Modelling of microstructural evolution in thermomechanical processing of structural steels

~

~ '~

MATERIALS SCIENCE & ENGINEERING

~*

E LS EVI E R

lvIaterials Science and Engineering A223 (I997) 134-145

Modelling of microstructural evolution in thermomechanical processing of structural steels Y. Saito I Department of Materials Science and 3/Ietalho~y, University of Cambridge, Pembroke Street, Cambridge CB2 3QZ, UK

Abstract Computational models have been widely applied to the optimum design of chemistry and manufacturing process of structural steels. The metallurgical phenomena in thermomechanical treatment of steel, such as austenite grain growth, recrystallization and growth, carbonitride precipitation and austenite to ferrite phase transformation can be predicted by the model. This review describes the outline of the computer simulation model of microstructural evolution on the basis of chemical thermodynamics and classical nucleation and gro~th theory. The concept of the modelling, fundamental equations and the techniques and algorithm for the modelling of microstructure are introduced. © 1997 Elsevier Science S.A. Keywords: Structural steels; lVlicrostructural evolution; Computer simulation

1. Introduction High-strength low-alloy (HSLA) steels with good low-temperature toughness, low weld cracking susceptibility and high corrosion resistance are demanded for structural use. The application of thermomechanical control process (TMCP) [1] is essential for producing the above-mentioned high-quality steel. The mechanical properties of steel plates produced by T M C P are highly influenced by manufacturing conditions. Consequently, control of microstructure through an optimization of chemistry and manufacturing process is important for the best use of T M C P and the improvement of mechanical properties of HSLA steels. The conventional way to develop a new structural steel is to make laboratory-scale experiments and then to perform production-scale trials prior to mass production. Nowadays, the increase in the cost, and the time needed for the development, of new structural steel has become a serious problem. Computer simulation of microstructural evolution in thermomechanical processing is considered to be one of the most important tools for the efficient development of high-quality steels. The computer simulation model Present address" Department of Materials Science and Engineering, Waseda University, 3-4-I Okubo, Shinjuku-ku~ Tokyo 169, Japan. 0921-5093/97/$I7.00 © 1997 Elsevler Science S.A. All rights reserved. PII S 0 9 2 1 - 5 0 9 3 ( 9 6 ) 1 0 4 9 0 - 1

based on chemical thermodynamics and classical nucleation and growth theory has been successfully applied to the prediction of metallurgical phenomena, such as austenite grain growth, carbonitride coarsening and dissolution, recrystallization and grain growth after hot deformation, carbonitride precipitation and austenite to ferrite phase transformation [2-10]. For the adoption of the computer simulation model to the design of chemistry and the manufacturing process, it is necessary to describe evolutions of these metallurgical phenomena in terms of alloying elements and processing variables, such as reheating, rolling and cooling conditions. The synergetic effects of the two or three phenomena must be taken into consideration in the modelling. The present paper describes the outline of the recent development of computer simulation models and their applications.

2. Modelling of metallurgical phenomena The purpose of the mathematical model is to determine the optimum chemistry and manufacturing process required to obtain steel with the desired property. Table 1 shows controlling factors and microstructural changes in the thermomechanical control process (TMCP). The prediction of the grain size, the carbonitride precipitation and the phase of the transformed

Y. Saito /Materials Science alzd Engmeer#zg A223 (I997) I 3 4 - I 4 5

I35

Table 1

Controlling factors and microstructural changes in the thermomechamcalcontrol process Process

Reheating

Rolling

Coohng

Recrystallization region Controlling fac- Reheating rate tor

Non-recrystaliization ),~y. two-phase region region

Rolling temperature

Coohng start temperature

Reheating tern- Reduction/pass perature Reheating time Interpass/time Microstructural change

--+), p h a s e transformation 7 grain growth

Recrystallization and growth Carbonitrideprecipitation

Cooling rate Finish cooling temperature FoHnation of de:~ nucleation and growth formed structure Carbonitride precipi- Formation of deformed tation structure in c~

u. nucleation and growth Formation of transformation structure

Dissolution of carbonitride

microstructure is essential for the design of the chemistry and manufacturing process.

is the volume at time t of the stable nucleus formed at time t'. The term, ~6 Y(t')V(t, t')dt', is called the extended volume.

2.1. Fundamental equatiotzs 2.1.2. Modelling of mtcleatiotz kinetics 2.1. I. ~Iodelling of nucleatiotz and growth phenometm by generalized Kohnogorov-dohnson-Meh/-Avrami equation The Avrami-type equations [11] are widely used for the prediction of nucleation and growth phenomena such as recrystallization, precipitation of carbonitrides and phase transformation. In the case of phase transformation, the variation of fraction transformed, Xv(t), with the time, t, is described as:

AT(t) = 1 -- exp( -- At x)

(1)

where A and K are constants which are dependent on steel chemistry and temperature. These constants are determined by experimental data. The Avrami-type equations are valid for isothermal phenomena if the values of the constants in equations are properly determined. However, the Avrami-type equation cannot be applied to the complicated thermomechanical cycle as T M C P because the equation assumes constant nucleation and growth rates. Equations which incorporate the temperature- and time-dependent nucleation and growth kinetics are required. The generalized Kolmogorov [12], J o h n s o n - M e h l [13] and Avrami [11] (KJMA) equation is applied for the present simulation. The generalized K J M A equation is given as:

[;o

X(t) = 1 -- exp --

J(t') V(t, t') dr'

The fundamental process of nucleation is not well known. Hence, the rigorous treatment of nucleation on the basis of statistical mechanics of first-order phase transformation is left as a future problem. The most prominent approach for the calculation of the nucleation rate, J(t), is to use classical nucleation theory [14]. The nucleation rate, J(t), described by the classical nucleation theory is g v e n by:

1

where N(z) is the number of nucleation siteS per unit volume, fi* is the rate at which single atoms join the critical nucleus, Z is the Zeldvitch non-equilibrium factor, 2xG* is the free energy of activation for formation of the critical nucleus, r is the incubation time, T is the absolute temperature and/c is the Bolzmann factor. The above parameters depend on the morphology of the nucleus, grain structure and chemistry of steel. The allotriomorphic critical nucleation model is applied for the calculation of grain boundary precipitation of carbonitrides. The parameters in Eq. (3), fl*, Z, AG* and r, are expressed as follows [14]:

fl* = 16~r~/~D:.z/~L/(a 4 2xG~.)

(4b)

AG* = 16~zKa)/j/(~A ~ , G,.) 2 (2)

where X(t) is the fraction of phase transformation or recrystallization, or the amount of carbonitride precipitation, J(t') is the nucleation rate at time t' and V(t, t')

(4a)

v = 8]cTKo~Fa4/[D.,z/~V~

(4c) " AG~,L]

(4d)

where c,/j is the matrix/precipitates interfaciaI energy, D.. is the diffusion coefficient of the carbonitride-forming element. 2~Gv is the change of free energy associated

Y. Saito /Materials Science and Engineering A223 (I997) 134-145

136

with the precipitation, V~ is the molar volume of the precipitated phase, a is the lattice constant of the matrix and K and L are shape factors of the nucleus. The driving force of precipitation of the MC-type carbide is given by

(kT),[-aM(x~a)-] =

In

-

. [-ac(x~)-]

-

+

In

-

.D V 2 C =

~C/~t

fl* = 8rc~r~:.D.,,Z/~g/(a4 AG y" >'+ ~)

(6a)

Z = V~(AG y" y + ~)2/[4~(3kTe)l/2o-~r ]

(6b)

d (7* -- 4rce(o-e:,)2/(A(77-+r+ ~)2

(6c)

r = 12kTKcr~ya4/[D,.zt~ V~(AG >'-r+ ~)2]

(6d)

where s is given by a function of the interfacial energies, o-~, cr~;b and Gv, as (see Fig. 1): ~r>,:,

(7)

The parameters determined by Enomoto and Aaronson [15] are as follows: e = 5 . 5 erg cm -2, o'e:.=60 erg cm -2. The number of nucleation sites. N(t), is calculated by a function of austenite grain size. Various thermodynamic models have been proposed to calculate the change of free energy associated with the nucleation of proeutectoid ferrite, zXGy~?'+~ [16-18]. Softwares such as T H E R M O C A L C [19] are very useful for efficient calculation of driving force.

0
(9a)

C(r= R, O)= C~a,

r >1R

(9b)

C(r=

0 < t < co

(%)

co, t) = CM,

where C(r, t) is the concentration at the position 7"and the time t, D is the diffusion coefficient of the alloying element by which the kinetics is controlled, C~ is the concentration at the matrix/precipitate phase interface and CM is the bulk concentration. The flux balance at the interface is given by: (Cp

-

C 0 dR/dr = D(SR/St), =R

c aT e aT cb

T

O" aT

O'TT

Fig. I. Pill box critical nucleus model [15].

(10)

where Cp is the composition of precipitated phase.

2.1.4. lVlodelling of intelfacial migration Allen and Cahn [21] proposed general equation of interfacial motion as v(,~, t) = M'K(,~, t)

(ii)

where v(,7, t) is the velocity of interface migration at a point ~ on the interface, M is the kinetic coefficient and K(& t) is the mean curvature of the interface. The determination of the velocity v(& t) enables the estimation of the volume of stable nucleus, V(t, t'), in Eq. (2).

2.2. Evolution of austenite grain and prediction of ferrite size In order to estimate the grain size of steel produced by the thermomechanical control process, it is necessary to simulate austenite grain growth, recrystallization and grain growth after hot deformation, the formation of deformed structure in austenite by deformation below recrystallization temperature, and austenite-ferrite phase transformation. The explicit form of Eq. (11) for the variation of austenite grain size, R, with the time, t, during reheating is given by Hillert [22] as: d~ =M~c~

o

(8)

C(r=R,t)=C~,

-

where aM(X~) and ac(x~) are activities of M element and carbon of equilibrium concentration, and aM(XM) and a¢(¥c) are activities of M element and carbon in the matrix. Unknown parameters in Eqs. 3 and (4a-d) are the number of nucleation sites, N(t), interfacial energy, ~/~, and the shape factors of the nucleus, K and L,, These parameters are determined so that good agreement between the computed and observed results is obtained (i.e., the values of these parameters employed for simulation of Nb(C, N) precipitation are as follows: N(t) = 5 x 1 0 ~7 cm -B, ¢~4=540 erg cm -2, K = 0 . 2 6 4 and L = 0.455 [5]). Nucleation of ferrite is predicted by the pill box critical nucleus model [15]. In the pill box critical nucleus model, the parameters in the equation describing nucleation rate (Eq. (3)) are given by:

c 4- o - cb e = ~,>,_ .:,-

2. i. 3. Modelling of diffi¢sion-l#nited growth The diffusion-limited growth rate is given by the solution of the diffusion equation (Eq. (8)) with boundary and initial conditions (Eqs. (9a-c)) [20]:

r

~-~2

(12)

where M is the mobility of the grain boundary, ~ is a dimensionless constant, ~r is the grain boundary energy, Re; is related to the radius of the grain with curvature of 0, and Z is the pinning force of precipitates given by a function of the volume fraction of precipitates, f, and the radius of precipitates, r, as [23]:

Z = 3f 4r

(13)

Y. Saito /Materials Science and Engineering ,4223 (I997) 134-I45

137

Table 2 Coefficients in Eq. (14a-e) for C - M n , Nb, T i - V , N b - T i steels Steel

C-Mn

Nb

Ti-V

Nb-Ti

Ref. k x B (,um --~)

[25] 2 0.5 2.5x 10 -19

[26] 1.7 0.5 5 x 10-Is

[27] 2 0.25 1.5 x 10 -I9, exp(30[Nb])

So m

0 4 2 300

[25] 2 0.5 2.52 x I0-1~*** 5.94 x 10 -3s**** 9.24 x I0-19,**** 0 4 2 325*** 780**** 130"**** 0.9 0 0.67 0.67 -0 2.4x 1067

0.058 5 2 280

0.025 2.8 2 3OO

195.7 4.3 0.57 0.I5 350 0.11

1.i 0 0.67 0.67

n

Qsa (kJ mol - I )

C (#m 1 - 0 D (#m) r u QHW (kJ tool -1)

0.5 0 I 0.37

v

0

E (]zm-1o s - i) Qgg (kJ m o l - i )

3.87 X 1032* 5.02 x 1053** 400

0

1420

The effect of alloying element on the mobility of the grain boundary, M, is known as the solute drag effect [24] and the pinning force, Z, can be calculated by the model of dissolution of carbonitride and Ostwald ripening of precipitates. The variation of average grain size during reheating is predicted by the numerical solution of Eq. (3). Theoretical treatment of recrystallization and grain growth after hot deformation is very difficult at present. Sellars and Whiteman [25] proposed the empirical equations which describe the variation of austenite grain size of C - M n steel during rolling. Similar types of equations are proposed for microaltoy steels [25-27]. The volume fraction recrystallized, X, the time for the fraction recrystallized x(0 < x < 1), tx, the size of recrystatlized grain, dsa, and the size of austenite, tiT, can be calculated in terms of initial grain size do, deformation temperature, T, strain s, using empirical equations such

of 20 and 300 /zm [30]. The austenite grain size is virtually identical in both cases after three passes. The grain growth after the completion of recrystallization is described in Eq. (t4e). The value of the exponent in this equation is much higher than the expected value of 2. Komatsubara et al. [29] assumed time-dependent mobility of grain boundary migration and received the value of exponent of 2 for grain growth after recrystallization.

TEMPERATURE, 'C 1200 1100

1300

3oo!

STEEL'0 01TP 0 08V-0 013N [] 1150" 1100" 1060" 1030"

\

20% \

250

\

\

200

X = 1 - exp[ - A ( t / t , . ) q

(14a)

A = ln[1/(1 -- x)]

(14b)

t x = B ( s - So)-"d'~ e x p ( Q s a / R T )

(14c)

dsa = Co-rd~ e x p ( Q H w / R T ) -~ + D

(14d)

d./t° _- d[O + Et exp( - Q g J R T ) ,

(14e)

where m, n, p, r, u, v, B, C, D, E, Qsr~, QHw and Qgg are constants dependent on steel chemistry. Table 2 shows constants of Eqs. (14a-e) for C - M n , Ti-V, Nb and N b - T i alloys. Fig. 2 shows the variation of austenite grain size during rolling in T i - V steels with initial grain

N__

150

\

\

\

g

-m

4'

50

I

20%

20%

\ \

100 z

20%

Cootlng rote l " C / s "~ [] measured grain size II ¢a[r.ulnted gram size

\

as:

1000

As ceheafed

o

\

\

~..o

_

20 I

PASS No

Fig. 2. Effect of initial grain size on the microstructure development during hot rolling [28].

138

K Sa#o / Materiah Science and Enghmer#zg A223 (1997) 134- I45 2.3. Carbonitrides dissolution and precipitation

100 0.07C - 1.5 Mn -0.035 Nb steel Austenite grain size: 50/zm

E

2.3. J. CarbonitHde dissolution By solving Eqs. (8)-(10), the fraction undissolved of carbonitrides, f~d, is given by a hyperfunction as: nI n [ f ~ s + O~,-

.N

I/2rPl/6 -~ ,ud -1- o,

10

-

ki72/777)J

2p

o Observed Calculated

(17)

The parameters in Eq. (17), p and :, are given by a function of the initial radius of carbonitride R0:

-

ii

,

.1

,

,

, , , , , I

T

,

,

t

i,,,I

10 Cooling rate °C/s 1

,

,

,

,

lO = ( icl4= ) Ii2

(I8a)

z

(18b)

,,,,

100

(](DIRo)2t

=

lc = 2 ( q - CM)/(Cp

Fig. 3. Average ferrite grain size of Nb steel %s. the cooling rate from austenite [5].

The formation of deformed structure is described by the variation of residual strain [30] or dislocation density [4]. Saito et al. [30] formulated the effect of deformation on the substructure of austenite by the residual strain, which is closely related to dislocation density. The residual strain is a function of the amount of deformation, the deformation temperature and the interdeformation time. With the increase of the accumulated strain, austenite to ferrite phase transformation is accelerated and the ferrite grain size is refined. Saito et at. [7] proposed an empirical equation describing ferrite grain size, d,, in terms of austenite grain size, @, cooling rate, Ca, and the residual strain prior to ;' to phase transformation, Ae~, as:

-

(18c)

q)

Fig. 4 shows the variation of the fl'action dissolved of Nb(C, N) with the initial size of 0.1 # m in 0.08C-Nb steels with time. The dissolution kinetics is very slow near the equilibrium dissolution temperature. 2.3.2. Precipitation o f carbonitride Nucleation of carbonitrides can be calculated by classical nucleation theory as shown in Section 2.1.2. The diffusion equation (Eq. (8)) with the boundary and the initial conditions is solved by the invariant field approximation [20]. This is an attempt to simplify the diffusion equation by setting 3 C / & = 0 and solving time-independent Laplace equation V2C= 0. One obtains C(r) and thus OC/3t from the Laplace equation and then d R / d r from Eq. (10). If ( C p - CM) >>(CM C~), reasonable accuracy is obtained with use of the invariant field approximation. The radius of the precipitates is given by:

R = [2(CM -- Ci)(q, - q)t/2(Dt)U2

(19)

In d~ = 0.92 + 0.44 In ct, - 0.77 In C~ - 0.88 tanh(10Ae~) (15) A similar type of empirical equation was given by Siwecki [28]. The average ferrite grain size is obtained by a function of the total number of nucleation, %~, which can be calculated by classical nucleation theory, as [31]: ct~ = (2/3n~) t/3

0.08C - Nb Steel

00,

?

/

0 . 0 3 % Nb

~

0.02 . . . . .

0.02%NbJ/,~_ ~ &

(16)

The average ferrite grain size versus the cooling rate from the austenite region is shown in Fig. 3. The ferrite grain size decreases with the increase of the cooling rate. The computed ferrite grain sizes are in good agreement with those observed. The migration of the ferrite grain boundary is neglected and the topology of the grain structure is not taken into consideration in the above equation. However, the form of the equation is so simple that it is suitable for practical application.

o ._= 0.01 Z~"

127/

/

o

o.oo

10

/

yy 100

.....

1150"C ,

1000

. . . . . . .

10000

Time, s

Fig. 4. Nb(C, N) dissolution kinetics during reheating of 0.08C-Nb steels. The initial size of precipitate is 0.1 #m [5].

139

Y. Saito /Materzals Sczence and Engineering .4223 (I997) 134-i45

40

30

++

',.

,+

0.08C-1 .SMn-O.O35Nb steel ., Reheating temp.: 1150 °C ,+ Reduction/pass: 15%

E

100 .N

I

4~

(a)

1 50

t

"~ 20

i

Fraction precipitated

e.-o

'~

~10

Austenite grain s)~_j

50

4~ 4~ 1,9

t.t_

0 700

800

900

1000

0

O0

Temperature, °C

40 0 . 0 8 C - 1.5 Mn- 0.04 Nb steel 30

(b)

>q e-

20

l&.

10

0

0 10 20 30 40 50 60 70 80 90100 Nb(C,N) size, A

Fig. 5. Computer simulationof Nb(C, N) precipitation during rolling: (a) variation o£ the amount of Nb(C, N) precipitation and austenite grain size with time; (b) the size distribution prior to phase transformation. The kinetics of precipitation is predicted by using the above equations in the following manner [32]: the initial states of precipitates are determined by the reheating condition. The thermomechanical process is divided into infinitesimal time steps. The temperature of steel at each step is calculated by the numerical solution of the Fourier equation of heat conduction. The degree of supersaturation is calculated by the solubility data of carbonitride. The nucleation rate at the time step is predicted by Eqs. (3)-(5). The growth of the nucleus is estimated by Eq. (19). The amount of precipitation is calculated by the K J M A equation Eq. (2). The above calculation is repeated until the completion of precipitation or the end of the time steps. Fig. 5 shows an example of the computer simulation of Nb(C, N) precipitation during rolling. The kinetics

of precipitation together with the variation of austenite grain size is plotted in this figure. The variation of the amount of Nb(C, N) precipitation with time is shown in Fig. 5(a). The size distribution prior to phase transformation is shown in Fig. 5(b). The precipitation reaction is very slow at the high-temperature region, where the degree of supersaturation is small. The amount of Nb(C, N) apparently increases at the temperature range below 850°C. About 70% of Nb(C, N) precipitates size falls within the range 30-60 A. Akamatsu et al. [33] simulated the synergetic effect of Nb(C, N) precipitation and recrystallization after hot deformation. Fig. 6 [33] shows the effect of strain on precipitation kinetics on NbC. The hot deformation has a distinctive effect on acceleration of Nb(C, N). In the high temperature range (i.e., T > 1000°C), the recrystallization is accelerated by hot deformation and the corn-

]I. Saito / Materials Science and Eng#wering A223 (1997) I34-145

140

pletion of recrystallization is attained before the start of Nb(C, N) precipitation. Therefore, the effect of strain on Nb(C, N) precipitation is very small in this temperature range. It has been reported that the precipitation behaviour of Ti(C, N) [34], MoC [35] and (Fe, Cr)23C 6 [36] can be predicted with good accuracy.

~mAJ F ~" ]

2.4. Austenite to ferrite phase transformation

....... ~=

(

700

"

I

~,,nL

3" I'" "~[;"4

f

c

I I | I [ \\\'~ / =

~

== =

",.\~\\

-"~N_-~'~.

Proeutectoid ferrite starts by the formation of coherent pill-box-type nucleus. Nucleation rate can be calculated by Eqs. (3), (6) and (7). The growth of ferrite is predicted by a diffusion-limited mechanism. Under growth conditions of ferrite usually encountered, equilibrium is sustained only with respect to carbon (para mode) [37]. By solving the diffusion equation (Eq. (8)) for carbon, the migration of the austenite/ferrite interface is described as [20] (20)

R = 2 ( D c t ) 1/2

where Dc is the diffusion coefficient of carbon in austenite and 2 satisfies the following equation:

a)

1100

1000

e =

(

,(,,.',,, ~ \

800-

fNbC ( ~ ) : 5 I I

~without \deformation 50

95 !

I

I

b) e = 0.51

1 100

-

I

I

i00

I0 ~

I

10 ~ time, s

,9 Fec0'4 C-1 ~dn-1Ni 10 a l0 ~ 10,~

Fig. 7. Calculated TTT curves for hypothetical steels [41].

2 -0exp(22)[exp( - 22) - )o(7c)1/2 erfc(2)] - ,r(x~ ~ z-- ~x~=) 2(x~ - x~ )

(21)

where x~, x~~ and .,c~~ are the carbon concentrations in the austenite matrix, the austenite side at the austenite/ ferrite interface and the ferrite side at the austenite/ferrite interface, respectively. The carbon concentrations at the interfaces are calculated by the thermodynamics model. Ferrite transformation kinetics is predicted by the K J M A equation (Eq. (2)) using Eqs. (3), (6), (7), (20) and (21). 2.4.2. Pearlite transformation

0.36

-

900 -

|

200

7 Fe-0,4 C-2Cr

8 Fe-0.4 C - 1 M n - I C r

|

2.4.1. Proeutectoid ferrite

-

.~e--0.4C-~M~ :'~__~..~">.~ r, I,'~-0.4C-2Mn

300 ~-: . . . . . . . . . . . . .

If the carbon concentration in austenite reaches the eutectoid carbon concentration, x e, pearlite phases are formed. The nucleation site of pearlite is austenite/ferrite boundaries and the nucleation rate is quite high. Pearlite transformation is controlled by the migration of interphase boundary. Two mechanisms of pearlite growth are proposed: volume diffusion mechanism and rate control process [38-40]. The mathematical treatments in the volume diffusion model are easier than those in the rate control process. In the volume diffusion mechanism [38], the transformation rate, v, is described as v = ( 2 D c / S R T x ¢ ) [ ( A H / T e ) ( T ~ - T) - 2¢ Vp/S]

1000 -

10 22 90(2

I

-

28

~z , / \\/~,, \x. 42 ~x "\ ~ ' \

\ k
800-

fNbc(°6) :5 I

l

_

2Fe-0.4C-2Si 3 Fe- 0 . 4 0 - 1 N i 4 Fe-0,4 C-2 Ni

I

~ deformation

50 1

95 f

2 3 4 holding time:log t (s)

I

(22)

where S is the pearlite lamellar spacing, R is the gas constant, AH is the latent heat for pearlite transformation, T¢ is the eutectoid temperature and Vp is the molar volume of pearlite. The extended volume in the K J M A equation is calculated by the product of the total austenite/ferrite interfacial area and the pearlite thickness.

5

Fig. 6. Effect of strain on precipitation kinetics of NbC calculated. Condition used for calculation: 0.05C-0.041Nb steel, strain rate 10 1/s, and initial grain size 100/zm [33].

2.4.3. BaOffte transformation

If the free energy change associated with the nudeation of the phase with the same composition as austenite, AG ~-~, exceeds 400 J tool-1 [41], bainite phase is

Y. Saito / Materials Science and Engineering ,4223 (1997) 134-145

formed. Bhadeshia [41] proposed an equation to predict the start of bainite transformation. Fig. 7 shows calculated TTT for various steels by Bhadeshia. The growth of bainite can be described by an equation similar to pearlite transformation.

2.4.4. Simulation procedure [5] The kinetics of phase transformation is predicted through use of the above models in the following manner: the initial condition of austenite is predicted by the models of austenite recrystallization and grain growth, the accumulated strain and carbonitride precipitation. The equilibrium temperature, Teq , is calculated from steel chemistry. The cooling process is into infinitesimal time steps. The temperature of steel at each step is calculated by the numerical solution of the Fourier equation of heat conduction. If the temperature of steel is lower than T~q, the following calculation of ferrite transformation is executed at each time step. The changes in free energy associated with the nucleation, AG ~'+~+~ and AG~-*% and the eutectoid carbon concentration are calculated. The nucleation rate at the time step is predicted by Eqs. (3), (6) and (7). The growth of the nucleus is estimated by Eqs. (20) and (21). The fraction transformed is calculated by the K J M A equation (Eq. (2)). The above calculation is repeated until the carbon concentration in austenite reaches the eutectoid carbon concentration. If the value of AG 7-~ exceeds 400 J mol - t, bainite is formed. Computed continuous-transformation-time (CTT) diagram of 0.8C-1.50Mn-0.035Nb steel is shown in Fig. 8. The computed CTT diagram is in good agreement with that observed.

800

Calculated ..... ,. ":',L! + : . : ' " .: " .~" ' - ' . - . 'J-_.a + ._

.....

700 -

Table 3 Thermomechanical conditions employed for the simulation Reheating condition Reheating temperature

'":::'.'.,

'%, '"., " , . F "'

780°C

Cooling condition Start cooling temperature Cooling rate Finish cooling temperature

740°C IOoCs -I 480°C

',

600

'~',", ,%,

"

b

3.i. Effect of processing variables on microstructure of steel produced by TMCP Accelerated cooling after controlIed rolling is very useful technology for producing high-strength steel plates [1,42]. The most important point in accelerated cooling is to control the transformed microstructure. The influences of steel chemistry and the thermomechanical condition on the transformed microstructure of Nb steel are simulated using the model described above. Table 3 shows the standard thermomechanical conditions employed for simulation.

3.1. I. b~fluence of the rollhzg condition on the transformed microstructure of Nb steel Austenite-to-ferrite transformation is accelerated with the refinement of austenite grain size and/or the increase of the amount of deformation in the low-temperature austenite region. The grain boundaries and deformed substructures in the grain interior act as nucleation sites of ferrite. Hence, the effective austenite interfacial area per unit volume, Sv, is closely related to the kinetics of transformation from deformed austenite

Observed

30

'

10

~

'

B

"+, ,._

~:Q

°

o+ 4 0 0 ~-

','~ .,,

,.

300+

~

,A

I

l'l I',

I1',

100

,

,,

• ,,

,,,,,I

I0

',

lIP

~,

~, ,

,

1

I

,

,

I'tttl+

,

1 O0

I. ++ tI i

I

..

i,

,.

i

i,

i I

',

,

ij

L

,,

i.

l +

~

+,

i,

~

I

I i

,,.

',

,~

,.

i

I [

,,

t

i,

I

Ii

L ",

~,

.a

~,,M

I

200

.~

v, ~

60% 75%

3. Application of the simulation model

"B +,

1I O 0 ° C

Rolling condition Total reduction in recrystallized region Total reduction in recrystallized region Finish rolling temperature

+,

",,

I4t

'1

;

,

~

,

+ ', ',

I

II +

, tttr,+

1000

I

l

, ', ',

I

I +

I1

,i

t

20

ca

15

"i

10

~-

5

p

p

2nd phase ~

5

.-= .~

, p

i,

P

~-

I

Ferrite 0.10

C-

grain size

1 . 5 Mn - O . 0 3 5 N b

.~

steel

,,~

I ,,1,1

10000

Time, s

Fig. 8. Computed continuous-transformation-time (CTT) diagram of 0.08C-I.50Mn-0.035Nb steel. The steel is reheated at 1150°C and deformed to 35% at 1020°C. The steel is subsequently deformed to 42% at 820°C and 34% at 800°C.

0

zOO

,

I

400

I

,

600

Effective T

r

I

800

,

I

,

I000

interfacial area, mm

0

120u -I

Fig. 9. Influence of the value of S, on the volume fraction of second phase and the ferrite grain size of 0.1C-1.5NIn-0.035Nb steel.

142

Y. Saito /Materials Scielzce and Engineering A223 (1997) I34-145 25 p

10

BIo

B

O.

[]

m

E

2nd phase

c-~

~5 .~

15

Ferrite grain size O.

2o ~e-

20

ca

B

15

B

Ferrite grain size ='-'----=------=

10

.~

z&

O

IO

2nd phase

.N

15

E

B

B

ed

[]

o"

.N

10 E

0.06 C - 1 2 5 Mn - 0.035 Nb steel 5

5

0.10C - 1.5 Mn - O.035Nb steel T

o 6

I

8

r

I

10

r

I

,

12

I

14

,

N

14-

0 6

0

Cooling rate, °C/s Fig. 10. Influenceof the cooling rate on the volume fraction of second phase and the ferrite grain size of 0.1C-I.5Mn-0.035Nb steel. [43]. The influence of the effective austenite interfacial area on transformed microstructure is simulated for 0 . 1 C - 1 . 5 M n - 0 . 0 3 5 N b steels. Fig. 9 shows the influence of the value of Sv on the volume fraction of the second phase and the ferrite grain size. At low Sv values, i.e., light controlled rolling, the transformed microstructure consists of ferrite and bainite phases and the volume fraction of the second phase is large. The volume fraction of the second phase decreases with the increase of the S,, value by the heavy controlled rolling. The second phase at smaller Sv values is bainite. The ferrite grain size decreases with the increase of S,. These simulation results are in good agreement with those observed [43]. 3.I.2. h~uence of cooling condition on transformed microstructure o f Nb steel As is well known, the transformed microstructure of steel is highly influenced by the accelerated cooling condition. The influences of the cooling rate and cooling start temperature on the transformation microstructures of two Nb steels, 0 . 1 C - 1 . 5 M n - 0 . 0 3 5 N b and 0.06C-1.25Mn-0.035Nb steels (from here on, these are called high C - h i g h Mn and low C - l o w Mn steels), are predicted by the model. Fig. 10 shows the influence of the cooling rate on the volume fraction of the second phase and the ferrite grain size of high C - h i g h Mn steel. At iower cooling rates, the transformed microstructure consists of ferrite and pearlite phases and the volume fraction of the second phase is small. The retained C concentration in austenite is higher in high C - h i g h Mn steel than that of low C - l o w Mn steel at the same fraction transformed. The formation of pearlite is easier in high C - h i g h Mn steel. With the increase of the cooling rate, the volume fraction increases. The transformed microstructure at higher cooling rates is changed to a ferrite-bainite structure. The influence of accelerated cooling rate on ferrite grain size is small.

f

,

6

,

8

I

,

I

,

I

10 12 Cooling rate, °C/s

,

14

o 16

Fig. 11. Influences of the cooling start temperature on the volume fraction of second phase and the ferrite grain size of 0.06C-1.25Mn0.035Nb steel. The influence of the cooling rate on the volume fraction of the second phase and the ferrite grain of low C - l o w Mn steel with a start cooling temperature of 700°C is shown in Fig. 11. The transformed microstructure consists of ferrite and bainite phases, even at lower cooling rates. The effect of the cooling rate on the volume fraction of the second phase is smaller than that of high C - h i g h Mn steel. The overall transformation kinetics is controlled by the transformation behaviour prior to accelerated cooling. Hence, the influence of the accelerated cooling rate on the transformed microstructure is very small. On the other hand, the transformed microstructure of low C - l o w Mn steel is highly influenced by the cooling start temperature. Fig. 12 shows the influence of the cooling start temperature on the volume fraction of the second phase and the ferrite grain size of low C - l o w Mn steel. The volume fraction of the second 25

15

Ferrite grain size

20

E

ihase ~=

I0

15

.N 0

._~

Od

.8 14--

lO 0.06 C - 1.25 Mn - 0.035 Nb steel s

0

b80

[.l.

,

I

700

,

I

720

,

I

740

,

I

760

,

0

78U

Cooling s t a r t temperature,°C

Fig. 12. Influence of the cooling start temperature on the volume fraction of second phase and the ferrite grain size of 0.06C- 1.25Mn0.035Nb steel.

143

Y. Saito / 3Saterials Science and Engineer#zg A223 (I997) i 3 4 - I 4 5

phase decreases with the fall of the cooling start temperature. The coarser ferrite grain is obtained at a lower cooling start temperature. As shown above, the effect of processing variables in thermomechanical treatment on the transformed microstructure can be predicted by the model. The computer simulation model can be applied to the design of chemistry and the manufacturing conditions for steel plates according to the demands. In low C - l o w Mn steel, the influences of rolling condition and cooling rate on transformed microstructure are smaller, so that the one can find the manufacturing conditions for producing steel plates with uniform microstructure and mechanical properties. On the other hand, the transformed microstructure of high C - h i g h Mn steel is apparently influenced by processing variables in thermomechanical treatment. The manufacturing conditions for the best use of the thermomechanical control process can be determined for the steel.

3.2. Prediction of strength of steel plate and optimization of the manufacturing process Various types of structure-property relationship have been reported [2,3,9,44-48]. The relation between strength and ferrite grain size is expressed by the wellknown Hall-Petch equation [49,50]. The law of mixture is useful to describe the strength of multi-phase steel. The strength, or, is described as cf = ~o + k f d 2 ~P-+ k',d2b ~12+ y, K,_t~

(23)

where d~ is the ferrite grain size, dsb is the subgrain size and kr, k'r and K, are constants. The term 0% is ~ven by

o-0 = 0% + Ass + Appt -}- Adls

(24)

where ~; is constant and A~, App t and Ad~s a r e solid solution hardening, precipitation hardening and dislocation hardening, respectively. The terms of Eqs. (23) and (24) can be predicted by the method described above and the constants can be determined by regression analysis. In the accelerated cooling process of steel plate, the interpass time and the time after rolling to start water cooling is long enough to on-line control of the manufacturing condition. Saito et al. [7] proposed a model for optimum control of accelerated cooling. In their model, tensile strength, TS, is assumed to be the sum of the tensile strength of air-cooled steel under corresponding conditions, T S a o and the increment of strength by accelerated cooling, ATS: TS = T S A c + ATS

(25)

The value of T S a c is calculated by chemical composition of the steel and its rolling condition. The empirical formula to predict the value of ATS is given by [7]:

i

~

,

i

;07C-1.5Mn-003Nb

--:

,

60

15O

40

I00 O~

20

"~

,

st~'eI

2.d phaso

5o

~ P q - B p P.÷ B

'

I[°

13: Baimte

°'-"-°--£-"~

6

Pearl

. .0.02 . . . 0 04

0

0 b6

PT

F~g. 13. Effect of control cooling parameter, -PT" on transformed structure and increase of tensile strength by water cooling, ATS [3].

I,, Act~ ATS = - 4.0(MPa mm '-) ~ + 1.0 A l/p + 2.5 VB

+ 13.2 ~q,~s

(26)

where Vp. 2.5;rG and ]rqs are volume fractions (%) of pearlite, bainite and martensite, respectively. If the right hand side of Eq. (26) is expressed by the cooling condition of steel plate, it is very useful for optimum control. The increment of tensile strength of steel plate by accelerated cooling is given by ATS = aoPT (PT > 0.02) ]aT = a~C~'-

exp{

l

as(F s

l

(27a)

Ts)H(Fs

l

tanh[a4(B~ - Tf) + 1]}

T~) (27b)

where F~ and Bs are ferrite and bainite start temperature, T~ and Tr are start and finish cooling temperature, and a0, al, a2, a3 and a 4 are constants dependent on steel chemistry. H(x) is Heaviside's function. If the value of PT is higher than 0.02, the linear relationship between PT and the volume fraction of the second phase and ATS is as shown in Fig. 13. If the increment of strength by accelerated cooling, f, is defined as the function of CR and Tr, and the required increment of strength is f(0), the optimum cooling condition is determined so that the following test function has a minimum,

~(cR, T0 = ] f - f ( 0 ) ] + [Af(C~, ;5)1,

(28)

where Af is the variation of strength due to fluctuation in cooling condition and is described as follows:

kkaC.) \arU Here, ACR(CR, Tt-) and ATf(Ca, Tr) are fluctuations in cooling rate and finish-cooling temperature, respectively. The optimum cooling condition is sought using the steepest descent method [51]. Curved surfaces in Fig. 14 represent TS(CR, Tr) and Y(Ca, Yp). As shown in this figure, the optimum condition is determined by, at most, several steps.

144

Y. Saito / Materials Science and Engineering A223 (1997) 134-145

Ts ( a i m e d ) : 5 2 0 (MPa) Cooling start temp.:800"C S:Start of search E:End o~ search

~

I I

O

5o 500

[a-

I

? /

7

15

,

I0

5

Cooling rate ('C/sac)

//?

/4£ ,,~6'~" ~"

/'g'

15 Cooling rate ('C/sec)

Fig. 14. Search for optimum cooling condition by steepest descent method [7]. 4. S u m m a r y The computer simulation model of microstructural evolution on the basis o f chemical t h e r m o d y n a m i c s a n d classical nucleation and g r o w t h t h e o r y has been developed. T h e m e t a l l u r g i c a l p h e n o m e n a in t h e r m o m e c h a n i cal t r e a t m e n t o f steel, such as austenite grain growth, recrystallization a n d growth, c a r b o n i t r i d e p r e c i p i t a t i o n a n d austenite-to-ferrite p h a s e t r a n s f o r m a t i o n are form u l a t e d as m a t h e m a t i c a l equations. W i t h use o f these equations, the effects o f chemistry a n d m a n u f a c t u r i n g c o n d i t i o n s on the m i c r o s t r u c t u r e o f steel plate can be p r e d i c t e d with g o o d accuracy.

Acknowledgements

This w o r k was carried out u n d e r the auspices o f A t o m i c A r r a n g e m e n t Design a n d C o n t r o l Project, which is a c o l l a b o r a t i v e venture between the University o f C a m b r i d g e a n d the R e s e a r c h a n d D e v e l o p m e n t Corp o r a t i o n o f Japan. T h e a u t h o r w o u l d like to express his sincere g r a t i t u d e to D r H . K . D . t t . B h a d e s h i a at the U n i v e r s i t y o f C a m b r i d g e for m a n y useful suggestions a n d encouragements.

References

[1] I. Tamura, H. Sekine, T. Tanaka and C. Ouchi, Therrnomechanicat Processing of High Strength Low Alloy Steel, Butterworth, London, 1988.

[2] 131st and 132rid Nishiyarna Memorial Lectures, Iron and Steel Institute of Japan, Tokyo, 1989. [3] Y. Saito, Tetsu to Hagane, 74 (1988) 609 [in Japanese]. [4] T. Senuma, M. Suehiro and H. Yada, ISIJ Int. J., .32 (1992) 423. [5] Y. Saito and C. Shiga, ISIJ Int. J., 32 (I992) 414. [6] J.S. Karkaldy, B.A. Thomson and E. Baganis, Hardenability Concept with Application to Steels, ASM, Materials Park, OH, I978, p. 82. [7] Y. Saito, M. Tanaka, T. Sekine and H. Nishizaki, Proe. hzt. Conf. HSLA steel, University of Wollongong, Wollongong, Australia, i985, p. 28. [8] H. Yada, Sosei to Kako, 28 (1987) 413 [in Japanese]. [9] P. Choquet, A. LeBon and C. Predix, 7th Int. Conf Strength of Metals and Alloys, Montreal, Canada, 1982, p. 1025.

[10] E. Anelli, M. Ghersi, M. Maseanzoni, M. Paolicci, A. Apprile, A. DeVito and F. DeMitri, 7th h~t. Conf. Strength of Metals and Alloys, Montreal, Canada, 1982, p. I031. [11] M. Avrami, J. Chem. Phys., 7 (1939) 1103; 8 (1940) 212; 9 (1941) 177. [12] A.N. Kolmogorov, [zv. Akad. Nauk, Ser. Mater., I (1937) 355. [I3] W.A. Johnson and R.F. MehI, Trans. AIME, 135 (I939) 416. [14] H.L Aaronson and J.K. Lee, Lecture on the Theory of Phase Transformation, ASM, Warrendale, PA, 1975, p. 82. [15] M. Enomoto and H.I. Aaronson, Metall. TraJ~s. A, 17 (1986) i385. [16] M. Hitlert and L.I. Staffanson, Acta Chem. Stand., 24 (1970) 3618. [I7] R.H. Fowler and E.A. Guggenheim, Proc. R. Soc. A, I50 (1935) 552. [18] M. Enomoto and H.I. Aaronson, Calphad, 9 (1985) 43. [19] B. Sundman, B. Jansson and J.-O, Andersson, Calphad, 9 (I985) 153. [20] H.B. Aaron, D. Fainstein and G.R. Kotler, J. Appl. Phys., 41 (I 97O) 4404. [2I] S.M. Allen and LW. Cahn, Acta Metall., 27 (1979) 1085. [22] M. Hiltert, Acta Metall., 13 (1965) 227. [23] C. Zener, quoted by C. Smith, Trans. AIME, 175 (1948) 15.

!I. Saito / Materials Science and Engineer#zg A223 (1997) I 3 4 - I 4 5

[24] K. L~cke and K. Detart, Aeta Metall., 5 (1957) 628. [25] C.M. Sellars and J.A. Whiteman, Met. Sci., i3 (1979) I87. [26] W. Robert, A. Sandberg, T. Siwecki and T. Werefers, Proc. Int. Conf. HSLA Steels Technology and Application, Philadelphia, PA, I993, The MetalIurgical Society of AIME, Warrendale, PA, 1993, p. 619. [27] J.G. Williams, C.R. Killmore and G.R. Harris, Proc. Int. Conf Physical Metallurgy of Thermomechanieal Process#zg of Steels and Other Alloys, Tokyo, Japan, 1988, Iron and Steel Institute of Japan, Tokyo, 1988, p. 224. [28] T. Siwecki, ISIJ Int. J., 32 (1992) 368. [29] N. Komatsubara, S. Okaguchi, K. Kunishige, T. Hashimoto, I. Tamura and M. Umemoto, CAMP-ISIJ, 2 (1989) 676. [30] Y. Saito, T. Enami and T. Tanaka, Trans. ISIJ, 25 (1985) 1146. [31] M. Umemoto, H. Ohtsuka and I. Tamura, Tetsu to Hagane, 70 (1984) 557 [in Japanese]. [32] Y. Saito, C. Shiga and T. Enami, Proe. Int. Co1~ Physical Metallurgy of Thermomeehanieal Processing of Steels and Other Alloys, Tokyo, Japan, I988, Iron and Steel Institute of Japan, Tokyo, 1988, p. 753. [33] S. Akamatsu, Y. Matsumura, T. Senuma, H. Yada and S. Ishikawa, Tetsu to Hagane, 75 (1989) 933 [in Japanese]. [34] W.J. Liu and 3.J. Jones, Metal. Trans. A, 20 (1989) 689. [35] J.D. L'Ecuyar and G. L'Esperane, Acta Metall., 37 (1989) 1023. [36] Y. Saito, A. Matsuzaki and C. Shiga, Computer hmovation o f New Materials, Elsevier/North-Holland, Amsterdam, 1991, p.

145

819. [37] J.R Bradley and H.I. Aaronson, Metall. Trans. A, I2 (I981) 1729.

[38] M. Hillert, Metall. Trans. A, 6 (I975) 5. [39] D. Turnbui1, Acta Metall., 3 (I955) 55. [40] K. Hashlguchi and J.S. Kirkaldy, Scand. J. Metall., I5 (I984). [41] H.K.D.H. Bhadeshia, Met. Sei., I6 (1982) 159. [42] K. Tsukada, K. Matsumoto, K Hirabe and K. Takashige, Iron Steehnaker, 9 (1982) 21. [43] I. Kozasu. C. Ouchi, T. Sampei and T. Okita, Microalloy#zg 75, Union Carbide Corporation, New York, 1977, p 120. [44] F.B. Picketing, Physical Metallurgy and The Design of Steels, Elsevier Applied Science, Barking, UK, 1978. [45] K. Esaka, J. Wakita, M. Takahashi, O. Kawano and S. Harada, Seitetsu Kenkyu, 32 (1986) 92 [in Japanese]. [46] H. Yada, Proc. Accelerated Cooling of Rolled Steels, CIM, 1988, p. 105. [47] Y. Tomota, M. Umemoto, N. Komatsubara, A. Hiramatsu, N. Nakajima. A. Moriya, T. Watanabe, S. Nanba, K. Kunishige, Y. H1go and M. Miyahara, ISIJ hzt. J., 32 (1992) 343. [48] O. Kwon, ISfJ hzt. J., 32 (1992) 350. [49] E.O. Hall, Proc. Phys. Soe. (London) B, 64 (I95I) 747. [50] N.J. Petch, J[SL i74 (I953) 25. [5I] Y. Nishikawa, N. San-nomiya and T. Ibaragi, Optimization (1982) [Iwanami Shoten].