.\input texp.tex
\centerline {\large Estimation of The Mechanical Properties of Ferritic
Steel Welds:}
\centerline {\large Part I: Yield and Tensile Strength}
\centerline{S. H. Lalam, H. K. D. H. Bhadeshia and D. J. C. Mackay$^{\dag}$}\medskip
\centerline{ University of Cambridge}
\centerline{ Department of Materials Science and Metallurgy}
\centerline{ Pembroke Street, Cambridge CB2 3QZ, U.K.}
\medskip
\centerline{ $^{\dag}$Cavendish Laboratory}
\centerline{Madingley Road, Cambridge CB3 0HE, U.K.}
\bigskip \singlespace
\sec {ABSTRACT}
{\parindent=20pt \narrower \medskip
\x The yield strength and ultimate tensile strength of ferritic steel
weld metal have been expressed as functions of chemical
composition, the heat input during welding, and of the heat treatment
given after welding is completed. The method involved a neural network
analysis of a vast and quite general database assembled from
publications on weld metal properties. The outputs of the model have
been assessed in a variety of ways, including specific studies of model
predictions for the so--called carbon--manganese and $2{1\over4}$Cr1Mo
systems. Where possible, comparisons have also been made against
corresponding methods which use simple physical metallurgical
principles. The models created are believed to have been trained on the
largest weld metal database to date, and are shown to capture vital
metallurgical trends. The computer programs associated with the work
have been made freely available on the world--wide--web.
\medskip}
\sec {INTRODUCTION}
It has been possible for some time to estimate the
microstructure of ferritic steel weld metals from their chemical
composition and welding parameters [1], although the full effects of
heat--treatment have yet to be modelled. This has been useful
in the development of alloys, given a broad understanding of what
constitutes a good microstructure. The methodology cannot, however, be
used directly in engineering design because that requires specific
values of the mechanical properties.
The yield strength is one of the simpler of mechanical properties and
to a small extent it has been possible to apply physical metallurgy
principles towards its estimation for weld metals [1]. The method begins
with the assumption that the yield strength of
steel microstructures can be factorised into a number of intrinsic
components:
$$\sigma = \sigma_{Fe} + \sum_i x_i\sigma_{SS_i} +
x_C\sigma_C + K_L(\overline{L})^{-1} + K_D \rho^{0.5}_D
\numeqn $$
where $x_i$ is the concentration of a substitutional solute
which is represented here by a subscript $i$. The other terms
in this equation can be listed as follows:
\medskip
\settabs=10 \columns
\+ & $K_L$ & coefficient for strengthening due to \lq grain'
size, ${\rm 115\,MN\,m^{-1}}$\cr
\+ & $K_D$ & coefficient for strengthening due to
dislocations, ${7.34 \times 10^{-6}\,\rm MN\,m^{-1}}$ \cr
\+ & $\sigma_{Fe}$ & strength of pure, annealed iron,
${219\,\rm MN\,m^{-2}}$ at 300 K \cr
\+ &$\sigma_{SS_i}$ & substitutional solute ($i$)
strengthening \cr
\+& $\sigma_C$ & solid solution
strengthening due to carbon \cr
\+ & $ \rho_D$ & dislocation density, typically
$10^{16}\,{\rm m^{-2}}$ \cr
\+ & $\overline L$ & measure of the ferrite plate
size, typically 0.2 ${\rm \mu m}$ \cr
\+ \cr
\x The individual strengthening contributions are
not described here, suffice it to say that it is possible to obtain
reasonable estimates for all of the coefficients from data which are
independent of welding. Equation~\nnumeqn\ covers just one
microstructure; similar equations are needed for each of the phases
present. The contributions from each phase then has to be
appropriately summed to obtain the overall strength [1].
The application of such work is, obviously, limited since it fails
to address the complexity that is inherent in real welds, \ie the
number of variables influencing the strength is in fact far greater
than implied above. For example, many welds are heat--treated after
fabrication and there is no satisfactory theory available to deal with
this. Many other complexities exist, as will become evident during the
course of this paper.
Linear regression analysis is frequently used to overcome these
difficulties [2]. In this work, we apply a much more general form of
regression, \ie neural network analysis, to enable the estimation of
the yield and tensile strength of ferritic weld metals. These
represent by far the largest group of welding materials. The present
work builds on an earlier study [3] which was more limited in scope. In a future paper we shall attempt to deal with the elongation and
Charpy impact properties of ferritic steel welds.
\sec{THE METHOD}
A neural network is a general method of regression analysis in
which a very flexible non--linear function is fitted to experimental
data, the details of which have been extensively reviewed [4]. It is,
nevertheless, useful to present some salient features in order to
place the technique in context.
The flexibility of the non--linear function scales
with the number of hidden nodes
$i$. Thus, the dependent variable $y$ is given in the present work by
$$ y = \sum_i w_i^{(2)} h_i + \theta^{(2)} \numeqn $$ where
$$h_i = \tanh \bl(\sum_j w^{(1)}_{ij}x_j + \theta^{(1)}_i\br)
\numeqn$$ where $x_j$ are the $j$ variables on which the output $y$
depends, $w_i$ are the weights (coefficients) and
$\theta_i$ are the biases (equivalent to the constants in linear
regression analysis). The combination of equation~\nnumeqn\ with a set of
weights, biases, value of $i$ and the minimum and maximum values of the
input variables defines the network completely. The availability of a sufficiently complex and flexible function means that
the analysis is not as restricted as in linear regression where the form
of the equation has to be specified before the analysis.
The neural network can capture interactions between the inputs because the hidden units are nonlinear. The nature of these interactions is implicit in the values of the weights, but the weights may not always be easy to interpret. For example, there may exist more than just pairwise interactions, in which case the problem becomes difficult to visualise from an examination of the weights. A better method is to actually use the network to make predictions and to see how these depend on various combinations of input.
\ssec{Error Estimates}
The input parameters are generally assumed in the analysis to be
precise and it is normal to calculate an overall error by comparing the
predicted values $(y_j)$ of the output against those measured $(t_j)$, for
example,
$$ E_D \propto \sum_j (t_j - y_j)^2 \numeqn $$ $E_D$ is expected to increase if important input variables have been
excluded from the analysis. Whereas $E_D$ gives an overall perceived
level of noise in the output parameter, it is, on its own, an unsatisfying
description of the uncertainties of prediction.
MacKay has developed a particularly useful treatment of neural networks
in a Bayesian framework [5], which allows the calculation of error bars
representing the uncertainty in the fitting parameters. The
method recognises that there are many functions which can be fitted or
extrapolated into uncertain regions of the input space, without unduly
compromising the fit in adjacent regions which are rich in accurate data.
Instead of calculating a unique set of weights, a probability distribution of
sets of weights is used to define the fitting uncertainty. The error bars
therefore become large when data are sparse or locally noisy.
In this context, a very useful measure is the log predictive error
because the penalty for making a wild prediction is reduced if that wild
prediction is accompanied by appropriately large error bars [6]:
$$LPE=
\sum_n
\bigg [ {1\over 2} {
\bigg ( t^{(n)}-y^{(n)}
\bigg ) } ^2 \bigg / \sigma_y^{{(n)}^2} + log \big ( \root \of {2 \pi
\sigma_y^{(n)} } \big ) \bigg ] $$
Note that a larger value of the log predictive error implies a better
model.
\ssec{Overfitting}
A potential difficulty with the use of powerful non--linear regression
methods is the possibility of overfitting data. To avoid this,
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. The
training error tends to decrease continuously as the model complexity
increases. It is the minimum in the test error which enables that model
to be chosen which generalises best on unseen data [6]. There are other important features in the control of complexity which are discussed elsewhere [6].
Finally, it should be noted that the analysis uses normalised values
of the variables in the range
$\pm$0.5 as follows:
$$x_N={x-x_{min}\over{x_{max}-x_{min}}}-0.5$$
where $x$ is the original value from the database, $x_{max}$ and
$x_{min}$ are the respective maximum and minimum of each variable in the
original data and $x_N$ is the normalised value. This step is not
essential to the running of the neural network but is a
convenient way of comparing the effect of different variables on the
output.
\fagg\ shows the general structure of the simple three layer neural
network.
\sec{THE DATABASE}
All of the data collected are from multirun weld deposits designed for
low--dilution to enable specifically the measurement of all--weld
metal properties. Furthermore, they all represent electric arc
welds made using one of the following processes: manual metal arc (MMAW),
submerged arc (SAW) and tungsten inert gas arc (TIG). The welding process
itself was represented only by the level of heat input. This is
because a large number of published papers did not specify welding
parameters in sufficient detail to enable the creation of a dataset
without missing values. Missing values cannot be tolerated in the
method used here. If the effect of a welding process is not properly
represented by the heat input and chemical composition, then neglect
of any other parameters would make the
predictions more noisy. As will be seen later, the noise in the
output was found to be acceptable; the greater uncertainty arises
from the lack of a uniform coverage of the input space. The sources of
all data are listed in references [7--74].
The aim of the neural network analysis was to predict
the yield and tensile strength as a function of a large number of
variables, including the chemical composition, the welding heat input
and any heat treatment. The databases for the yield and ultimate
tensile strength (UTS) are different because the UTS database also
included the oxygen concentration since tensile failure should depend
on inclusions which nucleate voids. As a consequence, the yield
strength database consists of 2002 separate experiments whereas the UTS
database is slightly smaller at 1972 experiments since the oxygen
concentration was not always reported \footnote \dag {This compares with previous work which was based on 770 yield strength and 520 UTS experiments [3]. }. Our method cannot cope with
missing values of any of the variables. In 14 cases the sulphur and phosphorus concentrations were not available. Since these impurities might be important, it would not be satisfactory to set them to zero. Missing values of sulphur and phosphorus were therefore set at the average of the database.
\ssec{The Yield Strength Database}
\tablaa\ shows the range, mean and standard deviation of each variable
including the output (yield strength). The purpose here is simply to
list the variables and provide an idea of the range covered. It is
emphasised however, that unlike linear regression analysis, the
information in \tablee\ cannot be used to define the range of
applicability of the neural network model. This is because the inputs
are in general expected to interact. We shall see later that it is the
Bayesian framework of our neural network analysis which allows the
calculation of error bars which define the range of useful
applicability of the trained network. A visual impression of the
spread of data is shown in \fagg. It can be concluded from {\figg} that the effect on yield strength of carbon, manganese, silicon, nickel, molybdenum and heat input have been systematically studied. Hence, future experiments could focus on examining the effect of chromium in the range 3--8 wt\%, vanadium (0.1--0.2 wt\%), cobalt at all concentrations but in greater variety of alloy systems, tungsten at low and high concentrations, titanium and boron in high strength weld. The effect of tempering temperature in the range 250--500\deg also needs to be studied.
\topinsert
\thicksize=0.7pt \thinsize=0.5pt
\tablewidth=6.4truein {\smalltype
\begintable
{\bf Input element } | {\bf Minimum } |{\bf Maximum } | {\bf Mean } | {\bf Standard Deviation } \cr
Carbon (wt\%) \hfill | 0.01 \hfill | 0.22 \hfill | 0.072 \hfill | 0.025 \nr
Silicon (wt\%) \hfill | 0.01 \hfill | 1.63 \hfill | 0.344 \hfill | 0.138\nr
Manganese (wt\%) \hfill | 0.27 \hfill | 2.31 \hfill | 1.192 \hfill | 0.41 \nr
Sulphur (wt\%) \hfill | 0.001 \hfill | 0.14 \hfill | 0.009 \hfill | 0.006 \nr
Phosphorus (wt\%) \hfill | 0.001 \hfill | 0.25 \hfill | 0.012 \hfill | 0.009 \nr
Nickel (wt\%) \hfill | 0.0 \hfill | 4.79 \hfill | 0.43 \hfill | 0.888 \nr
Chromium (wt\%) \hfill | 0.0 \hfill | 12.1 \hfill | 0.808 \hfill | 1.952 \nr
Molybdenum (wt\%) \hfill | 0.0 \hfill | 2.4 \hfill | 0.221 \hfill | 0.368 \nr
Vanadium (wt\%) \hfill | 0.0 \hfill | 0.32 \hfill | 0.026 \hfill | 0.06 \nr
Copper (wt\%) \hfill | 0.0 \hfill | 2.18 \hfill | 0.063 \hfill | 0.185 \nr
Cobalt (wt\%) \hfill | 0.0 \hfill | 2.8 \hfill | 0.007 \hfill | 0.115 \nr
Tungsten (wt\%) \hfill | 0.0 \hfill | 3.86 \hfill | 0.091 \hfill | 0.427\nr
Titanium (ppm wt.) \hfill | 0.0 \hfill | 900 \hfill | 64.9 \hfill | 112.14 \nr
Boron (ppm wt.) \hfill | 0.0 \hfill | 195 \hfill | 5.8 \hfill | 19.08\nr
Niobium (ppm wt.) \hfill | 0.0 \hfill | 1770 \hfill | 69.6 \hfill | 168.13 \nr
Heat input (kJ mm$^-$$^1$) \hfill | 0.55 \hfill | 7.9 \hfill | 1.6 \hfill | 1.234\nr
Interpass temperature ($^\circ$C) \hfill | 20 \hfill | 375 \hfill | 207.8 \hfill | 52.67 \nr
Tempering temperature ($^\circ$C) \hfill | 20 \hfill | 780 \hfill | 358.3 \hfill | 249.29 \nr
Tempering time (h) \hfill | 0.0 \hfill | 50 \hfill | 6.5 \hfill | 6.45 \nr
Yield Strength (MPa) \hfill | 288 \hfill | 1003 \hfill | 533.9 \hfill | 113.64
\endtable }
{\tabtit{\tablee : } {The inputs for the yield strength model. }}
\endinsert
\ssec{The UTS Database}
\tablaa\ shows the range, mean and standard deviation of each variable
including the output (ultimate tensile strength). The corresponding
visual impression of the UTS database is similar to that of the yield strength. The UTS contains an extra input variable oxygen \figg, the effect of which at higher concentrations (above 900 ppm wt.) needs to be studied.
\thicksize=0.7pt \thinsize=0.5pt
\tablewidth=6.4truein {\smalltype \begintable
{\bf Input element }\hfill | {\bf Minimum }\hfill |{\bf Maximum }\hfill | {\bf Mean }\hfill | {\bf Standard Deviation } \cr
Carbon (wt\%) \hfill | 0.01 \hfill | 0.22 \hfill | 0.072 \hfill | 0.024 \nr
Silicon (wt\%) \hfill | 0.01 \hfill | 1.63 \hfill | 0.345 \hfill | 0.142 \nr
Manganese (wt\%) \hfill | 0.27 \hfill | 2.31 \hfill | 1.191 \hfill | 0.410 \nr
Sulphur (wt\%) \hfill | 0.001 \hfill | 0.14 \hfill | 0.009 \hfill | 0.006 \nr
Phosphorus (wt\%) \hfill | 0.001 \hfill | 0.25 \hfill | 0.012 \hfill | 0.009 \nr
Nickel (wt\%) \hfill | 0.0 \hfill | 4.79 \hfill | 0.426 \hfill | 0.900 \nr
Chromium (wt\%) \hfill | 0.0 \hfill | 12.1 \hfill | 0.748 \hfill | 1.810 \nr
Molybdenum (wt\%) \hfill | 0.0 \hfill | 2.4 \hfill | 0.219 \hfill | 0.370 \nr
Vanadium (wt\%) \hfill | 0.0 \hfill | 0.32 \hfill | 0.0252\hfill | 0.060 \nr
Copper (wt\%) \hfill | 0.0 \hfill | 2.18 \hfill | 0.053 \hfill | 0.160 \nr
Cobalt (wt\%) \hfill | 0.0 \hfill | 2.8 \hfill | 0.008 \hfill | 0.110 \nr
Tungsten (wt\%) \hfill | 0.0 \hfill | 3.86 \hfill | 0.093 \hfill | 0.500 \nr
Oxygen (ppm wt.) \hfill | 0.0 \hfill | 1650 \hfill | 362 \hfill | 200.8 \nr
Titanium (ppm wt.) \hfill | 0.0 \hfill | 900 \hfill | 67 \hfill | 116.5 \nr
Boron (ppm wt.) \hfill | 0.0 \hfill | 195 \hfill | 6 \hfill | 19.3 \nr
Niobium (ppm wt.) \hfill | 0.0 \hfill | 1770 \hfill | 66 \hfill | 163.6 \nr
Heat input (kJ mm$^-$$^1$) \hfill | 0.55 \hfill | 7.9 \hfill | 1.56 \hfill | 1.17 \nr
Interpass temperature ($^\circ$C) \hfill | 20 \hfill | 375 \hfill | 209 \hfill | 51.8 \nr
Tempering temperature ($^\circ$C) \hfill | 20 \hfill | 770 \hfill | 368 \hfill | 241.8 \nr
Tempering time (h) \hfill | 0.0 \hfill | 50 \hfill | 6.9 \hfill | 6.5 \nr
Ultimate Tensile Strength (MPa)\hfill | 440 \hfill | 1151 \hfill | 624 \hfill | 117.5
\endtable }
{\tabtit{\tablee : } {The inputs for the ultimate tensile strength
model.}}
\sec { THE YIELD STRENGTH MODEL}
Some eighty yield strength neural network models were trained on a
training dataset which consisted of a random selection of half
the data (1001) from the yield strength dataset. The remaining 1001 data
formed the test dataset which was used to see how the model
generalises on unseen data. Each model contained the 19 inputs listed in
Table~1 but with different numbers of hidden units or the random seeds
used to initiate the values of the weights.
\fagg\ shows the results. As expected, the perceived level of noise
($\sigma_\nu$) in the normalised yield strength decreases as the model
becomes more complex, \ie the number of hidden units increases. This
is not the case for the test error, which goes through a minimum at
three hidden units, and for the log predictive error which reaches a
maximum at 6 hidden units.
The error bars presented throughout this work represent a
combination of the perceived level of noise $\sigma_\nu$ in the output
and the fitting uncertainty estimated from the Bayesian framework. It
is evident that there are a few outliers in the plot of the predicted
versus measured yield strength for the test dataset. Each of these
outliers has been investigated and found to represent unique data
which are not represented in the training dataset. For example, there
is a weld with a sulphur concentration of 0.15 wt.\%\ and another with
a phosphorus concentration of 0.25 wt.\%, both extremely high and
unusual level of impurities in weld metals.
It is possible that a committee of models can make a more reliable
prediction than an individual model [75]. The best models are ranked
using the values of the log predictive errors (\figg). Committees are
then formed by combining the predictions of the best $L$ models, where
$L = 1, 2, \ldots$; the size of the committee is therefore given by the
value of $L$. A plot of the test error of the committee versus its size
gives a minimum which defines the optimum size of the committee, as
shown in \figg.
The test error associated with the best single model is clearly greater
than that of any of the committees (\figg). The committee with twenty eight
models was found to have an optimum membership with the smallest test
error. The committee was therefore retrained on the entire data set without changing the complexity of any of its member models. The final comparison between the predicted and measured values of the yield strength for the
committee of twenty eight is shown in \fagg. The details of the twenty eight members of the optimum committee are presented in \tablaa.
\midinsert \thicksize=1pt \thinsize=0.8pt
\tablewidth=5.2truein \begintable
{\bf Model }& {\bf Hidden units }& {$\sigma_{\nu}$ } | {\bf Model }& {\bf Hidden units } & { $\sigma_{\nu}$ }\cr
{\bf 1 } & 6 & 0.044062 | {\bf 15 } & 5 & 0.066848 \nr
{\bf 2 } & 5 & 0.048167 | {\bf 16 } & 2 & 0.064810 \nr
{\bf 3 } & 6 & 0.043382 | {\bf 17 } & 3 & 0.059811 \nr
{\bf 4 } & 7 & 0.040949 | {\bf 18 } & 9 & 0.036538 \nr
{\bf 5 } & 4 & 0.053474 | {\bf 19 } & 2 & 0.066146 \nr
{\bf 6 } & 4 & 0.052702 | {\bf 20 } & 3 & 0.060487 \nr
{\bf 7 } & 3 & 0.058641 | {\bf 21 } & 2 & 0.066151 \nr
{\bf 8 } & 3 & 0.060561 | {\bf 22 } & 9 & 0.035645 \nr
{\bf 9 } & 5 & 0.047489 | {\bf 23 } & 10 & 0.033591 \nr
{\bf 10 } & 4 & 0.047489 | {\bf 24 } & 5 & 0.045975 \nr
{\bf 11 } & 5 & 0.046503 | {\bf 25 } & 9 & 0.034897 \nr
{\bf 12 } & 7 & 0.039873 | {\bf 26 } & 4 & 0.051630 \nr
{\bf 13 } & 3 & 0.064206 | {\bf 27 } & 7 & 0.039677 \nr
{\bf 14 } & 2 & 0.067036 | {\bf 28 } & 8 & 0.036436
\endtable \tabtit{\tablee : } {The 28 members of the optimum committee. $\sigma_{\nu}$ is the perceived level of noise in the yield strength and the complexity of the model increases with the number of hidden units.} \endinsert
\tablee\ indicates the significance ($\sigma_w$) of each of the
input variables, as perceived by first five neural network models in the committee. The $\sigma_w$ value represents the extent to which a particular input explains the variation in the output, rather like a partial correlation coefficient in linear regression analysis.
The post-weld heat treatment temperature on the whole explains a large
proportion of the variation in the yield strength \fagg. All of the
variables considered are found to have a significant effect on the
output indicating a good choice of inputs.
\sec {THE ULTIMATE TENSILE STRENGTH MODEL}
The models were trained
on 1972 individual experimental measurements, of which a random half
of the data formed the training dataset and the other half the test
dataset. The procedures are otherwise identical to those described
for the yield strength model, resulting in the characteristics
illustrated in {\fagg} and the performance of the optimum committee
of best models is illustrated in \fagg . The perceived significance of first five models are shown in \fagg. Here the additional input varibale oxygen shows more significance along with post-weld heat treatment variables.
\midinsert{\smalltype\thicksize=1pt \thinsize=0.8pt
\tablewidth=5.2truein \begintable
{\bf Model }& {\bf Hidden units } & { $\sigma_{\nu}$ }
| {\bf Model }& {\bf Hidden units } & { $\sigma_{\nu}$ } \cr
{\bf 1 } & 8 & 0.029323 | {\bf 21 } & 7 & 0.027016 \nr
{\bf 2 } & 9 & 0.026503 | {\bf 22 } & 8 & 0.060074 \nr
{\bf 3 } & 6 & 0.027177 | {\bf 23 } & 3 & 0.028483 \nr
{\bf 4 } & 10 & 0.025500 | {\bf 24 } & 3 & 0.039529 \nr
{\bf 5 } & 9 & 0.031034 | {\bf 25 } & 2 & 0.065098 \nr
{\bf 6 } & 6 & 0.026604 | {\bf 26 } & 2 & 0.052619 \nr
{\bf 7 } & 4 & 0.027218 | {\bf 27 } & 2 & 0.029716 \nr
{\bf 8 } & 10 & 0.026010 | {\bf 28 } & 2 & 0.045941 \nr
{\bf 9 } & 6 & 0.025530 | {\bf 29 } & 4 & 0.053212 \nr
{\bf 10 } & 5 & 0.031240 | {\bf 30 } & 9 & 0.053929 \nr
{\bf 11 } & 8 & 0.026471 | {\bf 31 } & 4 & 0.033334 \nr
{\bf 12 } & 9 & 0.026485 | {\bf 32 } & 3 & 0.034551
\endtable}\tabtit{\tablee : } {The hidden units and $\sigma_{\nu}$ in the
optimum UTS committee model.}
\endinsert
\sec {APPLICATION TO C--Mn WELDS}
Carbon--manganese weld metals refer to a popular class of
ferritic steels in which the substitutional solutes other than silicon
and manganese are generally kept to low concentration levels. They are
interesting because there is a great deal already known about them,
making it easy to interpret the physical significance of the neural
network model. Furthermore, there exists an alternative
semi--empirical model for the estimation of the yield and tensile
strengths of such multirun welds [76] enabling a further comparison.
The semi--empirical model is henceforth referred to as the \lqq
physical model" or PM for short. The basic values of the variables used
in applying the model to carbon--manganese welds are listed in
\tablaa. The specified low--temperature heat treatment is simply a
standard hydrogen removal treatment (250\deg for 14 h) applied to most welds before mechanical testing.
\midinsert
\vskip 0.5 in
\thicksize=0.7pt \thinsize=0.5pt
\tablewidth=3.0truein {\smalltype \begintable
Carbon (wt\%) \hfill | 0.06 \nr
Silicon (wt\%) \hfill | 0.50 \nr
Manganese (wt\%) \hfill | 1.50 \nr
Sulphur (wt\%) \hfill | 0.006 \nr
Phosphorus (wt\%) \hfill | 0.008 \nr
Nickel (wt\%) \hfill | 0.0 \nr
Chromium (wt\%) \hfill | 0.0 \nr
Molybdenum (wt\%) \hfill | 0.0 \nr
Vanadium (wt\%) \hfill | 0.0 \nr
Copper (wt\%) \hfill | 0.0\nr
Cobalt (wt\%) \hfill | 0.0\nr
Tungsten (wt\%) \hfill | 0.0 \nr
Oxygen (ppm wt.) \hfill | 300 \nr
Titanium (ppm wt.) \hfill | 0.0 \nr
Boron (ppm wt.) \hfill | 0.0 \nr
Niobium (ppm wt.) \hfill | 0.0 \nr
Heat input (kJ mm$^-$$^1$) \hfill | 1.00 \nr
Interpass temperature ($^\circ$C) \hfill | 175 \nr
Tempering temperature ($^\circ$C) \hfill | 250 \nr
Tempering time (h) \hfill | 14.0
\endtable}
\tabtit{\tablee : }{ The inputs relevant for a typical \lqq
carbon--manganese" weld metal made using the manual metal arc welding
process.}
\endinsert
The results as a function of the carbon and manganese
concentrations are illustrated in
\fagg\ for a variety of interesting cases. The calculated yield
strength is in all cases found to be consistent with that expected from
the physical model, although there are systematic differences at high
yield strength values for all cases other than at the highest manganese
concentration. However, the deviations are all within the error bounds
of the neural network model for yield strength. The major
discrepancies arise with the UTS especially at high UTS values. It is
believed that the physical model is poorly constructed since the UTS
is essentially taken arbitrarily to be linearly related to a single
variable, the yield strength, \fagg\ shows the comparision between the measured and strength estimation by physical model. Physical models at higher strength values behaved very poorly, it estimated the strength higher than the measured.
An interesting feature of strengthening due to substitutional solutes
is the synergistic effect with carbon. {\fagg} shows that the
dependence of the strengthening effect of molybdenum on the carbon
concentration is particularly large, the effect of molybdenum in strengthening the weld is greaterthan that of Cr or Mn. This is consistent with published literature [76]. Elements such as molybdenum and vanadium are associated with strong secondary hardening effects which frequently trigger a reduction in toughness. In ordinary carbon--manganese multirun welds, the secondary microstructure, \ie regions of weld metal which are tempered by subsequent weld runs, lose most of their microstructural
strength. This is not necessarily the case in weld metal containing
strong carbide formers. For example, it is well--established that the
yield strength calculated using equation~1 is always underestimated
with molybdenum--containing welds, the degree of underestimation
increasing with the molybdenum concentration [1]. The behaviour
observed in {\figg} is not therefore surprising.
The sensitivity to carbon concentration, and the net magnitude of the
strengthening effect decreases for the ultimate tensile strength
(\figg ). This is expected since the UTS is measured at large plastic
strains whereas the yield strength is much more sensitive to the
initial microstructure.
The predicted dependence of the strengthening effect of niobium on the
carbon concentration is shown in \figg . The strength increment plotted on the vertical axis is based on average effect on niobium in the concentration range 0-1500 parts per million by weight, for any given carbon concentrations. The increment per weight percent of niobium is obviously very large and this may be reason why niobium is generally found to be determinantal to toughness [77].
\fagg\ shows other predictions; although there are no surprises, it is
worth noting the error bars. These error bars can be used to identify
regions of the input space where further experiments would be useful.
For example, the prediction uncertainties associated with niobium, or
with large heat inputs, are much larger than say with changes in the
manganese concentration. This is where future experiments could be
focussed.
\sec{APPLICATION TO $2{1\over4}$Cr1Mo WELDS}
The $2{1\over4}$Cr1Mo weld metal system is designed
primarily for applications where the components will
serve at elevated temperatures (450--565\degg) for
long periods of time ($\simeq$30 years). This in
contrast to carbon--manganese weld metals which are used in
structural applications such as buildings and bridges which are
essentially at ambient temperature. Consequently, the post--weld
heat treatment is of vital importance to $2{1\over4}$Cr1Mo weld
metals, not only to relieve residual stresses but also to generate
a stable microstructure in which the carbides hinder creep
deformation. The basic values of the variables
used in applying the models to $2{1\over4}$Cr1Mo welds are listed in
\tablaa. The specified high--temperature heat treatment is a typical
post--weld heat treatment (PWHT).
\topinsert
\vskip 0.5 in
\thicksize=0.7pt \thinsize=0.5pt
\tablewidth=3.0truein {\smalltype \begintable
Carbon (wt\%) \hfill | 0.11 \nr
Silicon (wt\%) \hfill | 0.20 \nr
Manganese (wt\%) \hfill | 0.80 \nr
Sulphur (wt\%) \hfill | 0.002 \nr
Phosphorus (wt\%) \hfill | 0.005 \nr
Nickel (wt\%) \hfill | 0.20 \nr
Chromium (wt\%) \hfill | 2.25 \nr
Molybdenum (wt\%) \hfill | 1.0 \nr
Vanadium (wt\%) \hfill | 0.0 \nr
Copper (wt\%) \hfill | 0.0\nr
Cobalt (wt\%) \hfill | 0.0\nr
Tungsten (wt\%) \hfill | 0.0 \nr
Oxygen (ppm wt.) \hfill | 300 \nr
Titanium (ppm wt.) \hfill | 0.0 \nr
Boron (ppm wt.) \hfill | 0.0 \nr
Niobium (ppm wt.) \hfill | 0.0 \nr
Heat input (kJ mm$^-$$^1$) \hfill | 1.5 \nr
Interpass temperature ($^\circ$C) \hfill | 200 \nr
Tempering temperature ($^\circ$C) \hfill | 690 \nr
Tempering time (h) \hfill | 8.0
\endtable}
\tabtit{\tablee : } {The inputs relevant for a typical
2$1\over4$Cr - 1Mo wt\% weld metal.}
\endinsert
It is notable from the predictions illustrated in \fagg\ that there are
greater uncertainties (larger error bars) associated with the estimation
of mechanical properties for the $2{1\over4}$Cr1Mo system when
compared with the carbon--manganese welds. This is largely because
there are fewer data available for $2{1\over4}$Cr1Mo welds.
Another striking feature is that the sensitivity of the strength to
alloying elements, in the PWHT condition, is far smaller than in the
as--welded condition. This is not surprising given the severe nature
of the post--weld heat treatment at 690\degg\ for 8 hours. It is
emphasised that although the yield and tensile strengths are not
particularly sensitive to composition in the PWHT condition, this will
not be the case for creep properties where the tempering heat
treatment is essential for the generation of alloy carbides and to
provide a microstructure which has long term stability.
\sec{SOFTWARE}
We have examined a vast number of other trends predicted by the
models. It is not reasonable to present all the results in this paper,
suffice it to say that the trends have been found to be reasonable from
a metallurgical point of view.
Models like these cannot be fully studied because the number of
possibilities is very large indeed. In addition, although the
carbon--manganese and $2{1\over4}$Cr1Mo have been illustrated in the
present paper, the work is much more widely applicable since the
database from which the neural network models were created covers a
vast range of alloys.
The software capable of doing these calculations can be obtained
freely from
$$http://www.msm.cam.ac.uk/map/map.html$$
\sec{SUMMARY \&\ CONCLUSIONS}
The yield strength and ultimate tensile strength of ferritic steel
weld metal have been analysed using a neural network method within a
Bayesian framework. The data used were mostly obtained from the
published literature and represent a wide cross--section of alloy
compositions and arc--welding processes.
Trends predicted by the models appear to be consistent with those
expected metallurgically, although it must be emphasised that only the
simplest of trends have been examined since the number of variables
involved is very large. The models can be applied widely because the
calculation of error bars whose magnitude depends on the local
position in the input space is an inherent feature of the neural
network used. The error bar is not simply an estimate of the perceived
level of noise in the output but also includes an uncertainty
associated with fitting the function in the local region of input
space. This means that the method is less dangerous in extrapolation
or interpolation since it effectively warns when experimental data are
lacking or are exceptionally noisy. The work has clearly identified regions of the input space where further experiments should be encouraged.
\sec {ACKNOWLEDGEMENTS}
The authors are thankful to the to the Cambridge Commonwealth
Trust for funding this work and to Kenneth Mitchell of National Power
for further financial support. We would like to acknowledge Professor
Alan Windle of the University of Cambridge for the provision of
laboratory facilities.
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\def\ref#1#2#3#4#5#6#7{\hoffset=6.0mm \parindent=-6.0mm
\singlespace \rightskip=6.0mm \singlespace {#1.} #2: {\it
#4} {\bf #5} (#3) #6\vskip 0.5truemm}
{\parskip=0mm \parindent=10pt \narrower
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\medskip}
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{\tabtit{Fig. 1 : }{ Schematic illustration of the input, hidden and output layers of the neural network model utilised here.}}
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{\tabtit{Fig. 2 : } {The database values of each variable versus the
yield strength.}}
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{\tabtit{Fig. 3 : }{Characteristics of the yield strength model.}}
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{\tabtit{Fig. 4 : }{Comparison of the predicted and measured values
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\vskip 3.5 in
\special{psfile=best_YS_sigma_w.eps hoffset=0 voffset=-50 hscale=80 vscale=80}
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{\tabtit{Fig. 6 : }{Characteristics of the ultimate tensile strength model.}}
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\topinsert
\vskip 2.5 in
\special{psfile=UTSCOM_RESULT.eps hoffset=60 voffset=-10 hscale=45 vscale=45}
{\tabtit{Fig. 7 : }{Comparison of the predicted and measured values
of ultimate tensile strength for the optimum committee model.}}
\vskip 3.5 in
\special{psfile=best_UTS_sigma_w.eps hoffset=0 voffset=-55 hscale=80 vscale=80}
{\tabtit{Fig. 8 : }{The perceived significance $\sigma_w$ values of first best five UTS models for each of the inputs.}}
\endinsert
\midinsert
\vskip 2.3 in
\special{psfile=0.1Mn_YUC_pwht.eps hoffset=-60 voffset=0 hscale=40 vscale=40}
\special{psfile=0.3Mn_YUC_pwht.eps hoffset=108 voffset=0 hscale=40 vscale=40}
\special{psfile=0.5Mn_YUC_pwht.eps hoffset=280 voffset=0 hscale=40 vscale=40}
\vskip 2.3 in
\special{psfile=1.1Mn_YUC_pwht.eps hoffset=-60 voffset=-10 hscale=40 vscale=40}
\special{psfile=1.5Mn_YUC_pwht.eps hoffset=108 voffset=-10 hscale=40 vscale=40}
\special{psfile=1.9Mn_YUC_pwht.eps hoffset=280 voffset=-10 hscale=40 vscale=40}
\tabtit{Fig. 9 : }{A series of calculations for carbon--manganese
welds, using both the neural network committee models and an
alternative published model [76].}
\vskip 2.3 in
\special{psfile=EXP_PHY.eps hoffset=108 voffset=0 hscale=40 vscale=40}
\tabtit{Fig. 10 : }{Comparision between measured and published physical model calculations.}
\endinsert
\topinsert
\vskip 2.6 in
\special{psfile=Stg_coeff_YS.eps hoffset=-20 voffset=-10 hscale=45 vscale=45}
\special{psfile=Stg_coeff_UTS.eps hoffset=200 voffset=-10 hscale=45 vscale=45}
\bigskip
\vskip 2.6 in
\special{psfile=Stg_coeff_V.eps hoffset=-20 voffset=-10 hscale=45 vscale=45}
\special{psfile=Stg_coeff_Nb.eps hoffset=200 voffset=-10 hscale=45 vscale=45}
\bigskip
\tabtit{Fig. 11 : }{Change in strength and the YS/UTS ratio as
a function of a wt\% of substitutional solute content in carbon--manganese steel
welds. The error bars are not included for clarity, but the
maximum values are 60.}
\endinsert
\midinsert
\vskip 2.3 in
\special{psfile=C_MnYUMn_pwht.eps hoffset=-60 voffset=0 hscale=40 vscale=40}
\special{psfile=C_MnYUCr_pwht.eps hoffset=108 voffset=0 hscale=40 vscale=40}
\special{psfile=C_MnYUMo_pwht.eps hoffset=280 voffset=0 hscale=40 vscale=40}
\bigskip
\vskip 2.3 in
\special{psfile=C_MnYUNb_pwht.eps hoffset=-60 voffset=-10 hscale=40 vscale=40}
\special{psfile=C_MnYUNi_pwht.eps hoffset=108 voffset=-10 hscale=40 vscale=40}
\special{psfile=C_MnYUSi_pwht.eps hoffset=280 voffset=-10 hscale=40 vscale=40}
\bigskip
\vskip 2.3 in
\special{psfile=C_MnYUHI_pwht.eps hoffset=20 voffset=-15 hscale=40 vscale=40}
\special{psfile=C_MnYUPWHT_t_600.eps hoffset=200 voffset=-20 hscale=40 vscale=40}
\bigskip
\tabtit{Fig. 12 : } {Variations in the yield and ultimate
tensile strengths of carbon--manganese weld metal as a
function of alloying elements and heat treatment.}
\endinsert
\topinsert
\vskip 2.35 in
\special{psfile=Cr_YUC.eps hoffset=-20 voffset=-10 hscale=45 vscale=45}
\special{psfile=Cr_YUC_pwht.eps hoffset=200 voffset=-10 hscale=45 vscale=45}
\bigskip
\vskip 2.35 in
\special{psfile=Cr_YUMo.eps hoffset=-20 voffset=-10 hscale=45 vscale=45}
\special{psfile=Cr_YUMo_pwht.eps hoffset=200 voffset=-10 hscale=45 vscale=45}
\bigskip
\vskip 2.35 in
\special{psfile=Cr_YUCr.eps hoffset=-20 voffset=-10 hscale=45 vscale=45}
\special{psfile=Cr_YUCr_pwht.eps hoffset=200 voffset=-10 hscale=45 vscale=45}
\bigskip
\tabtit{Fig. 13 : } {The effect of carbon, molybdenum and chromium
concentrations on the strength of $2{1\over4}$Cr1Mo welds in
the as--welded and PWHT (690$\,^\circ$C, 8~h) conditions.}
\endinsert
\vfil \eject
\end