\input texp 
\def\hhh {$\,{\rm K\,min^{-1}}$}

{\large
\title{Power Plant Steels: Remanent Life Assessment }
\title{and the Approach to Equilibrium} \bigskip
\bigskip \singlespace
\centerline{H. K. D. H. Bhadeshia}
\centerline{University of Cambridge}
\centerline{Department of Materials Science and Metallurgy}
\centerline{Pembroke Street, Cambridge CB2 3QZ, U.
K.~~hkdb@cus.cam.ac.uk}} \singlespace

\sec {ABSTRACT}

{\parindent=20pt \narrower \medskip
\x The steels used in the power generation industry are almost always
given a severe tempering heat--treatment before they enter service.
This would, in most contexts, give them a highly stable microstructure
which is close to equilibrium. In fact, they undergo many changes
over long periods of time. This paper is a review of some of the
methods which exploit the changes in order to estimate 
the life that remains in alloys which are only partly exhausted.
\medskip}


\sec{INTRODUCTION}

\x Many of the safety--critical components in power plant are made of
steels developed to resist deformation when used  in the   range
480--565~\deg and 15--90~MPa. They are expected to serve reliably for a
period of about 30 years, giving a  maximum tolerable creep strain rate
of about
$3\times 10^{-11}{\rm \,s^{-1}}$ (approximately 2\%\ elongation over the
30 years). The   design stress must be set to be small enough to prevent
creep rupture over the intended life of the plant.


The steels are able to survive for such long periods because the
operating temperature  is only about half of the absolute melting
temperature, making the migration of atoms very slow indeed. Creep
therefore depends on the ability of dislocations to overcome obstacles
with the help of thermal energy. The obstacles are mainly carbide
particles which are dispersed throughout the microstructure.

Suppose that the microstructure and the operating conditions do not
change during service. The accuracy with which component life might then
be predicted would depend only on the quality of the experimental data.
The so--called \lqq safety factors" common in design could then be
greatly reduced with obvious benefits. Of course, this never happens in
practice; the steels are always heterogeneous and the service
conditions vary over a range of scales and locations. The design life is
therefore set conservatively to account for the fact that measured
creep data follow a Gaussian distribution with a significant width.
In spite of this, experience has shown that  decommissioned plant could
have been kept in service without sacrificing safety. To take
advantage of this observation requires methods for the reliable
estimation of  the
{\it remaining life}. The techniques used for this purpose are summarised
in \tablaa. Of the properties listed, no single measurement is
comprehensive enough to  describe the steel with all requisite
completeness. The implementation of a life--extension procedure must
consequently  be based on broad considerations backed by more frequent
inspections.

\midinsert \thicksize=1pt \thinsize=0.8pt
\tablewidth=5.7truein \begintable
Property   &  Example references \cr
Damage summation \hfill & \hfill Evans, 1984 [1] \nr
Hardness\hfill & \hfill Goto, 1984 [2] \nr
Tensile test \hfill & \hfill  Cane and Williams, 1987 [3] \nr
Interparticle spacing \hfill & Askins and Menzies, 1985 [4] \nr
Cavitation parameter \hfill & \hfill Gooch \ett, 1989 [5] \nr
Number density of cavities \hfill & \hfill Dyson and McLean, 1972 [6]\nr
Fraction of cavities \hfill & \hfill Walker and Evans [7] \nr
Impact toughness \hfill & \hfill Wignarajah \et [8] 

\endtable \tabtit{\tablee : } {Some of the methods used in the
estimation of remaining life.}\endinsert

\sec{DAMAGE SUMMATION}

\x A satisfactory way of representing creep damage $(C)$ is to use a
parameter ($\omega$) which is normalised by its value at failure
($\omega_r$). The magnitude of $\omega_r$ will depend on the precise
values of stress
$(\sigma)$, temperature $(T)$ and any other variable which influences the
creep process. Since these variables are not necessarily constant, the
extent of damage is often written [1,9]
$$ C = \sum_i {{\omega_i}\over{(\omega_r)_i}} \numeqn $$
$\omega$ is typically the time or the creep strain. Failure occurs when
the sum achieves a value of unity.

 Evans [1] argues that it is more appropriate to use the strain rather
than the time, since the latter is not considered as a \lqq state
variable".  In the context of thermodynammics, the state of a system can
in principal be specified completely by a number of state variables (such
as temperature, pressure) such that its properties do not depend at all on
the path by which those variables were achieved. This clearly cannot be
the case even for the creep strain. This is because the extent of
damage {\it is} expected to depend on the path by which a given value of
strain is achieved, for example, whether the strain is localised at grain
boundaries or uniformly distributed. This necessarily means that
equation~\nnumeqn\ is an approximation; as Evans states, it should be a
reasonable approximation if the mechanism of creep does not change
between the components of the summation. Thus, Cane and Townsend [10]
conclude that the use of the life fraction rule in taking
account of temperature variations is  more justified than for
variations in stress. This is because for the latter case, the
dislocation networks become finer (relative to the carbide
spacings) at large stresses. The network nodes then do not
coincide with carbide particles, thus changing the mechanism of
deformation. This is not the case with variations in temperature
because the dislocation network then scales with the particle
spacing. The failure of the life fraction rule is sometimes
accommodated by empirically setting the limiting value of
$C$ to some positive value which is not unity.

\sec{HARDNESS, INTERPARTICLE SPACING}

\x The hardness can be used as an indicator for the state of the
steel in its life cycle. Changes in hardness occur due to  recovery,
coarsening of carbide particles, and recrystallisation. All
creep--resistant power plant steels are severely tempered before they
enter service. They are therefore  beyond the state where secondary
hardening is expected and the hardness can, during service, be expected
to decrease monotonically. In these circumstances, an Avrami equation
adequately represents the changes in hardness,
$$\xi = 1 - \exp\{-k_A t^n \}\numeqn $$
where $t$ is the time, $k_A$ and $n$ rate constants and $\xi$ is given
by 
$$\xi\{t\} = {{H_0 - H\{t\}}\over{H_0 - H_\infty}}\numeqn$$ 
where $H_0$ is the initial hardness,  
$H_\infty$ is its hardness at the end of useful life and $H\{t\}$ the
hardness at time $t$. 

The hardness at the point where the
microstructure is exhaustively annealed is likely to be around
$H_\infty=150-190$~HV for most power plant steels. Its main components
include the intrinsic strength of iron and solid solution strengthening.
The starting hardness is likely to be in the range $H_0=220-300$~HV.
Therefore, all that can be expected is a change in hardness of about
30--70~HV over a period of some 30 years. Thus, Roberts and Strang
[11] have shown that the hardness can decrease by about 20\%\ in
the stressed regions of long--term creep test specimens; this
is consistent with an approximately 25\%\ reduction found by Maguire
and Gooch [12].
\fagg\ shows the nature of the changes in hardness to be expected
typically, as reported by Maguire and Gooch [12] for a 1CrMoV
steel which was tempered at 700 \deg for 18~h prior to the ageing
at temperatures in the range 600--640
\degg. 

\picture{hardness}{190}{127}{700}{\tabtit{\figg : }{Changes in the
hardness of a 1CrMoV steel during ageing in the temperature range
600--640 \degg. The samples were tempered at 700 \deg for 18 h prior to
testing; the effect of the 700 \deg treatment is not included in the
calculation of the time--temperature parameter plotted on the
horizontal axis. The open circles are data from the grips of a creep test
specimen, and the filled circles represent measurements from the gauge
length. The large filled circles correspond to samples whose hardness
values were measured at failure,
\ie after the complete exhaustion of life. The stress applied during each
of the tests was 93~MPa. Data from Maguire and Gooch.}}

Precipitates impede the motion of dislocations and  any strength in
excess of $H_\infty$ is often  related to the spacing
($\lambda$) between the particles  (Cane \ett, 1986, 1987):
$$ H-H_\infty \propto {{1}\over{\lambda}} \propto t^{-{1\over3}}
\numeqn $$ where it is assumed that $\lambda \propto t^{1\over3}$ in
order to be consistent with coarsening theory. 

It is well known that differences in hardness develop in the grip and
gauge length of a creep test specimen (\figg). The strain in the gauge
length leads to accelerated softening. In fact, the hardness reduction
that occurs in the gauge length is roughly proportional to that in
the grip [12]. Tack
\et [13] have therefore taken $H$ to be a function of strain:
$$ {{H\{t\}-H\{t,\epsilon\}}\over{H\{t\}-H_\infty}} = b_1 +
b_2\ln\{\epsilon\} \numeqn $$
where $b_1$ and $b_2$ are empirical constants. This illustrates the fact
that   hardness is a crude indicator of remaining life. Furthermore,   
it's ability to account for creep damage in the form of voids is not
represented in any theory. Hardness tests can therefore only serve a
useful purpose in the regime of steady--state creep, before the onset
of gross damage. This is evident from \figg\ where it is seen that
specimens with the same hardness are at different life--fractions. There
is a further complication, that the hardness of welded regions is likely
to be inhomogeneous even when the welds are made with matching
compositions. The potential location of failure is then difficult to
identify since creep ductility, creep strength and creep strain may vary
with position. The weld metal always  has a larger oxide content than
the parent steel, and hence has a lower creep ductility. It cannot
therefore be assumed that failure will always occur in the softest part
of the joint.  A harder weld  may be needed to ensure the same rupture
life as the parent steel (\fagg). 

\picture{ruptureHV} {190}{127} {500}{\tabtit{\figg : } {Data for
$2{1\over4}$Cr1Mo steel and matching weld metal. The curve
represents the locus of all points along which the weld metal and parent
metal have equal rupture lives. Note that for a given life, the weld
must be harder than the parent steel (after Tack
\ett, [13]).}}

We have seen that hardness can be related inversely to the spacing
between precipitate particles; this is illustrated in \fagg\ for a
1Cr${1\over 2}$Mo steel [4]. The spacings are typically measured using
transmission electron microscopy at a magnification of about $\times
20,000$ with approximately 100 fields of view covering 30$\,{\rm \mu
m}^2$ taken at random. The actual measurement involves counting the
number of particles per unit area ($N_A$) and it is assumed that 
 $\lambda = N_A^{-{1\over2}}$. The amount of material examined in any
transmission microscope experiment is incredibly small, so care has to
exercised in choosing representative samples of steel. In some cases,
the microstructure may be inherently inhomogeneous. One example is the
12Cr and 9Cr type steels where there is a possibility of regions of
$\delta$--ferrite where the precipitation is quite different from the
majority tempered martensite microstructure.

 Obviously,   hardness tests are  much  simpler to conduct when
compared with the effort required to properly measure particle spacings.
A further complication is that there is frequently a mixture of many
kinds of particles present, some of which continue precipitation during
service whereas others dissolve. Thus, Battaini \et [14] found
that  in a 12CrMoV steel,  precipitation continues to such an
extent during  service that there is a monotonic  {\it decrease}
in
$\lambda$. In fact, the distribution of particles was bimodal with
peaks at 30~nm and 300~nm diameters. It is strange that they were only
able to correlate the hardness against the changes in the coarser
particles.  For another steel (12CrMoVW), Battaini \et found an even more
complex variation in the interparticle spacing with a maximum value in
$\lambda$ for the coarse particles. 

It should be emphasised that the standard error in $\lambda$
measurements is quite large, frequently larger than the variations
observed. Given the difficulties of interpretation, and the
experimental error, it is unlikely that $\lambda$ measurements can be
used as a satisfactory general measure of remaining life. The data can
nevertheless be of use in the design of physically based creep models.

\picture{interparticle}{134}{110}{700}{\tabtit{\figg : }{ The hardness
as a function of the near neighbour spacing of carbides following creep
tests at 630 \deg for a variety of time periods. After Askins and
Menzies, 1985 [4].}}
 
\sec{EQUILIBRIUM PHASES}

\x The simplest assumption in kinetic theory is to take a \lq flux' to be
proportional to the \lq force', where the magnitude of the latter
depends on the deviation from equilibrium. It follows that to understand
the microstructural changes that happen in a   steel during service, it is
necessary first to consider the   equilibrium state. \tablaa\ lists some
of the common power plant alloys together with quotes from material
specifications. It seems that the specifications are compiled by agreement
between many vested interests and therefore tend to be unrealistically
broad. Thus, many of the steels have only a maximum carbon
concentration specified! The actual alloys are purchased by users to
much tighter ranges than those given in \tablee.



\midinsert {\smalltype \thicksize=1pt \thinsize=0.2pt
\tablewidth=6.9truein \begintable

Designation | C & Si & Mn & Ni & Mo & Cr & V & Others \cr
| & & & & & & &  \nr
$1\over 4$CrMoV \hfill | 0.15 & 0.25 & 0.50 & 0.05 & 0.50 & 0.30 & 0.25 &   \nr
 range \hfill | $<$ 0.18 & 0.10-0.60 & 0.40-0.65 & -- & 0.45-0.65 & 0.25-0.35 & 0.20-0.30 &   \cr

1CrMoV  \hfill | 0.25 & 0.25 & 0.75 & 0.70 & 1.00 & 1.10 & 0.35 &  \nr
 range \hfill | 0.24-0.31 & 0.17-0.27 & 0.74-0.81 & 0.60-0.76 & 0.65-1.08 & 0.98-1.15 & 0.27-0.36 &   \cr

2$1\over 4$Cr1Mo \hfill | 0.15 & 0.25 & 0.50 & 0.10 & 1.00 & 2.30 & 0.00 &  \nr
 range \hfill | $<$ 0.16 & $<$ 0.5 & 0.3-0.6 & -- & 0.9-1.1 & 2.0-2.5 & -- &  \cr

 Mod. 2$1\over 4$Cr1Mo \hfill | 0.1 & 0.05 & 0.5 & 0.16 & 1.00 & 2.30 & 0.25 & Ti=0.03 \hfill \nr
| & & & & & & & B=0.0024 \hfill \cr
 
3.0Cr1.5Mo \hfill | 0.1 & 0.2 & 1.0 & 0.1 & 1.5 & 3.0 & 0.1 &   \nr
 range \hfill | $<$ 0.16 & $<$ 0.5 & 0.30-0.60 & -- & 0.45-0.65 & 4.0-6.0 & -- &  \cr 
 
3.5NiCrMoV \hfill | 0.24 & 0.01 & 0.20 & 3.50 & 0.45 & 1.70 & 0.10  &  \nr
 range \hfill | $<$ 0.29 & $<$ 0.11 & 0.20-0.60 & 3.25-4.00 & 0.25-0.60 & 1.25-2.00 & 0.05-0.15 &  \cr
 
9Cr1Mo \hfill | 0.10 & 0.60 & 0.40 & -- & 1.00 & 9.00 & -- &   \nr
range \hfill | $<$0.15 & 0.25--1.00 & 0.30--0.60 & -- & 0.90--1.10 & 8.00--10.00 & -- &  \cr

Mod. 9Cr1Mo \hfill | 0.1 & 0.35 & 0.40 & 0.05 & 0.95 & 8.75 & 0.22 & Nb=0.08 N=0.05 \nr
range \hfill | 0.08--0.12 & 0.20--0.50 & 0.30--0.60 & $<$0.2 & 0.85--1.05 & 8.00--9.50 & 0.18--0.25 & Nb 0.06--0.10 \hfill\nr
| & & & & & & &  N 0.03--0.07 \hfill\nr
| & & & & & & & Al $<$0.04 \hfill \cr

9Cr$1\over2$MoWV \hfill | 0.11 & 0.04 & 0.45 & 0.05 & 0.50 & 9.00 & 0.20 & W=1.84 Nb=0.07 \nr
  | & & & & & & &  N=0.05  \hfill\nr
range \hfill | 0.06--0.13 & $<$ 0.50 & 0.30--0.60 & $<$ 0.40 & 0.30--0.60 & 8.00--9.50 & 0.15--0.25 & Nb 0.03--0.10 \hfill\nr
| & & & & & & &  N 0.03--0.09 \hfill\nr
| & & & & & & &  Al $<$0.04 \hfill\cr

12CrMoV \hfill | 0.20 & 0.25 & 0.50 & 0.50 & 1.00 & 11.25 & 0.30 & \nr
range \hfill | 0.17--0.23 & $<$0.5  & $<$1.00  & 0.30--0.80 & 0.80--1.20 & 10.00--12.50 & 0.25--0.35 & \cr

12CrMoVW \hfill | 0.20 & 0.25 & 0.50 & 0.50 & 1.00 & 11.25 & 0.30 & W=0.35 \hfill \nr
range \hfill | 0.17--0.23 & $<$0.5  & $<$1.00 & 0.30--0.80 & 0.80--1.20 & 10.00--12.50 & 0.25--0.35 & W$<$0.70 \hfill \cr

12CrMoVNb \hfill | 0.15 & 0.20 & 0.80 & 0.75 & 0.55 & 11.50 & 0.28 & Nb 0.30 \hfill \nr
| & & & & & & &  N 0.06 \hfill


\endtable }\tabtit{\tablee : }{Typical compositions (wt.\% ) of
creep--resistant steels  used in the power generation and
petrochemical industries.   The range of compositions stated
lie within the technical  specification, but are really
intended to reflect the  variations observed in practice. The
sulphur concentration  is usually within the range 0.005--0.02
wt.\%, and that of  phosphorus within the range 0.005--0.025
wt.\% }
\endinsert  

The results of thermodynamic calculations which give the phase fractions
and compositions as a function of the overall alloy composition and
temperature, are given in \tablaa\ for the common power plant steels.
The calculations have been done using the {\it MTDATA} computer
program and {\it SGTE} database, taking into account the carbide
phases and Laves phase listed. The chemical elements considered
are carbon, silicon, manganese, chromium, nickel, molybdenum, vanadium
and niobium; nitrogen is not included but is known to be important in
forming MX particles. Note also that some very careful work has
recently identified an ${\rm M_5C_2}$ in 1Cr--0.5Mo steels [15];
this phase has not been included in the analysis.



\midinsert {\smalltype \thicksize=1pt \thinsize=0.1pt
\tablewidth=6.9truein \begintable

Designation | M$_2$X | M$_3$C  | M$_7$C$_3$ | M$_{23}$C$_6$ | M$_6$C | Laves | NbC | NbN | VN \cr

$1\over 4$CrMoV \hfill | 0.0053 |  | | 0.0247 | | | | |   \cr

1CrMoV  \hfill |  0.0089 |  | | 0.0412 | | | | |   \cr

2$1\over 4$Cr1Mo \hfill |   |  | | 0.0335 | | | | |   \cr

 Mod. 2$1\over 4$Cr1Mo \hfill |  |  | | 0.0211 | 0.0019 | | | |   \cr

3.0Cr1.5Mo \hfill |   |  | | 0.0185 | 0.0057 | | | |   \cr

3.5NiCrMoV \hfill |  0.0009 |  | 0.0161 | 0.0285 | | | | |   \cr
 
9Cr1Mo \hfill |  |  | | 0.0222 | | | | |   \cr

Mod. 9Cr1Mo \hfill | |  | | 0.0222 | | | | 0.0009 | 0.0030    \cr

9Cr$1\over2$MoWV \hfill | |  | | 0.0248 | | 0.0135 | | 0.0008 | 0.0032   \cr

12CrMoV \hfill | |  | | 0.0443 | | | | |   \cr

12CrMoVW \hfill | |  | | 0.0444 | | 0.0007 | | |   \cr

12CrMoVNb \hfill |  |  | | 0.0318 | |  | 0.0006 | 0.0029 | 0.0018  

\endtable }\tabtit{\tablee : }{The mole fractions of precipitate phases 
in the steels listed in Table~2, when at equilibrium at 565\deg (838~K)}
\endinsert  



\sec{CHANGES IN CARBIDE CHEMICAL COMPOSITIONS}

\x The cementite that forms in association with bainite or martensite
has at first, a chemical composition which is far from its equilibrium
composition. It therefore enriches with elements such as chromium
during service in power plant, and might be used as a built--in
recorder for the thermal energy felt by the steel [16--25]. 



\ssec{Virgin Cementite}

\x There strong evidence  to show that cementite and other
iron--based transition carbides grow by a paraequilibrium, displacive
transformation mechanism when the precipitation is from martensite or
bainite. This subject has been reviewed recently [26]. In this mechanism,
the substitutional lattice is displaced into the new structure at a rate
controlled by the diffusion of carbon to the transformation front. This
is not terribly surprising given the very small distances over which the
susbtitutional atoms can diffuse during the precipitation event (\fagg).
Cementite, for example, precipitates in a matter of seconds during the
tempering of martensite.

\picture{diffusion}{190}{127}{700}{\tabtit{\figg : }{An illustration
of the extraordinarily small  diffusion distances at the temperatures
where some carbides  precipitate.}}

Of course, not all cementite in power plant steels comes from the
bainite or martensite reactions. Many low--alloy steels contain small
quantities of pearlite. The cementite which is a constituent of
pearlite grows by a reconstructive transformation mechanism in which
all of the atoms diffuse in a way which minimises strain energy. But
the diffusion at the same time allows species of atoms to partition
into phases where they are most comfortable. Consequently, the
cementite within pearlite always has a chemical composition which is
closer to equilibrium. This is illustrated in \fagg a, where the
cementite when it first forms,   is seen to be richer in
chromium for pearlite than for bainite. This is the case even when
the pearlite and bainite grow at the same temperature [27]. 

\picture{chance}{170}{212}{700}{\tabtit{\figg : }{ (a) The partition 
coefficient for cementite in bainite and pearlite, together 
with equilibrium data [27]. The partition
coefficient is the ratio of the concentration in
cementite to that in the ferrite.(b) The concentration profile 
that develops during the enrichment of a cementite particle.  }}

Naturally, the cementite will tend towards its equilibrium chemical
composition when exposed to  elevated temperatures for prolonged
periods of time. This generally means that its substitutional solute
content increases with time. It can be shown [20] that this change
is given by:
$$ t = {{\pi[x^{\theta}(\overline c - 
c^{\theta})]^2}\over{16D_{\alpha}(c^{\alpha\theta} - 
\overline c)^2}} \numeqn $$
where it is assumed that the particle is in the form of a thin plate
of thickness $x^\theta$, $t$ is the time at the annealing temperature, 
$c^{\theta}$ is the average cementite ($\theta$) composition 
at time $t$. $D_{\alpha}$ is the diffusion coefficient of the 
substitutional solute in ferrite, $c^{\alpha\theta}$ is the 
concentration in ferrite which is in equilibrium with 
cementite, and $\overline c$ is the starting composition of 
the carbide (\figg b). Note that  $c^{\theta}$ is not the
concentration $c^{\theta\alpha}$ in cementite which is in {\it
equilibrium} with ferrite. The diffusion flux in the ferrite is so small
that the concentration in the cementite at the boundary cannot build up
to the equilibrium level since any enrichment at the interface is
relieved by diffusion within the cementite. This is because  
$(\overline c - c^{\alpha\theta})$ which drives the diffusion within
the ferrite is much smaller than $(c^{\theta\alpha}-\overline c)$, \ie
the solubility of elements such as chromium is much larger in cementite
than in ferrite. It follows that in spite of the diffusion flows, the
concentration distribution in the cementite is always likely to be quite
uniform. This is a useful result since it makes it reasonable to
determine the particle composition using microanalysis techniques which
gather information from the entire particle. Thus the concentration in
the cementite rises uniformly from $\overline c$ to $c^{\theta\alpha}$.

Equation \nnumeqn\ indicates that the change in composition ($c^{\theta}
- \overline c$) should vary with $t^{1\over 2}$ during isothermal 
annealing; this is described as a {\it parabolic} law. This contrasts
with the common assumption made even today, that the change should
follow $t^{1\over 3}$ kinetics. It is often not possible in practice to
distinguish between  the two exponents; the experimental data give
reasonable  linear fits with either relationship. However, values of
the  diffusion coefficient extracted from such plots are only  
correct in the case for parabolic kinetics [28]. This confirms the
physical significance of equation~\nnumeqn, which can 
extrapolated with more confidence than  the
$t^{1/3}$ relationship. An additional  advantage is that the effect of
the steel chemistry is  explicitly included in equation~\nnumeqn\ via the
equilibrium and  average compositions, so that data from many different
alloys  can be interpreted together.


\fagg\ shows experimental measurements of the composition of cementite
as a function of tempering time, for a microstructure which was
initially fully bainitic [29,30]. The steel composition is
Fe--0.1C--0.24Si--0.48Mn--0.84Cr--0.48Mo wt.\%; the reason why the
cementite composition is richer than the average is because a
stress--relief heat treatment was given prior to ageing. There are a
number of important features in this graph other than the fact that the
parabolic law is verified. It is obvious that the largest changes are
observed for chromium, which would therefore be a good element to
monitor. This is because the solubility of Cr in the cementite is much
larger than that of Mn or Mo (\tablaa). Notice that the solubility
varies with temperature, becoming smaller with increasing temperature.


\midinsert \thicksize=1pt \thinsize=0.8pt
\tablewidth=6.2truein \begintable
| \multispan2 \hfill Equilibrium  wt.\%, 703 \degg \hfill |
\multispan2 \hfill Equilibrium   wt.\%, 550 \degg \hfill
\nr | $c^{\alpha\theta}$ & $c^{\theta\alpha}$ | $c^{\alpha\theta}$
& $c^{\theta\alpha}$ \cr
Cr | 0.52 & 21.3 | 0.30 & 36.0 \nr
Mo | 0.40 & 5.8 | 0.34 & 9.3 \nr
Mn | 0.38 & 6.7 | 0.29 & 13.2 

\endtable \tabtit{\tablee : }{Calculated equilibrium
concentrations for Fe--0.1C--0.24Si--0.48Mn--0.84Cr--0.48Mo
wt.\%\ alloy used by Afrouz \ett\ These calculations are
specific to this steel. }\endinsert


It should also be emphasized that the equilibria listed in \tablee\ are
steel specific and will vary greatly within the range of power plant
alloys used typically in industry. A simple method for estimating the
values is to write the partition coefficient for element $Z$
 as follows:
$$k_Z ={{c^{\theta\alpha}}\over{c^{\alpha\theta}}}\simeq \exp\bl\{  {{A +
BT}\over {RT}}\br\} \numeqn $$ where the parameters $A$ and $B$ are
listed in
\tablaa\ [31]. The partition coefficient only gives a ratio -- to
calculate the actual concentrations requires the following further
relationship which derives from mass balance:
$$ \overline c = c^{\theta\alpha}V_\theta +
{{c^{\theta\alpha}}\over{k_Z}} 
- {{c^{\theta\alpha} V_\theta}\over{k_Z}} \numeqn $$
$V_\theta$ can be estimated using the lever rule with carbon, so that the
equilibrium compositions of the two phases can easily be calculated as a
function of heat treatment and average composition.


\midinsert \thicksize=1pt \thinsize=0.8pt
\tablewidth=5.2truein \begintable
Element $Z$ | A / J mol$^{-1}$ &  B / J mol$^{-1}\,$K$^{-1}$ \cr
Cr | 47028 & -17.45 \nr
Mn | 42844 & -20.21 \nr
Mo | 27363 & -5.86 \nr
Ni | -2619 & -2.80 \nr
Si | 0 & -25.10 

\endtable \tabtit{\tablee : }{Compilation of parameters used for the
calculation of partition coefficients [31].}\endinsert


\picture{Afrouz}{190}{127}{700}{\tabtit{\figg : }{Measured changes in the
chemical composition of cementite particles as a function of the square
root of time, during ageing at 550 \degg. The steel composition is
Fe--0.1C--0.24Si--0.48Mn--0.84Cr--0.48Mo wt.\%. It was heat treated to
give a fully bainitic microstructure, stress--relieved at 730 \deg for
one hour and then tempered at 550 \deg for the periods illustrated.
Data from Afrouz, Collins and Pilkington [19].}}



Another feature highlighted by the parabolic law
is the dependence of enrichment on the particle size $x^\theta$. Smaller 
particles enrich faster and saturate earlier because they are 
smaller reservoirs for solute (\fagg).


\picture{size}{92}{88}{1000}{\tabtit{\figg : }{Size dependence 
of cementite enrichment (annealed at 565 \deg for 4 weeks 
[29,30]. }}


The model described above predicts parabolic kinetics, which can be
considered to have been verified by published experimental data.
Although the data refer to time periods as long as 10,000~h, the time
scales of interest are much larger, of the order of 260,000~h. Data can
of course be obtained from steels which have been in service but they can
be quite unreliable because the history is then uncertain. It would not be
safe to extrapolate the parabolic law to very long times, because the
analytical expression does not account for a phenomenon known as
{\it soft--impingement}.  In this, the diffusion fields of adjacent
particles begin to overlap and significantly interfere, slowing down
the enrichment process until it stops completely when equilibrium is
achieved. Such a process cannot easily be treated analytically.
Bhadeshia has used a finite difference method instead; some results are
presented in \fagg, for precisely the experiments reported by Afrouz
\ett. The parabolic law is a good approximation during the \lq early\rq\
stages but  the effect of soft--impingement is to cause  deviations  
with a complete halt when the carbide saturates with the solute
concerned. Manganese and molybdenum do so quite rapidly for this
particular example. Thus, the numerical method (finite difference) can be
used to extrapolate to very long times.




\picture{Afrouz2}{139}{110}{700}{\tabtit{\figg : }{Calculated changes in
the chemical composition of cementite particles as a function of the
square root of time, during ageing at 550 \degg. The steel composition is
Fe--0.1C--0.24Si--0.48Mn--0.84Cr--0.48Mo wt.\%. The calculations are
done using a finite--difference method [20].}}


\ssec{Pearlite}

\x It has been emphasised that to ratio of substitutional solute
to iron atom ratio is the same everywhere in a bainitic
or martensitic microstructure as in the alloy as a whole. This is not the
case for pearlite where the reconstructive mechanism of transformation
ensures that the cementite is enriched with substitutional solutes, though
often not to the extent expected from equilibrium. This makes it
impossible to predict the initial composition of the cementite in
pearlite. It turns out that for  calculation purposes, this is not a
serious problem for long  ageing times, since the degree of enrichment
then tends to be  much larger than the starting solute level (\fagg).


\picture{pearlite}{134}{112} {700}{\tabtit{\figg : }{Data for the
enrichment of cementite associated  with pearlite, and calculated curves
assuming zero enrichment  at $t=0$ (continuous curve), and a finite
enrichment at $t=0$  (dotted line) [32].}}

\ssec{Effect of Carbon}

\x There are two effects which depend on the carbon 
concentration of the steel. The ternary Fe--Cr--C phase 
diagram on the $(Fe,X)_3C/\alpha$ field shows that an 
increase in the carbon concentration is accompanied by a 
decrease in the equilibrium concentration of chromium in the 
carbide. Thus, the carbide enrichment rate is expected to 
decrease. A further effect is that the volume fraction of 
cementite increases, in general leading to an increase in 
particle thickness and volume fraction. The thickness 
increase retards the rate of enrichment (equation~6). If the 
carbide particles are closer to each other then 
soft--impingement occurs at an earlier stage, giving a slower 
enrichment rate at the later stages of annealing.

Local variations in carbon concentration may have a similar 
effect to changes in average concentration. Such variations 
can be present through solidification induced segregation, or 
because of microstructure variations caused by differences in 
cooling rates in thick sections. It is well known that the 
microstructure near the component surface can be fully 
bainitic with the core containing a large amount of 
allotriomorphic ferrite in addition to bainite. In the latter 
case, the bainite which grows after the allotriomorphic 
ferrite, transforms from high carbon austenite. The 
associated carbides are then found to enrich at a slower rate 
(\fagg a). This discussion emphasises the role of carbon, an 
element which unfortunately is not tightly specified in creep 
resistant alloys.



\picture{mixed}{153}{231}{600}{\tabtit{\figg : }{  (a)
$2{1\over4}$Cr1Mo steel,  cementite  enrichment in a fully bainitic
microstructure and one which  is a mixture of allotriomorphic ferrite
and bainite [29,30]. (b)  Rapid saturation of cementite in a
12Cr1MoV steel aged at 700  \degg, as calculated for a variety of
particle sizes [33].  }}


\ssec{High--Alloy Steels, Alloy Carbides}

\x The vast majority of steels used in the power generation 
industry have a total alloy concentration less than 5 wt.\%, 
but there are richer martensitic alloys such as the 9Cr1Mo, 
12Cr1MoV destined for more stringent operating conditions. 
For low alloy steels, the initial carbide phase found 
immediately after the stress--relief heat treatment 
(typically several hours at 700 \degg) is cementite. The 
cementite is very slow to enrich and to convert to alloy 
carbides, thus allowing the process to be followed over a 
period of many years during service. As the concentration of 
carbide forming elements is raised, the kinetics of cementite 
enrichment and alloy carbide precipitation become more rapid 
(\figg b). Thus, relatively stable alloy carbides tend to 
dominate the microstructure after short periods in service or 
even after the stress relief heat treatment [33]. The alloy 
carbides form with virtually their equilibrium compositions, 
and hence show no change on annealing. \fagg\ shows
microanalysis data from vanadium--rich carbides for two
samples with average composition
$$\hbox{ Fe-0.11C-0.3Mn-0.28Mo-0.23V-0.023N~wt.\%}$$
The \lqq virgin" sample was produced by tempering at 700
\deg for an unspecified time period, following normalisation
at 950 \deg for an unspecified period. The exposed sample
had in addition spent $10^5$ hours at 540~\deg and 50~MPa.
It is clear that there is no significant difference between
the carbide compositions from the two samples. There is
therefore  little prospect of estimating the remaining life of
high  alloy steels, or of steels in general, by following changes
in the chemical compositions of alloy carbides. As a matter of interest,
\figg b shows the chromium and molybdenum concentrations of the most
prominent alloy carbide found in most power plant steels (see also
Tables~2,3).

\picture{vn} {189} {229} {700}{\tabtit{\figg : }{(a) The chemical
compositions of vanadium--rich carbides in virgin material and
in samples which had been in service for ten years [22]. (b) The
concentrations of Cr and Mo in M$_{23}$C$_6$ in the power plant steels
listed in Table~2, for 565 \degg. The data are calculated using the
{\it MTDATA} program.}}

One possibility is to follow the detailed microstructural changes.
\fagg\ is a simplified presentation of the evolution of carbide and
Laves phases during the ageing of a 9CrMoVW steel {\it (NF616)} which
is tempered from its martensitic state at a temperature of 600 \deg
over six orders of magnitude of time. Obviously, an identification of the
fractions of the phases could help to fix the state of the steel. Some
of the precipitation reactions are sensitive to temperature. For
example, the driving force for the precipitation of Laves phases
decreases greatly as the temperature increases.

\picture{nf616} {164}{135} {700}{\tabtit{\figg : }{Calculated evolution
of the phase fractions in a 9CrMoVW steel ({\it NF616}) during
tempering at 600 \deg [34].}}

\sec{CAVITATION DAMAGE}

\x Deformation by creep is far from homogeneous on a microstructural
scale. One manifestation of this is the
formation of cavities at the grain boundaries between differently
oriented crystals. Cavitation plays a critical role in the events leading
to ultimate failure and hence is a direct measure of remaining life.

The cavities tend to form on boundaries with normals parallel to the
principal tensile stress, by a process of nucleation and growth. Thus,
at any instant of life, only a fraction of the transverse boundaries
have cavitation damage. There are  constraints to cavity formation when
only some of the transverse boundaries are cavitated, and this has
important consequences on the creep rupture life [35]. In
\fagg a, all the transverse boundaries have identical cavities. The
regularity of the structure allows the cavities to grow unconstrained,
  the extension along the stress axis being accommodated by grain
boundary sliding. This is not the case when, as illustrated in
\figg b, only some of the boundaries have cavities. The extension due
to the  cavities at $wx$ results in sliding which causes the adjacent
grains to overlap at $yz$. This matter in the shaded region at $yz$ must
be carried away by diffusion before the cavities at $wx$ can continue to
grow. This in effect imposes a constraint on the growth of the cavities
when compared with the situation in \figg a.  The physical basis of the
retardation is that cavity growth now
becomes controlled by the rate of matrix creep. The retardation is very
significant, as seen in \figg c.

\picture{constraint}{182}{225}{700}{\tabtit{\figg : }{The effect of
constraint on the growth rate of cavities. (a) All transverse
boundaries cavitated. (b) Non--uniform cavitation. (c) The reduced
cavity growth rate when cavitation is not uniform [35].}}



The  cavitation is characterised by an \lq $A$--parameter\rq\ determined
as the number fraction of cavitated grain boundaries traversed by a scan
parallel to the stress axis of a creep test specimen, whilst avoiding
the necked region [36].  There is considerable work in the
published literature dealing with the kinetics of the constrained
growth of cavities growth [1] and this has been used to
express the parameter $A$ as follows:
$$ A = 1- \br(1-{{t}\over{t_r}}\bl)^{(1-{{\epsilon}\over{\epsilon_r}})/n}
\numeqn $$ where $t$ is the time, $t_r$ the time to rupture, $\epsilon$
the creep strain, $\epsilon_r$ the creep rupture strain and $n$ a
fitted constant.
\fagg\ shows both experimental data  and a curve calculated
using equation~\nnumeqn\ and assumed values of
$\epsilon_r/\epsilon=2.5$ and $n=3$ [36]. The data seem
follow the form indicated by the equation but there are other results
[12,37] which suggest that the
relationship remains linear up to the point of life--exhaustion,
although the degree of scatter in the $A$--parameter increases
significantly for a life fractions greater than 0.96. 

The $A$--parameter is defined in terms of the number of transverse
boundaries which are cavitated. Many models for creep associate
fracture with a complete loss of area available to support the load as
the cavities coalesce along the appropriate plane [7,38,39]. A
strict measure of the damage would then be {\it area} of
transverse boundaries that is decorated by cavities.  The
$A$--parameter on the other hand could reached unity before the
transverse boundaries are completely detached. However, this may
not be a problem in practice if the rate of damage increase in
the final stages is very high.

\picture{cavitation}{172}{114}{700} {\tabtit{\figg : }{The fraction of
cavitating grain boundaries as a function of the life fraction consumed
[36].}}

Another method of assessing the remaining life using cavitation damage
involves the measurement of the number density of cavities, which seems
to relate directly to the creep strain during uniaxial loading [6]. This
may not in fact amount to a different method (compared with the
$A$--parameter) in the context of steels where the important cavities
tend to form at the grain boundaries.

The volume fraction of cavities increases approximately with the
strain and can be used as a measure of remaining life [7], although
differences in the  distribution of cavities (due to differences in
microstructure) could introduce variations in creep life at constant
cavity fraction.

\sec{FRACTURE TOUGHNESS}

\x It is well established that the fracture toughness of many
power plant steels deteriorates during service for two reasons.
Firstly, the carbide particles, particularly those located at
boundaries, coarsen and hence provide easier sites for crack or
void nucleation. Secondly, the segregation of impurities to
interfaces has an opportunity to proceed to its equilibrium
extent during service. This tendency to embrittle during service
is sometimes simulated by an accelerated step cooling experiment
in which the sample is held at a variety of decreasing
temperatures over the embrittlement range, and then allowed to
cool to ambient temperature at a very slow rate. Unfortunately,
although the procedure does allow the sensitivity of the sample
to embrittlement phenomena to be tested, the data cannot be used
to discover the state of the steel in its ageing process (a
remaining life assessment).

\fagg a\ shows how the fracture assessed impact transition
temperature (FATT) varies with the mean carbide particle size in
a 2${1\over4}$Cr1Mo steel [8]. \figg b shows the complete
transition curves for both the virgin steel (\ie at
implementation into service) and for an ex--service sample which
had experienced 88,000~h at 813~K. The points represent samples
of virgin steel subjected to 873~K for 10,000~h in an attempt to
simulate prolonged service at 813~K. These conditions were
chosen in order to achieved the same Larson--Miller parameters
for the ex--service and simulated samples. It is clear that the
agreement between the simulation and service exposed steel is
excellent, allowing the method to be added to the panoply of
tools available for remaining life assessment.

\picture{impact}{273}{129}{500}{\tabtit{\figg : }{Impact test
data on 2${1\over4}$Cr1Mo steel. (a) Change in the
fracture assessed impact transition temperature with the mean
carbide size as a function of ageing. (b) Comparison of the
impact energies of the steel in its virgin condition, after
service exposure at 813~K for 88,000~h and for a
service--simulated sample (873~K for 10,000~h). After Wignarajah
\et [8].}}


\sec{CONCLUSIONS}

\x The present work, which is limited by deadlines, is
by no means a comprehensive review of the available literature.
But it should be obvious that there is a vast array of   methods
available for the assessment of remaining life in power plant
steels. Impressive as the methods are, they must be used with
experience and caution since the number of variables (and indeed
the variability) is quite large.


\sec{ACKNOWLEDGMENTS}


\x It is a pleasure to acknowledge David Gooch
(National Power plc.), the late Brian Greenwell (NEI Parsons), and Roger
Conroy (NEI Parsons) for their generous support of our research over a
period of more than eleven years. I am also
grateful to the  Royal Society  for a Leverhulme Trust Senior Research
Fellowship and to Professor Alan Windle for the provision of laboratory
facilities at the University of Cambridge. 
 
 
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\def\ref#1#2#3#4#5#6{\item{#1.} #2: {\it #3}  {\bf #4} {(#5)} {#6}\vskip
0.5truemm} 

\medtype
\baselineskip=9.5pt minus 0.75pt


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