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\centerline{\large Final Report GR/K23263}
\centerline{\large Quantitative Alloy Design Tools for Ni--Base Superalloys}
\bigskip \singlespace



\sec{INTRODUCTION \&\ ORIGINAL OBJECTIVES}

\x Alloy design for critical aeroengine components such as 
turbine blades and discs is a difficult, time--consuming and 
expensive process. The development period prior to 
application in an engine is typically of the order of ten 
years. Airframe manufacturers usually announce a new design 
only five years prior to flight certification. This leaves 
the engine designers only half the current period necessary 
for materials development. 

Despite decades of research on the $\gamma/\gamma'$ nickel 
base superalloy system, new alloys have in the past been investigated by
making  as many as a hundred different variants. Each of these has to 
be cast or made into powder form, thermo--mechanically 
processed, assessed for the presence of deleterious phases 
and for processability, and tested on a laboratory scale. 
There is a further stage of optimisation before a small 
selection of alloys is tested to commercial standards. 
Predictive modelling, at any stage of this empirical alloy 
design procedure, would obviously reduce the cost and the 
time involved in the development of new materials. It was the intention of this
project to create some of the tools needed for the modelling of nickel--base
superalloys.

\x The objectives as listed in the original proposal are as follows:

\zz{1. }To develop quantitative strength prediction models  
$\gamma/\gamma'$ Ni--base  superalloys. 

\zz{2. } To develop quantitative models for other mechanical properties,
particularly fatigue

\zz{3. } To produce these models in the form of usable alloy design tools.



We shall now describe the range of models that has been developed in order
to enable weld design.

\sec{SOLID SOLUTION STRENGHTENING OF $\gamma$}

\x Nickel base superalloys are essentially mixtures of 
$\gamma'$ precipitates ($L1_2$ structure) in a disordered 
$\gamma$ (cubic--close packed structure) matrix. The yield strength of
multicomponent nickel solid solution alloys has not been modelled in the past
with respect to the effects of temperature and chemical composition. There
have been investigations of the effect on the yield stress of solutes in binary
systems at a fixed temperature, but not for multicomponent solutions or as a
function of the test temperature. Solid solution theory has been used to
solve this [1].

Two forms of the \lqq trough" model, in which dislocations reside in energy
wells, were  used to determine the most appropriate model. The yield strength
of three binary nickel--chromium and three ternary nickel alloys were
determined experimentally over a range of temperatures. The data were then
modelled using the Feltham equation (a trough model). The constants
determined by fitting the Feltham equation to the experimental data were
then applied to other experimental solid solution alloys and also to published
information on commercial solid solution nickel alloys. It was thus
demonstrated that the yield strength of $\gamma$ solid solutions can be
modelled successfully using the Feltham equation [1].




\sec{THE NEURAL NETWORK METHOD} 

\x Many mechanical properties are so complex in their dependance on
material characteristics that there are no theories
available to make quantitative predictions of the kind necessary in engineering
design. The neural network method is ideal in such circumstances since it
thrives in complexity, and when combined with experience from physical
metallurgy,   can be enormously useful both in the design of new materials
and in the definition of critical experiments.

 Neural networks are parameterized non--linear models used for empirical
regression and classification modelling. Stated simply, this represents a method
for the quantitative recognition of patterns in data, without any {\it a priori}
specification of the nature of the relationship between the input and output
variables. They can model relationships of almost arbitrary complexity.  

 The outcome of neural network training is a set of
coefficients (called weights) and a specification of the functions which in
combination with the weights relate the input to the output. The training
process involves a search for the optimum non--linear relationship between the
inputs and the outputs and is computer intensive. Once the network is trained,
estimation of the outputs for any given inputs is very rapid. 

	There are methods, such as that of  MacKay [2], which implement a  Bayesian
framework on the neural network; this helps in the determination of the
relevance of individual inputs.  Furthermore,  the error bars then  depend
on the specific position in input space, reducing the   dangers of extrapolation
and interpolation. These methods and associated references are   described
fully in [2]. We have found throughout  the present work that the method is
capable of revealing interesting metallurgical trends.



\sec{OVERALL STRENGTH} 

The yield and ultimate tensile strength of nickel--base superalloys with
$\gamma/\gamma'$ microstructures has been modelled [3,4] using the neural
network method, as a function of the Ni, Cr, Co, Mo, W, Ta, Nb, Al, Ti, Fe, Mn,
Si, C, B, and Zr concentrations, and of the test temperature. The analysis is
based on data selected from the published literature. The trained models were
subjected to a variety of metallurgical tests. As expected, the test temperature
(in the range 25--1100 \degg) was found to be the most significant variable
influencing the tensile properties, both via the temperature dependence of
strengthening mechanisms and due to variations in the $\gamma'$ fraction
with temperature. Since precipitation hardening is a dominant strengthening
mechanism, it was encouraging that the network recognised Ti, Al and Nb to
be key factors controlling the strength. The physical significance of the neural
network was apparent in all the interrogations we performed. 

One example illustrating this last point is presented in \fagg. The softening of
the $\gamma$ matrix is offset by the remarkable reversible increase in the
strength of the $\gamma'$ with increasing temperature. 

\picture{joyjones}{120}{88}{500}{\tabtit{\figg : }{Predicted temperature
dependence of the yield strength of a $\gamma/\gamma'$ superalloy.}}

A further revelation from the neural network analysis came from the error
estimates, which demonstrated clearly that there are great uncertainties in
the experimental data on the effect of large concentrations of molybdenum
on the tensile properties. This has identified a region where careful
experiments are needed; these will be conducted in future studies since
molybdenum is known to have a large influence on the $\gamma/\gamma'$
lattice misfit.

The methodology for tensile properties has already been exploited in
Rolls--Royce to reduce the number of variants involved in experimental alloy
design programmes.

\sec{FATIGUE PROPERTIES}

An extensive literature review was carried out to assess methods for
predicting the fatigue crack growth rates. Physical models were examined but
where either found not to be generally applicable or gave only qualitative
indications of the trends [5].  Selected experiments were conducted  [6] to
clarify the factors controlling the initiation and propagation of cracks in
turbine--disc superalloys. These results failed to clarify how fatigue theory could
be used to make quantitative predictions.

A neural network method was therefore used after identifying some 51
variables that could be expected to influence the fatigue behaviour of nickel
base superalloys [7]. The variables included the stress intensity range $\Delta
K$, chemical composition, temperature, grain size, heat treatment,
frequency, load waveform, atmosphere, $R$--ratio, the distinction between
short and long crack growth, sample thickness and yield strength.  The
analysis was conducted on some 1894 data collected from the published
literature.

The model, unlike any experimental approach, could be used to study the
effect of each variable in isolation. This gave interesting results. For example,
it was verified that an increase in the grain size should lead to a decrease in
the fatigue crack growth rate, when the grain size is varied without affecting
any other input.  This cannot be done in experiments because the change in
grain size is achieved by altering the heat treatment, which in turn influences
other features of the microstructure (\fagg). It was also possible to confirm
that $\log\{\Delta K\}$ is more strongly linked to the fatigue crack growth rate
than to $\Delta K$, as expected from the Paris law. There are many other
metallurgical trends revealed [7]. The method has been used in the context of
a Rolls--Royce disc alloy development programme, where it was demonstrated
quantitatively that the uncertainties of current knowledge are so large that
experiments were definitely necessary.

\picture{junk}{120}{88}{700}{\tabtit{\figg : }{Astroloy: (a) effect of grain
size alone; (b) effect of heat treatment alone [7].}}

In another approach [8] a different neural computing approach was used to
focus on stage II of the Paris regime, where the growth rate should depend
mainly on the stress intensity range, Young's modulus and yield strength. The
model was used successfully in estimating new test data.  The effect of the
ultimate tensile strength and phase stability was also investigated; although
this proved promising, it is probable that the results will be more convincing
when a greater range of data become available.

\sec{CREEP}

The creep rupture life of nickel base superalloys has been modelled as a
function of 42 variables including Cr, Co, C, Si, Mn, P, S, Mo, Cu, Ti, Al, B, N,
Nb, Ta, Zr, Fe, W, V, Hf, Re, Mg, La and ${\rm ThO_2}$ [9]. Other variables
include four heat treatment steps (characterised by temperature, duration and
cooling rate), the sample shape and the solidification method.

The results have been interpreted using physical metallurgy principles
where this is possible, and the model is currently being used in our Technology
Foresight Programme.



\sec{LATTICE PARAMETERS}

The lattice constants of the $\gamma$ and $\gamma'$ phases of nickel
superalloys have been modelled using a neural network within a Bayesian
framework [10]. The analysis was based on  new X--ray  
measurements and peak separation techniques, for a number of alloys and
as a function of temperature. These data were supplemented using the
published literature. 

The lattice parameters of the two phases were expressed as a non--linear
function of eighteen variables including the chemical composition and
temperature. It was possible to estimate the uncertainties and the method
has proved to be extremely useful in understanding both the effect of solutes
on the lattice mismatch, and on how this mismatch changes with temperature.
The method has been used in the development of new single--crystal alloys
(TMS series) where rafting is an important phenomenon.

\sec{OTHER MATERIALS \&\ PROCESSES}

Our experience with neural networks has been very positive so the method
has been applied outside of the scope of the original project.

The treatment of iron--base superalloys using both neural network and
physical modelling is described in [11,12]; a description of the
modelling of constitutive relations obtained by torsion testing is
in [13]; the modelling of steel plate processing using more than
100 variables is in [14].

\sec{DISSEMINATION}

A lot of the work and data are available from the materials algorithms
project world wide web (www) site on
$$http://www.msm.cam.ac.uk/map/mapmain.html$$ and specifically on
MAP\_LATTMISIFT (lattice misfit),  MAP\_FAT\_Ni\_MAT (fatigue properties),
MAP\_STEEL\_NEURAL (iron base superalloys),  MAP\_NEURAL\_MASTEEL
(iron base superalloys). 

The work, including the weights, has been published widely (see list) and has
been presented at a variety of conferences and colloquia.


\vfill\eject
\sec{PUBLICATIONS}




{\singlespace \parskip=0.5mm

\newcount\refno \refno=0 \def\ref#1#2#3{\global\advance\refno 
by1 \medskip\medbreak \vbox{\item {\the\refno. } #1 
\itemitem{} #2 \itemitem{} #3}} 


\ref{\lqq Modelling solid solution strengthening in nickel alloys"}
{Metallurgical and Materials Transactions A, Vol. 28A (1997) 1329--1335.}
{H. A. Roth, C. L. Davis and R. C. Thomson}

\ref{\lqq  Bayesian non--linear modelling with neural networks"} {Mathematical
Modelling of Weld Phenomena III, eds H. Cerjak and H. K. D. H. Bhadeshia,
Institute of Materials, London, (1997) 359--389.} {D. J. C. MacKay}

\ref{\lqq Neural network modelling of the mechanical properties of nickel base
superalloys"} {8th Int. Symposium on Superalloys, Seven Springs,
Pennsylvania, USA, September '96, eds R. D. Kissinger \ett, published by
TMS, (1996) 417--424.}{J. Jones and D. J. C. MacKay}

\ref{\lqq The strength of nickel--base superalloys: a Bayesian neural network
analysis"} {Proc. 4th International Symposium on Advanced Materials, eds
Anwar ul Haq \ett, A. Q. Kahn Research Laboratories, Pakistan (1995)
659--666.}
{J. Jones, D. J. C. MacKay and H. K. D. H. Bhadeshia}

\ref{\lqq The modelling of fatigue in nickel base alloys"}{Ph.D. Thesis,
University of Cambridge}{J. M. Schooling}

\ref{\lqq The application of neural computing methods to the modelling of
fatigue in nickel--base superalloys"} {8th Int. Symposium on Superalloys, Seven
Springs, Pennsylvania, USA, September '96, eds R. D. Kissinger \ett, published
by TMS, (1996) p. 409--416.}{J. M. Schooling and P. A. S. Reed}

\ref{\lqq Bayesian neural network analysis of fatigue crack growth rate in
nickel--base superalloys} {ISIJ International, Vol.~36 (1996) 1373--1382.} {H.
Fujii, D. J. C. MacKay and H. K. D. H. Bhadeshia}

\ref{\lqq The effect of load ratio on near--threshold fatigue crack
propagation in a nickel--base superalloy"} {Proc. 4th International Symposium on
Advanced Materials, eds Anwar ul Haq
\ett, A. Q. Kahn Research Laboratories, Pakistan (1995) 555--561.}
{J. M. Schooling and P. A. S. Reed}

\ref{\lqq Estimation of creep rupture strength in nickel--base superalloys"}
{6th Leige Conference, Materials for Advanced Power Engineering, October
1998, Leige, Belgium, in press} {H. Fujii, D. J. C. MacKay, H. K. D. H. Bhadeshia,
H. Harada and K. Nogi}

\ref {\lqq Estimation of the $\gamma$ and $\gamma'$ lattice parameters in
nickel--base superalloys using neural network analysis"} {ISIJ International,
Vol.~38 (1998) 495--502.} {S. Yoshitake, V. Narayan, H. Harada, H. K. D. H.
Bhadeshia and D. J. C. MacKay}

\ref{\lqq Tensile Properties of
Mechanically Alloyed Oxide Dispersion Strengthened Iron Alloys.
Part 1 - Neural Network Models"} {Materials Science and Technology,
Vol. 14, 1998, 793--809.} {A. Y. Badmos,   H. K. D. H. Bhadeshia and
D. J. C. MacKay}

\ref{\lqq Yield Strength of Mechanically Alloyed ODS Iron Alloys.
Part 2: Physical Interpretation"} {Accepted for publication in
Materials Science and Technology, 1997} {A. Y. Badmos and
H. K. D. H. Bhadeshia}

\ref{\lqq Estimation of Hot Torsion Stress--Strain curves in Iron
Alloys Using a Neural Network Analysis"}
{Submitted to ISIJ International, 1998}
{V. Narayan, R. Abad--Lera, B. Lopez, H. K. D. H. Bhadeshia and D.
J. C. MacKay}

\ref{\lqq Neural Network Analysis of Steel Plate Processing"}
{Accepted for publication in Ironmaking and Steelmaking, 1998}
{S. B. Singh, H. K. D. H. Bhadeshia, D. J. C. MacKay, H. Carey and I.
Martin}


}


\vfill\eject

\sec{SUMMARY}

\x{\bf Introduction: } {\parindent=15 pt  \narrower \medskip
\x The development of new nickel alloys for aeroengine applications is a
difficult taks, frequently achieved by trial and experience. The purpose of this
work was to enable a significant proportion of the development procedure to be
done by computation. A variety of methods have been used towards this end,
ranging from physical models to the power technique of neural network
analysis.

A more general method of regression is neural network
analysis. As before, the input data  $x_j$ are multiplied by weights,
but the sum of all these products forms the argument of a hyperbolic
tangent. The output $y$ is therefore a non--linear function of
$x_j$, the function usually chosen being the hyperbolic tangent
because of its flexibility. The exact shape of the hyperbolic
tangent can be varied by altering the weights. Further
degrees of non--linearity can be introduced by combining several of
these hyperbolic tangents, so that the neural network
method is able to capture almost arbitrarily non--linear
relationships. For example, it is well known that the effect of
chromium on the microstructure of steels is quite different at large
concentrations than in dilute alloys. Ordinary regression analysis
cannot cope with such changes in the form of relationships.


A potential difficulty with the use of powerful regression methods is
the possibility of overfitting  data. For example, it is
possible to produce a neural network model for a
completely random set of data. To avoid this difficulty, the
experimental data can be divided into two sets, a {\it training}
dataset and a {\it test} dataset. The model is produced using only the
training data. The test data are then used  to check that the
model behaves itself when presented with previously unseen data.

Neural network models in many ways  mimic human experience and
are capable of learning or being trained to recognize the correct
science rather than nonsensical trends. Unlike human experience, these
models can be  transferred readily  between generations and steadily
developed to make design tools of lasting value. These models also
impose a discipline on the digital storage of valuable experimental
data, which may otherwise be lost with the passage of time.

We have addressed solid solution strengthening, tensile properties, fatigue,
creep, lattice misfit in the context of nickel--base superalloys, and have
applied the method to other materials and processes.

\vskip 2cm

\x{\bf Achievements: }  A large number of quantitative models have been
produced, dealing with the microstructure and mechanical properties of
nickel--base superalloys. This is the first time it has become possible to
estimate properties as complex as the fatigue crack growth rate as a function
of a very large number of variables. The models have been tested successfully
against the known principles of physical metallurgy. They have been used
already, both in reducing the scale of experimental programmes and in
identifying regimes where experiments are essential.

\vfill\eject\bye

