Design of precipitated austenite for dispersed-phase transformation toughening in high strength CoNi steels

Materials Science and Engineering, A 128 (1990) 201-207

201

Design of Precipitated Austenite for Dispersed-phase Transformation Toughening in High Strength Co-Ni Steels M. GRUJICIC

Department of Mechanical Engineering, 318 Riggs Hall, Clemson University, Clemson, SC 29631 (U.S.A.) (Received December 11, 1989; in revised form February 13, 1990)

Abstract

Optimization of the austenite stabifity and the transformation volume change, both of which depend on the alloy compositions, is analyzed in the context of dispersed-phase transformation toughening associated with the precipitated austenite in high strength Co-Ni steels, such as AF1410. A computer-aided thermodynamics-based procedure has been devised to describe the effect of composition on austenite stability while the effect of composition on the transformation volume change is treated using Vegard's law and the two-7-states formalism to account for the magnetovolume effects in austenite. A new alloy matrix chemistry Fe-14wt. %Co-15wt. %Ni-2wt. %Cr is proposed, which has a substantially better combination of austenite stability and transformation volume change over that in AF1410 steel. I. Introduction It is well established [1-3] that significant toughening effects in high strength Co-Ni secondary-hardening steels, such as AF1410, can be obtained through the interaction of deformation-induced martensitic transformation in precipitated austenite with fracture-controlling processes. This toughening mechanism, called dispersed-phase transformation toughening, is controlled by the stability of the austenitic dispersion, i.e. the size and the chemical composition of the austenite particles, and by the austenite ~ martensite transformation volume change. For an effective transformation toughening, the stability of the austenite dispersion must be optimized with respect to the crack tip stress state. Such an optimum stability is generally quite high owing to a high triaxiality of the crack tip stress field which through effects of the stress state sen0921-5093/90/$3.50

sitivity of the martensitic transformation promotes the austenite-* martensite transformation at a lower effective stress. This condition then prescribes the microstructural requirements, i.e. average austenite particle size and composition, to obtain the required austenite stability. It is shown elsewhere [4, 5] that thermodynamic calculations of the driving force for austenite nucleation and of the equilibrium austenite composition can be used to design alloy compositions which ensure the required stability of the precipitated austenite. In addition to a high austenite stability, the transformation toughening requires a large positive transformation volume change. The effect of the transformation volume change on the transformation toughening could involve three phenomena: (a) the phenomenon of the stress state sensitivity of transformation plasticity, mentioned earlier, which leads to an enlargement of the plastic zone and reductions in crack tip stresses; (b) the direct P A V energy dissipation and (c) a modification of the crack tip stress state and effective stress intensity by the volume change. The thermodynamics-based alloy design procedure, outlined above, should then be subjected to a constraint that, at the desired level of austenite stability, the alloy compositions should maximize the transformation volume change. This is a formidable task in the case of precipitated austenite in high strength Co-Ni steels where magnetic changes, such as ferromagnetism and the Invar effect, tend to reduce the transformation volume change and hence the fracture toughness. In the present paper a computer-aided thermodynamics-based procedure is introduced and applied to high strength Co-Ni secondary-hardening steels to optimize the composition of the metallic matrix (tempered martensite plus precipitated austenite) for transformation toughening. © Elsevier Sequoia/Printed in The Netherlands

202

A companion design procedure of the M2C coherent carbide dispersion for strengthening of the same class of steels has been presented elsewhere [5, 6].

2. Austenite stability As mentioned above, the stability of the precipitated austenite dispersion is controlled by the chemical composition and the size of austenite particles. A quantitative characterization of the effect of austenite composition on its stability can be obtained using the free-energy change A G ch for martensitic transformation evaluated at room temperature (300 K). The THERMOCALC computer program [7] was used to calculate this quantity in the Fe-Ni-Co system. The martensitic phase was represented by the b.c.c, phase with the same composition as the f.c.c, phase since the transformation is diffusionless. Figure 1 shows the contours of the free energies of the f.c.c, and b.c.c, phases as a function of composition, evaluated at 300 K. The difference between these free energies is the free-energy change for the martensitic transformation,, namely A G ch= G b.c.c.- G f'c'c'. The more positive this quantity, the more stable is the austenite. Figure 2 shows contours of AG ch as a function of composition for the ternary Fe-Ni-Co system. Superimposed on the same plot is the isothermal section of Fe-Ni-Co at the typical temperature of 783 K (broken lines) with two tie-lines shown. This combined plot allows one to select alloy compositions for austenite stabilization. Starting with an alloy content X0,

the corresponding tie-line defines the austenite content Xy that precipitates at 783 K. This content then defines, from the A G ch contour plot, the chemical driving force at room temperature. Following this procedure, alloy compositions that maximize austenite stability can be selected. The effect of particle size on austenite stability is the result of the heterogeneous nature of martensite nucleation at pre-existing nucleation sites. That is, the probability that a particle will contain at least one nucleation site increases with increasing volume. Hence, the finer the average particle size, the more stable is the austenite dispersion. In addition, the transformation of fine austenite particles gives rise to the formation of the preferred fine martensitic structure. Hence, when designing precipitated austenite for dispersed-phase transformation toughening, one should strive for the finest austenitic particles. This can be achieved by maximizing the driving force for austenite nucleation, a quantity which can be readily calculated using the well-known parallel-tangent method [4]. 3. Transformational volume change The f.c.c.--"b.c.c, transformation volume change can be controlled through the composition dependence of the lattice parameters of austenite and martensite. Alloy compositions can then be selected that result in high A V/V and hence enhanced toughening. The volume change

Co

Co

I I It t l

--- BCC -- F C C

-I.5

Ill l

BCC

Xii , .:

I' ','d, "' ~/I//,/~1111 ll It

I

-3.5

Fe

---Z--- ~.s - - ~ Fe

Ni

Fig. 1. Free-energy contours (in kilojoules per mole) of the f.c.c, and b.c.c, phases in the F e - N i - C o system at 300 K.

0

0.5

~ 1.5 2

2.5

Ni

Fig. 2. Contours of the free-energy change (in kilojoules per mole) for the martensitic transformation at 300 K ( ) and superimposed isothermal section (- - -) at 783 K for the F e - N i - C o system. )Co represents the alloy content and Xr the austenite content.

203

resulting from the martensitic transformation f.c.c.--,b.c.c, is related to the lattice parameters ae.c.c and ab.c.c,of the two phases as AV

(ab.c.c.] 3

~-2

-1

(1)

\ afcc ] The lattice parameters are functions of the alloy composition, i.e. ae.¢.c.=fl(Xi), ab.c.c.=f2(Xi), i -----Fe, Co, Ni, Cr where X i is the mole fraction of element i (the carbon content of the precipitated austenite in A F 14 10 steels is negligibly small [3]). The form of the function fl for the f.c.c, phase is particularly important because of magnetic changes in austenite, such as volume magnetostriction and the Invar effect, which increase the value of aec.¢ and hence reduce AV/V. An attempt will be made here to develop analytical expressions for af ..... taking into account these magnetic effects. The two-y-states formalism developed by Tauer and Weiss [8] is used to describe the Invar effect on af.c.¢. In order to reconcile a large mag netic entropy term at high temperatures with the absence of a specific heat anomaly at the N6el temperature (low temperatures), Weiss and Tauer [9, 10] proposed two discrete electronic states of 7-Fe: Yl and 72- These two states coexist in equilibrium with each other in a manner corresponding to a two-level Schottky excitation [11]. Individual atoms are not considered to possess permanently either of the electronic configurations but have a probability of exhibiting both states on a time-transient basis according to the ratio

exp --~--~

f

-- gl

a=l-f

go

(2)

or

f= 1 + (go~g,) exp( + AEIRT )

where a is the probability that the upper energy state is occupied and f t h e corresponding fraction of iron atoms in the upper energy state. AE represents the energy difference between the two states, and go and gl are the degeneracies of the upper and lower states respectively. The properties of the Yl and Y2 states are presented in Table 1. In pure y-Fe, the antiferromagnetic (low volume) states is the (low energy) ground state. Addition of nickel, cobalt and chromium promotes the ferromagnetic (high volume) Y2 state, while addition of manganese stabilizes the antiferromagnetic (low volume) Yl state [ 10, 12]. The formulation of the compositional dependence of the lattice parameter af.c.c, is based on the description of two composition-dependent effects: (1) the effect of composition on the relative stability, i.e. occupancy of the Yl and Y2 states through the compositional dependence of AE, the energy difference between the two y states; (2) the effect of composition on the overall magnetic moment, which controls the magnitude of volume magnetostriction. These effects in the F e - N i - C o - C r system are studied here using the following procedure. An expression for the lattice parameter is derived in the Fe-Ni binary system which is adopted as a reference. Then the effect of chromium and cobalt on both the relative stability of Yl and Y2 states and the magnetic moment is considered, to derive the final expression for af.c.c.. Using Vegard's law, the lattice parameter for the Fe-Ni austenite is af.c.c = (1 - XNi)(ayfy2 + arf7,)+XNiaNi

(3)

where fr, and fr2 are the fractions of iron atoms in the yl and Y2 states respectively as defined by eqn. (2), and at, and a72 are the lattice parameters of the y~ and 72 states given in Table 1. XN~ and ~Ni (0.35 1 7 nm) are the mole fraction and lattice parameter of nickel respectively. According to data on the composition dependence of AE obtained by Weiss and Tauer [10], A E = 0 at XNi= 0.29.

TABLE 1 Approximate values of lattice parameter, spin per atom and Curie or N~el temperature of the two electronic structures of ),-Fe [12] Crystal structure

Lattice parameter

Magnetic structure

(nm) 7~ 72

f.c.c. f.c.c.

0.354 0.364

Antiferromagnetic Ferromagnetic

Spin per atom

Curie or N&I temperature

(/~B)

(K)

0.5 2.8

80 1800

204 Therefore for XNi > 0.29 the 72 state is more stable. In order to develop eqn. (3) further, an analytical expression for AE(XNi) is required. The following expression was assumed: A E = A + BXNi + CXNi 2 + OXNi 3 + EXNi 4

(4)

This expression was fitted to data on AE (J mol- 1) vs. XNi reported by Weiss and Tauer [10], which gives A = 3 4 3 2 , B = 2 8 0 0 , C = 4 6 4 9 8 and D = 259658 for XNi <~0.29, and A = - 4 . 4 8 x 105, B = 5.38 x 10 6, C = 2.39 x 10 7, D = 4.65 x 10 7 and E = - 3.36 x 107 for 0.29 < Xr~i< 0.5. Having established the compositional dependence of AE, the lattice parameter ae.~.~. (nm) in the Fe-Ni system is found to be 4~(1 -- XNi )" af ..... =0.35411 --41SNi-I1 + g/1.79 t for XN~< 0.29, g~ = 1.79 go arc... = 0 . 3 6 4 { 1 - 4X /N~

AE=A'+

(5a)

42'(1l +--gXN~)i

for XN~> 0.29, g~ = 1.0 go

(5b)

with g = e x p ( + A E / R T ) and 41=0.0075, 42 = 0.0293, 41 t = 0.0338 and 42' = 0.0273. AE is given by eqn. (4). Figure 3 presents the lattice parameter af.c.~,as a function of the mole fraction of nickel in the Fe-Ni system for 0.29 < SNi < 0.5. Austenites with XNi<0.29 do not possess the required stability and will not be analyzed any further. Stabilization of the high volume ferromagnetic ?e state with the addition of nickel, as depicted in Fig. 3, results in an increase in the lattice parameter in the nickel content gNi range

"E 0.359 c

E

0.30-0.40. However, with further addition of nickel, the 72 state becomes saturated and the effect of the low volume nickel becomes evident in reducing the lattice parameter above XN~= 0.40. The data presented here for the lattice parameter for the Fe-Ni system are in very good agreement with experimental data obtained by Ruhl [13]. Addition of chromium markedly reduces the saturation magnetization [14], which results in a decrease in the lattice parameter irrespective of nickel content. However, chromium stabilizes the ?z high volume state, promoting the transition f r o m ?1 to 72 to o c c u r at lower nickel contents (XNi < 0.29). The data of Miodownik [14] for AE (J mol-l) vs. XCr in the Fe-Ni-Cr system were fitted to an expression of the form B'Xfr + C'Xcr 2

yielding A ' = 209, B' = 3.84 x 10 4 and C ' = 2 . 1 2 x 105. Then, using Vegard's law and af.c.c.(Cr)=0.304 nm the composition dependence of af.~.~,in the Fe-Ni-Cr system becomes 22(1 -- gNi -- Scr)l af.c.c.= 0.364 [1 - 41XNi-I- ~ e ~ - p ~ l

-{-Scr{C1- C2(XNi -- C3)2 --I- C4(SNi - C3) 4}

(6) Equation (6) was fitted to experimental data of Pearson [15] for lattice parameters in the Fe-Ni-Cr system to give C 1= 0.3433, C2 = 0.2838, C3 = 0.506 and C4 = 16.2605. Cobalt stabilizes the 72 high volume state [16]. On the contrary, it also increases the saturation magnetic moment, which should lead to larger lattice parameters. Since A E = 0 at Xco = 0.4 in the Fe-Co system and the AE vs. Xco curve is similar to the AE vs. XNi curve, the effect of cobalt on the relative stability between the ?! and 72 states can be quantified by substituting XNi = Xco(0.29/0.40) in eqn. (4). The result is

O. 358

A E = A " + B"Xco + C"Xco 2 + D " S c o 3

O.

2

O.357 0.3

0.4 Mole Froction

0.5 Ni

Fig. 3. Lattice parameter af.c.c,for the Fe-Ni system.

(7)

where A ' = 3432, B "= 2488, C"= 3.37 x 10 4 and D "= 1.88 x 105. The magnetic moment (in Bohr magnetons) was calculated for the f.c.c, phase in the F e - N i - C o - C r system using the following equa-

205

tion: B = Z XiB (f.c.c., i, O) + Z Z xixi i

i~j

x {B (f.c.c., i, j, 0) + (Xi - X)B (f.c.c., i, j, 1) +(Xi-Xi)2B(f.c.c., i,j, 2)+ ...}

(8)

where i, j---Co, Cr, Fe, Ni and B (f.c.c., i, j) are defined in the T H E R M O C A L C database [7] and their values are given in Table 2. Figure 4 shows the variation in the magnetic moment with nickel content for the Fe-Ni, Fe-Ni-5wt.%Co, and Fe-Ni-5wt.%Cr systems evaluated using eqn. (8). The correlation between the magnetic moment and the austenite lattice parameter in Fe-Ni-Cr can be quantified with the aid of eqn. (6) and Fig. 4. For XNi=0.4 and Xcr=0.05, eqn. (6) gives af.~.c.=0.357 75 nm, while excluding the last term in braces (eqn. (6)) from the calculation (representing the effect of

TABLE 2 Coefficients B (f.c.c., i, j) in eqn. (8) for the magnetic moment defined in the THERMOCALC database

the magnetic moment) gives ctf.c.c.= 0.359 24 nm. Therefore the addition of 5 wt.% Cr reduces the lattice parameter by A a = 0.001459 nm. According to Fig. 4, this reduction in the lattice parameter is due to a reduction AB in the magnetic moment of 0.35/t B caused by the addition of 5 wt.% Cr. Therefore, A a = (0.001 495/0.35)AB or A a = 0 . 0 0 4 2 7 1 A B , where A a is in nanometres and AB is in Bohr magnetons. To find the corresponding effect of cobalt, the value of AB at XNi----0.40 was evaluated from the curves for Fe-Ni and Fe-Ni-5wt.%Co in Fig. 4. This gave AB=0.1. Assuming the relation between Act and AB above to hold for cobalt as well, we obtain Aa=O.OO4271X(O.1/O.O5)Xco or Aa--0.085 42Xco. This equation represents the effect of the magnetic moment on the lattice parameter due to addition of cobalt. Taking all the above into account and adding the Vegard's law contribution associated with o~f.c.c.(Co ) -~- 0.3544 nm, the lattice parameter ctf..... for the F e - N i - C r - C o system is given as af.c.c.= 0.364{1- 0.0338XNi-- 0.0238Xco + 0.0273

171 Magnetic moment

Value

B (f.c.c., Fe; 0) B (f.c.c., Ni; 0) B (f.c.c., Co; 0) B (f.c.c., Cr; 0) B (f.c.c., Fe, Ni; 0) B (f.c.c., Fe, Ni; 1) B (f.c.c., Fe, Ni; 2) B (f.c.c., Fe, Ni; 3) B (f.c.c., Co, Fe; 0) B (f.c.c., Co, Fe; 1) B (f.c.c., Co, Ni; 0) B (f.c.c., Co, Ni; 1) B (f.c.c., Cr, Ni; 0)

- 2.1 0.52 1.35 -2.1 9.55 7.23 5.93 6.18 9.74 -3.51 1.04 0.16 - 1.91

I

t.- ~. o .cO -2

t

-3 o.o

• • --

Fe*NI Fe-Ni-SCo

- - - - Fe-NI-SCr

o.a 0.4 0.6 o.a ko Mole Froetion Ni Fig. 4. Effect of nickel on the magnetic moment of Fe-Ni, Fe-Ni-5wt.%Co and Fe-Ni-5wt.%Cr systems evaluated using the T H E R M O C A L C database [7].

X

1 - XNi - Xcr - Xco1/ 1 + exp(AE/gT) J

-'1-Xcr { C 1 - C 2 ( X N i - C3) 2 Jr- C 4 ( X N i - C3) 4 }

+ 0.085 42Xco

(9)

AE in eqn. (9) is given by summing the contributions of nickel, chromium and cobalt: A E = A + BXNi + CXNi2 + DXNi3 + B'Xcr + C'Xcr 2 + B"Xco

+

CnXco 2 -.{-D,'Xco 3

where the coefficients have been defined above. Equation (9) is the final equation to be used for the evaluation of the lattice parameter af.c.c,in the F e - N i - C r - C o system. The value of the lattice parameter ab.c.c,for the product b.c.c, phase is calculated using Vegard's law: Otb.c.c.-~- (ZFeXFe "1- aNiXNi -t- 0{crXcr

(10)

where a o = 0.2885 nm for b.c.c, chromium from Barrett and Massalski [17]. Experimental data from Ruhl [13] on the Fe-Ni b.c.c, lattice parameters are used to develop eqn. (10)further. Ruhl's data for the compositional dependence of the b.c.c, lattice parameter for a nickel content higher

206

than 25 at.% can be represented as

The increment A(A G °) in the mechanical driving force due to an increment Ao in stress, can be calculated using the Patel-Cohen [19] criterion as

ab.c.c.= 0.286 64 -- 0.011 (XNi-- 0.25) Equation (10) then becomes

3(AG °) a(aGO)=ao , 3a

ab.¢.¢.= {0.286 64 -- 0.011 (Xr~i- 0.25)} (11)

x ( 1 - Xcr) + 0 . 0 2 8 8 4 6 X c r

Equation (11) gives the lattice parameter of the Fe-Ni-Cr b.c.c, system for XNi > 0.25. The effect of cobalt was not taken into account because of the lack of information on the lattice parameter of Fe-Co b.c.c, solid solutions. Equations (1), (9) and (11) are then used to evaluate the f.c.c. --"b.c.c. transformation volume change A V/V as a function of the austenite composition. The validity of this approach has been verified in the case of A F 1410 steel ( 14wt.%Co- 10wt.%Ni matrix) where the calculated value of A V/V=2.20% agrees quite well with the experimental value of 2.25% [1]. In addition, good agreement (within + 10%) has been obtained with experimental data reported by Magee and Davies [18] for 13 Fe-Ni-Co alloys containing between 20 and 30 wt.% Ni and between 5 and 30 wt.% Co. 4. Results and discussion

The procedure developed in Sections 2 and 3 is applied here to design several high Co-Ni steels with tensile strengths in the range 1700-2100 MPa corresponding to Rockwell C hardness levels of 50-55 HRC. These strength levels are higher by approximately 400 MPa than the corresponding strengths of the commercial AF1410 steel. Hence the additional mechanical driving force for martensitic transformation at these higher strengths must be compensated by a corresponding chemical stabilization of the precipitated austenite to ensure a similar toughening effect as for A F 1410 steel.

TABLE 3

Using equivalent stress 6, Olson [20] obtained 3(AG)/36= 1.42 J mo1-1 MPa -1 for the stress field ahead of a mode I crack tip. So, for A o = 400 MPa, A(AG°) = 568 J mol -a. The freeenergy change for the martensitic transformation evaluated at room temperature for equilibrium austenite precipitated at a tempering temperature of 783 °C will be adopted as a measure of stability. This quantity was calculated using the THERMOCALC software with a procedure described in Section 2. For AF1410 steel (14wt.%Co-10wt.%Ni matrix) the free-energy change AG Cnis 813 J m o 1 - 1 . Therefore, in order to compensate for the increment in the mechanical driving force calculated above, the free-energy change must increase to 1380 J mo1-1. The quantity AG ch was calculated for several alloy compositions, and the results are listed in Table 3. Also shown in Table 3 are the results of our calculation of the driving force AG" for austenite nucleation from a supersaturated matrix at 783 K. The equilibrium austenite contents are also shown in Table 3. As was mentioned earlier, the selection of alloy compositions is subject to the constraint of a large volume change. Therefore the transformation volume change was calculated for each of the alloys listed in Table 3 with the aid of eqns. (1), (9) and (11) in the previous section. Several of the alloys in Table 3 contain chromium. It is important to note that this is the chromium remaining in solution after complete precipitation of M2C at 783 K. The alloy compositions in Table 3 are therefore matrix compositions. The alloy composition in Table 3 that

Dependence of austenite stability and transformation volume change on alloy matrix compositions

Alloy

14Co-10Ni 16Co-15Ni 14Co-15Ni-2Cr 14Co-13Ni-3Cr 13Co-13Ni-3Cr

Austenite composition (783 K) (wt.%) Ni

Co

Cr

0.41 0.43 0.39 0.37 0.37

0.032 0.045 0.044 0.044 0.04

0 0 0.043 0.067 0.066

A G ch (300 K)

(A V/V ) (300K)

A G" (783 K)

(J mol- ')

(%)

(J mol- ')

813 1300 1427 1115 1045

2.20 2.24 3.30 4.1 3.88

1870 1790 1990 2020 1995

207

satisfies the stability requirement of 1380 J mol-1 is Fe- 14wt.%Co- 15wt.%Ni-2wt.%Cr which has AG ch = 1427 J mol- 1 and A V / V = 3.3%. Both the stability and the volume change are much higher than the corresponding values for the reference Fe- 14wt.%Co- 10wt.%Ni matrix. In addition, the driving force AGn(783 K) for austenite nucleation from the supersaturated matrix of 1990 J tool- t is significantly larger than that for A F 1410 steel, guaranteeing a smaller average particle size of the precipitated austenite. F e - 1 4 w t . % C o 15wt.%Ni-2wt.%Cr therefore represents the alloy matrix composition with the substantially improved combination of austenite stability and transformation volume change for enhanced transformation toughening at higher strength levels. Microstructural characterization and fracture toughness measurements of this alloy are under way [21]. It should be noted that the present analysis has been carried out with the objective of maximizing the room temperature toughness. Obviously, the efficiency of transformation toughening is affected by the service temperature. The effect of temperature on austenite stability can be readily examined using T H E R M O C A L C . However, the effect of temperature on relative stability of the two y states and hence on the transformation volume change is beyond the scope of this work.

5. Conclusions Dispersed-phase transformation toughening of ultrahigh strength steels can be potentially enhanced by increasing austenite stability and transformation volume change. Computer-aided thermodynamic analysis can be used to quantify the effect of composition on austenite stability and design austenite of an optimum chemistry. The effect of composition on the transformation volume change stems from the compositional dependence of the lattice parameters of austenite and martensite. While Vegard's law accounts quite well for the composition dependence of the martensite (b.c.c.) lattice parameter, the effect of alloy additions on the relative stability of the two y states and the magnitude of volume magnetostriction dominates the composition dependence of the austenite lattice parameter.

Acknowledgments This research has been supported by the National Science Foundation Grant DMR 8418718 as part of the Steel Research Group Program. The author thanks Professor G. B. Olson for many helpful discussions. References 1 G. N. Haidemenopoulos, Dispersed-phase transformation toughening in ultrahigh-strength steels, Doctoral Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1988. 2 G.N. Haidemenopoulos, G. B. Olson and Morris Cohen, Dispersed-phase transformation toughening in ultrahighstrength steels. In G. B. Olson and M. Azrin (eds.), Innovations in Steel Technology, Proc. 34th Sagamore Army Materials Research Conference, Lake George, NY, August 1987, American Society for Metals, Metals Park, OH, 1990, in the press. 3 C. C. Young, Sharp-crack transformation toughening in phosphocarbide strengthened austenitic steels, Doctoral Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1987. 4 M. Hillert, in H. I. Aaronson (ed.), Lectures on the Theory of Phase Transformations, Metallurgical Society of AIME, New York, 1957, p. 1. 5 M. Grujicic, Calphad, 14 (1990) 45. 6 M. Grujicic, Mater. Sci. Eng. A, 117(1989) 215. 7 B. Sundman, B. Jansson and J. O.-Anderson, Clalphad, 9 (1985) 153. 8 K. J. Tauer and R. J. Weiss, Bull. Am. Phys. Soc. 11, 6 (1986) 125. 9 R.J. Weiss and K. J. Tauer, Phys. Rev, 102(1956) 1490. 10 R. J. Weiss and K. J. Tauer, J. Phys. Chem. Solids, 4 (1958) 135. 11 L. Kaufman, E. W. Clougherty and R. J. Weiss, Acta Metall., 11 (1963) 323. 12 R.J. Weiss, Proc. Phys. Soc., London, 82(1963) 281. 13 R. Ruhl, Splat quenching of Fe-based alloys, Doctoral Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1967. 14 A. P. Miodownik, Actu MetalL, 18 (1970) 541. 15 W. B. Pearson, Handbook of Lattice Spacings, Pergamon, London, 1958. 16 A.P. Miodownik, Scr. Metall., 3 (1969) 931. 17 C. S. Barrett and T. B. Massalski, Structure of Metals, McGraw-Hill, New York, 1966. 18 C. L. Magee and R. G. Davies, Acta Metull., 20 (1972) 1031. 19 J. R. Patel and M. Cohen, Acta Metall., 1 (1953) 531. 20 G. B. Olson, Transformation plasticity and the stability of plastic flow. In Deformation Processing and Structure, American Society for Metals, Metals Park, OH, 1983, pp. 391-424. 21 G. B. Olson, Research in progress, Northwestern University, 1989.