\input texp \parindent=0pt \parskip=3mm


\title{The Bainite Transformation: Unresolved Issues}
\singlespace \bigskip
\centerline{ H. K. D. H. Bhadeshia }
\medskip
\centerline{ University of Cambridge}
\centerline{ Department of Materials Science and Metallurgy}
\centerline{ Pembroke Street, Cambridge CB2 3QZ, U.K.}
\centerline{hkdb@cus.cam.ac.uk~~~~www.msm.cam.ac.uk/phase--trans}
 
\bigskip 
\singlespace

\sec{ABSTRACT}

{\parindent=20 pt  \narrower \medskip
\x A great deal of research, over many decades since the discovery of
bainite, has revealed substantial information about the mechanism of the
bainite transformation in steels. Elements of the theory
are now routinely being used in many parts of the world in the design of
novel alloys and in the interpretation of a variety of experimental data.
However, there remain some difficulties which are holding up
technological progress. Two of these, the precipitation of carbides and
the formation of bainite from plastically deformed austenite, are
reviewed in the hope that further progress can be stimulated,
particularly with respect to the development of quantitative models. 
\medskip}

\sec{INTRODUCTION}

Science is essentially empirical; for example, there is no fundamental
justification for the conservation of energy other than the fact that
the principle has thus far stood the test of time.  A further
difficulty is that there are always loose ends in all known
theories. Consequently, the assessment of unresolved issues requires a
definition of what is meant by \lqq resolved", particularly when
considering the bainite transformation which has been the subject of
much discussion. A good scheme would be to follow Hawkins [1], who makes
the reasonable proposition that a  theory must satisfy at least two
criteria. Firstly, it must describe a large class of observations with
few arbitrary parameters. And secondly, it must make predictions which can
be verified or disproved. 


A simple theory, the  elements of which were first proposed by
Hehemann and coworkers [2], elegantly fits these criteria. It  has
subsequently been developed   and has survived the panoply of
experimental techniques available today. The details of the theory have
been reviewed extensively [3--5] so it is not the intention here to
repeat the story. Nevertheless, a brief introduction is presented in
order to set the scene for the discussion of unresolved topics where
research could be focussed.

\sec{OVERVIEW OF TRANSFORMATION MECHANISM}

Bainite nucleates like martensite but with the partitioning of
interstitial carbon [6]. The surface to volume ratio decreases beyond the
nucleation stage, making it possible for growth to occur without any
diffusion. The growth process is illustrated in
\fagg, [7]. A plate or lath of bainite grows by a martensitic mechanism
but because the transformation occurs at a relatively high temperature,
the excess carbon soon afterwards partitions into the residual
austenite. The carbon then precipitates from the austenite giving the
inter--plate carbides typical of an upper bainitic microstructure. This
post--transformation diffusion process slows down as the transformation
temperature is reduced, thereby providing an opportunity for some of the
carbon to precipitate from supersaturated bainitic ferrite. The
remainder partitions into the residual austenite, giving the classical
lower bainitic microstructure (\figg).

\picture{upperlower}{175}{161}{500}{\tabtit{\figg : }{An illustration of
the growth of bainite and the development of upper or lower bainite.}}

This theory explains a large class of observations, {\it some} of  which
are as follows:

{\parindent=9pt \narrower \medskip

\zz{1. } There is an invariant--plane strain shape deformation
associated with the formation of bainite [8--11]. This deformation has a
large shear component [10,11] and given the small diffusion coefficient
of iron, implies a coordinated movement of atoms. Such disciplined
movements cannot be sustained across grain boundaries of the parent
phase, so plates of bainite are confined to the grains in which they
grow. This is quite unlike reconstructive transformation products such
as allotriomorphic or massive ferrite and pearlite, which are not
hindered by austenite grain boundaries. 

\zz{2. } The shape deformation induces plastic accommodation in the
adjacent austenite, causing a loss of coherency at the transformation
interface and bringing the growth process to a premature halt, \ie
before the plate encounters  a hard obstacle such as a
grain boundary [12]. Subsequent growth then has to occur by the
nucleation of new plates, giving rise to the observed sheaf
microstructure [2,12]. Notice that there is no mechanism by which  a
phase that grows by a reconstructive mechanism can be halted in this way
before equilibrium is reached.

\zz{3. } Substitutional elements do not partition even on the finest
conceivable scale, as proven by direct atomic resolution images and
atomic resolution chemical analysis experiments [4]. These observations
are only consistent with a displacive transformation mechanism.

\zz{4. } In steels where carbide precipitation is prevented, the
transformation stops at a point where diffusionless growth becomes
thermodynamically impossible. It remains incomplete in the sense that
the chemical potentials are not uniform at the point where
the bainite reaction stops [4]. 

\zz{5. } The product grows in the form of thin plates in order to
minimise the strain energy due to the shape deformation.
Once the reaction has stopped, continued annealing of a mixture of
bainitic ferrite plates and residual austenite allows the formation of
ferrite at grossly retarded rates by a reconstructive mechanism in which
there is no  invariant--plane strain (IPS) shape deformation with a large
shear component [13].

\medskip}

The theory has also made a number of exciting quantitative predictions
which have subsequently been verified experimentally. For example, it
predicted that unlike the textbook story, both upper and lower bainite
need not form in all steels [7].  It is possible in high--carbon steels
to avoid completely the formation of upper bainite so that the pearlite
transformation gives way directly to lower bainite as the transformation
temperature is reduced. Similarly, in low carbon steels it is possible
for the upper bainite reaction to be replaced directly by martensite at
lower temperatures, without any intervening lower bainite!

Welds frequently contain intragranularly nucleated upper bainite (the
so--called acicular ferrite microstructure). The theory explains why
intragranularly nucleated lower bainite is not found in such welds. The
low carbon concentration of the vast majority of welding alloys prevents
the formation of lower bainite. An experiment in which the carbon
concentration was deliberately increased led to the first ever observation
of \lqq lower acicular ferrite" [14].

The nucleation part of the theory provides the only explanation for why
\wid\ ferrite does not form in all steels, and why some steels only
transform to martensite [6]. 

The response of the bainite transformation to  applied elastic stresses
can be modelled using standard deformation theory, reinforcing the
concept of a displacive transformation mechanism [4,15,16].

It is worth emphasising that the theory, in its quantitative form, has
been used in the design of novel steels which are produced
commercially. One example is the carbide--free bainitic rail steel with
outstanding wear resistance, rolling--contact fatigue strength and
low--temperature toughness [17,18]. There are many other developments in
progress, which take into account the fact that the reaction stops when
diffusionless transformation becomes thermodynamically impossible,
three of these can be found in the present proceedings (papers by
Kutsov, Jacques and Girault).

\sec{UNRESOLVED ISSUES}

There are a number of key issues which are not yet amenable to a full
quantitative treatment; of these, the following are probably the most
urgent:

{\parindent=10pt\narrower\medskip

\zz{$\bullet$ } The measured growth rate of individual plates of
bainite is far greater than expected from the
theory of diffusion--controlled growth [4,19]. It is clear that bainitic
ferrite forms with a large supersaturation of carbon, but the
possibility of some interstitial diffusion during growth cannot entirely
be ruled out. There are a number of theories about this [20,21,22] but
none of them are completely convincing for reasons discussed elsewhere
[4, page 167].

\zz{$\bullet$ } There exists kinetic theory capable of estimating the
fraction of bainite obtained during the isothermal transformation of
austenite in steels where the reaction is not accompanied by the
formation of carbides [23]. The model incorporates the mechanism of
transformation but the nature of the fitting constants needs to be
understood. This entails a better treatment of autocatalytic
nucleation.

\zz{$\bullet$ } The quantitative modelling of the precipitation of
carbides during the bainite transformation is a remarkably neglected
area of research. This is all the more surprising  given that bainite,
in the vast majority of steels, contains carbides. Indeed, there is no
quantitative theory for the tempering of martensite!

\zz{$\bullet$ } There are many features of the transformation of
plastically deformed austenite to bainite which are not understood.
This is important given the fact that there is large scale production
of bainitic steels by thermomechanical processing.

\medskip}
It is the intention in this paper to focus on the last two of the topics
listed above, since these   are probably the most important from the
point of view of contemporary technological developments.

\sec{MECHANISM OF CARBIDE FORMATION}

There is a great deal of evidence that
the carbides associated with these transformations form by displacive
mechanism without the partitioning of substitutional solute. The
mechanism must obviously involve the  diffusion of carbon, but not of
substitutional solutes (X) or  iron atoms.  Considerable experimental
data  show that the carbide precipitation associated with bainite and 
martensite does not lead to a partitioning of substitutional solutes 
[24--26]. The precipitation can occur under conditions where the
diffusion rates  of iron and substitutional atoms are incredibly small 
compared with the rate of precipitation (\fagg). 


\picture{diffdist}{190}{127}{500}{\tabtit{\figg : } {A $2\sqrt{Dt}$
estimate of the distance diffused in one hour, for iron and some
substitutional solutes in iron, as a function of temperature [4]. $D$
is the diffusion coefficient and $t$ the time.}}


 It has in fact been believed for some time
that  the cementite lattice may be generated by the deformation of 
the ferrite crystal structure, at a rate controlled by the diffusion of 
carbon into the appropriate sites. The Fe/X ratio thus 
remains constant everywhere and subject to that constraint, 
the carbon achieves equality of chemical potential; the 
cementite is then said to grow by {\it paraequilibrium} 
transformation [27]. The way in which the ferrite lattice 
could be deformed to produce the right arrangement of iron 
atoms needed to generate the cementite has been considered by 
Andrews [28] and Hume--Rothery \et [29], and the subject has 
been reviewed by Yakel [30]. High--resolution transmission electron
microscopy supports the idea that the carbide particles grow  by displacive
transformation (Sandvik [31],  Nakamura and Nagakura [32] and Taylor \et
[33,34]).


In a remarkable experiment, Babu \et [35] have shown using the
atom--probe technique that the cementite obtained by tempering
martensite is forced to inherit the silicon concentration of the
martensite. This is  in spite of the fact that the equilibrium
solubility of silicon in cementite is negligible. 

The number of carbide variants which grow in any given plate 
of martensite decreases when the virgin martensite is 
tempered under the influence of the stress [36,37]. This response of the
microstructure to an externally applied stress is consistent with a displacive
mechanism of transformation.

To summarise, it appears that substitutional solute atoms are trapped in
the cementite when the latter precipitates during the bainitic or
martensitic transformations in steels. That is, the cementite forms by a
paraequilibrium transformation mechanism. In silicon containing steels
the free energy change associated with the paraequilibrium precipitation
of cementite must be much smaller than when the cementite is free of
silicon. It is probable  that this is what leads to   suppression of
cementite in high--silicon bainitic or martensitic steels [38]. However,
a full theory is far from established and the work is urgent given the
technological significance of carbide--free bainitic steels.

A major omission from the array of experimental evidence is that no one
has as yet measured the shape deformation due to carbide precipitation.
This is because the particles are so incredibly fine, but atomic force
microscopy may yet prove possible as an aid to characterising the
surface relief.

\sec{KINETICS OF CARBIDE FORMATION}

Many of the carbides that form during the bainite transformation are
metastable [4]; annealing therefore leads to a series of precipitation
reactions which occur together, albeit at different rates. These
reactions have been difficult to model given the interactions between
the variety of precipitate phases. It is essential to
resolve this, both from the point of view of structural steels where
silicon is exploited as an alloying element, and for steels used at
elevated temperatures where the precipitates control the creep
behaviour. A theory is needed to treat the kinetics of simultaneous
reactions, and such a theory would naturally be useful in many
applications. For example, the tempering of martensite or the ageing of
aluminium alloys.

 A model for a single transformation   begins with the
calculation of the nucleation and growth rates using classical  theory,
but an estimation of the volume fraction requires impingement between
particles to be taken into account. This is generally done using the
extended volume concept of Johnson, Mehl, Avrami, and Kolmogorov [39] as
illustrated in \fagg. Suppose that two  particles exist at time $t$; a
small interval
$\delta t$ later, new regions marked $a$, $b$, $c$ \&\ $d$ are formed
assuming that they are able to grow unrestricted in extended space
whether or not the region into which they grow is already transformed.
However, only those components of
$a$, $b$, $c$ \&\ $d$ which lie in previously untransformed matrix can
contribute to a change in the real volume of the product phase (identified by
the subscript \lq 1\rq) so that :
$$ dV_1 = (1 - {{V_1}\over {V}})dV_1^e   \numeqn $$ where  it is assumed
that the microstructure develops randomly. The superscript
$e$ refers to extended volume, $V_1$ is the volume of phase $1$ and
$V$ is the total volume. Multiplying the change in extended volume
by the probability of finding untransformed regions has the effect of
excluding regions such as $b$, which clearly cannot contribute to the
real change in volume of the product. This equation can easily be
integrated to obtain the real volume fraction, 
$${{V_1}\over{V}} = 1-\exp\bl\{ -{{V^e_1}\over{V}}  \br\} \numeqn $$
Nucleation and growth rates can readily be substituted into
$V^e_1$, leading to the
familiar  Avrami equation.

\picture{extended}{164}{95}{500}{\tabtit{\figg : }  {The concept of
extended volume. Two precipitate particles have nucleated
and grown to a finite size in the time $t$. New regions $c$
and $d$ are formed as the original particles grow, but  
$a$ \&\ $b$ are new particles, of which $b$ has formed in a region
which is already transformed.}}


In practice, as stated earlier, there are many cases where several
transformations occur together. The different reactions interfere with
each other in a way which is seminal to the development of bainitic
steel microstructures. The principles involved are first illustrated
with an example in which $\beta$ and $\theta$ precipitate at the same
time from the parent phase which is designated $\alpha$ [40,41]. For the
sake of discussion it is assumed that the nucleation and growth rates do
not change with time and that the particles grow isotropically.

The increase in the extended volume due to particles nucleated in a time
interval $t=\tau$\ to $t=\tau + d\tau$ is, therefore, given by
$$dV_\beta^e = {4\over 3}\pi G_\beta^3(t-\tau)^3 I_\beta(V)~d\tau
\numeqn $$
$$ dV_\theta^e ={4\over 3}\pi G_\theta^3(t-\tau)^3I_\theta (V)~d\tau
\numeqn
$$ where $G_\beta {\rm ,\ }G_\theta {\rm ,\ }I_\beta {\rm
\ and\ }I_\theta
$ are the growth and nucleation rates of $\beta$ and
$\theta$ respectively, all of which are assumed here to be independent of time.
$V$ is the total volume of the system. For each phase, the increase in extended
volume will consist of three separate parts. Thus, for $\beta$:

{\parindent=15pt \narrower \medskip
\zz{(i) } $\beta$ which has formed in untransformed
$\alpha$.

\zz{(ii) } $\beta$ which has formed in regions which are already
$\beta$.

\zz{(iii) } $\beta$ which has formed in regions which are already
$\theta$.

\medskip}

\x Only $\beta$ formed in untransformed $\alpha$ will contribute to the real
volume of $\beta$. On average a fraction $\left( 1 - {V_\beta + V_\theta
\over V} \right) $ of the extended volume will be in previously untransformed
material. It follows that the increase in real volume of $\beta$ is given by
$$dV_\beta =\left( 1 - {V_\beta + V_\theta\over V}
\right) dV_\beta ^e \numeqn $$  and similarly for $\theta$,
$$dV_\theta = \left( 1 - {V_\beta + V_\theta \over V}
\right) dV_\theta ^e
\numeqn $$

\x Generally $V_\beta$ will be some complicated function of $V_\theta$ and it is
not possible to integrate these expressions analytically to  find the
relationship between the real and extended volumes.  Numerical
integration is straightforward and offers the opportunity to change
the boundary conditions for nucleation and growth as
transformation proceeds, to account for the change in the matrix
composition during the course of reaction. The method can in
principle be applied to any number of simultaneous reactions.


{\bf Complex reactions: } The multiple reactions found in bainitic
and martensitic power plant steels have important complications which can
all be dealt with in the scheme of simultaneous transformations as
presented above. The phases interfere with each other not only by
reducing the volume available for transformation, but also by removing
solute from the matrix and
 thereby changing its composition. This change in matrix
composition affects the growth and nucleation rates of the phases.
 The main features of the application of the theory to power
plant steels are summarised below; a full description is given
in references [40,41].


{\parindent=15pt \narrower \medskip
\zz{$\bullet$ }{The model allows for the simultaneous precipitation of
\mtwox, \mtwentythreecsix, \msevencthree, \msixc\ and Laves
phase.
\mthreec\ is assumed to nucleate instantaneously with the
paraequilibrium composition. Subsequent enrichment of
\mthreec\ as it approaches its equilibrium composition is accounted
for.}

\zz{$\bullet$ }{All the phases, except \mthreec, form close to their
equilibrium composition. The driving forces and compositions of the
precipitating phases are calculated using standard thermodynamic
methods.}

\zz{$\bullet$ }{The interaction between the precipitating phases is accounted
for by considering the change in the average solute level in the matrix as each
phase forms.}

\zz{$\bullet$ }{The model does not require prior knowledge of the precipitation
sequence.}

\zz{$\bullet$ }{Dissolution of non--equilibrium phases is incorporated
as a natural event.}

\zz{$\bullet$ }{A single set of fitting parameters for the nucleation equations
(site densities and surface energies) has been found which is applicable to a
wide range of power plant steels.}

\medskip}

The compositions of three power plant alloys used here for
illustration purposes, are shown in
\tablaa. These three  alloys, whilst of quite different chemical
compositions, show similar precipitation {\it sequences} 
but with vastly different rates.  For example, at 600$\,$\degg\ the
time taken before \mtwentythreecsix\ is observed is  1$\,$h in the
10CrMoV steel, 10$\,$h in the 3Cr1.5Mo alloy  and in  excess of
1000$\,$h in the
2.25Cr1Mo steel [40]. These differences have never before been explained
[40,42,43].

\midinsert \thicksize=1pt \thinsize=0.8pt \tablewidth=3.2 truein {
\smalltype \begintable
 {} | C & N & Mn & Cr & Mo & Ni & V & Nb  \cr 

{2.25Cr1Mo} | 0.15 & -- & 0.50 & 2.12 & 0.9 & 0.17 & -- & --    \nr
{3Cr1.5Mo} | 0.1 & -- & 1.0 & 3.0 & 1.5 & 0.1 & 0.1 & --    \nr
{10CrMoV} | 0.11 & 0.056 & 0.50 & 10.22 & 1.42 & 0.55 & 0.20 & 0.50 
\endtable }
\tabtit {\tablee : }{ Concentration (in weight\%) of the major alloying elements in the
steels used to demonstrate the model.}
\endinsert

\sec{Microstructure Calculations}

A plot showing the predicted variation of volume fraction of each precipitate as  a
function of time at 600$\,$\degg\ is shown in \fagg . Consistent
with experiments, the precipitation kinetics of \mtwentythreecsix\
are  predicted to be much slower in the 2.25Cr1Mo \ steel compared to the
10CrMoV and  3Cr1.5Mo alloys. One contributing factor is that in the
2.25Cr1Mo \ steel a relatively  large volume fraction of \mtwox\ and
\msevencthree\ form prior to \mtwentythreecsix. These deplete
the matrix and therefore suppress 
\mtwentythreecsix\ precipitation. The volume fraction of \mtwox\
which forms in the  10CrMoV steel is relatively small, and there
remains a considerable excess of solute  in the matrix, allowing
\mtwentythreecsix\  to precipitate rapidly. Similarly, in the 3Cr1.5Mo steel
the volume fractions of \mtwox\   and \msevencthree\   are insufficient to
suppress
\mtwentythreecsix\  precipitation to the same extent as in the 2.25Cr1Mo \ steel.

\mtwentythreecsix\  is frequently observed  in the form of coarse
particles which are less effective in hindering creep deformation.
Delaying its precipitation would have the effect of stabilising the
finer dispersions of \mtwox\   and MX to longer times with a
possible enhancement of creep strength. 

\picture{all}{142}{365}{500}{\tabtit{\figg : }{The predicted evolution
of precipitate volume fractions at 600$\,^\circ$C for three power
plant materials (a) 2.25Cr1Mo   (b) 3Cr1.5Mo and (c) 10CrMoV.}}

Calculations like these can be used to design microstructures
exploiting knowledge built up over decades concerning what is good
and bad for creep strength. It is often argued that Laves phase
formation is bad for creep resistance -- it leads to a reduction in the
concentration of solid solution strengthening elements; since the
Laves precipitates are few and coarse, they do not themselves
contribute significantly to strength. The model presented here can
be used to design against Laves phase formation. 

We note for the moment, that this is as far as microstructure
modelling has progressed. The models are not yet capable of giving
size distributions and even if that were to be possible, there are no
physical models of creep deformation which have sufficient
precision to make use of this information. A great deal of work also
needs to be done on realistic methods of taking into account the
details of the nucleation process and in tackling soft impingement
phenomena in an internally consistent manner.



\sec{PLASTIC DEFORMATION \&\ MECHANICAL STABILISATION}

\x We have emphasised that displacive transformations  involve the
coordinated movement of atoms, and that such movements cannot be
sustained against strong defects such as grain boundaries. Thus,
martensite plates, which form by a displacive mechanism, cannot cross
grain boundaries. Smaller defects such as isolated dislocations hinder
the progress of such transformations, but  can
often be incorporated into the martensite lattice. However, it is well
established [44--49] that the heavy deformation of austenite prior
to its transformation hinders the growth of martensite, causing a
reduction in the fraction of transformation in spite of an increased
rate of  heterogeneous nucleation rate due to the greater number density
of defects. Another way of visualising this mechanical stabilisation is
in terms of the structure of the transformation interface. Displacive
transformations are accomplished by the advance of glissile interfaces
which can be rendered sessile by the accumulation of dislocation debris.
Thus, whereas an appropriate stress can stimulate displacive
transformation in the same way that it enables normal deformation
[50,51], mechanical stabilisation actually retards the decomposition of
the austenite.

Most of the work on mechanical stabilisation  effects has been on steels,
and has focused on the martensite transformation with very few
studies on bainite. Some early experimental data based on hot--rolling
experiments [52] can be interpreted to show that
bainitic transformation is retarded in deformed austenite, as
illustrated in \fagg\ [4]. More direct evidence comes from the work of
Tsuzaki \et\ [53] who found that although deformed austenite transformed
faster, the net volume fraction of bainite that formed decreased when
compared with undeformed austenite. 

\picture{davenport} {110}{122}{500}{\tabtit{\figg : }{Plots of
temperature versus time, for samples undergoing bainitic transformation
during cooling. The deviations from linearity indicate the onset of
transformation. The reaction is retarded in the more severely deformed
austenite, a manifestation of mechanical stabilisation.}}

 Huang \et\ [54] showed that deformed austenite transforms to bainite at
a higher temperature during continuous cooling and at a faster rate, as
long as the cooling occurred sufficiently fast to suppress recovery of
the austenite. Similar effects were reported by Smith and Siebert
[55,56].

Maki [57] found that ausforming markedly increases the rate of
transformation of upper bainite, with a smaller effect on the lower
bainite reaction (the progress of which was eventually retarded). The
plates of bainitic ferrite were thinner and contained a larger
dislocation density when they grew from deformed austenite. It was
concluded that the bainite inherited the defect structure of the
austenite. Edwards and Kennon [58] had earlier reported these same
observations and in addition, demonstrated nucleation of bainite from
slip bands. Freiwillig \et\ [59] also reported an initial acceleration
of transformation to bainite from deformed austenite.

Other studies are more difficult to  interpret. In Bhattacharyya and
Kehl�s experiments [60] the bainite was formed from deformed austenite,
but under the influence of an applied stress; the latter can complicate
interpretation [16]. In microalloyed steels, strain induced precipitation
of carbides, carbonitrides or borides can influence the subsequent
transformation of austenite [61]. 

In recent work [62] it has been demonstrated using metallography, that
the bainite transformation can be mechanically stabilised in a manner
identical to the mechanical stabilisation of martensite in steels
(\fagg). The mechanism appears to be that the growth of bainite is
retarded by the deformation debris in the austenite. Heterogeneous
nucleation becomes more frequent as defects are introduced into the
austenite, but their growth by a displacive mechanism is stifled as the
interface encounters forests of dislocations. Deformed austenite
therefore transforms to a smaller quantity of bainite than undeformed
austenite, and any bainite that forms is more refined. 

\picture{}{100}{150}{500}{\tabtit{\figg : }{Optical micrographs
showing the large effect of mechanical stabilisation in refining the
microstructure and in reducing the amount of bainitic transformation.
(a) Transformation from undeformed austenite. (b) Transformation from
plastically deformed austenite.}}

The mechanical stabilisation is also evident in quantitative
experiments, for example, when the transformation is monitored using
dilatometric techniques [63], \fagg. There are two puzzling features
illustrated, first that transformation from deformed austenite leads to
a smaller terminal fraction of bainite. The second feature is that
although the transformation rate is at first is accelerated by
deformation, it is eventually retarded. If this initial acceleration is
explained by increasing the number density of nuclei then it is not
possible to reach a smaller terminal fraction given that each nucleus
transforms a fixed volume of austenite. On the other hand, if it is
assumed that the smaller limiting fraction  is due to the reduction in
volume transformed per nucleus then it is not possible to explain the
initial acceleration. There are other complications described
elsewhere, all of which can only be resolved by arguing that the
austenite is inhomogeneously deformed [64]. The lightly deformed regions
transform more rapidly relative to undeformed austenite because of the
increase in the defect density. The nucleation rate is larger in the
heavily deformed regions but the overall rate of transformation is
reduced because each nucleus then transforms to a smaller volume due to
mechanical stabilisation of the interface.

This qualitative theory needs to be developed in much more detail, and
has to be backed by direct metallographic observations of the
distribution of bainite sub--unit sizes. Such a distribution should be
bimodal. The importance of such work cannot be underemphasised given
the increasing use of thermomechanical processing of bainitic steels.

\picture{stab}{153}{122}{500}{\tabtit{\figg : }{Change in radial
dilatation during isothermal transformation to bainite as a function
of time and prestrain (the strain in the austenite prior to
transformation); values of the prestrain are indicated next to individual
curves.}}

Mechanical stabilisation has been found to occur for all of the
plate--shaped ferritic phases that occur in steels. This includes \wid\
[65], bainite and martensite, all of which cause an invariant--plane
strain shape deformation with a large shear component on transformation.
Indeed, mechanical stabilisation has been shown to occur in materials
as diverse as lithium and brass (see other papers in this conference).
On the other hand, reconstructive transformations are without
exception accelerated when they occur from the deformed parent phase.
This is because there is not only an increase in the number density of
nucleation sites in deformed austenite, but the defects are destroyed
by the growth of the new phase, akin to what happens in
recrystallisation. This is unlike displacive transformations where the
defects are inherited by the growing phase. The defects therefore add to
the driving force for reconstructive transformation. With these
general observations it is possible to define a disarmingly simple
criterion to distinguish these two mechanisms of transformation:

\setbox4=\vbox{\hsize 23pc \noindent \strut There is no mechanism by
which plastic deformation can retard reconstructive transformation.
Likewise, only displacive transformations can be mechanically
stabilised. \strut}
$$ \boxit{\boxit{\box4}}$$

Finally, there are other effects to do with stress and strain
affected transformations for which there are only qualitative
explanations. One of the most important criterion in steel design is the
grain size, in the case of bainite the plate thickness. A recent
empirical analysis [66] based on the neural network technique has
identified, in descending order of importance, the austenite strength,
the driving force for nucleation and the temperature as the   variables 
controlling the thickness of bainite plates (\fagg). 

\picture{sigmaw}  {140.0} {100.0} {500}
{\tabtit{\figg : }{The model perceived significance
of the input variables in influencing the bainite
plate thickness. A large value of the parameter $\sigma_w$ implies
that the variable concerned is most effective in explaining the
variation in the bainite plate thickness.}}


\sec{CONCLUSIONS}

\x We have seen that the theory for the bainite transformation is at a
sufficiently advanced stage to allow its use in the design of novel
steels. It is the personal opinion of the author that the next major
advances will be in: (a) the understanding of the carbide precipitation
reactions that are associated with bainite and (b) the interaction of
defects induced by plastic deformation on the kinetics and
microstructure of bainite. It was in 1989 that I was asked to present a
few unresolved problems in the theory of phase transformations in the
context of high--resolution microanalytical techniques [67]. I believe
that three of the four problems presented then have now essentially been
solved. I therefore look forward to 2009 to assess the value of the work
presented here!

\sec{ACKNOWLEDGMENTS}

\x I would like to express my heartfelt thanks  to Drs
Marcos Sade, Manfred Ahlers and the team from the Centro Atomico  
Bariloche, for their kindness and care during my stay in
Argentina. This alone went a long way towards making ICOMAT '98 a
memorable event. As a bonus, the technical content of the
meeting was as breathtaking as the local scenery.

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\medskip}
\vfill\eject
\sec{FIGURE CAPTIONS}

\parskip=6mm \doublespace

Figure 1: An illustration of
the growth of bainite and the development of upper or lower bainite.

Figure 2: A $2\sqrt{Dt}$
estimate of the distance diffused in one hour, for iron and some
substitutional solutes in iron, as a function of temperature [4]. $D$
is the diffusion coefficient and $t$ the time.

Figure 3: The concept of
extended volume. Two precipitate particles have nucleated
and grown to a finite size in the time $t$. New regions $c$
and $d$ are formed as the original particles grow, but  
$a$ \&\ $b$ are new particles, of which $b$ has formed in a region
which is already transformed.

Figure 4: The predicted evolution
of precipitate volume fractions at 600$\,^\circ$C for three power
plant materials (a) 2.25Cr1Mo   (b) 3Cr1.5Mo and (c) 10CrMoV.

Figure 5: Plots of
temperature versus time, for samples undergoing bainitic transformation
during cooling. The deviations from linearity indicate the onset of
transformation. The reaction is retarded in the more severely deformed
austenite, a manifestation of mechanical stabilisation.

Figure 6: Optical micrographs
showing the large effect of mechanical stabilisation in refining the
microstructure and in reducing the amount of bainitic transformation.
(a) Transformation from undeformed austenite. (b) Transformation from
plastically deformed austenite.

Figure 7: Change in radial
dilatation during isothermal transformation to bainite as a function
of time and prestrain (the strain in the austenite prior to
transformation); values of the prestrain are indicated next to individual
curves.

Figure 8: The model perceived significance
of the input variables in influencing the bainite
plate thickness. A large value of the parameter $\sigma_w$ implies
that the variable concerned is most effective in explaining the
variation in the bainite plate thickness.


\bye