of transformed pearlite using the Avrami equation. The required parameters in the equation were obtained from the Time-Temperature-Transformation (TTT) ...
Modeling of the Liquid/Solid and the Eutectoid Phase Transformations in Spheroidal Graphite Cast Iron SULI CHANG, DONGKAI SHANGGUAN, and DORU M. STEFANESCU A model of phase transformations in spheroidal graphite (SG) cast iron has been developed to quantitatively describe the microstructural evolution during solidification and the subsequent solid-state phase transformations (eutectoid reaction) during continuous cooling and to predict some of the microstructural characteristics of final phases formed in SG iron castings. Such characteristics include phase fractions, phase spacings, and grain dimensions. In the model, the nucleation and growth of primary dendrites and eutectics were described based on existing theories, whereas the mathematical formulation for the eutectoid reaction, i.e., the formation of pearlite and ferrite from the as-cast austenite, was developed based on theories as well as physical evidence obtained from the experimental work. The Johnson-Mehl equation and the Avrami equation were used to calculate the fraction of transformed phases under continuous cooling conditions. The role of the grain impingement factor used in these two equations and the significance of the additivity principle in treating nonisothermal transformations were briefly discussed. The latent heat method was used for the numerical treatment of the release of latent heat during phase transformations. A two-dimensional finite element code which can be used in either Cartesian or cylindrical coordinates (ALCAST-2D) was used to solve the time-dependent temperature distribution throughout the metal/mold system. Numerical predictions were validated against experimental results, and good agreement was obtained. I.
BACKGROUND
S P H E R O I D A L graphite (SG) cast iron is a competitive engineering material. The spheroidal shape of the graphite provides higher mechanical properties than those of gray (lamellar graphite) iron. Machinability is better than that of steel. Under certain circumstances, the probability of shrinkage formation can be decreased because of the expansion caused by graphite precipitation during solidification.t1] At the present time, computer modeling of phase transformations and computer-aided design are perceived by the metal casting industry as major tools in improving the quality of SG iron castings through process control and product design. Manufacturing times and costs will also be saved when these modern tools are fully implemented. One of the main goals of computer modeling of the liquid/solid and the solid-state phase transformations of SG cast iron is to predict the mechanical properties of the material. If the microstructure of an SG iron casting can be related quantitatively, through computer simulation, to the processing conditions and the configuration of the casting and, ultimately, to the mechanical properties in various locations in the casting, it would be possible to improve, or at least to control, casting properties by altering either the production process or the heat treatment. Accordingly, the development of an appropriate computer model for predicting the fractions and length
DONGKAI SHANGGUAN, previously Assistant Research Engineer, The University of Alabama, is Manufacturing Engineer, Electronics Division, Ford Motor Co, Dearborn, MI 48121. SULI CHANG, Graduate Research Assistant, and DORU M. STEFANESCU, University Research Professor and Director, are with the Solidification Laboratory, Department of Metallurgical and Materials Engineering, The University of Alabama, Box 870202, Tuscaloosa, AL 35487-0202. Manuscript submitted May 25, 1991. METALLURGICAL TRANSACTIONS A
scales of the final phases can be a significant contribution to the production of quality SG cast iron. Figure 1 shows a typical temperature-time curve (cooling curve) for an SG iron continuously cooled from the liquid state down to room temperature. Schematic microstructures of transformation products are superimposed on the curve. The transformation process includes three distinct stages: solidification of primary dendrites, the eutectic solidification, and the eutectoid reaction. The computer model should therefore describe all of these three transformations. Examination of the phenomenology of solidification of SG iron is conducive to the following sequence o f solidification, t2] (1) For a hypoeutectic alloy, primary austenite dendrites begin to form as the temperature decreases below the liquidus temperature. Even for an alloy of eutectic composition, solidification will start with the formation of austenite dendrites due to nonequilibrium solidification. (2) At the eutectic temperature, austenite dendrites and graphite spheroids nucleate independently in the liquid. (3) Very limited growth of spheroidal graphite occurs in contact with the liquid (maximum 10 to 15 /xm, or 0.2 of the final radius). (4) Flotation or convection then determines the collision of graphite spheroids with the austenite dendrites. (5) Graphite encapsulation in austenite can occur before or immediately after these collisions. (6) The bulk of graphite growth occurs by carbon diffusion through the austenite shell only after graphite spheroids have attached themselves to austenite dendrites. (7) Austenite dendrites grow partly due to carbon diffusion (from the eutectic reaction) and partly because of melt undercooling and supersaturation (as a primary phase). This sequence is illustrated schematically in Figure 2. Note that the eutectic austenite is dendritic and can hardly be discerned from primary austenite dendrites. The eutectoid transformation of SG iron is rather well documented in the literature, ttj It consists of VOLUME 23A, APRIL 1992--1333
1400
1250
o
2100
•
Liquid+ Dendrite
~
~
~
Liquid+ Dendrite + Eutectio
gso
~Grephite
eO0
650
500
i 200
, 400
i 800 Time
| 800
i lO00
* 1200
i400
(see)
Fig. 1 - - A typical cooling curve with schematic transformation microstructures during the solidification of SG iron.
transformation of austenite into pearlite and ferrite. The pearlite/ferrite ratio depends mainly on the cooling rate during transformation. In the recent past, solidification modeling of SG iron has been the subject of several studies. It was generally assumed that the growth of the SG-austenite eutectic is controlled by the diffusion of carbon through the austenite shell, and the growth equation was then derived from Fick's law.[3-6] Various investigators have also proposed general models for the growth of primary equiaxed dendrites tT-]~ and for the solid-state transformation. I4,H] Stefanescu and Kanetkar t41 assumed that the formation of ferrite was due to the diffusion of carbon through the ferrite shell under steady-state conditions. The growth rate was formulated from Fick's law, which is analogous to the equation for the growth rate of the SG-austenite eutectics. They also adopted an equation derived by Frye et al. I]2j for the growth rate of pearlite in Mn steel. The nucleation rate was calculated by the classic nucleation law. However, the nucleation rate of pearlite during isothermal transformation was reported [~3] to vary with time, and thus, this issue merits further study. Venugopalan [HI developed a model to calculate the isothermal transformation rate of ferrite in SG iron. This model is similar to that proposed earlier, [4] except that the back diffusion of carbon into the austenite matrix was taken into account. Although the predicted results were in agreement with experimental observations, the applicability of the model is rather limited, since SG iron castings are mostly produced under continuous cooling conditions and the eutectoid transformation always in-
Fig. 2--Schematic illustration of the progression of growth of the austenite-SG eutectic. 1334--VOLUME
23A,
APRIL
1992
volves two reactions: the formation of ferrite and of pearlite. Different approaches for modeling the transformation of pearlite in carbon steel have been presented by Agarwal and Brimacombe, t~aJ Kuban et al., [15] and U m e m o t o et al. [~6] They emphasized the importance of additivity of transformed pearlite in the calculation of the volume of transformed pearlite using the Avrami equation. The required parameters in the equation were obtained from the Time-Temperature-Transformation (TTT) diagram. It was recognized that the austenite grain could grow during the eutectoid transformation, which could, in turn, result in a variation in the transformation rate. U m e m o t o et al. t~6] took this effect into account in the equation to calculate the volume fraction of pearlite. However, in SG iron, because of the presence of graphite nodules, austenite grains cannot grow considerably. Based on the present state of the art, the issues that must be addressed in order to obtain a correct model for the liquid/solid and the solid/solid (eutectoid) transformations of SG iron are as follows: (1) the solidification model should include dendritic and eutectic solidification; (2) a model consistent with continuous cooling through the eutectoid range must be developed; (3) the dependence of the transformation rate of pearlite on temperature should be correctly treated; and (4) the numerical scheme should solve simultaneously the heat-transfer problem and phase transformation kinetics. II.
MATHEMATICAL
FORMULATION
Solidification is a coupled process of heat/mass transfer and phase transformation kinetics; therefore, accurate modeling of solidification must correctly describe this coupling, which is achieved through the release of latent heat. The governing equation for heat transfer can be obtained from the generalized equation of energy. The resultant partial differential equation has to be solved subject to boundary conditions appropriate to the experimental system under consideration, as shown in Figure 3. Due to the complex boundary conditions resuiting from the existence of two interfaces, i.e., the metal/mold interface and the mold/air interface, an analytical solution is impossible to obtain. Hence, a previously developed finite element code (Alcast-2D) was used to calculate the time-dependent temperature distribution throughout the system. A. The Heat-Transfer (Macro-) M o d e l 1. Derivation o f the governing equation When considering a transport problem with heat transfer and fluid flow, the equation of energy can be constructed based on the conservation of energy in the system. The accumulation rate of thermal energy should be equal to the sum of the net energy change due to (1) heat conduction, (2) mechanical work from phase changes, (3) fluid motion, (4) heat generated by chemical reactions or phase changes, and (5) heat convection and/or radiation. The main assumptions used in the Alcast-2D code are as follows: METALLURGICAL
TRANSACTIONS
A
convection will produce uniform temperature across the liquid. Therefore, the initial temperature of the metal is assumed to be uniform and equal to the highest temperature recorded by the thermocouple placed in the casting. The initial temperature of the mold is assumed to be equal to room temperature. 4. Latent heat release
Fig. 3 - - T h e arrangement of the casting and mold with thermocouples.
(1) the physical properties of the metal and the mold are isotropic; (2) the energy yielded by mechanical work can be ignored due to the small volume change involved; (3) fluid flow in the liquid can be neglected; and (4) the effect of the air gap formed between the casting and the mold wall during solidification can be included by an effective heat-transfer coefficient at the metal/mold interface. In cylindrical coordinates, the governing heat-transfer equation can be written as pCp-- = K -at r Or
r
+ K
~z 2
+ Q.
[1]
where T is temperature, t is time, Cp is specific heat, K is thermal conductivity, p is density, Q is the heat generation rate, r and z are the cylindrical coordinates. 2. B o u n d a r y conditions
Three types of boundary conditions are commonly u s e d I171 at the metal/mold and mold/air interfaces:
B1. A T=
prescribed
temperature
at
the
interface,
To.
B2. A prescribed heat flux at the interface, q = q0. B3. Newton's law of cooling, q = h ( T - Ts), where h is the heat-transfer coefficient and (T - TI) is the temperature difference across the interface. In the system under investigation, the third type of boundary condition is appropriate to specify the heat transfer between the metal and the mold wall and between the mold wall and the surrounding air. The physical constants used in the numerical calculation are listed in the Appendix.
During phase transformations upon cooling, latent heat is released. This affects the temperature field, which, in turn, influences the microstructural evolution. In 1946, a simple and intuitive method, the temperature recovery method, was proposed t18J to take this effect into account in solidification modeling. In this approach, the temperature of a node, Ti, is reset to the melting temperature, Tm, once Ti falls below Tm, until the accumulation of the difference between T~ and T,, reaches the value of the latent heat divided by the specific heat, L / C v , when the nodal temperatmre is allowed to fall down again. This method has recently been used by Goettsch and Dantzig t~91 to combine heat transfer and solidification kinetics and was applied to the analysis of the solidification process of lamellar graphite cast iron. Other procedures used to account for the release of the latent heat are the specific heat method t2~ and the enthalpy method. I2~JIt is assumed that the fraction of solid is evolved over a fictitious temperature range. Then some functional dependency of the fraction of solid on temperature is assumed. Such typical assumptions are, for example, linear dependency for eutectics and Scheil equation for dendritic alloys. The heat generation term in Eq. [1] can then be expressed as a function of temperature. Thus, a homogeneous equation can be obtained and then solved numerically. When using these two methods, the fictitious temperature range has to be known a priori. However, the nonequilibrium solidification temperature range is not a constant but, rather, depends on several factors and, in principle, on cooling conditions. The enthalpy method has been modified to couple heat transfer and solidification kinetics in the microenthalpy method, t221 The recently developed latent heat method, t6,23,24J also used for coupling heat transfer and solidification kinetics, can realistically predict the occurrence of recalescence, which is found on the cooling curves of many cast alloys. It is also believed ~25jto be more accurate than other approaches used to treat the heat generation term, such as the microenthalpy method, since it does not use any (sometimes arbitrarily) prescribed functions for the enthalpy or the fraction of solid as a function of temperature; instead, the evolution of the fraction of solid is calculated based on the nucleation and growth kinetics. In this work, the latent heat method was employed. The rates of heat generation due to growth of primary dendrites, eutectics, and eutectoids are written, respectively, as --
0feut
O.L/S = Of~ L~ + - - Leut Ot Ot
[21
(2sis = ~ L~ +
[3]
3. The initial condition
During pouring, the cold mold absorbs heat from the superheated liquid metal, resulting in a temperature gradient in the melt. For the casting under consideration, however, mold filling is rapid, and it is assumed that METALLURGICAL TRANSACTIONS A
L~
where L is the latent heat and the subscripts are L, liquid; VOLUME 23A, APRIL 1992--1335
S, solid; % austenite; eut, eutectic; a, ferrite; and pc, pearlite. The evolution of the fraction of austenite, eutectic, ferrite, and pearlite with time is determined by the phase transition kinetics and can be obtained through microscopic models. These models are introduced in Section B.
B. The Phase Transformation Kinetics (Micro-) Models As mentioned previously, the release of latent heat, Q, is determined by phase transformation kinetics. During continuous cooling from the liquid state down to room temperature (Figure 1), SG iron has to go through both the liquid/solid transformation, i.e., solidification, and the solid-state phase transformation, i.e., the eutectoid reaction. The final microstructure in an SG iron casting can be a mixture of pearlite and ferrite, fully pearlitic, or fully ferritic. In the case of a pearlitic-ferritic structure, the graphite spheroids are surrounded by ferrite shells (Figure 4). The release of latent heat due to these phase transformations affects the temperature field, which, in turn, affects phase transformations. This interplay is reflected in Eq. [1] via Eqs. [2] and [3]. In Eqs. [2] and [3], OfJOt, Ofeut/~t, ~fJOt, and OfpJOt are the rates of formation of various phases and are determined by the kinetics of each individual phase transformation involved. To model these phase transformations accurately, the kinetics of these phase transformations have to be established and properly incorporated in the model through Eqs. [2] and [3].
1. The liquid~solid transformation As discussed, in a hypoeutectic SG iron, as the temperature of the melt decreases below the liquidus temperature, primary austenite dendrites are formed. Then, as the eutectic temperature is reached, eutectic grains are formed through nucleation and growth. Experimental observations suggest that both the dendritic and the eutectic grains are equiaxed. In this work, all grains formed are assumed to be equiaxed; i.e., the eutectic grains are assumed to be spherical in shape, and the dendritic grains are assumed to be enveloped within a sphere (Figure 5).
Fig. 5 - - M o d e l s for equiaxed grains used in HT-SK modeling.
a. Fraction of solid Calculation offs from the nucleation and growth laws is based on three basic assumptions: (1) All grains nucleated grow to the end of the process; i.e., coalescence and dissolution of the growing grains are ignored. Then, for a volume element in which the average volume of a grain is Vgrain: fs = Ngrain Vgrain
[41
(2) Once nucleated, the grains remain in fixed positions (role of fluid flow is ignored). (3) Grains are spherical in shape, which is valid for equiaxed grains. Under these assumptions, at a given time, t, and a location, x, in the casting, the local average fraction of solid, fs, is 4 f, (x, t) = N(x, t) 3 r
t)
[5]
where N(x, t) is grain density (number of grains per unit volume) and R(x, t) is grain radius. It can be seen from Figure 5 that for eutectic grains, R is simply the radius of the grain, while for dendritic grains, the radius of an equivalent sphere having the same mass as the dendrite, Re, must be considered. Then, assuming a constant grain density throughout solidification, from Eq. [5] one can obtain:
Ofs at
4r
OR at
[6]
where OR/at is the growth rate. To account for grain impingement, the effective area between the solid grains and the liquid is weighted by (1 - fs). This is similar to the Johnson-Mehl approximation. I261 Thus, after grain impingement, the fraction of solid is calculated as
Ofs = (1 - f ~ ) 4~rNR2 __OR Ot Ot
[6a]
In obtaining Eq. [6a], the grain impingement factor was assumed to be (1 - f~), which may not be true in some cases, as will be discussed later.
Fig. 4--Microstructure of a pearlitic-ferritic hypoeutectic SG iron. 1336--VOLUME 23A, APRIL 1992
b. Nucleation In the calculations, the instantaneous nucleation model t241 was used. This means that the grain densities of dendrites and eutectics are assumed to be constant METALLURGICAL TRANSACTIONS A
during transformation. The value of the grain density depends on the nucleation potential of the melt. c. The dendritic g r o w t h model
The dendritic growth model used in this research is based on the models proposed by Rappaz and Thevoz [9] and Kanetkar and Stefanescu. ~176As shown schematically in Figure 5, an equiaxed dendritic grain is assumed to be enveloped within a sphere, the growth rate of which is determined by the growth rate of the dendrite tip. The equivalent sphere is a sphere of the total mass of the dendrite, and the growth rate of the equivalent sphere can be related to supersaturation by [1~ OR e
"D m --
Ot
(C i -
Ca) 2
[7]
~-2F(k- 1)C0
where Re is the radius of the equivalent sphere, D,, is the diffusion coefficient of carbon in liquid iron, F is the Gibbs-Thompson coefficient, k is the equilibrium partition coefficient of the alloy, Co is the initial carbon concentration, C~ is the carbon concentration at the liquid/ solid interface, and Ca is the average carbon concentration in the interdendritic liquid. d. The eutectic growth m o d e l
When the temperature decreases below the stable eutectic temperature, eutectic grains are nucleated and grow in competition with the primary dendrites. The spherulitic austenite/graphite eutectic aggregate consists of an austenite shell surrounding a graphite nodule (Figure 6). It is generally accepted that the growth of this type of eutectics is controlled by the diffusion of carbon through the anstenite shell. [2,3,41 The growth rate of the austenite shell, under steady-state conditions, is given by TM dR~
RG
C G/~ - C L/~
dt = D~ (Re - R~)R~ C ~/L - C L/~
(Figure 6). The radius of the austenite shell is then calculated through time integration of Eq. [8]. Once R r is known, RG is calculated from solute balance: 4 Pc ~ r
) - (R~)3]Cc
4 =
p, 3
,n-LLt(,>,
)
--
[9]
where pi is the density of component i, and C~ is the carbon concentration in graphite (Co ~ 1). The new radius of the graphite nodule obtained from Eq. [9] will then be used in Eq. [8] to calculate the austenite growth rate at (t + At). 2. The solid-state transformation
The eutectoid transformation of SG iron is a competitive growth process. It consists of two reactions: the formation of ferrite, which is a stable phase, and the formation of pearlite, which is a metastable phase. Upon completion of solidification, the carbon solubility in austenite decreases with temperature down to the stable eutectoid temperature, At. Then, the rejected carbon migrates toward the "carbon sink," i . e . , the existing graphite spheroids, which results in the formation of carbondepleted zones in austenite. This provides favorable sites for the nucleation of ferrite, which then grows as a shell around the graphite spheroids. If the transformation is not completed before the metastable eutectoid temperature, A1, is reached, pearlite is nucleated and grows competitively with ferrite. a. The f e r r i t e g r o w t h m o d e l
Ferrite can result either from the breakdown of pearlite or from the direct decomposition of austenite, as shown below: /~G+a
[8]
where D~ is the diffusion coefficient of carbon in austenite, R~ is the radius of component i (graphite or austenite), and C ~/2 is the carbon concentration at the i / j interface. The carbon concentration at various interfaces is calculated from the equilibrium phase diagram
t+At 3
v
",a G + pE To produce a model for the growth of ferrite, the following assumptions were made: (1) ferrite formation results only from the decomposition of austenite; (2) ferrite grows as a shell surrounding the graphite spheroids and confined within the austenite grain, and the number of ferrite shells is equal to the number of graphite spheroids (Figure 7) (note that a ferrite shell may include several ferrite grains); (3) the austenite-to-ferrite transformation is controlled by carbon diffusion and occurs at steady state; (4) local equilibrium exists at the G / a and a / y interfaces, defined by the equilibrium solvus lines extended below the equilibrium eutectoid temperature (Figure 7); and (5) no carbon diffusion occurs from the ferrite/austenite interface toward austenite, because the diffusion coefficient and the concentration gradient in austenite are small compared with those in f e r r i t e . [27] Based on the above assumptions, the growth rate of ferrite can be calculated as
Fig. 6 - - Schematic diagram of the eutectic growth model with carbon concentration at various interfaces as a function of temperature. METALLURGICAL TRANSACTIONS A
dR~ dt
Ro C vm - C cl~ D c (R~ - RG)R~ C ~/~ - C ~/~
[10]
VOLUME 23A, APRIL 1992-- 1337
pearlite at each time interval at a series of constant temperatures can be simply added to give the continuous transformation rate. To avoid this difficulty, the Avrami equation was used to calculate the fraction of pearlite formed: fpr = 1 - exp [ - C ( T , d ) t "(r'd>]
[12]
where t is the transformation time, n(T, d) is a function of the grain size, d, and the temperature, T, and C(T, d) is the shape equation of the T T T curve (Figure 8), which can be described by a simple equation:
C(T, d) = exp [a(d)T 2 + b(d)T + c(d)]
[13]
where a(d), b(d), and c(d) are functions of the grain size. For a given material, these coefficients can be determined experimentally, as described in Section I I I - C .
c. The pearlite lamellar spacing Fig. 7 - - Schematic diagram of the eutectoid growth model with carbon concentration at various interfaces as a function of temperature.
where notations have meanings similar to those in Eq. [8]. Since the number of ferrite shells was assumed to be a constant, the rate of formation of ferrite can be calculated with an equation similar to Eq. [6]. The rejected carbon will be absorbed by the graphite spheroids, resulting in the concomitant growth of graphite and ferrite. The radius of the graphite spheroid is calculated from mass conservation, as follows:
4
V~Az~ = constant
[14]
where V~ is the local growth rate of the pearlite front and Ape is the local lamellar spacing. It was shown H2,3H that the growth velocity of pearlite can be described by
P~ ~ ~'[(~t~+a') ~ -- (R~)3]C~ rlnt+At x3 (R~)3]CV/~ = P'~3"n'tLn~ ) --
The structure of pearlite has been described as a type of in situ composite. The mechanical properties of SG iron are remarkably affected by the fineness of pearlite (i.e., the lamellar spacing). The mechanism of lamellar growth has been studied by Brandt, [3~ Zener, [311 and Hillert.[321 They assumed that this is a volume diffusioncontrolled reaction with all the diffusion events occurring in front of the interface, and that local equilibrium is achieved at the growth front. The relationship between the growth rate and the lamellar spacing is given by
Vr~ = / ~
[ 1 1]
(AT) 2
[15]
where p, is the density of ferrite.
where /Xp~ is a growth constant and AT is the interface undercooling. From Eq. [14] it follows that
b. The pearlite growth model
Wpe,tA 2pe,t = Vp~,t+a~pe,t+at 2
When pearlite grows from austenite, it can nucleate either at the austenite grain boundaries or at the austeniteferrite boudaries. Following nucleation, the pearlite colonies grow either as hemispheres or spheres. By the movement of high mobility (low interface energy) incoherent interfaces, colonies can grow edgewise or sidewise into the austenite, in competition with ferrite, until the austenite grains are totally consumed. [28] In order to calculate the evolution of the fraction of pearlite with temperature, appropriate nucleation and growth laws need to be known. However, measuring the nucleation rate of pearlite under isothermal conditions is a rather difficult matter, because the nucleation rate varies with time and pearlite grains impinge on one another in the early stage at a relatively high cooling rate. It has been suggested by .Cahn [29] that the nucleation rate of pearlite in steel, N, varies with time, according to N(t) = kt', where k and n are constants. It can be expected that k may be affected by the austenite grain size and pearlite nucleation site (comer, edge, or boundary). The nucleation rate of pearlite in the Fe-C system has been measured under isothermal conditions by Brown and Ridley. [13] However, the nucleation rate for the continuous cooling process cannot be calculated straightforwardly, as it is questionable whether the transformed 1338--VOLUME 23A, APRIL 1992
[ 16]
where A, and At+A t a r e the lamellar spacings at time t and t + At, respectively. This is assuming that diffusion is fast, compared with the growth velocity, such that the local lamellar spacing approaches the steady-state value
oo
log (t) Fig. 8 - - S c h e m a t i c TTT diagram for SG cast iron. METALLURGICAL TRANSACTIONS A
under the same interface undercooling. Eq. [15] into Eq. [16] leads to AT, hr~.,+A/ = A p e . , - AT,+a,
Substituting
[17]
where AT, and mzt+at are the interface undercoolings at time t and t + At, respectively. From Eq. [17], the evolution of the lamellar spacing can be calculated. It is expected that it will follow the cooling curve. Indeed, the pearlite colony with the largest lamellar spacing nucleates first at a small undercooling, and as the temperature decreases, colonies with finer spacings are developed. From Figure 9, it can be seen that in a given section of the casting, there are several pearlite colonies with different lamellar spacings. From this figure, by measuring the lamellar spacing on different colonies, an average lamellar spacing can be determined. Although this is a two-dimensional measurement, it will be describing a correct average spacing for the three-dimensional microvolume, since statistically the metallographic plane of incidence can have angles from 0 to 90 deg.
3. The grain impingement factor Grain impingement can occur at any time, depending on the distribution of nuclei (random or aggregated) and the manner of growth (shape of grains and growth velocity). Grain impingement can significantly delay the growth process and affect the temperature evolution during transformation. The effect of grain impingement has been treated by Johnson and Mehl t261 and A v r a m i . {33,34] Assuming random nucleation sites, they derived that the real volume fraction, f, can be calculated from the extended volume fraction, fex, which is the volume fraction yielded under the condition of free growth (i.e., with no impingement). The relationship between the real volume fraction and the extended volume fraction was given as [34]
df = (1 - f )
dfex
[18]
where (1 - f ) is the grain impingement factor, and the extended volume fraction is equal to
4 fex = - - arN/~3t4 3
119]
The Johnson-Mehl-Avrami equation has four restrictions under isothermal conditions, i.e., random nucleation site, constant nucleation rate, N, constant growth rate, R, and small incubation time, r. It is to be noted that many phase transformations nucleate at grain or phase boundaries, and as such, the impingement factor ( 1 - f ) may not be applicable. A modified expression for the impingement factor was proposed by Hillert: psi
df = (1 - f ) ' df, x
[20]
where the value of i is greater than unity if the transformed phases are aggregated and less than unity if they are random. In the present model, dendritic, eutectic, and ferritic grains are assumed to nucleate randomly and instantaneously, and hence, the extent of impingement is not severe or even negligible at the early stage of transformarion. Accordingly, the impingement factor should be included only at high volume fractions of transformed phases. The critical fraction of transformed phases where grains begin to impinge upon one another depends strongly on the shape of grains and the type of packing. Theoretically, this value is 0.74 for spherical grains with closepacking. However, in the as-cast structure, the ferrite grains may impinge upon one another at a relatively low fraction. Therefore, an alternative factor, (1 - f)Y, which implies that the extent of impingement becomes more pronounced as the fraction of transformed phases increases, was used. The influence of these different treatments of grain impingement on the evolution of the fraction of solid and on the cooling curve during the eutectic transformation is shown in Figure 10. It can be seen that case 1 results in a sudden deflection on the temperature curve. Cases 2 and 3 give almost identical results, except for the difference in the arrest temperature. In case 4, the fraction of solid only reaches 0.93 after 400 seconds, which is physically unrealistic.
1350
1.0
t250 / *""
:
~
/
0.8 ~.-o.-~>
.- _
E9
o.a ~.
o
'0 . 4 1050 0.2
95O
50 100 150 200 250 300 350 40~"0 T~
Fig. 9 - - T y p i c a l microstructure of the experimental SG iron casting showing the pearlite lamellar structure. METALLURGICAL TRANSACTIONS A
(see)
Fig. 1 0 - - T h e temperature evolution during the eutectic transformation with different impingement factors. VOLUME 23A, APRIL 1992-- 1339
4. Additivity of transformed phases Generally, the amount of phase change during phase transformations can be calculated by the Johnson-Mehl or the Avrami equation. They are, however, only valid for isothermal transformations. Under continuous cooling conditions, it is essential to examine whether the phase transformed isothermally at different temperatures under different kinetics can be simply added to yield the amount of continuously transformed phases. The pioneering work on additivity was done by Scheil, r36] who assumed that when the sum of the time taken at different temperatures, At, divided by the incubation time, ri, is equal to unity, that is,
f0 t --=
1
[211
ri
the transformation begins. The additivity rule was originally derived to predict the start of transformation. It was then extended and defined in detail by Avrami r34j (isokinetic range), Cahn t37~ (site saturation), and Kuban et al. t151 (effective site saturation). A description of additivity criteria is given in Table I. According to these additivity criteria, the additivity of transformed phases can be determined. Additivity applies when the transformation rate is a function of both temperature and amount of transformed phase. For the case of pearlite, V is constant during the isothermal transformation, since it only depends on interface undercooling. The nucleation issue is more complicated. As previously mentioned, N is a function of time. Nevertheless, for small cooling rates, as in the present experiments, and therefore for small undercoolings, it can be assumed that site saturation occurs in the early stages of transformation. Then, either the "site saturation" or the "effective site saturation" criterion applies. Indeed, as shown by Kuban et al., ~15~ for the case of austenite to pearlite transformation in steel, the effective site saturation criterion is valid. Thus, Eq. [12] can be used to describe transformation during continuous cooling of SG iron.
ment method was used in the modeling because of its ability to handle complicated shapes and boundary conditions. The variational approach t38j was used to transform the governing heat-transfer equation into the finite element form and then to solve the temperature field within the system explicitly. Due to axisymmetry and longitudinal symmetry of the casting configuration, twodimensional discretization for 1/4 of the longitudinal section of the system was performed, as shown in Figure 1 1. Dynamic allocation and banded matrices were used to save computational time and memory storage. While using a numerical scheme to solve partial differential equations, typically, two types of errors, i.e., discretization and running errors, may arise. Therefore, the stability, convergency, and accuracy of the numerical solution need to be examined. Basically, the accuracy of a numerical solution can be checked by comparing it either with the analytical solution or with experimental results, as was done in this work. The stability of the solution can be ensured by using small enough time steps, dt, and mesh sizes, dr, as long as the mathematical model and the boundary conditions are correct. To check the convergency of the numerical solution, results obtained using the mesh shown in Figure 1 1 were compared with those obtained using a finer mesh, and no significant difference was detected. III.
WORK
The experimental work performed includes: (1) production of SG iron bar castings for cooling curve acquisition at various locations in the casting; (2) metallographical examination to obtain microstructural data, i.e., the number of graphite spheroids
M E S H GENERAl]ON PLOT
C. The Numerical Scheme
N4~tX~i>
