VOLT - real
VOLT is the welding voltage (Volts).

EFF - real
EFF is the arc transfer efficiency.

VELOC - real
VELOC is the welding speed (ms-1).

JI3 - integer
If JI3 is set to 5, VOLFRA becomes an input parameter. The volume fraction of allotriomorphic ferrite is not recalculated by the program. No empirical correction is made to the volume fraction. Q is calculated using equaton 6(c) of reference 1.

PPMBSL - real
PPMBSL is the soluble boron concentration (ppm by weight).

JBOR - integer
If JBOR=1 the effect of boron on the volume fraction of Widmanstätten ferrite formed is taken into account.

FBORW - real
FBORW determines the strength of the interaction between boron and the formation of Widmanstätten ferrite. A value of 5.0 is generally used.

Output parameters

VACIC - real
VACIC is the volume fraction of acicular ferrite formed.

VOLW - real
VOLW is the volume fraction of Widmanstätten ferrite formed.

VOLFRA - real
VOLFRA is the volume fraction of allotriomorphic ferrite formed.

Q - real
Q is the half thickness of allotriomorphic ferrite (microns).

TIMC - real
TIMC is the duration of the Widmanstätten ferrite growth (seconds).

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Error Indicators


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No information supplied.

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Further Comments


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1. Program text

      WRITE(*,*) 'Input 6 temperature values:'
      READ (5,*) (X(I), I=1,6)
      WRITE(*,*) 'Input 6 values for the parabolic thicknening rate:'
      READ (5,*) (Y(I), I=1,6)
      WRITE(*,*) 'Input HIGHT,LOWT,L,VMAX:'
      WRITE(*,*) 'Input C1,C2,TINT:'
      READ (5,*) C1,C2,TINT
      WRITE(*,*) 'Input CURR,VOLT,EFF,VELOC:'
      WRITE(*,*) 'Input JBOR,PPMBSL,FBORW:'
      JI3 = 0
    1 FORMAT('Volume fraction allotriomorphic ferrite   = ',F7.3/
     &       'Volume fraction Widmanstatten ferrite     = ',F7.3/
     &       'Volume fraction acicular ferrite          = ',F7.3/
     &       'Half thickness of allotriomorphic ferrite = ',F7.3,
     &       ' microns'/
     &       'Duration of Widmanstatten ferrite growth  = ',F7.3,
     &       ' seconds')

2. Program data

 Input 6 temperature values:
 640.0 680.0 720.0 760.0 800.0 840.0
 Input 6 values for the parabolic thicknening rate:
 3.02E-04 3.32E-04 3.32E-04 3.06E-04 2.22E-04 5.54E-05
 722.0 621.0 168.0 1.98E-04
 Input C1,C2,TINT:
 1325.0 1.6 200.0
 180.0 34.0 0.775 4.0E-03
 1 1.0 5.0

3. Program results

Volume fraction allotriomorphic ferrite   =   0.338
Volume fraction Widmanstatten ferrite     =   0.080
Volume fraction acicular ferrite          =   0.582
Half thickness of allotriomorphic ferrite =  13.538 microns
Duration of Widmanstatten ferrite growth  =   0.211 seconds

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Auxiliary Routines


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steel welds, acicular, Widmanstatten, allotriomorphic, ferrite, volume fraction

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Download source code

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