PHGN 450 / EGGN 502
Fall 2005
Project 3: Quench Kinetics of Carbon Steels
Version: 09/17/05
Due Sept. 21
I.
INTRODUCTION
The goal of this project is to model the decomposition of austenite in a small piece of
steel that is subjected to a known cooling pro…le. The steel is an alloy with the following
composition in weight percent:
Mn Si
Ni Cr Mo
0.84 0.22 0.15 0.8 0.04
The carbon content, often treated as a function of position and time, is taken to be
C = 0:39 wt. %.
There are four product phases to be considered: ferrite, pearlite, bainite and martensite.
The …rst three are di¤usive in nature–i.e. the rates of transformation depend on the rate at
which iron and carbon atoms can move in response to thermal ‡uctuations. The martensite
reaction, though, is referred to as an athermal (or displacive) transformation because the
amount of martensite present is a function only of the temperature and the atomic fraction
of the other phases.
Although the internal state variable method is largely phenomenological, it must be
consistent with critical temperatures that can be derived from …rst principles. Here the
temperature ranges over which each product phase can form:
Ferrite: TB < T < Ae3
Pearlite: TB < T < Ae1
Bainite: Ms < T < TB
Martensite: T < Ms
1
For this type of steel, the ferrite start, pearlite start and bainite start temperatures are
given by (in Celsius):
279C + 90:0C 3
Ae3 = 894
20:6Cr = 763
Ae1 = 726 + 5:79C + 17:3Cr2
TB = 656
57:7 C
+15:3 N i
13:6M n
35:0 M n
34:0 Cr
(1)
15:1N i = 725
75:0 Si
41:2 M o = 518
The martensite start temperature is given by:
Ms = 512
453 C
217 C 2
16:9 N i + 15:0 Cr
71:5 C M n
67:6 C Cr
9:5 M o +
(2)
7:50 Si = 331
The ferrite is the …rst phase to form upon cooling, but carbon is pushed out of the
austenite as the ferrite forms. This carbon enriches the austenite, and a point is reached at
which pearlite will start to form instead of ferrite. The atomic fraction at which this occurs is
called the Equilibrium Fraction of Ferrite, and it can be calculated directly from a knowledge
of how the critical temperatures, above, vary with carbon content. The calculation uses the
wt % of carbon in cementite–Ccem = 6:67: The Equilibrium Fraction is then approximated
as:
'f;eq (T ) =
8
>
>
>
<
0:5(Ae3 T )
;
Ae3 Ae1
0:5(T TB )
;
Ae1 TB
>
>
>
: 0;
Ae3 < T < Ae1
T < Ae1
otherwise
Another piece of thermodynamic guidance is that the rate at which a given phase forms
should be proportional to geometric function that characterizes the manner in which phase
boundaries interact. This geometric function should be small when there is only a small
amount of the phase present because a number of small nuclei are growing with a boundary
speed that is roughly constant. At values of product atomic fraction near unit, the geometric
factor should again be small because continued growth is based on the transformation of
small pockets of retained austenite; the reverse situation to that of initial phase growth.
The proportionality parameter (the mobility) between the rate of growth and the geometric factor should depend on temperature in such a way that it roughly parabolic with
a positive second derivative. This form of the mobility re‡ects the fact that mobility is
2
high due to thermal ‡uctuations at high temperatures and is relatively high again at low
temperatures due to a large thermodynamic driving force on individual phase boundaries.
Finally, the rate of formation of the product phases should not be an explicit function of
time.
A.
Kinetic Equations
Denote the atomic fraction of the phases as 'a (austenite) , 'f (ferrite), 'p (pearlite),
'b (bainite) and 'm (martensite). A set of di¤usive kinetics equations for predicting microstructure are:
8
<
a
b
f
f
'f;eq 'f ; Bs < T < Ae3
d'f
f (T )'f 'a
=
: 0; otherwise
dt
8
< (T )'app ' 1 ' bp 1 ; B < T < A and ' > 0:9'
d'p
s
e1
p
p
f
f;eq
a
=
: 0; otherwise
dt
8
< (T )'ab ' (1 ' )bb 1 ; M < T < B
d'b
b
s
s
a
b
b
=
: 0; otherwise
dt
(3)
where 'f;eq the temperature and carbon dependent equilibrium atomic fraction of ferrite.
Note that, in order for pearlite to form, the ferrite must be near its equilibrium atomic
fraction.
The mobility terms, i (T ), have the following form:
8
h
<
gf (T ) = ln
f 0 exp [ (! f 1 + ! f 2 gf (T )) gf (T )] ;
f (T ) =
: 0; otherwise
8
h
<
gp (T ) = ln
p0 exp [ (! p1 + ! p2 gp (T )) gp (T )] ;
p (T ) =
: 0; otherwise
8
h
<
gb (T ) = ln
b0 exp [ (! b1 + ! b2 gf (T )) gb (T )] ;
b (T ) =
: 0; otherwise
with coe¢ cients
i0 ,
T Tf 1
Tf 2
T Tp1
Tp2
Tb1 T
Tb2
i2
i2
i2
;
T > Tf 1
;
T > Tp1
;
T < Tb1
(4)
! i1 and ! i2 non-negative. These mobility functions are non-negative
and are bell shaped with maximum values at T = Tf 1 + Tf 2 , T = Tp1 + Tp2 , and T =
3
Tb1 Tb2 , respectively. The coe¢ cients ! i1;2 are used to adjust the asymmetry of the mobility
dependence on temperature.
The martensite reaction is athermal:
8
am
'm )bm
d'm <
m 'm (1
=
: 0; otherwise
dT
1
'a ;
T < Ms
(5)
The kinetic parameters for the di¤usive kinetic equations are:
f0
0.190
p0
0.283
b0
0.172
! f 1 8.73 ! p1 5.00 ! b1 0.283
! f 2 9.86 ! p2 5.00 ! b2 0.0527
Tf 1 510 Tp1
528
Tb1
557
Tf 2 125.7 Tp2 94.383 Tb2 41.0
af 0.639 ap 0.939 ab 0.633
bf 0.223 bp
7.83
bb 0.340
while the parameters for the martensite kinetics are:
am = 0:712
bm = 1:89
m
= 0:076
The units of each parameter follow directly from the equations in which they appear.
B.
Lattice Parameters
In order to be able to make a quantitative link between dilatometry measurements and
product phase fraction, the lattice parameters of each phase must be known precisely. In
lattice parameter estimates found in the literature, two di¤erent measures of the carbon
content in steel are used: weight percent denoted C; atomic percent denoted C. These
metrics are equivalent as is clear from the two relations given below:
C (at)
100 C
C
WC ( WCC + 100
)
WF e
The relation is given as approximate only because we have disregarded the very small weights
of alloying elements. Here Wi is the atomic weight per cent of chemistry element, with
WC = 12:01 and WF e = 55:84:
4
Phase
Ferr.
Cem.
(nm3 )
1 + (T + 273
100C (at)
100+C (at)
1000) 24:9
100C (at)
100+C (at)
0:61
10
V = 0:159 + 7:83
6
a3
4
v =
a3
2
v =
V
12
10 T
0:000280C (at)
vm =
cm = (1 + km ) 0:287 + 0:00256C (at)
km = 1:156
v =
6
a = 0:289 [1 + 1:59 10 5 (T + 273 800)]
2
(0:289 0:000279C (at) ) (0:287+0:00250C (at) ) 0:2863
1+
3 0:2872
am = (1 + km ) 0:287
Mart.
Atom Fe
(nm3 )
a = 0:363 + 0:000752
Aust.
Vol. per
Lattice Parameters (nm) and Unit Cell Volumes
10 5 (T
a2m cm
2
25)
Table. Lattice parameters. Temperature is in Celsius.
For a carbon level of 0.39 wt. %, these formulae reduce to (in units of nm3 ) :
v = 0:25(0:358 + 8:68
10 6 T )3
v = 0:0122(1:0 + 1:59
10 5 (T
v = 0:437(0:159 + 7:83
vm = 0:0119(0:358 + 1:156
(6)
527))3
10 6 T )
10 5 (T
25))3
The lattice relations given thus far must be further processed in order to obtain strain
relations for pearlite and bainite because these phases are themselves combinations of ferrite
and cementite. Within pearlite, the relative fraction of iron atoms associated with cementite
5
( ) and ferrite( ) is given by:
Cenriched (T )
Ccem
(rel)
(rel)
' _in_pearlite = 1 ' _in_pearlite
C
Cenriched (T ) =
1 'f;eq (T )
'
(rel)
_in_pearlite
(7)
=
Here the term Cenriched (T ) gives the carbon in the austenite just prior to the start of perlite
formation. Also, the value of C in the numerator of Equation (7)3 is just the wt. % of
carbon in our steel–i.e. 0.39 wt. %. Finally, the value of Ccem in the numerator of Equation
(7)1 is 6:67 wt. %. A reasonable assumption is that these relations can also be applied
to determine the relative proportion of ferrite and cementite in bainite. Then the volume
per atom of iron of a mixture of phases can be estimated to be a linear combination of the
relative volume of each phase–i.e.
vavg (T; 'f ; 'p ; 'b ; 'm ) = v 'a + v
h
'f + ('p + 'b )'
(rel)
_in_pearlite
1 = 'a + 'f + 'p + 'b + 'm
i
+v
h
('p + 'b )'
(rel)
_in_pearlite
The …nal step is to turn this into a linear strain using the small strain approximation:
vavg(T;'f ;'p ;'b ;'m )
1
1
(9)
linear (T; 'f ; 'p ; 'b ; 'm ) =
3 vavg (900; 'f ; 'p ; 'b ; 'm )
This gives the linear strain as a function of temperature and phase composition.
II.
APPLICATION
Apply the austenite decomposition model to consider the kinetics of a material with
carbon level of 0.39 wt. %. Use a temperature function given by:
T (t) = 25 + 875e
rt
(10)
where the temperature, T, is in Celsius, the time, t, is in seconds, and r is a parameter that
controls the rate of cooling.
A.
Step 1 (20 points)
Use the formulae provided to generate the linear strain of each phase (relative to the
volume at room temperature). Show these curves on a single plot. Here linear strain is
de…ned above in Equation (9).
6
i
+ vm '
(8)
m
B.
Step 2 (20 points)
Implement the kinetics model for an arbitrary cooling rate (your choice of the parameter
r). Use a Runge-Kutta algorithm to accomplish this. Provide a copy of your code.
On a single plot, show the atomic fraction of each phase as a function of temperature. Give
an initial atomic fraction for each product phase of 0.001. Why is this necessary?
C.
Step 3 (20 points)
Using the results of Step 2 and the linear strain formula of Equation (9), determine the
linear strain of a specimen as a function of temperature. Reference this so that the strain
is equal to zero at 900 C. Create a plot of linear strain v. temperature for the cooling rate
chosen in Step 2.
D.
Step 4 (20 points)
Use your model to determine the four cooling rates that yield the largest atomic presence
of ferrite, pearlite, bainite and martensite. Show plots that support your results.
E.
Step 5 (20 points)
Apply the routine of Step 3 to show, on a single plot, the linear strain v. temperature
results associated with the four cooling rates of Step 4. Reference the system so that the
strain is zero, in each case, at a temperature of 900 C. Label your curves. Provide an
interpretation of these results.
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