Empirical Rant

In metallurgy we often term very simple models to be `empirical models’ in contrast to `physical models’. I really wish there was a better name for the `empirical models` – since physical models are more empirical, and `empirical models’ are actually less empirical. Use of such equations can be very useful because they do provide a summary of observations with-in some range of observed behaviour. Even when a physical model exists these simple models are often still preferred because of the ease with which they can be used.

The source of my confusion is the now contradictory uses of the word empirical…

Physical models incorporate more physical understanding, are based on a theoretical understanding. Any theory can only be based on, and validated against, observations. (Edit: i.e. empirical observations)

A better description for our `empirical models’ would be Ad-hoc, make-do, summary or arbitrary.

Comparison of empirical and physical models
This is best described by an example. The martensite start temperature (MS) is often described by an equation of the form; MS = A*XC + B*XMn + C*XCr…

MS(C) = 521 – 353.C – 225.Si – 24.3.Mn – 27.4.Ni 0 17.7.Cr – 25.8.Mo

Another example is the use of various ‘carbon equivilant’s.
Carbon Equivilant = CE = C + Mn/5 + Mo / 5 + Cr/10 + Ni/50

Thomas Sourmail and Carlos Garcia-Mateo have written a paper on prediciton of M_S by various methods,
(Critical assessment of models for predicting the Ms temperature of steels, T. Sourmail and C. Garcia-Mateo Comp. Mater. Sci., 2005:34, p323-334) it is available on Thomas’s webpage;Predicting the martensite start temperature (Ms) of steels.

Ms/ K, all compositions in wt%
[8] 772-316.7C-33.3Mn-11.1Si-27.8Cr-16.7Ni-11.1Mo-11.1W
[9] 811-361C-38.9Mn-38.9Cr-19.4Ni-27.8Mo
[10] 772-300C-33.3Mn-11.1Si-22.2Cr-16.7Ni-11.1Mo
[11] 834.2-473.9C-33Mn-16.7Cr-16.7Ni-21.2Mo
[12] 812-423C-30.4Mn-12.1Cr-17.7Ni-7.5Mo
[12] 785-453C-16.9Ni-15Cr-9.5Mo+217(C)2-71.5(C)(Mn)-67.6(C)(Cr)

Potency of Elements on MS temperature (Change per weight percent).

N C Ni Co Cu Mn W Si Mo Cr V Al
-450 -450 -20 +10 -35 -30 -36 -50 -45 -20 -46 -53 P-1976
  • P-1976 F.B. pickering, `Physical metallurgy of stainless steel developments’, Int. Met. Rev., 21, pp 227-268, 1976.
  • 8 P. Payson and C. H. Savage. Trans. ASM, 33:261-281, 1944.
  • 9 R. A. Grange and H. M. Stewart. Trans. AIME, 167:467-494, 1945.
  • 10 A. E. Nehrenberg. Trans. AIME, 167:494-501, 1945.
  • 11 W. Steven and A. G. Haynes. JISI, 183:349-359, 1956.
  • 12 K. W. Andrews. JISI, 203:721-727, 1965.
  • 13 C. Y. Kung and J. J. Rayment. Metall. Trans. A, 13:328-331, 1982.

Neural network models have been developed to predict both martensite start and bainite start temperatures. It is also possible to calculate these using ‘physically’ based models based on thermodynamics.

They’re all just maths! 🙂

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Mixed Up Microstructure




Mixed Up Microstructure

Originally uploaded by asuma.

I think this micrograph is pretty. Optical microscopy is useful to see what is really happening, it’s no use just going straight to high resolution without seeing what is happening on the millimeter and micrometer scale.

A mixed microstructure of martensite and bainite. Red needles are bainite in this micrograph, larger brown plates are martensite, the white bits are the untrasformed austenite. The contrast is from viewing the etched surface (nital) using differential interference contrast.

Multicomponent carbides trap Hydrogen

A new paper by Shingo Yamasaki and Harry Bhadeshia published in the Proceedings of the Royal Society (A 462 (2006) 2315-2330) shows that by controlling the lattice parameters of the carbide and the matrix via the steel composition and heat treatment it is possible to enhance hydrogen trapping. In Fe-C-Mo-V the cohency of the M4C3 with the matrix is shown to influence the ability to trap hydrogen, this effects the resistance of the steel to corrosion and reduction of mechanical properties.

Once again I was called into action to perform an interview of Harry, which you can find along with the paper. Interviewing seems to be getting easier, but I wish I’d remembered to mention Shingo’s name in the introduction. Also in future I have to find a way to track people down and do the interviews over the phone, otherwise every interview will be with myself and Harry 🙂

You can also find Shingo’s Phd thesis and presentation slides on hydrogen trapping on the phase transformations website.

This paper has been selected as the materials science and metallurgy department’s paper of the month for June 2006.

Bainite Interviews

I have a new job in my research group now, I get to interview people when they publish a paper, and the interview will go online with the paper on the group website. I think this was partially my idea when podcasting first became hot, but I’m not sure I imagined my supervisor would be using me to do the interviews. Anyway I hope people like it.

Interviewing Sourabh Chatterjee about cracking in martensite.

Interviews

  1. Cracking of Bainite paper by Chatterjee and Bhadeshia. Interview with Sourabh Chatterjee.
  2. Stabilisation of austenite by Chatterjee, Wang, Yang and Bhadeshia. Interview with Sourabh Chatterjee.
  3. Roughness of Bainite paper by Kang and Bhadeshia. Interview with Harry Bhadeshia.

mucg73.f

MUCG73 is a program available from map, which is written in FORTRAN. The program was originally written by HKDH (Harry) Bhadeshia and using the methods described in the 1982 paper, “Thermodynamic analysis of Isothermal transformation diagrams”. PDF available here: Thermodynamics analysis of isothermal transformation diagrams from phase transformations group.

I’ve been making some changes to the program since I at first wanted to increase the range of the calculation to lower temperatures, and also wanted to decrease the temperature step size from 20 degrees to 1 degree. This was to allow me to calculate the kinetics below 200 degrees Celcius which is the temperature range I am interested in for the ‘low temperature’ bainite. Calculating with a temperature step of 1 degree allows me to more easily calculate a CCT diagram using Scheil’s Additive law, otherwise it is necessary to interpolate between each 20 degrees step.

The most useful change I made so far was to replace the regression routine used for calculating the intersection of the free energy and the stored energy for bainite and for martensite. This was necessary after changing either the range or the step size used in the calculation because the regression previously was calculated by using the energy calculated for each temperature except the last 10 points. This corresponds to the range, range 200-480, which you can see in the figure graph of T vs FTO below contains a change in gradient. Finding the intersection can be made more robust/general by replacing this with a comparison of the free energy at each temperature, when the energy reaches the level for bainite or martensite I then interpolate to find the critical temperature. Since the energy in the program varies almost linearly with temperature this means the solution no longer depends upon the temperature step, it should always work find the same answer as long as the lines intersect within the temperature range calculated.

Criterion for calculation of martensite and bainite start temperatures.
[Download gnuplot script to reproduce this graph here: gnuplot gallery.

Professor Bhadeshia recommended replacing the ENERGY subroutine with the original routine from MUCG46 for the time being, since the constants determined in the rest of the program where calculated using the original version of this function, therefore any fitting was to these values. The ENERGY2 routine written by Suresh Babu should be more correct, but to use it various parameters need to be recalculated to reproduce the TTT diagram. The next step in this project is to compare these two functions, and also I also to take a look at the calculation of the Widmanstatten start temperature, which does also change slightly with the step size. I think the functions should be extended to work below 0 degrees Celcius if possible to allow calculation of the martensite start temperature in steels with higher alloy contents.

hmmm… also I have to work out a way to avoid getting NaN for a result at low temperatures.

Here are some snipets of FORTRAN code.

The changes in the main part of the program (it’s also necessary to declare the existance of subroutines).

C CALL MAP_UTIL_ANALY(J8,10,CONST,SLOPE,CORR,DT4,DDFTO)
C
C BS=(-400.0-CONST)/SLOPE
C MS=(-1120.0D+00-10568.0D+00*X1+94.1D+00-CONST)/SLOPE
BS=MAT_BS(J8,DT4,DDFTO)
MS=MAT_MS(J8,DT4,DDFTO,X1)
C

The subroutine for calculating the martensite start temperature.

C***************************************************************************
C MATHEW JAMES PEET, 21 APRIL 2006
C UNIVERSITY OF CAMBRIDGE
C
DOUBLE PRECISION FUNCTION MAT_MS(IMAX,T,G,X1)
IMPLICIT NONE
INTEGER I,IMAX,IMS
DOUBLE PRECISION T(1000),G(1000)
DOUBLE PRECISION MCOND
DOUBLE PRECISION X1
MCOND=-1120.0D+00-10568.0D+00*X1+94.1D+00
C
C WRITE(*,*) "MCOND, X1"
C WRITE(*,*) MCOND,X1
C
DO 1 I=1,IMAX
IF(G(I) .LT. MCOND) THEN
MAT_MS = T(I)
IMS = I
C WRITE (*,*) MAT_MS,I
C
ENDIF
C
1 CONTINUE
C WRITE (*,*) MAT_MS
C MAT_MS = 0.5*(T(IMS)+T(IMS-1))
MAT_MS = T(IMS-1)+(T(IMS)-T(IMS-1))*
& ((G(IMS-1)-MCOND)/(G(IMS-1)-G(IMS)))
RETURN
END
C******************************************************************

It would be nice to get hold of the data used to train the models. A similar problem exists in calculating TTT diagrams with the MAP program MTTTDATA, which uses MTDATA, which as far as I understand is also incomplete for calculating TTT diagrams.

Of course most of this wont be necessary when we can solve the Shrodinger Equation.